Econometrics

Michael Creel
Department of Economics and Economic History
Universitat Autnoma de Barcelona
version 1.00, October 2012

Contents
1 About this document

1.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Licenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.4 Obtaining the materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.5 An easy way to use LYX and Octave today . . . . . . . . . . . . . . . . . . . . . . . .

12

2 Introduction: Economic and econometric models

13

3 Ordinary Least Squares

16

3.1 The Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.2 Estimation by least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

3.3 Geometric interpretation of least squares estimation

. . . . . . . . . . . . . . . . .

18

3.4 Influential observations and outliers . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.5 Goodness of fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.6 The classical linear regression model . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.7 Small sample statistical properties of the least squares estimator . . . . . . . . . . .

25

3.8 Example: The Nerlove model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

4 Asymptotic properties of the least squares estimator

4.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34
34

4.2 Asymptotic normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

4.3 Asymptotic efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

5 Restrictions and hypothesis tests

37

5.1 Exact linear restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

5.2 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

5.3 The asymptotic equivalence of the LR, Wald and score tests . . . . . . . . . . . . .

45

5.4 Interpretation of test statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

5.5 Confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

5.6 Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

5.7 Wald test for nonlinear restrictions: the delta method . . . . . . . . . . . . . . . . .

50

5.8 Example: the Nerlove data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

6 Stochastic regressors

57

6.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

6.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

6.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

6.4 When are the assumptions reasonable? . . . . . . . . . . . . . . . . . . . . . . . . .

60

6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

7 Data problems

62

7.1 Collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

7.2 Measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

7.3 Missing observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

8 Functional form and nonnested tests

79

8.1 Flexible functional forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

8.2 Testing nonnested hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

9 Generalized least squares

88

9.1 Effects of nonspherical disturbances on the OLS estimator . . . . . . . . . . . . . .

88

9.2 The GLS estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

9.3 Feasible GLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

9.4 Heteroscedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

9.5 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
10 Endogeneity and simultaneity

123

10.1 Simultaneous equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

10.2 Reduced form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
10.3 Estimation of the reduced form equations . . . . . . . . . . . . . . . . . . . . . . . 126
10.4 Bias and inconsistency of OLS estimation of a structural equation . . . . . . . . . . 128
10.5 Identification by exclusion restrictions . . . . . . . . . . . . . . . . . . . . . . . . . 129
10.6 Note about the rest of this chaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10.7 2SLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
10.8 Testing the overidentifying restrictions . . . . . . . . . . . . . . . . . . . . . . . . . 137
10.9 System methods of estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
10.10Example: 2SLS and Kleins Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
11 Numeric optimization methods

146

11.1 Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

11.2 Derivative-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
11.3 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
11.4 Examples of nonlinear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
11.5 Numeric optimization: pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
11.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
11.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
12 Asymptotic properties of extremum estimators

164

12.1 Extremum estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

12.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
12.3 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
12.4 Example: Consistency of Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . 170
12.5 Example: Inconsistency of Misspecified Least Squares . . . . . . . . . . . . . . . . . 171
12.6 Example: Linearization of a nonlinear model . . . . . . . . . . . . . . . . . . . . . . 171
12.7 Asymptotic Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

12.8 Example: Classical linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

12.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
13 Maximum likelihood estimation

177

13.1 The likelihood function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

13.2 Consistency of MLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
13.3 The score function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
13.4 Asymptotic normality of MLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
13.5 The information matrix equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
13.6 The Cramr-Rao lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
13.7 Likelihood ratio-type tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
13.8 Example: ML of Nerlove model, assuming normality . . . . . . . . . . . . . . . . . 188
13.9 Example: Binary response models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
13.10Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
13.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
14 Generalized method of moments
194
14.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
14.2 Definition of GMM estimator

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

14.3 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

14.4 Asymptotic normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
14.5 Choosing the weighting matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
14.6 Estimation of the variance-covariance matrix . . . . . . . . . . . . . . . . . . . . . . 201
14.7 Estimation using conditional moments . . . . . . . . . . . . . . . . . . . . . . . . . 203
14.8 Estimation using dynamic moment conditions . . . . . . . . . . . . . . . . . . . . . 205
14.9 A specification test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
14.10Example: Generalized instrumental variables estimator . . . . . . . . . . . . . . . . 206
14.11Nonlinear simultaneous equations

. . . . . . . . . . . . . . . . . . . . . . . . . . . 212

14.12Maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

14.13Example: OLS as a GMM estimator - the Nerlove model again . . . . . . . . . . . . 215
14.14Example: The MEPS data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
14.15Example: The Hausman Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
14.16Application: Nonlinear rational expectations . . . . . . . . . . . . . . . . . . . . . . 221
14.17Empirical example: a portfolio model . . . . . . . . . . . . . . . . . . . . . . . . . . 223
14.18Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
15 Models for time series data

227

15.1 ARMA models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

15.2 State space models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
15.3 VAR models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
15.4 ARCH and GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
15.5 Nonstationarity and cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
15.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
16 Bayesian methods

241

16.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

16.2 Philosophy, etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
16.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
16.4 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
16.5 Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

16.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

16.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
17 Introduction to panel data

248

17.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

17.2 Static issues and panel data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
17.3 Estimation of the simple linear panel model . . . . . . . . . . . . . . . . . . . . . . 250
17.4 Dynamic panel data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
17.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
18 Quasi-ML

256

18.1 Consistent Estimation of Variance Components . . . . . . . . . . . . . . . . . . . . 257

18.2 Example: the MEPS Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
18.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
19 Nonlinear least squares (NLS)

267

19.1 Introduction and definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

19.2 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
19.3 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
19.4 Asymptotic normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
19.5 Example: The Poisson model for count data . . . . . . . . . . . . . . . . . . . . . . 270
19.6 The Gauss-Newton algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
19.7 Application: Limited dependent variables and sample selection . . . . . . . . . . . 272
20 Nonparametric inference

275

20.1 Possible pitfalls of parametric inference: estimation . . . . . . . . . . . . . . . . . . 275

20.2 Possible pitfalls of parametric inference: hypothesis testing . . . . . . . . . . . . . . 279
20.3 Estimation of regression functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
20.4 Density function estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
20.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
20.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
21 Simulation-based estimation

297

21.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

21.2 Simulated maximum likelihood (SML) . . . . . . . . . . . . . . . . . . . . . . . . . 301
21.3 Method of simulated moments (MSM) . . . . . . . . . . . . . . . . . . . . . . . . . 303
21.4 Efficient method of moments (EMM) . . . . . . . . . . . . . . . . . . . . . . . . . . 305
21.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
21.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
22 Parallel programming for econometrics

313

22.1 Example problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

23 Final project: econometric estimation of a RBC model

318

23.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

23.2 An RBC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
23.3 A reduced form model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
23.4 Results (I): The score generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
23.5 Solving the structural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

24 Introduction to Octave

323

24.1 Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

24.2 A short introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
24.3 If youre running a Linux installation... . . . . . . . . . . . . . . . . . . . . . . . . . 325
25 Notation and Review

326

25.1 Notation for differentiation of vectors and matrices . . . . . . . . . . . . . . . . . . 326

25.2 Convergenge modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
25.3 Rates of convergence and asymptotic equality . . . . . . . . . . . . . . . . . . . . . 329
26 Licenses

331

26.1 The GPL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

26.2 Creative Commons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
27 The attic

343

27.1 Hurdle models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

List of Figures
1.1 Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.2 LYX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

3.1 Typical data, Classical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

3.2 Example OLS Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

3.3 The fit in observation space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

3.4 Detection of influential observations . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.5 Uncentered R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.6 Unbiasedness of OLS under classical assumptions . . . . . . . . . . . . . . . . . . .

26

3.7 Biasedness of OLS when an assumption fails . . . . . . . . . . . . . . . . . . . . . .

26

3.8 Gauss-Markov Result: The OLS estimator . . . . . . . . . . . . . . . . . . . . . . . .

28

3.9 Gauss-Markov Resul: The split sample estimator . . . . . . . . . . . . . . . . . . . .

29

5.1 Joint and Individual Confidence Regions . . . . . . . . . . . . . . . . . . . . . . . .

48

5.2 RTS as a function of firm size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

7.1 s() when there is no collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

7.2 s() when there is collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

7.3 Collinearity: Monte Carlo results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

7.4 with and without measurement error . . . . . . . . . . . . . . . . . . . . . . .

75

7.5 Sample selection bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

9.1 Rejection frequency of 10% t-test, H0 is true. . . . . . . . . . . . . . . . . . . . . .

90

9.2 Motivation for GLS correction when there is HET . . . . . . . . . . . . . . . . . . .

98

9.3 Residuals, Nerlove model, sorted by firm size . . . . . . . . . . . . . . . . . . . . . 100

9.4 Residuals from time trend for CO2 data

. . . . . . . . . . . . . . . . . . . . . . . . 104

9.5 Autocorrelation induced by misspecification . . . . . . . . . . . . . . . . . . . . . . 105

9.6 Efficiency of OLS and FGLS, AR1 errors . . . . . . . . . . . . . . . . . . . . . . . . . 110
9.7 Durbin-Watson critical values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
9.8 Dynamic model with MA(1) errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9.9 Residuals of simple Nerlove model . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.10 OLS residuals, Klein consumption equation . . . . . . . . . . . . . . . . . . . . . . . 118
11.1 Search method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
11.2 Increasing directions of search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
11.3 Newton iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
11.4 Using Sage to get analytic derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 152
11.5 Dwarf mongooses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
11.6 Life expectancy of mongooses, Weibull model . . . . . . . . . . . . . . . . . . . . . 158
11.7 Life expectancy of mongooses, mixed Weibull model . . . . . . . . . . . . . . . . . 160
11.8 A foggy mountain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6

14.1 Asymptotic Normality of GMM estimator, 2 example . . . . . . . . . . . . . . . . . 199

14.2 Inefficient and Efficient GMM estimators, 2 data . . . . . . . . . . . . . . . . . . . 201
14.3 GIV estimation results for , dynamic model with measurement error . . . . . . 211
14.4 OLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
14.5 IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
14.6 Incorrect rank and the Hausman test . . . . . . . . . . . . . . . . . . . . . . . . . . 220
15.1 NYSE weekly close price, 100 log differences . . . . . . . . . . . . . . . . . . . . . 238
16.1 Bayesian estimation, exponential likelihood, lognormal prior . . . . . . . . . . . . . 243
16.2 Chernozhukov and Hong, Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 244
16.3 Metropolis-Hastings MCMC, exponential likelihood, lognormal prior . . . . . . . . 247
20.1 True and simple approximating functions . . . . . . . . . . . . . . . . . . . . . . . . 276
20.2 True and approximating elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
20.3 True function and more flexible approximation . . . . . . . . . . . . . . . . . . . . 278
20.4 True elasticity and more flexible approximation . . . . . . . . . . . . . . . . . . . . 278
20.5 Negative binomial raw moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
20.6 Kernel fitted OBDV usage versus AGE . . . . . . . . . . . . . . . . . . . . . . . . . . 293
20.7 Dollar-Euro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
20.8 Dollar-Yen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
20.9 Kernel regression fitted conditional second moments, Yen/Dollar and Euro/Dollar . 295
22.1 Speedups from parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
23.1 Consumption and Investment, Levels . . . . . . . . . . . . . . . . . . . . . . . . . . 318
23.2 Consumption and Investment, Growth Rates . . . . . . . . . . . . . . . . . . . . . . 319
23.3 Consumption and Investment, Bandpass Filtered . . . . . . . . . . . . . . . . . . . 319
24.1 Running an Octave program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

List of Tables
17.1 Dynamic panel data model. Bias. Source for ML and II is Gouriroux, Phillips and
Yu, 2010, Table 2. SBIL, SMIL and II are exactly identified, using the ML auxiliary
statistic. SBIL(OI) and SMIL(OI) are overidentified, using both the naive and ML
auxiliary statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
17.2 Dynamic panel data model. RMSE. Source for ML and II is Gouriroux, Phillips and
Yu, 2010, Table 2. SBIL, SMIL and II are exactly identified, using the ML auxiliary
statistic. SBIL(OI) and SMIL(OI) are overidentified, using both the naive and ML
auxiliary statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
18.1 Marginal Variances, Sample and Estimated (Poisson) . . . . . . . . . . . . . . . . . 259
18.2 Marginal Variances, Sample and Estimated (NB-II) . . . . . . . . . . . . . . . . . . 262
18.3 Information Criteria, OBDV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
27.1 Actual and Poisson fitted frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . 344
27.2 Actual and Hurdle Poisson fitted frequencies . . . . . . . . . . . . . . . . . . . . . . 348

Chapter 1

About this document

1.1

Prerequisites

These notes have been prepared under the assumption that the reader understands basic statistics,
linear algebra, and mathematical optimization. There are many sources for this material, one are
the appendices to Introductory Econometrics: A Modern Approach by Jeffrey Wooldridge. It is the
students resposibility to get up to speed on this material, it will not be covered in class
This document integrates lecture notes for a one year graduate level course with computer
programs that illustrate and apply the methods that are studied. The immediate availability of
executable (and modifiable) example programs when using the PDF version of the document is a
distinguishing feature of these notes. If printed, the document is a somewhat terse approximation
to a textbook. These notes are not intended to be a perfect substitute for a printed textbook. If
you are a student of mine, please note that last sentence carefully. There are many good textbooks
available. Students taking my courses should read the appropriate sections from at least one of the
following books (or other textbooks with similar level and content)
Cameron, A.C. and P.K. Trivedi, Microeconometrics - Methods and Applications
Davidson, R. and J.G. MacKinnon, Econometric Theory and Methods
Gallant, A.R., An Introduction to Econometric Theory
Hamilton, J.D., Time Series Analysis
Hayashi, F., Econometrics
A more introductory-level reference is Introductory Econometrics: A Modern Approach by Jeffrey
Wooldridge.

1.2

Contents

With respect to contents, the emphasis is on estimation and inference within the world of stationary
data. If you take a moment to read the licensing information in the next section, youll see that
you are free to copy and modify the document. If anyone would like to contribute material that
expands the contents, it would be very welcome. Error corrections and other additions are also
welcome.
The integrated examples (they are on-line here and the support files are here) are an important
part of these notes. GNU Octave (www.octave.org) has been used for most of the example programs, which are scattered though the document. This choice is motivated by several factors. The
9

Figure 1.1: Octave

first is the high quality of the Octave environment for doing applied econometrics. Octave is similar
R and will run scripts for that language without modification1 .
to the commercial package Matlab ,

The fundamental tools (manipulation of matrices, statistical functions, minimization, etc.) exist
and are implemented in a way that make extending them fairly easy. Second, an advantage of free
software is that you dont have to pay for it. This can be an important consideration if you are at a
university with a tight budget or if need to run many copies, as can be the case if you do parallel
computing (discussed in Chapter 22). Third, Octave runs on GNU/Linux, Windows and MacOS.
Figure 1.1 shows a sample GNU/Linux work environment, with an Octave script being edited, and
the results are visible in an embedded shell window. As of 2011, some examples are being added
using Gretl, the Gnu Regression, Econometrics, and Time-Series Library. This is an easy to use
program, available in a number of languages, and it comes with a lot of data ready to use. It runs
on the major operating systems. As of 2012, I am increasingly trying to make examples run on
Matlab, though the need for add-on toolboxes for tasks as simple as generating random numbers
limits what can be done.
The main document was prepared using LYX (www.lyx.org). LYX is a free2 what you see is what
you mean word processor, basically working as a graphical frontend to LATEX. It (with help from
1 Matlab is
R
a trademark of The Mathworks, Inc. Octave will run pure Matlab scripts. If a Matlab script calls an
extension, such as a toolbox function, then it is necessary to make a similar extension available to Octave. The examples
discussed in this document call a number of functions, such as a BFGS minimizer, a program for ML estimation, etc. All of
this code is provided with the examples, as well as on the PelicanHPC live CD image.
2 Free is used in the sense of freedom, but L X is also free of charge (free as in free beer).
Y

Figure 1.2: LYX

other applications) can export your work in LATEX, HTML, PDF and several other forms. It will run
on Linux, Windows, and MacOS systems. Figure 1.2 shows LYX editing this document.

1.3

Licenses

All materials are copyrighted by Michael Creel with the date that appears above. They are provided
under the terms of the GNU General Public License, ver. 2, which forms Section 26.1 of the notes,
or, at your option, under the Creative Commons Attribution-Share Alike 2.5 license, which forms
Section 26.2 of the notes. The main thing you need to know is that you are free to modify and
distribute these materials in any way you like, as long as you share your contributions in the same
way the materials are made available to you. In particular, you must make available the source
files, in editable form, for your modified version of the materials.

1.4

Obtaining the materials

The materials are available on my web page. In addition to the final product, which youre probably
looking at in some form now, you can obtain the editable LYX sources, which will allow you to create
your own version, if you like, or send error corrections and contributions.

1.5

An easy way to use LYX and Octave today

The example programs are available as links to files on my web page in the PDF version, and
here. Support files needed to run these are available here. The files wont run properly from your
browser, since there are dependencies between files - they are only illustrative when browsing. To
see how to use these files (edit and run them), you should go to the home page of this document,
since you will probably want to download the pdf version together with all the support files and
examples. Then set the base URL of the PDF file to point to wherever the Octave files are installed.
Then you need to install Octave and the support files. All of this may sound a bit complicated,
because it is. An easier solution is available:
The PelicanHPC distribution of Linux is an ISO image file that may be burnt to CDROM. It contains a bootable-from-CD GNU/Linux system. These notes, in source form and as a PDF, together
with all of the examples and the software needed to run them are available on PelicanHPC. PelicanHPC is a live CD image. You can burn the PelicanHPC image to a CD and use it to boot your
computer, if you like. When you shut down and reboot, you will return to your normal operating
system. The need to reboot to use PelicanHPC can be somewhat inconvenient. It is also possible
to use PelicanHPC while running your normal operating system by using a virtualization platform
such as Virtualbox 3
The reason why these notes are integrated into a Linux distribution for parallel computing will
be apparent if you get to Chapter 22. If you dont get that far or youre not interested in parallel
computing, please just ignore the stuff on the CD thats not related to econometrics. If you happen
to be interested in parallel computing but not econometrics, just skip ahead to Chapter 22.

3 Virtualbox is free software (GPL v2). That, and the fact that it works very well, is the reason it is recommended
here. There are a number of similar products available. It is possible to run PelicanHPC as a virtual machine, and to
communicate with the installed operating system using a private network. Learning how to do this is not too difficult, and
it is very convenient.

Chapter 2

Introduction: Economic and

econometric models
Heres some data: 100 observations on 3 economic variables. Lets do some exploratory analysis
using Gretl:
histograms
correlations
x-y scatterplots
So, what can we say? Correlations? Yes. Causality? Who knows? This is economic data, generated
by economic agents, following their own beliefs, technologies and preferences. It is not experimental data generated under controlled conditions. How can we determine causality if we dont
have experimental data?

13

Without a model, we cant distinguish correlation from causality. It turns out that the variables
were looking at are QUANTITY (q), PRICE (p), and INCOME (m). Economic theory tells us that
the quantity of a good that consumers will puchase (the demand function) is something like:
q = f (p, m, z)
q is the quantity demanded
p is the price of the good
m is income
z is a vector of other variables that may affect demand
The supply of the good to the market is the aggregation of the firms supply functions. The market
supply function is something like
q = g(p, z)
Suppose we have a sample consisting of a number of observations on q p and m at different time
periods t = 1, 2, ..., n. Supply and demand in each period is
qt = f (pt , mt , zt )
qt = g(pt , zt )
(draw some graphs showing roles of m and z)
This is the basic economic model of supply and demand: q and p are determined in the market
equilibrium, given by the intersection of the two curves. These two variables are determined jointly
by the model, and are the endogenous variables. Income (m) is not determined by this model, its
value is determined independently of q and p by some other process. m is an exogenous variable.
So, m causes q, though the demand function. Because q and p are jointly determined, m also
causes p. p and q do not cause m, according to this theoretical model. q and p have a joint causal
relationship.
Economic theory can help us to determine the causality relationships between correlated
variables.
If we had experimental data, we could control certain variables and observe the outcomes
for other variables. If we see that variable x changes as the controlled value of variable y is
changed, then we know that y causes x. With economic data, we are unable to control the
values of the variables: for example in supply and demand, if price changes, then quantity
changes, but quantity also affect price. We cant control the market price, because the market
price changes as quantity adjusts. This is the reason we need a theoretical model to help us
distinguish correlation and causality.
The model is essentially a theoretical construct up to now:
We dont know the forms of the functions f and g.
Some components of zt may not be observable. For example, people dont eat the same
lunch every day, and you cant tell what they will order just by looking at them. There
are unobservable components to supply and demand, and we can model them as random
variables. Suppose we can break zt into two unobservable components t1 and t2 .

An econometric model attempts to quantify the relationship more precisely. A step toward an
estimable econometric model is to suppose that the model may be written as
qt = 1 + 2 pt + 3 mt + t1
qt = 1 + 2 pt + t1
We have imposed a number of restrictions on the theoretical model:
The functions f and g have been specified to be linear functions
The parameters (1 , 2 , etc.) are constant over time.
There is a single unobservable component in each equation, and we assume it is additive.
If we assume nothing about the error terms t1 and t2 , we can always write the last two equations,
as the errors simply make up the difference between the true demand and supply functions and the
assumed forms. But in order for the coefficients to exist in a sense that has economic meaning,
and in order to be able to use sample data to make reliable inferences about their values, we need
to make additional assumptions. Such assumptions might be something like:
E(tj ) = 0, j = 1, 2
E(pt tj ) = 0, j = 1, 2
E(mt tj ) = 0, j = 1, 2
These are assertions that the errors are uncorrelated with the variables, and such assertions may
or may not be reasonable. Later we will see how such assumption may be used and/or tested.
All of the last six bulleted points have no theoretical basis, in that the theory of supply and
demand doesnt imply these conditions. The validity of any results we obtain using this model will
be contingent on these additional restrictions being at least approximately correct. For this reason,
specification testing will be needed, to check that the model seems to be reasonable. Only when
we are convinced that the model is at least approximately correct should we use it for economic
analysis.
When testing a hypothesis using an econometric model, at least three factors can cause a statistical test to reject the null hypothesis:
1. the hypothesis is false
2. a type I error has occured
3. the econometric model is not correctly specified, and thus the test does not have the assumed
distribution
To be able to make scientific progress, we would like to ensure that the third reason is not contributing in a major way to rejections, so that rejection will be most likely due to either the first
or second reasons. Hopefully the above example makes it clear that econometric models are necessarily more detailed than what we can obtain from economic theory, and that this additional
detail introduces many possible sources of misspecification of econometric models. In the next few
sections we will obtain results supposing that the econometric model is entirely correctly specified.
Later we will examine the consequences of misspecification and see some methods for determining
if a model is correctly specified. Later on, econometric methods that seek to minimize maintained
assumptions are introduced.

Chapter 3

Ordinary Least Squares

3.1

The Linear Model

Consider approximating a variable y using the variables x1 , x2 , ..., xk . We can consider a model
that is a linear approximation:
Linearity: the model is a linear function of the parameter vector 0 :
y

= 10 x1 + 20 x2 + ... + k0 xk + 

or, using vector notation:

y = x0 0 + 
0

The dependent variable y is a scalar random variable, x = ( x1 x2 xk ) is a k-vector of

0
explanatory variables, and 0 = ( 10 20 k0 ) . The superscript 0 in 0 means this is
the true value of the unknown parameter. It will be defined more precisely later, and usually
suppressed when its not necessary for clarity.
Suppose that we want to use data to try to determine the best linear approximation to y using
the variables x. The data {(yt , xt )} , t = 1, 2, ..., n are obtained by some form of sampling1 . An
individual observation is
yt = x0t + t
The n observations can be written in matrix form as
(3.1)

y = X + ,

0

0
is n 1 and X = x1 x2 xn .
y1 y2 yn
Linear models are more general than they might first appear, since one can employ nonlinear

where y =

transformations of the variables:

0 (z) =

1 (w) 2 (w)

p (w)

where the i () are known functions. Defining y = 0 (z), x1 = 1 (w), etc. leads to a model in the
form of equation 3.4. For example, the Cobb-Douglas model
z = Aw22 w33 exp()
1 For

example, cross-sectional data may be obtained by random sampling. Time series data accumulate historically.

16

Figure 3.1: Typical data, Classical Model

10
data
true regression line

-5

-10

-15
0

10
X

12

14

16

18

20

can be transformed logarithmically to obtain

ln z = ln A + 2 ln w2 + 3 ln w3 + .
If we define y = ln z, 1 = ln A, etc., we can put the model in the form needed. The approximation
is linear in the parameters, but not necessarily linear in the variables.

3.2

Estimation by least squares

Figure 3.1, obtained by running TypicalData.m shows some data that follows the linear model
yt = 1 + 2 xt2 + t . The green line is the true regression line 1 + 2 xt2 , and the red crosses are
the data points (xt2 , yt ), where t is a random error that has mean zero and is independent of xt2 .
Exactly how the green line is defined will become clear later. In practice, we only have the data,
and we dont know where the green line lies. We need to gain information about the straight line
that best fits the data points.
The ordinary least squares (OLS) estimator is defined as the value that minimizes the sum of
the squared errors:
=

arg min s()

where

s()

n
X

(yt x0t )

t=1

(y X) (y X)

= y0 y 2y0 X + 0 X0 X
= k y X k2

(3.2)

This last expression makes it clear how the OLS estimator is defined: it minimizes the Euclidean
distance between y and X. The fitted OLS coefficients are those that give the best linear approximation to y using x as basis functions, where best means minimum Euclidean distance. One
could think of other estimators based upon other metrics. For example, the minimum absolute
Pn
distance (MAD) minimizes t=1 |yt x0t |. Later, we will see that which estimator is best in terms
of their statistical properties, rather than in terms of the metrics that define them, depends upon
the properties of , about which we have as yet made no assumptions.
To minimize the criterion s(), find the derivative with respect to :
D s()

2X0 y + 2X0 X

Then setting it to zeros gives

= 2X0 y + 2X0 X 0
D s()
so
= (X0 X)1 X0 y.
To verify that this is a minimum, check the second order sufficient condition:
= 2X0 X
D2 s()
Since (X) = K, this matrix is positive definite, since its a quadratic form in a p.d. matrix
(identity matrix of order n), so is in fact a minimizer.

= X.
The fitted values are the vector y
The residuals are the vector = y X
Note that
y

X +

X +

Also, the first order conditions can be written as

X0 y X0 X = 0


X0 y X
= 0
X0 =

which is to say, the OLS residuals are orthogonal to X. Lets look at this more carefully.

3.3

Geometric interpretation of least squares estimation

In X, Y Space
Figure 3.2 shows a typical fit to data, along with the true regression line. Note that the true line and
the estimated line are different. This figure was created by running the Octave program OlsFit.m .
You can experiment with changing the parameter values to see how this affects the fit, and to see
how the fitted line will sometimes be close to the true line, and sometimes rather far away.

Figure 3.2: Example OLS Fit

15
data points
fitted line
true line
10

-5

-10

-15
0

10
X

12

14

16

18

20

In Observation Space
If we want to plot in observation space, well need to use only two or three observations, or well
encounter some limitations of the blackboard. If we try to use 3, well encounter the limits of my
artistic ability, so lets use two. With only two observations, we cant have K > 1.
We can decompose y into two components: the orthogonal projection onto the Kdimensional
and the component that is the orthogonal projection onto the nK
space spanned by X, X ,
subpace that is orthogonal to the span of X, .
Since is chosen to make as short as possible, will be orthogonal to the space spanned
by X. Since X is in this space, X 0 = 0. Note that the f.o.c. that define the least squares
estimator imply that this is so.

Projection Matrices
X is the projection of y onto the span of X, or
X = X (X 0 X)

X 0y

Therefore, the matrix that projects y onto the span of X is

PX = X(X 0 X)1 X 0
since
X = PX y.

Figure 3.3: The fit in observation space

Observation 2

e = M_xY

S(x)

x
x*beta=P_xY

Observation 1

is the projection of y onto the N K dimensional space that is orthogonal to the span of X. We
have that
y X

y X(X 0 X)1 X 0 y


In X(X 0 X)1 X 0 y.

=
=

So the matrix that projects y onto the space orthogonal to the span of X is
= In X(X 0 X)1 X 0

MX

= I n PX .
We have
= MX y.
Therefore
y

= PX y + M X y
= X + .

These two projection matrices decompose the n dimensional vector y into two orthogonal components - the portion that lies in the K dimensional space defined by X, and the portion that lies in
the orthogonal n K dimensional space.
Note that both PX and MX are symmetric and idempotent.
A symmetric matrix A is one such that A = A0 .
An idempotent matrix A is one such that A = AA.
The only nonsingular idempotent matrix is the identity matrix.

3.4

Influential observations and outliers

The OLS estimator of the ith element of the vector 0 is simply

i

 0 1 0 
(X X) X i y

= c0i y
This is how we define a linear estimator - its a linear function of the dependent variable. Since
its a linear combination of the observations on the dependent variable, where the weights are
determined by the observations on the regressors, some observations may have more influence
than others.
To investigate this, let et be an n vector of zeros with a 1 in the tth position, i.e., its the
tth column of the matrix In . Define
ht

(PX )tt

= e0t PX et
so ht is the tth element on the main diagonal of PX . Note that
= k PX et k2

ht
so

ht k et k2 = 1
So 0 < ht < 1. Also,
T rPX = K h = K/n.
So the average of the ht is K/n. The value ht is referred to as the leverage of the observation. If
the leverage is much higher than average, the observation has the potential to affect the OLS fit
importantly. However, an observation may also be influential due to the value of yt , rather than
the weight it is multiplied by, which only depends on the xt s.
To account for this, consider estimation of without using the tth observation (designate this
estimator as (t) ). One can show (see Davidson and MacKinnon, pp. 32-5 for proof) that
(t) =

1
1 ht

(X 0 X)1 Xt0 t

so the change in the tth observations fitted value is

x0t

x0t (t)


=

ht
1 ht


t

While an observation may be influential if it doesnt affect its own fitted value,
is
 it certainly

ht
influential if it does. A fast means of identifying influential observations is to plot 1ht t (which
I will refer to as the own influence of the observation) as a function of t. Figure 3.4 gives an example
plot of data, fit, leverage and influence. The Octave program is InfluentialObservation.m. (note
to self when lecturing: load the data ../OLS/influencedata into Gretl and reproduce this). If
you re-run the program you will see that the leverage of the last observation (an outlying value of
x) is always high, and the influence is sometimes high.
After influential observations are detected, one needs to determine why they are influential.
Possible causes include:
data entry error, which can easily be corrected once detected. Data entry errors are very

Figure 3.4: Detection of influential observations

14
Data points
fitted
Leverage
Influence

12

10

0
0

0.5

1.5

2.5

3.5

common.
special economic factors that affect some observations. These would need to be identified
and incorporated in the model. This is the idea behind structural change: the parameters
may not be constant across all observations.
pure randomness may have caused us to sample a low-probability observation.
There exist robust estimation methods that downweight outliers.

3.5

Goodness of fit

The fitted model is

y = X +
Take the inner product:
y 0 y = 0 X 0 X + 20 X 0 + 0
But the middle term of the RHS is zero since X 0 = 0, so
y 0 y = 0 X 0 X + 0
The uncentered Ru2 is defined as
Ru2

=
=
=
=

0
y0 y

0 X 0 X
y0 y
k PX y k2
k y k2
cos2 (),

(3.3)

where is the angle between y and the span of X .

The uncentered R2 changes if we add a constant to y, since this changes (see Figure 3.5,
the yellow vector is a constant, since its on the 45 degree line in observation space). Another,
Figure 3.5: Uncentered R2

more common definition measures the contribution of the variables, other than the constant
term, to explaining the variation in y. Thus it measures the ability of the model to explain
the variation of y about its unconditional sample mean.
Let = (1, 1, ..., 1)0 , a n -vector. So
M

In (0 )1 0

In 0 /n

M y just returns the vector of deviations from the mean. In terms of deviations from the mean,
equation 3.3 becomes
y 0 M y = 0 X 0 M X + 0 M
The centered Rc2 is defined as
Rc2 = 1
where ESS = 0 and T SS = y 0 M y=

Pn

0
ESS
=1
0
y M y
T SS

t=1 (yt

y)2 .

Supposing that X contains a column of ones (i.e., there is a constant term),

X 0 = 0

t = 0

so M = . In this case
y 0 M y = 0 X 0 M X + 0

So
Rc2 =

RSS
T SS

where RSS = 0 X 0 M X
Supposing that a column of ones is in the space spanned by X (PX = ), then one can show
that 0 Rc2 1.

3.6

The classical linear regression model

Up to this point the model is empty of content beyond the definition of a best linear approximation
to y and some geometrical properties. There is no economic content to the model, and the regression parameters have no economic interpretation. For example, what is the partial derivative of y
with respect to xj ? The linear approximation is
y = 1 x1 + 2 x2 + ... + k xk + 
The partial derivative is
y

= j +
xj
xj
Up to now, theres no guarantee that


xj =0.

For the to have an economic meaning, we need

to make additional assumptions. The assumptions that are appropriate to make depend on the
data under consideration. Well start with the classical linear regression model, which incorporates
some assumptions that are clearly not realistic for economic data. This is to be able to explain
some concepts with a minimum of confusion and notational clutter. Later well adapt the results
to what we can get with more realistic assumptions.
Linearity: the model is a linear function of the parameter vector 0 :
y

= 10 x1 + 20 x2 + ... + k0 xk + 

(3.4)

or, using vector notation:

y = x0 0 + 
Nonstochastic linearly independent regressors: X is a fixed matrix of constants, it has rank
K equal to its number of columns, and
lim

1 0
X X = QX
n

(3.5)

where QX is a finite positive definite matrix. This is needed to be able to identify the individual
effects of the explanatory variables.
Independently and identically distributed errors:
 IID(0, 2 In )

(3.6)

is jointly distributed IID. This implies the following two properties:

Homoscedastic errors:
V (t ) = 02 , t

(3.7)

E(t s ) = 0, t 6= s

(3.8)

Nonautocorrelated errors:

Optionally, we will sometimes assume that the errors are normally distributed.
Normally distributed errors:
 N (0, 2 In )

3.7

(3.9)

Small sample statistical properties of the least squares estimator

Up to now, we have only examined numeric properties of the OLS estimator, that always hold. Now
we will examine statistical properties. The statistical properties depend upon the assumptions we
make.

Unbiasedness
We have = (X 0 X)1 X 0 y. By linearity,
=

(X 0 X)1 X 0 (X + )

= + (X 0 X)1 X 0
By 3.5 and 3.6
E(X 0 X)1 X 0 =

E(X 0 X)1 X 0

(X 0 X)1 X 0 E

so the OLS estimator is unbiased under the assumptions of the classical model.
Figure 3.6 shows the results of a small Monte Carlo experiment where the OLS estimator was
calculated for 10000 samples from the classical model with y = 1 + 2x + , where n = 20, 2 = 9,
and x is fixed across samples. We can see that the 2 appears to be estimated without bias. The
program that generates the plot is Unbiased.m , if you would like to experiment with this.
With time series data, the OLS estimator will often be biased. Figure 3.7 shows the results of
a small Monte Carlo experiment where the OLS estimator was calculated for 1000 samples from
the AR(1) model with yt = 0 + 0.9yt1 + t , where n = 20 and 2 = 1. In this case, assumption
3.5 does not hold: the regressors are stochastic. We can see that the bias in the estimation of 2 is
about -0.2.
The program that generates the plot is Biased.m , if you would like to experiment with this.

Normality
With the linearity assumption, we have = + (X 0 X)1 X 0 . This is a linear function of . Adding
the assumption of normality (3.9, which implies strong exogeneity), then
N , (X 0 X)1 02

since a linear function of a normal random vector is also normally distributed. In Figure 3.6 you
can see that the estimator appears to be normally distributed. It in fact is normally distributed,
since the DGP (see the Octave program) has normal errors. Even when the data may be taken
to be IID, the assumption of normality is often questionable or simply untenable. For example, if
the dependent variable is the number of automobile trips per week, it is a count variable with a

Figure 3.6: Unbiasedness of OLS under classical assumptions

0.1

0.08

0.06

0.04

0.02

0
-3

-2

-1

Figure 3.7: Biasedness of OLS when an assumption fails

0.12

0.1

0.08

0.06

0.04

0.02

0
-1

-0.8

-0.6

-0.4

-0.2

0.2

0.4

discrete distribution, and is thus not normally distributed. Many variables in economics can take
on only nonnegative values, which, strictly speaking, rules out normality.2

The variance of the OLS estimator and the Gauss-Markov theorem

Now lets make all the classical assumptions except the assumption of normality. We have =
= . So
+ (X 0 X)1 X 0 and we know that E()

V ar()



0 



E (X 0 X)1 X 0 0 X(X 0 X)1

(X 0 X)1 02

The OLS estimator is a linear estimator, which means that it is a linear function of the dependent
variable, y.
=

 0 1 0 
(X X) X y

= Cy
where C is a function of the explanatory variables only, not the dependent variable. It is also
unbiased under the present assumptions, as we proved above. One could consider other weights W
that are a function of X that define some other linear estimator. Well still insist upon unbiasedness.
Consider = W y, where W = W (X) is some k n matrix function of X. Note that since W is a
function of X, it is nonstochastic, too. If the estimator is unbiased, then we must have W X = IK :
E(W y)

E(W X0 + W )

W X0

WX

IK

The variance of is
= W W 0 2 .
V ()
0
Define
D = W (X 0 X)1 X 0
so
W = D + (X 0 X)1 X 0
Since W X = IK , DX = 0, so

V ()

=
=

0
D + (X 0 X)1 X 0 D + (X 0 X)1 X 0 02


1
DD0 + (X 0 X)
02

So
V ()

V ()
V ()
is a positive
The inequality is a shorthand means of expressing, more formally, that V ()
2 Normality may be a good model nonetheless, as long as the probability of a negative value occuring is negligable under
the model. This depends upon the mean being large enough in relation to the variance.

Figure 3.8: Gauss-Markov Result: The OLS estimator

0.12

0.1

0.08

0.06

0.04

0.02

0
0

0.5

1.5

2.5

3.5

semi-definite matrix. This is a proof of the Gauss-Markov Theorem. The OLS estimator is the best
linear unbiased estimator (BLUE).
It is worth emphasizing again that we have not used the normality assumption in any way to
prove the Gauss-Markov theorem, so it is valid if the errors are not normally distributed, as
long as the other assumptions hold.
To illustrate the Gauss-Markov result, consider the estimator that results from splitting the sample
into p equally-sized parts, estimating using each part of the data separately by OLS, then averaging
the p resulting estimators. You should be able to show that this estimator is unbiased, but inefficient
with respect to the OLS estimator. The program Efficiency.m illustrates this using a small Monte
Carlo experiment, which compares the OLS estimator and a 3-way split sample estimator. The data
generating process follows the classical model, with n = 21. The true parameter value is = 2.
In Figures 3.8 and 3.9 we can see that the OLS estimator is more efficient, since the tails of its
histogram are more narrow.

1
= and V ar()
= X0X
2 , but we still need to estimate the variance
We have that E()
0

of , 02 , in order to have an idea of the precision of the estimates of . A commonly used estimator
of 02 is
c2 =

0
This estimator is unbiased:

1
0
nK

Figure 3.9: Gauss-Markov Resul: The split sample estimator

0.12

0.1

0.08

0.06

0.04

0.02

0
0

c2

=
=

c2 )
E(
0

=
=
=
=
=
=

1
0
nK
1
0 M
nK
1
E(T r0 M )
nK
1
E(T rM 0 )
nK
1
T rE(M 0 )
nK
1
2 T rM
nK 0
1
2 (n k)
nK 0
02

where we use the fact that T r(AB) = T r(BA) when both products are conformable. Thus, this
estimator is also unbiased under these assumptions.

3.8

Example: The Nerlove model

Theoretical background
For a firm that takes input prices w and the output level q as given, the cost minimization problem
is to choose the quantities of inputs x to solve the problem
min w0 x
x

subject to the restriction

f (x) = q.
The solution is the vector of factor demands x(w, q). The cost function is obtained by substituting
the factor demands into the criterion function:
Cw, q) = w0 x(w, q).
Monotonicity Increasing factor prices cannot decrease cost, so
C(w, q)
0
w
Remember that these derivatives give the conditional factor demands (Shephards Lemma).
Homogeneity The cost function is homogeneous of degree 1 in input prices: C(tw, q) =
tC(w, q) where t is a scalar constant. This is because the factor demands are homogeneous
of degree zero in factor prices - they only depend upon relative prices.
Returns to scale The returns to scale parameter is defined as the inverse of the elasticity of
cost with respect to output:

=

C(w, q)
q
q
C(w, q)

1

Constant returns to scale is the case where increasing production q implies that cost increases
in the proportion 1:1. If this is the case, then = 1.

Cobb-Douglas functional form

The Cobb-Douglas functional form is linear in the logarithms of the regressors and the dependent
variable. For a cost function, if there are g factors, the Cobb-Douglas cost function has the form
C = Aw11 ...wgg q q e
What is the elasticity of C with respect to wj ?
eC
wj


=

C
WJ



wj 
C
1

= j Aw11 .wj j

..wgg q q e

wj

1
Aw1 ...wg g q q e

= j
This is one of the reasons the Cobb-Douglas form is popular - the coefficients are easy to interpret,
since they are the elasticities of the dependent variable with respect to the explanatory variable.
Not that in this case,
eC
wj



C
WJ

xj (w, q)

sj (w, q)

wj 
C

wj
C

the cost share of the j th input. So with a Cobb-Douglas cost function, j = sj (w, q). The cost shares
are constants.

Note that after a logarithmic transformation we obtain

ln C = + 1 ln w1 + ... + g ln wg + q ln q + 
where = ln A . So we see that the transformed model is linear in the logs of the data.
One can verify that the property of HOD1 implies that
g
X

g = 1

i=1

In other words, the cost shares add up to 1.

The hypothesis that the technology exhibits CRTS implies that
=

1
=1
q

so q = 1. Likewise, monotonicity implies that the coefficients i 0, i = 1, ..., g.

The Nerlove data and OLS

The file nerlove.data contains data on 145 electric utility companies cost of production, output
and input prices. The data are for the U.S., and were collected by M. Nerlove. The observations
are by row, and the columns are COMPANY, COST (C), OUTPUT (Q), PRICE OF LABOR (PL ),
PRICE OF FUEL (PF ) and PRICE OF CAPITAL (PK ). Note that the data are sorted by output level
(the third column).
We will estimate the Cobb-Douglas model
ln C = 1 + 2 ln Q + 3 ln PL + 4 ln PF + 5 ln PK + 

(3.10)

using OLS. To do this yourself, you need the data file mentioned above, as well as Nerlove.m (the
estimation program), and the library of Octave functions mentioned in the introduction to Octave
that forms section 24 of this document.3
The results are

*********************************************************
OLS estimation results
Observations 145
R-squared 0.925955
Sigma-squared 0.153943
Results (Ordinary var-cov estimator)
constant
output
labor
fuel
capital

estimate
-3.527
0.720
0.436
0.427
-0.220

st.err.
1.774
0.017
0.291
0.100
0.339

t-stat.
-1.987
41.244
1.499
4.249
-0.648

p-value
0.049
0.000
0.136
0.000
0.518

*********************************************************
Do the theoretical restrictions hold?
Does the model fit well?
3 If

you are running the bootable CD, you have all of this installed and ready to run.

 What do you think about RTS?

While we will most often use Octave programs as examples in this document, since following the
programming statements is a useful way of learning how theory is put into practice, you may be
interested in a more user-friendly environment for doing econometrics. I heartily recommend
Gretl, the Gnu Regression, Econometrics, and Time-Series Library. This is an easy to use program,
available in English, French, and Spanish, and it comes with a lot of data ready to use. It even has
an option to save output as LATEX fragments, so that I can just include the results into this document,
no muss, no fuss. Here is the Nerlove data in the form of a GRETL data set: nerlove.gdt . Here the
results of the Nerlove model from GRETL:
Model 2: OLS estimates using the 145 observations 1145
Dependent variable: l_cost
Variable

Coefficient

Std. Error

3.5265

t-statistic

p-value

1.77437

1.9875

0.0488

l_output

0.720394

0.0174664

41.2445

0.0000

l_labor

0.436341

0.291048

1.4992

0.1361

l_fuel

0.426517

0.100369

4.2495

0.0000

0.219888

0.339429

0.6478

0.5182

const

l_capita

Mean of dependent variable

S.D. of dependent variable
Sum of squared residuals
Standard error of residuals (
)
Unadjusted R
2
Adjusted R

1.72466
1.42172
21.5520
0.392356
0.925955
0.923840

F (4, 140)

437.686

Akaike information criterion

145.084

Schwarz Bayesian criterion

159.967

Fortunately, Gretl and my OLS program agree upon the results. Gretl is included in the bootable
CD mentioned in the introduction. I recommend using GRETL to repeat the examples that are
done using Octave.
The previous properties hold for finite sample sizes. Before considering the asymptotic properties of the OLS estimator it is useful to review the MLE estimator, since under the assumption of
normal errors the two estimators coincide.

3.9

Exercises

1. Prove that the split sample estimator used to generate figure 3.9 is unbiased.
2. Calculate the OLS estimates of the Nerlove model using Octave and GRETL, and provide
printouts of the results. Interpret the results.
3. Do an analysis of whether or not there are influential observations for OLS estimation of the
Nerlove model. Discuss.
4. Using GRETL, examine the residuals after OLS estimation and tell me whether or not you believe that the assumption of independent identically distributed normal errors is warranted.
No need to do formal tests, just look at the plots. Print out any that you think are relevant,
and interpret them.

5. For a random vector X N (x , ), what is the distribution of AX + b, where A and b are

conformable matrices of constants?
6. Using Octave, write a little program that verifies that T r(AB) = T r(BA) for A and B 4x4
matrices of random numbers. Note: there is an Octave function trace.
7. For the model with a constant and a single regressor, yt = 1 + 2 xt + t , which satisfies the
classical assumptions, prove that the variance of the OLS estimator declines to zero as the
sample size increases.

Chapter 4

Asymptotic properties of the least

squares estimator
The OLS estimator under the classical assumptions is BLUE1 , for all sample sizes. Now lets see
what happens when the sample size tends to infinity.

4.1

Consistency
=
=

(X 0 X)1 X 0 y
(X 0 X)1 X 0 (X + )

= 0 + (X 0 X)1 X 0
 0 1 0
XX
X
= 0 +
n
n
Consider the last two terms. By assumption limn

X0X
n

= QX limn

X0X
n

1

= Q1
X ,

since the inverse of a nonsingular matrix is a continuous function of the elements of the matrix.
0
Considering Xn ,
n
X 0
1X
=
xt t
n
n t=1
Each xt t has expectation zero, so

E

X 0
n


=0

The variance of each term is

V (xt t )

xt x0t 2 .

As long as these are finite, and given a technical condition2 , the Kolmogorov SLLN applies, so
n

1X
a.s.
xt t 0.
n t=1
1 BLUE

best linear unbiased estimator if I havent defined it before

application of LLNs and CLTs, of which there are very many to choose from, Im going to avoid the technicalities.
Basically, as long as terms that make up an average have finite variances and are not too strongly dependent, one will be
able to find a LLN or CLT to apply. Which one it is doesnt matter, we only need the result.
2 For

34

This implies that

a.s.
0 .

This is the property of strong consistency: the estimator converges in almost surely to the true
value.
The consistency proof does not use the normality assumption.
Remember that almost sure convergence implies convergence in probability.

4.2

Asymptotic normality

Weve seen that the OLS estimator is normally distributed under the assumption of normal errors.
If the error distribution is unknown, we of course dont know the distribution of the estimator.
However, we can get asymptotic results. Assuming the distribution of is unknown, but the the
other classical assumptions hold:

0 + (X 0 X)1 X 0

0 = (X 0 X)1 X 0
 0 1 0


XX
X

n 0
=
n
n
Now as before,
Considering

0
X
,
n

X0X
n

1

Q1
X .

the limit of the variance is


lim V

X 0


=

lim E

X 0 0 X
n

= 02 QX
The mean is of course zero. To get asymptotic normality, we need to apply a CLT. We assume
one (for instance, the Lindeberg-Feller CLT) holds, so


X 0 d
N 0, 02 QX
n
Therefore,





d
n 0 N 0, 02 Q1
X

(4.1)

In summary, the OLS estimator is normally distributed in small and large samples if is
normally distributed. If is not normally distributed, is asymptotically normally distributed
when a CLT can be applied.

4.3

Asymptotic efficiency

The least squares objective function is

s()

n
X
t=1

(yt x0t )

Supposing that is normally distributed, the model is

y = X0 + ,

N (0, 02 In ), so


n
Y
2t
1

f () =
exp 2
2
2 2
t=1
The joint density for y can be constructed using a change of variables. We have = y X, so

y 0

= In and | y
0 | = 1, so

f (y) =

n
Y
t=1



(yt x0t )2
.
exp
2 2
2 2
1

Taking logs,
n
2
X

(yt x0t )
ln L(, ) = n ln 2 n ln
.
2 2
t=1

Maximizing this function with respect to and gives what is known as the maximum likelihood
(ML) estimator. It turns out that ML estimators are asymptotically efficient, a concept that will be
explained in detail later. Its clear that the first order conditions for the MLE of 0 are the same
as the first order conditions that define the OLS estimator (up to multiplication by a constant), so
the OLS estimator of is also the ML estimator. The estimators are the same, under the present
assumptions. Therefore, their properties are the same. In particular, under the classical assumptions
with normality, the OLS estimator is asymptotically efficient. Note that one needs to make an
assumption about the distribution of the errors to compute the ML estimator. If the errors had
a distribution other than the normal, then the OLS estimator and the ML estimator would not
coincide.
As well see later, it will be possible to use (iterated) linear estimation methods and still achieve
asymptotic efficiency even if the assumption that V ar() 6= 2 In , as long as is still normally
distributed. This is not the case if is nonnormal. In general with nonnormal errors it will be
necessary to use nonlinear estimation methods to achieve asymptotically efficient estimation.

4.4

Exercises

1. Write
an Octave
program that generates a histogram for R Monte Carlo replications of


n j j , where is the OLS estimator and j is one of the k slope parameters. R
should be a large number, at least 1000. The model used to generate data should follow the
classical assumptions, except that the errors should not be normally distributed (try U (a, a),
t(p), 2 (p)p, etc). Generate histograms for n {20, 50, 100, 1000}. Do you observe evidence
of asymptotic normality? Comment.

Chapter 5

Restrictions and hypothesis tests

5.1

Exact linear restrictions

In many cases, economic theory suggests restrictions on the parameters of a model. For example, a
demand function is supposed to be homogeneous of degree zero in prices and income. If we have
a Cobb-Douglas (log-linear) model,
ln q = 0 + 1 ln p1 + 2 ln p2 + 3 ln m + ,
then we need that
k 0 ln q = 0 + 1 ln kp1 + 2 ln kp2 + 3 ln km + ,
so
1 ln p1 + 2 ln p2 + 3 ln m

= 1 ln kp1 + 2 ln kp2 + 3 ln km
=

(ln k) (1 + 2 + 3 ) + 1 ln p1 + 2 ln p2 + 3 ln m.

The only way to guarantee this for arbitrary k is to set

1 + 2 + 3 = 0,
which is a parameter restriction. In particular, this is a linear equality restriction, which is probably
the most commonly encountered case.

Imposition
The general formulation of linear equality restrictions is the model
y
R

= X +
= r

where R is a Q K matrix, Q < K and r is a Q 1 vector of constants.

We assume R is of rank Q, so that there are no redundant restrictions.
We also assume that that satisfies the restrictions: they arent infeasible.

37

Lets consider how to estimate subject to the restrictions R = r. The most obvious approach is
to set up the Lagrangean
min s() =

1
0
(y X) (y X) + 20 (R r).
n

The Lagrange multipliers are scaled by 2, which makes things less messy. The fonc are
)

D s(,

0
2X 0 y + 2X 0 X R + 2R0

D s(,

RR r 0,

which can be written as

"

X 0X

R0

#"

"

X 0y

We get
"

"
=

X 0X

R0

#1 "

X 0y

#
.

Maybe youre curious about how to invert a partitioned matrix? I can help you with that:
Note that
"

(X 0 X)

R (X 0 X)

#"

IQ

X 0X

R0

AB
"

=
"

IK

(X 0 X)

R (X 0 X)

IK

R0

(X 0 X)

R0

R0
#

0
C,

and
"

IK

(X 0 X)1 R0 P 1

P 1

#"

IK
0

(X 0 X)

R0

#
DC
= IK+Q ,

so
DAB

= IK+Q

DA = "
B 1
#
#"
1
IK (X 0 X)1 R0 P 1
(X 0 X)
0
1
B
=
1
0
P 1
R (X 0 X)
IQ
"
#
1
1
(X 0 X) (X 0 X)1 R0 P 1 R (X 0 X)
(X 0 X)1 R0 P 1
=
,
1
P 1 R (X 0 X)
P 1

If you werent curious about that, please start paying attention again. Also, note that we have
made the definition P = R (X 0 X)
"

"
=

(X 0 X)

R0 )
1

(X 0 X)1 R0 P 1 R (X 0 X)

(X 0 X)1 R0 P 1

P 1 R (X 0 X)
P 1


(X 0 X)1 R0 P 1 R r




P 1 R r
"
"
#

#
0
1 0 1
IK (X 0 X)1 R0 P 1 R
(X
X)
R
P
r
+
P 1 R
P 1 r

#"

X 0y

are linear functions of makes it easy to determine their distributions, since

The fact that R and
the distribution of is already known. Recall that for x a random vector, and for A and b a matrix
and vector of constants, respectively, V ar (Ax + b) = AV ar(x)A0 .
Though this is the obvious way to go about finding the restricted estimator, an easier way, if the
number of restrictions is small, is to impose them by substitution. Write

R1

R2

"

1
2

y
#

= X1 1 + X2 2 +
= r

where R1 is Q Q nonsingular. Supposing the Q restrictions are linearly independent, one can
always make R1 nonsingular by reorganizing the columns of X. Then
1 = R11 r R11 R2 2 .
Substitute this into the model
y

y X1 R11 r

X1 R11 r X1 R11 R2 2 + X2 2 +


X2 X1 R11 R2 2 +

or with the appropriate definitions,

yR = XR 2 + .
This model satisfies the classical assumptions, supposing the restriction is true. One can estimate by
OLS. The variance of 2 is as before
1
0
V (2 ) = (XR
XR ) 02

and the estimator is

1 2
0
V (2 ) = (XR
XR )

where one estimates 02 in the normal way, using the restricted model, i.e.,

c2 =

c2
yR XR

0 

c2
yR XR

n (K Q)

To recover 1 , use the restriction. To find the variance of 1 , use the fact that it is a linear function
of 2 , so
V (1 )

R11 R2 V (2 )R20 R11

R11 R2 (X20 X2 )

0

0
R20 R11 02

Properties of the restricted estimator

We have that
R



= (X 0 X)1 R0 P 1 R r
= + (X 0 X)1 R0 P 1 r (X 0 X)1 R0 P 1 R(X 0 X)1 X 0 y
= + (X 0 X)1 X 0 + (X 0 X)1 R0 P 1 [r R] (X 0 X)1 R0 P 1 R(X 0 X)1 X 0

(X 0 X)1 X 0

(X 0 X)1 R0 P 1 [r R]

(X 0 X)1 R0 P 1 R(X 0 X)1 X 0

Mean squared error is
M SE(R ) = E(R )(R )0
Noting that the crosses between the second term and the other terms expect to zero, and that the
cross of the first and third has a cancellation with the square of the third, we obtain
M SE(R )

(X 0 X)1 2

(X 0 X)1 R0 P 1 [r R] [r R] P 1 R(X 0 X)1

(X 0 X)1 R0 P 1 R(X 0 X)1 2

So, the first term is the OLS covariance. The second term is PSD, and the third term is NSD.
If the restriction is true, the second term is 0, so we are better off. True restrictions improve
efficiency of estimation.
If the restriction is false, we may be better or worse off, in terms of MSE, depending on the
magnitudes of r R and 2 .

5.2

Testing

In many cases, one wishes to test economic theories. If theory suggests parameter restrictions, as in
the above homogeneity example, one can test theory by testing parameter restrictions. A number
of tests are available. The first two (t and F) have a known small sample distributions, when the
errors are normally distributed. The third and fourth (Wald and score) do not require normality of
the errors, but their distributions are known only approximately, so that they are not exactly valid
with finite samples.

t-test
Suppose one has the model
y = X +

and one wishes to test the single restriction H0 :R = r vs. HA :R 6= r . Under H0 , with normality
of the errors,
R r N 0, R(X 0 X)1 R0 02
so

R r
p

R(X 0 X)1 R0 02

R r
0

R(X 0 X)1 R0

N (0, 1) .

c2 in place of 2 , but
The problem is that 02 is unknown. One could use the consistent estimator
0
0
the test would only be valid asymptotically in this case.
Proposition 1.

N (0,1)
q

2 (q)
q

t(q)

as long as the N (0, 1) and the 2 (q) are independent.

We need a few results on the 2 distribution.
Proposition 2. If x N (, In ) is a vector of n independent r.v.s., then x0 x 2 (n, ) where
P
= i 2i = 0 is the noncentrality parameter.
When a 2 r.v. has the noncentrality parameter equal to zero, it is referred to as a central 2
r.v., and its distribution is written as 2 (n), suppressing the noncentrality parameter.
Proposition 3. If the n dimensional random vector x N (0, V ), then x0 V 1 x 2 (n).
Well prove this one as an indication of how the following unproven propositions could be
proved.
Proof: Factor V 1 as P 0 P (this is the Cholesky factorization, where P is defined to be upper
triangular). Then consider y = P x. We have
y N (0, P V P 0 )
but
V P 0P

In

P V P 0P

so P V P 0 = In and thus y N (0, In ). Thus y 0 y 2 (n) but

y 0 y = x0 P 0 P x = xV 1 x
and we get the result we wanted.
A more general proposition which implies this result is
Proposition 4. If the n dimensional random vector x N (0, V ), then x0 Bx 2 ((B)) if and only
if BV is idempotent.
An immediate consequence is
Proposition 5. If the random vector (of dimension n) x N (0, I), and B is idempotent with rank r,
then x0 Bx 2 (r).
Consider the random variable
0
02

=
=

0 MX
02
 0
 

MX
0
0
2 (n K)

Proposition 6. If the random vector (of dimension n) x N (0, I), then Ax and x0 Bx are independent if AB = 0.
Now consider (remember that we have only one restriction in this case)

Rr
R(X 0 X)1 R0
q
=
0

(nK)02

R r

c0

R(X 0 X)1 R0

This will have the t(n K) distribution if and 0 are independent. But = + (X 0 X)1 X 0
and
(X 0 X)1 X 0 MX = 0,
so

R r
R r
p
t(n K)
=

c0 R(X 0 X)1 R0

In particular, for the commonly encountered test of significance of an individual coefficient, for
which H0 : i = 0 vs. H0 : i 6= 0 , the test statistic is
i
t(n K)

Note: the t test is strictly valid only if the errors are actually normally distributed. If one has
nonnormal errors, one could use the above asymptotic result to justify taking critical values
d

from the N (0, 1) distribution, since t(n K) N (0, 1) as n . In practice, a conservative

procedure is to take critical values from the t distribution if nonnormality is suspected. This
will reject H0 less often since the t distribution is fatter-tailed than is the normal.

F test
The F test allows testing multiple restrictions jointly.
Proposition 7. If x 2 (r) and y 2 (s), then

x/r
y/s

F (r, s), provided that x and y are indepen-

dent.

Proposition 8. If the random vector (of dimension n) x N (0, I), then x0 Ax

and x0 Bx are independent if AB = 0.
Using these results, and previous results on the 2 distribution, it is simple to show that the
following statistic has the F distribution:


R r

0 

R (X 0 X)

F =

q
2

R0

1 

R r


F (q, n K).

A numerically equivalent expression is

(ESSR ESSU ) /q
F (q, n K).
ESSU /(n K)
Note: The F test is strictly valid only if the errors are truly normally distributed. The following tests will be appropriate when one cannot assume normally distributed errors.

Wald-type tests
The t and F tests require normality of the errors. The Wald test does not, but it is an asymptotic
test - it is only approximately valid in finite samples.
The Wald principle is based on the idea that if a restriction is true, the unrestricted model should
approximately satisfy the restriction. Given that the least squares estimator is asymptotically
normally distributed:





d
n 0 N 0, 02 Q1
X

then under H0 : R0 = r, we have





d
0
n R r N 0, 02 RQ1
X R
so by Proposition [3]

0



d
0 1
r
n R r
02 RQ1
R
R

2 (q)
X
2
Note that Q1
X or 0 are not observable. The test statistic we use substitutes the consistent estima0
tors. Use (X 0 X/n)1 as the consistent estimator of Q1
X . With this, there is a cancellation of n s,

and the statistic to use is



R r

0 

c2 R(X 0 X)1 R0

1 


d
R r 2 (q)

The Wald test is a simple way to test restrictions without having to estimate the restricted
model.
Note that this formula is similar to one of the formulae provided for the F test.

Score-type tests (Rao tests, Lagrange multiplier tests)

The score test is another asymptotically valid test that does not require normality of the errors.
In some cases, an unrestricted model may be nonlinear in the parameters, but the model is
linear in the parameters under the null hypothesis. For example, the model

y = (X) +
is nonlinear in and , but is linear in under H0 : = 1. Estimation of nonlinear models is a bit
more complicated, so one might prefer to have a test based upon the restricted, linear model. The
score test is useful in this situation.
Score-type tests are based upon the general principle that the gradient vector of the unrestricted model, evaluated at the restricted estimate, should be asymptotically normally
distributed with mean zero, if the restrictions are true. The original development was for ML
estimation, but the principle is valid for a wide variety of estimation methods.
We have seen that

so



1 
R(X 0 X)1 R0
R r


= P 1 R r

nP =



n R r

Given that





d
0
n R r N 0, 02 RQ1
X R

under the null hypothesis, we obtain

So

nP

0
nP N 0, 02 RQ1
X R

0

0
02 RQ1
X R

1 


d
nP 2 (q)

0
Noting that lim nP = RQ1
X R , we obtain,

R(X 0 X)1 R0
02

2 (q)

since the powers of n cancel. To get a usable test statistic substitute a consistent estimator of 02 .
This makes it clear why the test is sometimes referred to as a Lagrange multiplier test. It
may seem that one needs the actual Lagrange multipliers to calculate this. If we impose the
restrictions by substitution, these are not available. Note that the test can be written as


R0

0

(X 0 X)1 R0
02

2 (q)

However, we can use the fonc for the restricted estimator:

X 0 y + X 0 X R + R0
to get that

R0

= X 0 (y X R )
= X 0 R

Substituting this into the above, we get

0R X(X 0 X)1 X 0 R d 2
(q)
02
but this is simply
0R

PX
d
R 2 (q).
02

To see why the test is also known as a score test, note that the fonc for restricted least squares

X 0 y + X 0 X R + R0
give us
= X 0 y X 0 X R
R0
and the rhs is simply the gradient (score) of the unrestricted model, evaluated at the restricted
estimator. The scores evaluated at the unrestricted estimate are identically zero. The logic behind
the score test is that the scores evaluated at the restricted estimate should be approximately zero,
if the restriction is true. The test is also known as a Rao test, since P. Rao first proposed it in 1948.

5.3

The asymptotic equivalence of the LR, Wald and score tests

Note: the discussion of the LR test has been moved forward in these notes. I no longer teach the
material in this section, but Im leaving it here for reference.
We have seen that the three tests all converge to 2 random variables. In fact, they all converge to the same 2 rv, under the null hypothesis. Well show that the Wald and LR tests are
asymptotically equivalent. We have seen that the Wald test is asymptotically equivalent to

0



a
d
0 1
r
W = n R r
02 RQ1
R
R

2 (q)
X

(5.1)

Using
0 = (X 0 X)1 X 0
and
R r = R( 0 )
we get

nR( 0 )

=
=

nR(X 0 X)1 X 0
 0 1
XX
n1/2 X 0
R
n

Substitute this into [5.1] to get

a

=
a

=
a

1

0
RQ1
X X


1
0 X(X 0 X)1 R0 02 R(X 0 X)1 R0
R(X 0 X)1 X 0

1 0
0
2
= n1 0 XQ1
X R 0 RQX R

0 A(A0 A)1 A0
02
0
PR
02

where PR is the projection matrix formed by the matrix X(X 0 X)1 R0 .

Note that this matrix is idempotent and has q columns, so the projection matrix has rank q.
Now consider the likelihood ratio statistic
a

LR = n1/2 g(0 )0 I(0 )1 R0 RI(0 )1 R0

1

RI(0 )1 n1/2 g(0 )

Under normality, we have seen that the likelihood function is

0

1 (y X) (y X)
ln L(, ) = n ln 2 n ln
.
2
2

Using this,
1
ln L(, )
n
X 0 (y X0 )
n 2
X 0
n 2

g(0 ) D
=
=

(5.2)

Also, by the information matrix equality:

I(0 )

= H (0 )
lim D 0 g(0 )
X 0 (y X0 )
= lim D 0
n 2
X 0X
= lim
n 2
QX
=
2
=

so
I(0 )1 = 2 Q1
X
Substituting these last expressions into [5.2], we get
LR

=
a

=
a

0 X 0 (X 0 X)1 R0 02 R(X 0 X)1 R0

1

R(X 0 X)1 X 0

0 PR
02
W

This completes the proof that the Wald and LR tests are asymptotically equivalent. Similarly, one
can show that, under the null hypothesis,
a

qF = W = LM = LR
The proof for the statistics except for LR does not depend upon normality of the errors, as
can be verified by examining the expressions for the statistics.
The LR statistic is based upon distributional assumptions, since one cant write the likelihood
function without them.
However, due to the close relationship between the statistics qF and LR, supposing normality, the qF statistic can be thought of as a pseudo-LR statistic, in that its like a LR statistic in
that it uses the value of the objective functions of the restricted and unrestricted models, but
it doesnt require distributional assumptions.
The presentation of the score and Wald tests has been done in the context of the linear model.
This is readily generalizable to nonlinear models and/or other estimation methods.
Though the four statistics are asymptotically equivalent, they are numerically different in small
samples. The numeric values of the tests also depend upon how 2 is estimated, and weve already
seen than there are several ways to do this. For example all of the following are consistent for 2
under H0

0
nk
0
n
0R R
nk+q
0R R
n

and in general the denominator call be replaced with any quantity a such that lim a/n = 1.

It can be shown, for linear regression models subject to linear restrictions, and if
calculate the Wald test and

0R R
n

0
n

is used to

is used for the score test, that

W > LR > LM.

For this reason, the Wald test will always reject if the LR test rejects, and in turn the LR test
rejects if the LM test rejects. This is a bit problematic: there is the possibility that by careful choice
of the statistic used, one can manipulate reported results to favor or disfavor a hypothesis. A
conservative/honest approach would be to report all three test statistics when they are available.
In the case of linear models with normal errors the F test is to be preferred, since asymptotic
approximations are not an issue.
The small sample behavior of the tests can be quite different. The true size (probability of
rejection of the null when the null is true) of the Wald test is often dramatically higher than the
nominal size associated with the asymptotic distribution. Likewise, the true size of the score test is
often smaller than the nominal size.

5.4

Interpretation of test statistics

Now that we have a menu of test statistics, we need to know how to use them.

5.5

Confidence intervals

Confidence intervals for single coefficients are generated in the normal manner. Given the t statistic
t() =

a 100 (1 ) % confidence interval for 0 is defined by the bounds of the set of such that t()
does not reject H0 : 0 = , using a significance level:
C() = { : c/2 <

< c/2 }

The set of such is the interval

c c/2
A confidence ellipse for two coefficients jointly would be, analogously, the set of {1 , 2 } such
that the F (or some other test statistic) doesnt reject at the specified critical value. This generates
an ellipse, if the estimators are correlated.
The region is an ellipse, since the CI for an individual coefficient defines a (infinitely long)
rectangle with total prob. mass 1 , since the other coefficient is marginalized (e.g., can
take on any value). Since the ellipse is bounded in both dimensions but also contains mass
1 , it must extend beyond the bounds of the individual CI.
From the pictue we can see that:
Rejection of hypotheses individually does not imply that the joint test will reject.
Joint rejection does not imply individal tests will reject.

Figure 5.1: Joint and Individual Confidence Regions

5.6

Bootstrapping

When we rely on asymptotic theory to use the normal distribution-based tests and confidence
intervals, were often at serious risk of making important errors. If the sample
size is small and


errors are highly nonnormal, the small sample distribution of n 0 may be very different
than its large sample distribution. Also, the distributions of test statistics may not resemble their
limiting distributions at all. A means of trying to gain information on the small sample distribution
of test statistics and estimators is the bootstrap. Well consider a simple example, just to get the
main idea.
Suppose that
y

X0 +

IID(0, 02 )

X is nonstochastic
Given that the distribution of is unknown, the distribution of will be unknown in small samples.
However, since we have random sampling, we could generate artificial data. The steps are:
1. Draw n observations from with replacement. Call this vector j (its a n 1).
2. Then generate the data by yj = X + j
3. Now take this and estimate
j = (X 0 X)1 X 0 yj .
4. Save j
5. Repeat steps 1-4, until we have a large number, J, of j .
With this, we can use the replications to calculate the empirical distribution of j . One way to form
a 100(1-)% confidence interval for 0 would be to order the j from smallest to largest, and drop
the first and last J/2 of the replications, and use the remaining endpoints as the limits of the CI.
Note that this will not give the shortest CI if the empirical distribution is skewed.
for example a test
Suppose one was interested in the distribution of some function of ,
statistic. Simple: just calculate the transformation for each j, and work with the empirical
distribution of the transformation.
If the assumption of iid errors is too strong (for example if there is heteroscedasticity or
autocorrelation, see below) one can work with a bootstrap defined by sampling from (y, x)
with replacement.
How to choose J: J should be large enough that the results dont change with repetition of
the entire bootstrap. This is easy to check. If you find the results change a lot, increase J and
try again.
The bootstrap is based fundamentally on the idea that the empirical distribution of the sample
data converges to the actual sampling distribution as n becomes large, so statistics based on
sampling from the empirical distribution should converge in distribution to statistics based
on sampling from the actual sampling distribution.
In finite samples, this doesnt hold. At a minimum, the bootstrap is a good way to check if
asymptotic theory results offer a decent approximation to the small sample distribution.

 Bootstrapping can be used to test hypotheses. Basically, use the bootstrap to get an approximation to the empirical distribution of the test statistic under the alternative hypothesis, and
use this to get critical values. Compare the test statistic calculated using the real data, under
the null, to the bootstrap critical values. There are many variations on this theme, which we
wont go into here.

5.7

Wald test for nonlinear restrictions: the delta method

Testing nonlinear restrictions of a linear model is not much more difficult, at least when the model
is linear. Since estimation subject to nonlinear restrictions requires nonlinear estimation methods,
which are beyond the score of this course, well just consider the Wald test for nonlinear restrictions
on a linear model.
Consider the q nonlinear restrictions
r(0 ) = 0.
where r() is a q-vector valued function. Write the derivative of the restriction evaluated at as
D 0 r()| = R()
We suppose that the restrictions are not redundant in a neighborhood of 0 , so that
(R()) = q
about 0 :
in a neighborhood of 0 . Take a first order Taylors series expansion of r()
= r(0 ) + R( )( 0 )
r()
where is a convex combination of and 0 . Under the null hypothesis we have
= R( )( 0 )
r()
Due to consistency of we can replace by 0 , asymptotically, so

Weve already seen the distribution of

=
nr()

nR(0 )( 0 )

n( 0 ). Using this we get

d
0 2

nr()
N 0, R(0 )Q1
X R(0 ) 0 .

Considering the quadratic form

0 R(0 )Q1 R(0 )0
nr()
X
02

1

r()

2 (q)

under the null hypothesis. Substituting consistent estimators for 0, QX and 02 , the resulting
statistic is


1
0
0 R()(X

r()
X)1 R()
r()
c2

2 (q)

under the null hypothesis.

This is known in the literature as the delta method, or as Kleins approximation.

 Since this is a Wald test, it will tend to over-reject in finite samples. The score and LR tests
are also possibilities, but they require estimation methods for nonlinear models, which arent
in the scope of this course.
Note that this also gives a convenient way to estimate nonlinear functions and associated asymptotic confidence intervals. If the nonlinear function r(0 ) is not hypothesized to be zero, we just
have





d
0 2
r(0 )
n r()
N 0, R(0 )Q1
X R(0 ) 0

so an approximation to the distribution of the function of the estimator is

N (r(0 ), R(0 )(X 0 X)1 R(0 )0 2 )
r()
0
For example, the vector of elasticities of a function f (x) is
(x) =

x
f (x)

x
f (x)

where  means element-by-element multiplication. Suppose we estimate a linear function

y = x0 + .
The elasticities of y w.r.t. x are
(x) =

x
x0

(note that this is the entire vector of elasticities). The estimated elasticities are
b(x) =

x
x0

To calculate the estimated standard errors of all five elasticites, use

R()

(x)
0

x1 0

0 x

2
.
.
.
0

..

0
1 x21

..
0
. 0

x .

.
0
.
xk
0

2 x22
..

(x0 )2

0
..
.
0
k x2k

Note that the elasticity and the standard error

To get a consistent estimator just substitute in .
are functions of x. The program ExampleDeltaMethod.m shows how this can be done.
In many cases, nonlinear restrictions can also involve the data, not just the parameters. For
example, consider a model of expenditure shares. Let x(p, m) be a demand funcion, where p is
prices and m is income. An expenditure share system for G goods is
si (p, m) =

pi xi (p, m)
, i = 1, 2, ..., G.
m

Now demand must be positive, and we assume that expenditures sum to income, so we have the

restrictions
0
G
X

si (p, m) 1, i

si (p, m)

i=1

Suppose we postulate a linear model for the expenditure shares:

i
si (p, m) = 1i + p0 pi + mm
+ i

It is fairly easy to write restrictions such that the shares sum to one, but the restriction that the
shares lie in the [0, 1] interval depends on both parameters and the values of p and m. It is impossible to impose the restriction that 0 si (p, m) 1 for all possible p and m. In such cases, one
might consider whether or not a linear model is a reasonable specification.

5.8

Example: the Nerlove data

Remember that we in a previous example (section 3.8) that the OLS results for the Nerlove model
are

*********************************************************
OLS estimation results
Observations 145
R-squared 0.925955
Sigma-squared 0.153943
Results (Ordinary var-cov estimator)
constant
output
labor
fuel
capital

estimate
-3.527
0.720
0.436
0.427
-0.220

st.err.
1.774
0.017
0.291
0.100
0.339

t-stat.
-1.987
41.244
1.499
4.249
-0.648

p-value
0.049
0.000
0.136
0.000
0.518

*********************************************************
Note that sK = K < 0, and that L + F + K 6= 1.
Remember that if we have constant returns to scale, then Q = 1, and if there is homogeneity
of degree 1 then L + F + K = 1. We can test these hypotheses either separately or jointly.
NerloveRestrictions.m imposes and tests CRTS and then HOD1. From it we obtain the results that
follow:

Imposing and testing HOD1

*******************************************************
Restricted LS estimation results
Observations 145
R-squared 0.925652
Sigma-squared 0.155686
estimate

st.err.

t-stat.

p-value

constant
output
labor
fuel
capital

-4.691
0.721
0.593
0.414
-0.007

0.891
0.018
0.206
0.100
0.192

-5.263
41.040
2.878
4.159
-0.038

0.000
0.000
0.005
0.000
0.969

*******************************************************
Value
p-value
F
0.574
0.450
Wald
0.594
0.441
LR
0.593
0.441
Score
0.592
0.442
Imposing and testing CRTS
*******************************************************
Restricted LS estimation results
Observations 145
R-squared 0.790420
Sigma-squared 0.438861

constant
output
labor
fuel
capital

estimate
-7.530
1.000
0.020
0.715
0.076

st.err.
2.966
0.000
0.489
0.167
0.572

t-stat.
-2.539
Inf
0.040
4.289
0.132

p-value
0.012
0.000
0.968
0.000
0.895

*******************************************************
Value
p-value
F
256.262
0.000
Wald
265.414
0.000
LR
150.863
0.000
Score
93.771
0.000

Notice that the input price coefficients in fact sum to 1 when HOD1 is imposed. HOD1 is
not rejected at usual significance levels (e.g., = 0.10). Also, R2 does not drop much when
the restriction is imposed, compared to the unrestricted results. For CRTS, you should note that
Q = 1, so the restriction is satisfied. Also note that the hypothesis that Q = 1 is rejected by the
test statistics at all reasonable significance levels. Note that R2 drops quite a bit when imposing
CRTS. If you look at the unrestricted estimation results, you can see that a t-test for Q = 1 also
rejects, and that a confidence interval for Q does not overlap 1.
From the point of view of neoclassical economic theory, these results are not anomalous: HOD1
is an implication of the theory, but CRTS is not.
Exercise 9. Modify the NerloveRestrictions.m program to impose and test the restrictions jointly.
The Chow test

Since CRTS is rejected, lets examine the possibilities more carefully. Recall that

the data is sorted by output (the third column). Define 5 subsamples of firms, with the first group

being the 29 firms with the lowest output levels, then the next 29 firms, etc. The five subsamples
can be indexed by j = 1, 2, ..., 5, where j = 1 for t = 1, 2, ...29, j = 2 for t = 30, 31, ...58, etc. Define
dummy variables D1 , D2 , ..., D5 where

1
D1 =
0

1
D2 =
0

t {1, 2, ...29}
t
/ {1, 2, ...29}
t {30, 31, ...58}
t
/ {30, 31, ...58}

..
.

1
D5 =
0

t {117, 118, ..., 145}

t
/ {117, 118, ..., 145}

Define the model

ln Ct =

5
X

1 Dj +

j=1

5
X

j Dj ln Qt +

j=1

5
X

Lj Dj ln PLt +

j=1

5
X

F j Dj ln PF t +

j=1

5
X

Kj Dj ln PKt + t

j=1

(5.3)
Note that the first column of nerlove.data indicates this way of breaking up the sample, and provides and easy way of defining the dummy variables. The new model may be written as

y1

y2
..
.

y5

X1

0
..
.

X2

X3
X4

0
5
X5

1

2
..
.

5

(5.4)

where y1 is 291, X1 is 295, j is the 5 1 vector of coefficients for the j th subsample (e.g.,
1 = (1 , 1 , L1 , F 1 , K1 )0 ), and j is the 29 1 vector of errors for the j th subsample.
The Octave program Restrictions/ChowTest.m estimates the above model. It also tests the
hypothesis that the five subsamples share the same parameter vector, or in other words, that there
is coefficient stability across the five subsamples. The null to test is that the parameter vectors for
the separate groups are all the same, that is,
1 = 2 = 3 = 4 = 5
This type of test, that parameters are constant across different sets of data, is sometimes referred
to as a Chow test.
There are 20 restrictions. If thats not clear to you, look at the Octave program.
The restrictions are rejected at all conventional significance levels.
Since the restrictions are rejected, we should probably use the unrestricted model for analysis.
What is the pattern of RTS as a function of the output group (small to large)? Figure 5.2 plots RTS.
We can see that there is increasing RTS for small firms, but that RTS is approximately constant for
large firms.

Figure 5.2: RTS as a function of firm size

2.6
RTS
2.4

2.2

1.8

1.6

1.4

1.2

1
1

5.9

1.5

2.5

3.5

4.5

Exercises

1. Using the Chow test on the Nerlove model, we reject that there is coefficient stability across
the 5 groups. But perhaps we could restrict the input price coefficients to be the same but let
the constant and output coefficients vary by group size. This new model is
ln C =

5
X
j=1

j Dj +

5
X

j Dj ln Q + L ln PL + F ln PF + K ln PK + 

(5.5)

j=1

(a) estimate this model by OLS, giving R2 , estimated standard errors for coefficients, tstatistics for tests of significance, and the associated p-values. Interpret the results in
detail.
(b) Test the restrictions implied by this model (relative to the model that lets all coefficients
vary across groups) using the F, qF, Wald, score and likelihood ratio tests. Comment on
the results.
(c) Estimate this model but imposing the HOD1 restriction, using an OLS estimation program. Dont use mc_olsr or any other restricted OLS estimation program. Give estimated
standard errors for all coefficients.
(d) Plot the estimated RTS parameters as a function of firm size. Compare the plot to that
given in the notes for the unrestricted model. Comment on the results.
2. For the model of the above question, compute 95% confidence intervals for RTS for each of
the 5 groups of firms, using the delta method to compute standard errors. Comment on the
results.
3. Perform a Monte Carlo study that generates data from the model
y = 2 + 1x2 + 1x3 + 
where the sample size is 30, x2 and x3 are independently uniformly distributed on [0, 1] and
 IIN (0, 1)

(a) Compare the means and standard errors of the estimated coefficients using OLS and
restricted OLS, imposing the restriction that 2 + 3 = 2.
(b) Compare the means and standard errors of the estimated coefficients using OLS and
restricted OLS, imposing the restriction that 2 + 3 = 1.
(c) Discuss the results.

Chapter 6

Stochastic regressors
Up to now we have treated the regressors as fixed, which is clearly unrealistic. Now we will assume
they are random. There are several ways to think of the problem. First, if we are interested in an
analysis conditional on the explanatory variables, then it is irrelevant if they are stochastic or not,
since conditional on the values of they regressors take on, they are nonstochastic, which is the case
already considered.
In cross-sectional analysis it is usually reasonable to make the analysis conditional on the
regressors.
In dynamic models, where yt may depend on yt1 , a conditional analysis is not sufficiently
general, since we may want to predict into the future many periods out, so we need to
consider the behavior of and the relevant test statistics unconditional on X.
The model well deal will involve a combination of the following assumptions
Assumption 10. Linearity: the model is a linear function of the parameter vector 0 :
yt = x0t 0 + t ,
or in matrix form,

where y is n 1, X =

x1

x2

y = X0 + ,
0
, where xt is K 1, and 0 and are conformable.
xn

Assumption 11. Stochastic, linearly independent regressors

X has rank K with probability 1
X is stochastic


limn Pr n1 X 0 X = QX = 1, where QX is a finite positive definite matrix.

Assumption 12. Central limit theorem

d
n1/2 X 0 N (0, QX 02 )

Assumption 13. Normality (Optional): |X N (0, 2 In ):  is normally distributed

57

Assumption 14. Strongly exogenous regressors. The regressors X are strongly exogenous if
E(t |X)

0, t

(6.1)

Assumption 15. Weakly exogenous regressors: The regressors are weakly exogenous if
E(t |xt )

0, t

In both cases, x0t is the conditional mean of yt given xt : E(yt |xt ) = x0t

6.1

Case 1

Normality of , strongly exogenous regressors

In this case,
= 0 + (X 0 X)1 X 0

E(|X)

= 0 + (X 0 X)1 X 0 E(|X)
= 0

= , unconditional on X. Likewise,
and since this holds for all X, E()
N , (X 0 X)1 2
|X
0

If the density of X is d(X), the marginal density of is obtained by multiplying the conditional density by d(X) and integrating over X. Doing this leads to a nonnormal density for
in small samples.
,
However, conditional on X, the usual test statistics have the t, F and 2 distributions. Importantly, these distributions dont depend on X, so when marginalizing to obtain the unconditional distribution, nothing changes. The tests are valid in small samples.
Summary: When X is stochastic but strongly exogenous and is normally distributed:
1. is unbiased
2. is nonnormally distributed
3. The usual test statistics have the same distribution as with nonstochastic X.
4. The Gauss-Markov theorem still holds, since it holds conditionally on X, and this is true
for all X.
5. Asymptotic properties are treated in the next section.

6.2

Case 2

nonnormally distributed, strongly exogenous regressors

The unbiasedness of carries through as before. However, the argument regarding test statistics doesnt hold, due to nonnormality of . Still, we have
=
=
Now


0 + (X 0 X)1 X 0
 0 1 0
XX
X
0 +
n
n
X 0X
n

1

Q1
X

by assumption, and
n1/2 X 0 p
X 0

=
0
n
n
since the numerator converges to a N (0, QX 2 ) r.v. and the denominator still goes to infinity. We
have unbiasedness and the variance disappearing, so, the estimator is consistent:
p
0 .

Considering the asymptotic distribution



n 0
=
=
so

1 0
X 0X
X
n
n
 0 1
XX
n1/2 X 0
n



d
2
n 0 N (0, Q1
X 0 )

directly following the assumptions. Asymptotic normality of the estimator still holds. Since the
asymptotic results on all test statistics only require this, all the previous asymptotic results on test
statistics are also valid in this case.
Summary: Under strongly exogenous regressors, with normal or nonnormal, has the
properties:
1. Unbiasedness
2. Consistency
3. Gauss-Markov theorem holds, since it holds in the previous case and doesnt depend on
normality.
4. Asymptotic normality
5. Tests are asymptotically valid
6. Tests are not valid in small samples if the error is normally distributed

6.3

Case 3

Weakly exogenous regressors

An important class of models are dynamic models, where lagged dependent variables have an
impact on the current value. A simple version of these models that captures the important points

is
yt

= zt0 +

p
X

s yts + t

s=1

= x0t + t
where now xt contains lagged dependent variables. Clearly, even with E(t |xt ) = 0, X and are
not uncorrelated, so one cant show unbiasedness. For example,
E(t1 xt ) 6= 0
since xt contains yt1 (which is a function of t1 ) as an element.
This fact implies that all of the small sample properties such as unbiasedness, Gauss-Markov
theorem, and small sample validity of test statistics do not hold in this case. Recall Figure 3.7.
This is a case of weakly exogenous regressors, and we see that the OLS estimator is biased in
this case.
Nevertheless, under the above assumptions, all asymptotic properties continue to hold, using
the same arguments as before.

6.4

When are the assumptions reasonable?

The two assumptions weve added are

1. limn Pr n1 X 0 X = QX = 1, a QX finite positive definite matrix.
d

2. n1/2 X 0 N (0, QX 02 )
The most complicated case is that of dynamic models, since the other cases can be treated as nested
in this case. There exist a number of central limit theorems for dependent processes, many of which
are fairly technical. We wont enter into details (see Hamilton, Chapter 7 if youre interested). A
main requirement for use of standard asymptotics for a dependent sequence
n

{st } = {

1X
zt }
n t=1

to converge in probability to a finite limit is that zt be stationary, in some sense.

Strong stationarity requires that the joint distribution of the set
{zt , zt+s , ztq , ...}
not depend on t.
Covariance (weak) stationarity requires that the first and second moments of this set not
depend on t.
An example of a sequence that doesnt satisfy this is an AR(1) process with a unit root (a
random walk):
xt

xt1 + t

IIN (0, 2 )

One can show that the variance of xt depends upon t in this case, so its not weakly stationary.

 The series sin t + t has a first moment that depends upon t, so its not weakly stationary
either.
Stationarity prevents the process from trending off to plus or minus infinity, and prevents cyclical
behavior which would allow correlations between far removed zt znd zs to be high. Draw a picture
here.
In summary, the assumptions are reasonable when the stochastic conditioning variables have
variances that are finite, and are not too strongly dependent. The AR(1) model with unit
root is an example of a case where the dependence is too strong for standard asymptotics to
apply.
The study of nonstationary processes is an important part of econometrics, but it isnt in the
scope of this course.

6.5

Exercises

1. Show that for two random variables A and B, if E(A|B) = 0, then E (Af (B)) = 0. How is
this used in the proof of the Gauss-Markov theorem?
2. Is it possible for an AR(1) model for time series data, e.g., yt = 0 + 0.9yt1 + t satisfy weak
exogeneity? Strong exogeneity? Discuss.

Chapter 7

Data problems
In this section well consider problems associated with the regressor matrix: collinearity, missing
observations and measurement error.

7.1

Collinearity

Motivation: Data on Mortality and Related Factors

The data set mortality.data contains annual data from 1947 - 1980 on death rates in the U.S., along
with data on factors like smoking and consumption of alcohol. The data description is:
DATA4-7: Death rates in the U.S. due to coronary heart disease and their
determinants. Data compiled by Jennifer Whisenand
chd = death rate per 100,000 population (Range 321.2 - 375.4)
cal = Per capita consumption of calcium per day in grams (Range 0.9 - 1.06)
unemp = Percent of civilian labor force unemployed in 1,000 of persons 16 years and older
(Range 2.9 - 8.5)
cig = Per capita consumption of cigarettes in pounds of tobacco by persons 18 years and
olderapprox. 339 cigarettes per pound of tobacco (Range 6.75 - 10.46)
edfat = Per capita intake of edible fats and oil in poundsincludes lard, margarine and butter
(Range 42 - 56.5)
meat = Per capita intake of meat in poundsincludes beef, veal, pork, lamb and mutton
(Range 138 - 194.8)
spirits = Per capita consumption of distilled spirits in taxed gallons for individuals 18 and
older (Range 1 - 2.9)
beer = Per capita consumption of malted liquor in taxed gallons for individuals 18 and older
(Range 15.04 - 34.9)
wine = Per capita consumption of wine measured in taxed gallons for individuals 18 and
older (Range 0.77 - 2.65)

62

Consider estimation results for several models:

d = 334.914 + 5.41216 cig + 36.8783 spirits 5.10365 beer

chd
(58.939)

(5.156)

(7.373)

(1.2513)

+ 13.9764 wine
(12.735)

2 = 0.5528
T = 34 R

F (4, 29) = 11.2

= 9.9945

(standard errors in parentheses)

d = 353.581 + 3.17560 cig + 38.3481 spirits 4.28816 beer

chd
(56.624)

(4.7523)

2 = 0.5498
T = 34 R

(7.275)

(1.0102)

F (3, 30) = 14.433

= 10.028

(standard errors in parentheses)

d = 243.310 + 10.7535 cig + 22.8012 spirits 16.8689 wine

chd
(67.21)

(6.1508)

T = 34 R = 0.3198

(8.0359)

(12.638)

F (3, 30) = 6.1709

= 12.327

(standard errors in parentheses)

d = 181.219 + 16.5146 cig + 15.8672 spirits

chd
(49.119)

T = 34 R = 0.3026

(4.4371)

(6.2079)

F (2, 31) = 8.1598

= 12.481

(standard errors in parentheses)

Note how the signs of the coefficients change depending on the model, and that the magnitudes
of the parameter estimates vary a lot, too. The parameter estimates are highly sensitive to the
particular model we estimate. Why? Well see that the problem is that the data exhibit collinearity.

Collinearity: definition
Collinearity is the existence of linear relationships amongst the regressors. We can always write
1 x1 + 2 x2 + + K xK + v = 0
where xi is the ith column of the regressor matrix X, and v is an n 1 vector. In the case that
there exists collinearity, the variation in v is relatively small, so that there is an approximately exact
linear relation between the regressors.
relative and approximate are imprecise, so its difficult to define when collinearilty exists.
In the extreme, if there are exact linear relationships (every element of v equal) then (X) < K, so
(X 0 X) < K, so X 0 X is not invertible and the OLS estimator is not uniquely defined. For example,
if the model is
yt
x2t

= 1 + 2 x2t + 3 x3t + t
= 1 + 2 x3t

then we can write

yt

= 1 + 2 (1 + 2 x3t ) + 3 x3t + t
=

1 + 2 1 + 2 2 x3t + 3 x3t + t

(1 + 2 1 ) + (2 2 + 3 ) x3t

1 + 2 x3t + t

The 0 s can be consistently estimated, but since the 0 s define two equations in three 0 s, the
0 s cant be consistently estimated (there are multiple values of that solve the first order
conditions). The 0 s are unidentified in the case of perfect collinearity.
Perfect collinearity is unusual, except in the case of an error in construction of the regressor
matrix, such as including the same regressor twice.
Another case where perfect collinearity may be encountered is with models with dummy variables,
if one is not careful. Consider a model of rental price (yi ) of an apartment. This could depend
factors such as size, quality etc., collected in xi , as well as on the location of the apartment. Let
Bi = 1 if the ith apartment is in Barcelona, Bi = 0 otherwise. Similarly, define Gi , Ti and Li for
Girona, Tarragona and Lleida. One could use a model such as
yi = 1 + 2 Bi + 3 Gi + 4 Ti + 5 Li + x0i + i
In this model, Bi + Gi + Ti + Li = 1, i, so there is an exact relationship between these variables
and the column of ones corresponding to the constant. One must either drop the constant, or one
of the qualitative variables.

A brief aside on dummy variables

Dummy variable: A dummy variable is a binary-valued variable that indicates whether or not
some condition is true. It is customary to assign the value 1 if the condition is true, and 0 if the
condition is false.
Dummy variables are used essentially like any other regressor. Use d to indicate that a variable
is a dummy, so that variables like dt and dt2 are understood to be dummy variables. Variables like
xt and xt3 are ordinary continuous regressors. You know how to interpret the following models:
y t = 1 + 2 d t + t

yt = 1 dt + 2 (1 dt ) + t

y t = 1 + 2 d t + 3 x t + t
Interaction terms: an interaction term is the product of two variables, so that the effect of one
variable on the dependent variable depends on the value of the other. The following model has an
interaction term. Note that

E(y|x)
x

= 3 + 4 dt . The slope depends on the value of dt .

yt = 1 + 2 dt + 3 xt + 4 dt xt + t
Multiple dummy variables: we can use more than one dummy variable in a model. We will study
models of the form
yt = 1 + 2 dt1 + 3 dt2 + 4 xt + t

yt = 1 + 2 dt1 + 3 dt2 + 4 dt1 dt2 + 5 xt + t

Incorrect usage: You should understand why the following models are not correct usages of
dummy variables:
1. overparameterization:
yt = 1 + 2 dt + 3 (1 dt ) + t
2. multiple values assigned to multiple categories. Suppose that we a condition that defines 4
possible categories, and we create a variable d = 1 if the observation is in the first category,
d = 2 if in the second, etc. (This is not strictly speaking a dummy variable, according to our
definition). Why is the following model not a good one?
yt = 1 + 2 d + 
What is the correct way to deal with this situation?
Multiple parameterizations. To formulate a model that conditions on a given set of categorical
information, there are multiple ways to use dummy variables. For example, the two models
yt = 1 dt + 2 (1 dt ) + 3 xt + 4 dt xt + t
and
yt = 1 + 2 dt + 3 xt dt + 4 xt (1 dt ) + t
are equivalent. You should know what are the 4 equations that relate the j parameters to the j
parameters, j = 1, 2, 3, 4. You should know how to interpret the parameters of both models.

Back to collinearity
The more common case, if one doesnt make mistakes such as these, is the existence of inexact
linear relationships, i.e., correlations between the regressors that are less than one in absolute
value, but not zero. The basic problem is that when two (or more) variables move together, it is
difficult to determine their separate influences.
Example 16. Two children are in a room, along with a broken lamp. Both say I didnt do it!.
How can we tell who broke the lamp?
Lack of knowledge about the separate influences of variables is reflected in imprecise estimates,
i.e., estimates with high variances. With economic data, collinearity is commonly encountered, and
is often a severe problem.
When there is collinearity, the minimizing point of the objective function that defines the OLS
estimator (s(), the sum of squared errors) is relatively poorly defined. This is seen in Figures 7.1
and 7.2.
To see the effect of collinearity on variances, partition the regressor matrix as
X=

where x is the first column of X (note: we can interchange the columns of X isf we like, so theres
under the classical
no loss of generality in considering the first column). Now, the variance of ,
assumptions, is
= (X 0 X)1 2
V ()

Figure 7.1: s() when there is no collinearity

6
4

60
55
50
45
40
35
30
25
20
15

2
0
-2
-4
-6
-6

-4

-2

Figure 7.2: s() when there is collinearity

6
4
2
0
-2
-4
-6
-6

-4

-2

100
90
80
70
60
50
40
30
20

Using the partition,

"
0

XX=

x0 x

x0 W

W 0x

W 0W

and following a rule for partitioned inversion,

1

(X 0 X)1,1

1
x0 x x0 W (W 0 W )1 W 0 x
 
 1
0
=
x0 In W (W 0 W ) 1 W 0 x

1
= ESSx|W

where by ESSx|W we mean the error sum of squares obtained from the regression
x = W + v.
Since
R2 = 1 ESS/T SS,
we have
ESS = T SS(1 R2 )
so the variance of the coefficient corresponding to x is
V (x ) =

2
2
T SSx (1 Rx|W
)

(7.1)

We see three factors influence the variance of this coefficient. It will be high if
1. 2 is large
2. There is little variation in x. Draw a picture here.
3. There is a strong linear relationship between x and the other regressors, so that W can
explain the movement in x well. In this case, R2 will be close to 1. As R2 1, V (x )
x|W

x|W

.
The last of these cases is collinearity.
Intuitively, when there are strong linear relations between the regressors, it is difficult to determine the separate influence of the regressors on the dependent variable. This can be seen by
comparing the OLS objective function in the case of no correlation between regressors with the
objective function with correlation between the regressors. See the figures nocollin.ps (no correlation) and collin.ps (correlation), available on the web site.
Example 17. The Octave script DataProblems/collinearity.m performs a Monte Carlo study with
correlated regressors. The model is y = 1 + x2 + x3 + , where the correlation between x2 and
x3 can be set. Three estimators are used: OLS, OLS dropping x3 (a false restriction), and restricted
LS using 2 = 3 (a true restriction). The output when the correlation between the two regressors
is 0.9 is

octave:1> collinearity
Contribution received from node 0.

Received so far: 500

Contribution received from node 0.

Received so far: 1000

correlation between x2 and x3: 0.900000

descriptive
mean
0.996
0.996
1.008
descriptive
mean
0.999
1.905
descriptive
mean
0.998
1.002
1.002
octave:2>

statistics
st. dev.
0.182
0.444
0.436
statistics
st. dev.
0.198
0.207
statistics
st. dev.
0.179
0.096
0.096

for 1000 OLS replications

min
max
0.395
1.574
-0.463
2.517
-0.342
2.301
for 1000 OLS replications, dropping x3
min
max
0.330
1.696
1.202
2.651
for 1000 Restricted OLS replications, b2=b3
min
max
0.433
1.574
0.663
1.339
0.663
1.339

Figure 7.3 shows histograms for the estimated 2 , for each of the three estimators.
repeat the experiment with a lower value of rho, and note how the standard errors of the
OLS estimator change.

Detection of collinearity
The best way is simply to regress each explanatory variable in turn on the remaining regressors. If
any of these auxiliary regressions has a high R2 , there is a problem of collinearity. Furthermore,
this procedure identifies which parameters are affected.
Sometimes, were only interested in certain parameters. Collinearity isnt a problem if it
doesnt affect what were interested in estimating.
An alternative is to examine the matrix of correlations between the regressors. High correlations
are sufficient but not necessary for severe collinearity.
Also indicative of collinearity is that the model fits well (high R2 ), but none of the variables is
significantly different from zero (e.g., their separate influences arent well determined).
In summary, the artificial regressions are the best approach if one wants to be careful.
Example 18. Nerlove data and collinearity. The simple Nerlove model is
ln C = 1 + 2 ln Q + 3 ln PL + 4 ln PF + 5 ln PK + 
When this model is estimated by OLS, some coefficients are not significant (see subsection 3.8).
This may be due to collinearity.The Octave script DataProblems/NerloveCollinearity.m checks the
regressors for collinearity. If you run this, you will see that collinearity is not a problem with this
data. Why is the coefficient of ln PK not significantly different from zero?

Figure 7.3: Collinearity: Monte Carlo results

(a) OLS,2

(c) Restricted LS,2 , with true restriction2 = 3

(b) OLS,2 , dropping x3

Dealing with collinearity

More information
Collinearity is a problem of an uninformative sample. The first question is: is all the available
information being used? Is more data available? Are there coefficient restrictions that have been
neglected? Picture illustrating how a restriction can solve problem of perfect collinearity.
Stochastic restrictions and ridge regression
Supposing that there is no more data or neglected restrictions, one possibility is to change perspectives, to Bayesian econometrics. One can express prior beliefs regarding the coefficients using
stochastic restrictions. A stochastic linear restriction would be something of the form
R = r + v
where R and r are as in the case of exact linear restrictions, but v is a random vector. For example,
the model could be
y
R
!

= X +
= r+v
0

2 In

0nq

0qn

v2 Iq

This sort of model isnt in line with the classical interpretation of parameters as constants: according to this interpretation the left hand side of R = r + v is constant but the right is random. This
model does fit the Bayesian perspective: we combine information coming from the model and the
data, summarized in
y

= X +

N (0, 2 In )
with prior beliefs regarding the distribution of the parameter, summarized in
R N (r, v2 Iq )
Since the sample is random it is reasonable to suppose that E(v 0 ) = 0, which is the last piece of
information in the specification. How can you estimate using this model? The solution is to treat
the restrictions as artificial data. Write
#
"
y
r

"
=

X
R

"
+

This model is heteroscedastic, since 2 6= v2 . Define the prior precision k = /v . This expresses
the degree of belief in the restriction relative to the variability of the data. Supposing that we
specify k, then the model
"

y
kr

"
=

X
kR

"
+

kv

is homoscedastic and can be estimated by OLS. Note that this estimator is biased. It is consistent,
however, given that k is a fixed constant, even if the restriction is false (this is in contrast to the
case of false exact restrictions). To see this, note that there are Q restrictions, where Q is the
number of rows of R. As n , these Q artificial observations have no weight in the objective

function, so the estimator has the same limiting objective function as the OLS estimator, and is
therefore consistent.
To motivate the use of stochastic restrictions, consider the expectation of the squared length of

E(0 )

= E



+ (X X)

+ (X X)


= 0 + E 0 X(X 0 X)1 (X 0 X)1 X 0
1

= 0 + T r (X 0 X)
= 0 + 2

K
X

0 


X
0

i (the trace is the sum of eigenvalues)

i=1

>

0 + max(X 0 X 1 ) 2 (the eigenvalues are all positive, sinceX 0 X is p.d.

so
> 0 +
E(0 )

2
min(X 0 X)

where min(X 0 X) is the minimum eigenvalue of X 0 X (which is the inverse of the maximum eigenvalue of (X 0 X)1 ). As collinearity becomes worse and worse, X 0 X becomes more nearly singular,
so min(X 0 X) tends to zero (recall that the determinant is the product of the eigenvalues) and
tends to infinite. On the other hand, 0 is finite.
E(0 )
Now considering the restriction IK = 0 + v. With this restriction the model becomes
"

y
0

"

"
+

kIK

kv

and the estimator is

ridge

=
=

X0
0

kIK
2

X X + k IK

"

#!1

kIK

1

X0

IK

"

Xy

This is the ordinary ridge regression estimator. The ridge regression estimator can be seen to
add k 2 IK , which is nonsingular, to X 0 X, which is more and more nearly singular as collinearity
becomes worse and worse. As k , the restrictions tend to = 0, that is, the coefficients are
shrunken toward zero. Also, the estimator tends to
ridge = X 0 X + k 2 IK

1

X 0 y k 2 IK

1

X 0y =

X 0y
0
k2

0
so ridge
ridge 0. This is clearly a false restriction in the limit, if our original model is at all

sensible.
There should be some amount of shrinkage that is in fact a true restriction. The problem is to
determine the k such that the restriction is correct. The interest in ridge regression centers on the
fact that it can be shown that there exists a k such that M SE(ridge ) < OLS . The problem is that
this k depends on and 2 , which are unknown.
0
The ridge trace method plots ridge
ridge as a function of k, and chooses the value of k that
artistically seems appropriate (e.g., where the effect of increasing k dies off). Draw picture
here. This means of choosing k is obviously subjective. This is not a problem from the Bayesian
perspective: the choice of k reflects prior beliefs about the length of .
In summary, the ridge estimator offers some hope, but it is impossible to guarantee that it will
outperform the OLS estimator. Collinearity is a fact of life in econometrics, and there is no clear

solution to the problem.

7.2

Measurement error

Measurement error is exactly what it says, either the dependent variable or the regressors are
measured with error. Thinking about the way economic data are reported, measurement error is
probably quite prevalent. For example, estimates of growth of GDP, inflation, etc. are commonly
revised several times. Why should the last revision necessarily be correct?

Error of measurement of the dependent variable

Measurement errors in the dependent variable and the regressors have important differences. First
consider error in measurement of the dependent variable. The data generating process is presumed
to be
y

= X +

= y + v

vt

iid(0, v2 )

where y = y + v is the unobservable true dependent variable, and y is what is observed. We

assume that and v are independent and that y = X + satisfies the classical assumptions.
Given this, we have
y + v = X +
so
= X + v

= X +
iid(0, 2 + v2 )

As long as v is uncorrelated with X, this model satisfies the classical assumptions and can be
estimated by OLS. This type of measurement error isnt a problem, then, except in that the
increased variability of the error term causes an increase in the variance of the OLS estimator
(see equation 7.1).

Error of measurement of the regressors

The situation isnt so good in this case. The DGP is
yt

= x0
t + t

xt

= xt + vt

vt

iid(0, v )

where v is a K K matrix. Now X contains the true, unobserved regressors, and X is what is
observed. Again assume that v is independent of , and that the model y = X + satisfies the

classical assumptions. Now we have

yt

(xt vt ) + t

x0t vt0 + t

x0t + t

The problem is that now there is a correlation between xt and t , since

= E ((xt + vt ) (vt0 + t ))

E(xt t )

= v
where
v = E (vt vt0 ) .
Because of this correlation, the OLS estimator is biased and inconsistent, just as in the case of
autocorrelated errors with lagged dependent variables. In matrix notation, write the estimated
model as
y = X +
We have that
=

X 0X
n

1 

X 0y
n

and

plim

X 0X
n

1
= plim
=

(X 0 + V 0 ) (X + V )
n

(QX + v )

since X and V are independent, and

plim

V 0V
n

lim E

1X
vt vt0
n t=1

Likewise,

plim

X 0y
n


=
=

(X 0 + V 0 ) (X + )
n
QX
plim

so
plim = (QX + v )

QX

So we see that the least squares estimator is inconsistent when the regressors are measured with
error.
A potential solution to this problem is the instrumental variables (IV) estimator, which well
discuss shortly.
Example 19. Measurement error in a dynamic model. Consider the model
yt

+ yt1
+ xt + t

yt

yt + t

where t and t are independent Gaussian white noise errors. Suppose that yt is not observed, and
instead we observe yt . What are the properties of the OLS regression on the equation
yt = + yt1 + xt + t
? The Octave script DataProblems/MeasurementError.m does a Monte Carlo study. The sample size
is n = 100. Figure 7.4 gives the results. The first panel shows a histogram for 1000 replications
of , when = 1, so that there is significant measurement error. The second panel repeats
this with = 0, so that there is not measurement error. Note that there is much more bias with
measurement error. There is also bias without measurement error. This is due to the same reason
that we saw bias in Figure 3.7: one of the classical assumptions (nonstochastic regressors) that
guarantees unbiasedness of OLS does not hold for this model. Without measurement error, the
OLS estimator is consistent. By re-running the script with larger n, you can verify that the bias
disappears when = 0, but not when > 0.

Figure 7.4: with and without measurement error

(a) with measurement error: = 1

7.3

(b) without measurement error: = 0

Missing observations

Missing observations occur quite frequently: time series data may not be gathered in a certain
year, or respondents to a survey may not answer all questions. Well consider two cases: missing
observations on the dependent variable and missing observations on the regressors.

Missing observations on the dependent variable

In this case, we have
y = X +
or

"

y1

"
=

y2

X1

"

X2

where y2 is not observed. Otherwise, we assume the classical assumptions hold.

A clear alternative is to simply estimate using the compete observations
y1 = X1 + 1
Since these observations satisfy the classical assumptions, one could estimate by OLS.
The question remains whether or not one could somehow replace the unobserved y2 by a
predictor, and improve over OLS in some sense. Let y2 be the predictor of y2 . Now
("
=
=

X1

#0 "

X2
[X10 X1

X1

#)1 "

X2
1
X20 X2 ]

X1

#0 "

X2
[X10 y1

Recall that the OLS fonc are

X 0 X = X 0 y

X20 y2 ]

y1
y2

so if we regressed using only the first (complete) observations, we would have

X10 X1 1 = X10 y1.
Likewise, an OLS regression using only the second (filled in) observations would give
X20 X2 2 = X20 y2 .
Substituting these into the equation for the overall combined estimator gives
1

1
X10 X1 1 + [X10 X1 + X20 X2 ] X20 X2 2

[X10 X1 + X20 X2 ]

[X10 X1 + X20 X2 ]

X10 X1 1 + X20 X2 2

A1 + (IK A)2
where
1

A [X10 X1 + X20 X2 ]

X10 X1

and we use
[X10 X1 + X20 X2 ]

X20 X2

[X10 X1 + X20 X2 ]

[(X10 X1 + X20 X2 ) X10 X1 ]

1

= IK [X10 X1 + X20 X2 ]

X10 X1

= IK A.
Now,
 
= A + (IK A)E 2
E()
 
and this will be unbiased only if E 2 = .
The conclusion is that the filled in observations alone would need to define an unbiased
estimator. This will be the case only if
y2 = X2 + 2
where 2 has mean zero. Clearly, it is difficult to satisfy this condition without knowledge of
.
Note that putting y2 = y1 does not satisfy the condition and therefore leads to a biased
estimator.
Exercise 20. Formally prove this last statement.

The sample selection problem

In the above discussion we assumed that the missing observations are random. The sample selection problem is a case where the missing observations are not random. Consider the model
yt = x0t + t
which is assumed to satisfy the classical assumptions. However, yt is not always observed. What
is observed is yt defined as
yt = yt if yt 0
Or, in other words, yt is missing when it is less than zero.

Figure 7.5: Sample selection bias

25
Data
True Line
Fitted Line
20

15

10

-5

-10
0

10

The difference in this case is that the missing values are not random: they are correlated with
the xt . Consider the case
y = x +
with V () = 25, but using only the observations for which y > 0 to estimate. Figure 7.5 illustrates
the bias. The Octave program is sampsel.m
There are means of dealing with sample selection bias, but we will not go into it here. One
should at least be aware that nonrandom selection of the sample will normally lead to bias and
inconsistency if the problem is not taken into account.

Missing observations on the regressors

Again the model is
"

y1
y2

"
=

X1
X2

"
+

but we assume now that each row of X2 has an unobserved component(s). Again, one could just
estimate using the complete observations, but it may seem frustrating to have to drop observations
simply because of a single missing variable. In general, if the unobserved X2 is replaced by some
prediction, X2 , then we are in the case of errors of observation. As before, this means that the OLS
estimator is biased when X2 is used instead of X2 . Consistency is salvaged, however, as long as
the number of missing observations doesnt increase with n.
Including observations that have missing values replaced by ad hoc values can be interpreted
as introducing false stochastic restrictions. In general, this introduces bias. It is difficult
to determine whether MSE increases or decreases. Monte Carlo studies suggest that it is
dangerous to simply substitute the mean, for example.
In the case that there is only one regressor other than the constant, subtitution of x
for the
missing xt does not lead to bias. This is a special case that doesnt hold for K > 2.
Exercise 21. Prove this last statement.

 In summary, if one is strongly concerned with bias, it is best to drop observations that have
missing components. There is potential for reduction of MSE through filling in missing elements with intelligent guesses, but this could also increase MSE.

7.4

Exercises

1. Consider the simple Nerlove model

ln C = 1 + 2 ln Q + 3 ln PL + 4 ln PF + 5 ln PK + 
When this model is estimated by OLS, some coefficients are not significant. We have seen
that collinearity is not an important problem. Why is 5 not significantly different from zero?
Give an economic explanation.
2. For the model y = 1 x1 + 2 x2 + ,
(a) verify that the level sets of the OLS criterion function (defined in equation 3.2) are
straight lines when there is perfect collinearity
(b) For this model with perfect collinearity, the OLS estimator does not exist. Depict what
this statement means using a drawing.
(c) Show how a restriction R1 1 + R2 2 = r causes the restricted least squares estimator to
exist, using a drawing.

Chapter 8

Functional form and nonnested tests

Though theory often suggests which conditioning variables should be included, and suggests the
signs of certain derivatives, it is usually silent regarding the functional form of the relationship
between the dependent variable and the regressors. For example, considering a cost function, one
could have a Cobb-Douglas model
c = Aw11 w22 q q e
This model, after taking logarithms, gives
ln c = 0 + 1 ln w1 + 2 ln w2 + q ln q +
where 0 = ln A. Theory suggests that A > 0, 1 > 0, 2 > 0, 3 > 0. This model isnt compatible
with a fixed cost of production since c = 0 when q = 0. Homogeneity of degree one in input prices
suggests that 1 + 2 = 1, while constant returns to scale implies q = 1.
While this model may be reasonable in some cases, an alternative

c = 0 + 1 w 1 + 2 w 2 + q q +

may be just as plausible. Note that

x and ln(x) look quite alike, for certain values of the regres-

sors, and up to a linear transformation, so it may be difficult to choose between these models.
The basic point is that many functional forms are compatible with the linear-in-parameters
model, since this model can incorporate a wide variety of nonlinear transformations of the dependent variable and the regressors. For example, suppose that g() is a real valued function and that
x() is a K vector-valued function. The following model is linear in the parameters but nonlinear
in the variables:
xt

= x(zt )

yt

= x0t + t

There may be P fundamental conditioning variables zt , but there may be K regressors, where K
may be smaller than, equal to or larger than P. For example, xt could include squares and cross
products of the conditioning variables in zt .

8.1

Flexible functional forms

Given that the functional form of the relationship between the dependent variable and the regressors is in general unknown, one might wonder if there exist parametric models that can closely
approximate a wide variety of functional relationships. A Diewert-Flexible functional form is
79

defined as one such that the function, the vector of first derivatives and the matrix of second
derivatives can take on an arbitrary value at a single data point. Flexibility in this sense clearly
requires that there be at least


K = 1 + P + P 2 P /2 + P
free parameters: one for each independent effect that we wish to model.
Suppose that the model is
y = g(x) +
A second-order Taylors series expansion (with remainder term) of the function g(x) about the
point x = 0 is
g(x) = g(0) + x0 Dx g(0) +

x0 Dx2 g(0)x
+R
2

Use the approximation, which simply drops the remainder term, as an approximation to g(x) :
g(x) ' gK (x) = g(0) + x0 Dx g(0) +

x0 Dx2 g(0)x
2

As x 0, the approximation becomes more and more exact, in the sense that gK (x) g(x),
Dx gK (x) Dx g(x) and Dx2 gK (x) Dx2 g(x). For x = 0, the approximation is exact, up to the
second order. The idea behind many flexible functional forms is to note that g(0), Dx g(0) and
Dx2 g(0) are all constants. If we treat them as parameters, the approximation will have exactly
enough free parameters to approximate the function g(x), which is of unknown form, exactly, up
to second order, at the point x = 0. The model is
gK (x) = + x0 + 1/2x0 x
so the regression model to fit is
y = + x0 + 1/2x0 x +
While the regression model has enough free parameters to be Diewert-flexible, the question
= D2 g(0)?
remains: is plim
= g(0)? Is plim = Dx g(0)? Is plim
x

The answer is no, in general. The reason is that if we treat the true values of the parameters
as these derivatives, then is forced to play the part of the remainder term, which is a
function of x, so that x and are correlated in this case. As before, the estimator is biased in
this case.
A simpler example would be to consider a first-order T.S. approximation to a quadratic function. Draw picture.
The conclusion is that flexible functional forms arent really flexible in a useful statistical
sense, in that neither the function itself nor its derivatives are consistently estimated, unless
the function belongs to the parametric family of the specified functional form. In order to
lead to consistent inferences, the regression model must be correctly specified.

The translog form

In spite of the fact that FFFs arent really flexible for the purposes of econometric estimation and
inference, they are useful, and they are certainly subject to less bias due to misspecification of the
functional form than are many popular forms, such as the Cobb-Douglas or the simple linear in the

variables model. The translog model is probably the most widely used FFF. This model is as above,
except that the variables are subjected to a logarithmic tranformation. Also, the expansion point is
usually taken to be the sample mean of the data, after the logarithmic transformation. The model
is defined by
y

ln(c)
z 
= ln
z
= ln(z) ln(
z)

= + x0 + 1/2x0 x +

In this presentation, the t subscript that distinguishes observations is suppressed for simplicity.
Note that
y
x

=
=
=

+ x
ln(c)
(the other part of x is constant)
ln(z)
c z
z c

which is the elasticity of c with respect to z. This is a convenient feature of the translog model.
Note that at the means of the conditioning variables, z, x = 0, so

y 
=
x z=z
so the are the first-order elasticities, at the means of the data.
To illustrate, consider that y is cost of production:
y = c(w, q)
where w is a vector of input prices and q is output. We could add other variables by extending q
in the obvious manner, but this is supressed for simplicity. By Shephards lemma, the conditional
factor demands are
x=

c(w, q)
w

and the cost shares of the factors are therefore

s=

wx
c(w, q) w
=
c
w c

which is simply the vector of elasticities of cost with respect to input prices. If the cost function is
modeled using a translog function, we have
ln(c)

= + x + z + 1/2
0

h
0

"

11

12

012

22

#"

x
z

= + x + z + 1/2x 11 x + x 12 z + 1/2z 22

where x = ln(w/w)
(element-by-element division) and z = ln(q/
q ), and
"
11

=
"

11

12

12

22
#

13

12

22

= 33 .

23

Note that symmetry of the second derivatives has been imposed.

Then the share equations are just
s=+

11

12

"

Therefore, the share equations and the cost equation have parameters in common. By pooling the
equations together and imposing the (true) restriction that the parameters of the equations be the
same, we can gain efficiency.
To illustrate in more detail, consider the case of two inputs, so
"
x=

x1

#
.

x2

In this case the translog model of the logarithmic cost function is

ln c = + 1 x1 + 2 x2 + z +

11 2 22 2 33 2
x +
x +
z + 12 x1 x2 + 13 x1 z + 23 x2 z
2 1
2 2
2

The two cost shares of the inputs are the derivatives of ln c with respect to x1 and x2 :
s1

1 + 11 x1 + 12 x2 + 13 z

s2

2 + 12 x1 + 22 x2 + 13 z

Note that the share equations and the cost equation have parameters in common. One can do
a pooled estimation of the three equations at once, imposing that the parameters are the same.
In this way were using more observations and therefore more information, which will lead to
imporved efficiency. Note that this does assume that the cost equation is correctly specified (i.e.,
not an approximation), since otherwise the derivatives would not be the true derivatives of the
log cost function, and would then be misspecified for the shares. To pool the equations, write the
model in matrix form (adding in error terms)

ln c

s1
s2

x1

x2

x21
2

x22
2

z2
2

x1 x2

x1 z

= 0

x1

x2

x2

x1

x2 z

11

+ 2
22

3
33

12

13

23

This is one observation on the three equations. With the appropriate notation, a single obser-

vation can be written as

yt = Xt + t
The overall model would stack n observations on the three equations for a total of 3n observations:

y1

X1

y2
..
.

X2
..
.

2
..
.

Xn

yn

Next we need to consider the errors. For observation t the errors can be placed in a vector

1t

t = 2t
3t
First consider the covariance matrix of this vector: the shares are certainly correlated since they
must sum to one. (In fact, with 2 shares the variances are equal and the covariance is -1 times
the variance. General notation is used to allow easy extension to the case of more than 2 inputs).
Also, its likely that the shares and the cost equation have different variances. Supposing that the
model is covariance stationary, the variance of t won0 t depend upon t:

11

V art = 0 =

12

13

22

23
33

Note that this matrix is singular, since the shares sum to 1. Assuming that there is no autocorrelation, the overall covariance matrix has the seemingly unrelated regressions (SUR) structure.

V ar

2
..
.

0
..
.

0
..
.

..
.

0
..
.
0

= In 0
where the symbol indicates the Kronecker product. The Kronecker product of two matrices A
and B is

a11 B

AB =

a21 B
..
.
apq B

a12 B
..
.

a1q B

..

apq B

FGLS estimation of a translog model

So, this model has heteroscedasticity and autocorrelation, so OLS wont be efficient. The next
question is: how do we estimate efficiently using FGLS? FGLS is based upon inverting the estimated

 So we need to estimate .
error covariance .
An asymptotically efficient procedure is (supposing normality of the errors)
1. Estimate each equation by OLS
2. Estimate 0 using

X
0 = 1

t 0t
n t=1
0 will be singular
3. Next we need to account for the singularity of 0 . It can be shown that
when the shares sum to one, so FGLS wont work. The solution is to drop one of the share
equations, for example the second. The model becomes

"

ln c
s1

"

#
=

x1

x2

x21
2

x22
2

z2
2

x1 x2

x1 z

x1

x2

#
x2 z

"
#
1
11

+
2
22

33

12

13

23

or in matrix notation for the observation:

yt = Xt + t
and in stacked notation for all observations we have the 2n observations:

y1

X1

y2
..
.

X2
..
.

2
..
.

yn

Xn

or, finally in matrix notation for all observations:

y = X +
Considering the error covariance, we can define
"
0

= V ar
= In

2
0

as the leading 2 2 block of

0 , and form
Define
0
= In
0 .

This is a consistent estimator, following the consistency of OLS and applying a LLN.

4. Next compute the Cholesky factorization

 1
0
P0 = Chol
(I am assuming this is defined as an upper triangular matrix, which is consistent with the
way Octave does it) and the Cholesky factorization of the overall covariance matrix of the 2
equation model, which can be calculated as
= In P0
P = Chol
5. Finally the FGLS estimator can be calculated by applying OLS to the transformed model
0
P 0 y = P 0 X + P
or by directly using the GLS formula
F GLS =


1
 1
 1
0

0
X
0
X
X 0
y

It is equivalent to transform each observation individually:

P00 yy = P00 Xt + P00
and then apply OLS. This is probably the simplest approach.
A few last comments.
1. We have assumed no autocorrelation across time. This is clearly restrictive. It is relatively
simple to relax this, but we wont go into it here.
2. Also, we have only imposed symmetry of the second derivatives. Another restriction that the
model should satisfy is that the estimated shares should sum to 1. This can be accomplished
by imposing
1 + 2
3
X
ij

0, j = 1, 2, 3.

i=1

These are linear parameter restrictions, so they are easy to impose and will improve efficiency
if they are true.
3. The estimation procedure outlined above can be iterated. That is, estimate F GLS as above,
then re-estimate 0 using errors calculated as
= y X F GLS
These might be expected to lead to a better estimate than the estimator based on OLS , since
FGLS is asymptotically more efficient. Then re-estimate using the new estimated error
covariance. It can be shown that if this is repeated until the estimates dont change (i.e.,
iterated to convergence) then the resulting estimator is the MLE. At any rate, the asymptotic
properties of the iterated and uniterated estimators are the same, since both are based upon
a consistent estimator of the error covariance.

8.2

Testing nonnested hypotheses

Given that the choice of functional form isnt perfectly clear, in that many possibilities exist, how
can one choose between forms? When one form is a parametric restriction of another, the previously studied tests such as Wald, LR, score or qF are all possibilities. For example, the CobbDouglas model is a parametric restriction of the translog: The translog is
yt = + x0t + 1/2x0t xt +
where the variables are in logarithms, while the Cobb-Douglas is
yt = + x0t +
so a test of the Cobb-Douglas versus the translog is simply a test that = 0.
The situation is more complicated when we want to test non-nested hypotheses. If the two
functional forms are linear in the parameters, and use the same transformation of the dependent
variable, then they may be written as
M1 : y
t
M2 : y

= X +
iid(0, 2 )
= Z +
iid(0, 2 )

We wish to test hypotheses of the form: H0 : Mi is correctly specified versus HA : Mi is misspecified,

for i = 1, 2.
One could account for non-iid errors, but well suppress this for simplicity.
There are a number of ways to proceed. Well consider the J test, proposed by Davidson and
MacKinnon, Econometrica (1981). The idea is to artificially nest the two models, e.g.,
y = (1 )X + (Z) +
If the first model is correctly specified, then the true value of is zero. On the other hand, if
the second model is correctly specified then = 1.
The problem is that this model is not identified in general. For example, if the models
share some regressors, as in

M1 : yt

= 1 + 2 x2t + 3 x3t + t

M2 : yt

= 1 + 2 x2t + 3 x4t + t

then the composite model is

yt = (1 )1 + (1 )2 x2t + (1 )3 x3t + 1 + 2 x2t + 3 x4t + t
Combining terms we get
yt

((1 )1 + 1 ) + ((1 )2 + 2 ) x2t + (1 )3 x3t + 3 x4t + t

= 1 + 2 x2t + 3 x3t + 4 x4t + t

The four 0 s are consistently estimable, but is not, since we have four equations in 7 unknowns,
so one cant test the hypothesis that = 0.
The idea of the J test is to substitute in place of . This is a consistent estimator supposing
that the second model is correctly specified. It will tend to a finite probability limit even if the
second model is misspecified. Then estimate the model
y

(1 )X + (Z ) +

X +
y+

where y = Z(Z 0 Z)1 Z 0 y = PZ y. In this model, is consistently estimable, and one can show that,
p

under the hypothesis that the first model is correct, 0 and that the ordinary t -statistic for
= 0 is asymptotically normal:
t=

a
N (0, 1)

If the second model is correctly specified, then t , since

tends in probability to 1,
while its estimated standard error tends to zero. Thus the test will always reject the false
null model, asymptotically, since the statistic will eventually exceed any critical value with
probability one.
We can reverse the roles of the models, testing the second against the first.
It may be the case that neither model is correctly specified. In this case, the test will still reject
the null hypothesis, asymptotically, if we use critical values from the N (0, 1) distribution,
p

since as long as
tends to something different from zero, |t| . Of course, when we
switch the roles of the models the other will also be rejected asymptotically.
In summary, there are 4 possible outcomes when we test two models, each against the other.
Both may be rejected, neither may be rejected, or one of the two may be rejected.
There are other tests available for non-nested models. The J test is simple to apply when
both models are linear in the parameters. The P -test is similar, but easier to apply when M1
is nonlinear.
The above presentation assumes that the same transformation of the dependent variable is
used by both models. MacKinnon, White and Davidson, Journal of Econometrics, (1983)
shows how to deal with the case of different transformations.
Monte-Carlo evidence shows that these tests often over-reject a correctly specified model.
Can use bootstrap critical values to get better-performing tests.

Chapter 9

Generalized least squares

Recall the assumptions of the classical linear regression model, in Section 3.6. One of the assumptions weve made up to now is that
t IID(0, 2 )
or occasionally
t IIN (0, 2 ).
Now well investigate the consequences of nonidentically and/or dependently distributed errors.
Well assume fixed regressors for now, to keep the presentation simple, and later well look at the
consequences of relaxing this admittedly unrealistic assumption. The model is
y

= X +

E()

V ()

where is a general symmetric positive definite matrix (well write in place of 0 to simplify the
typing of these notes).
The case where is a diagonal matrix gives uncorrelated, nonidentically distributed errors.
This is known as heteroscedasticity: i, j s.t. V (i ) 6= V (j )
The case where has the same number on the main diagonal but nonzero elements off the
main diagonal gives identically (assuming higher moments are also the same) dependently
distributed errors. This is known as autocorrelation: i 6= j s.t. E(i j ) 6= 0)
The general case combines heteroscedasticity and autocorrelation. This is known as nonspherical disturbances, though why this term is used, I have no idea. Perhaps its because
under the classical assumptions, a joint confidence region for would be an n dimensional
hypersphere.

9.1

Effects of nonspherical disturbances on the OLS estimator

The least square estimator is

=
=

(X 0 X)1 X 0 y
+ (X 0 X)1 X 0

We have unbiasedness, as before.

88

 The variance of is
i
h
E ( )( )0



E (X 0 X)1 X 0 0 X(X 0 X)1

(X 0 X)1 X 0 X(X 0 X)1

(9.1)

Due to this, any test statistic that is based upon an estimator of 2 is invalid, since there isnt
any 2 , it doesnt exist as a feature of the true d.g.p. In particular, the formulas for the t,
F, 2 based tests given above do not lead to statistics with these distributions.
is still consistent, following exactly the same argument given before.
If is normally distributed, then
N , (X 0 X)1 X 0 X(X 0 X)1

The problem is that is unknown in general, so this distribution wont be useful for testing
hypotheses.
Without normality, and with stochastic X (e.g., weakly exogenous regressors) we still have


n

n(X 0 X)1 X 0
 0 1
XX
=
n1/2 X 0
n

Define the limiting variance of n1/2 X 0 (supposing a CLT applies) as


lim E

so we obtain

X 0 0 X
n


= , a.s.





d
1
n N 0, Q1
X QX . Note that the true asymptotic distribution

of the OLS has changed with respect to the results under the classical assumptions. If we
neglect to take this into account, the Wald and score tests will not be asymptotically valid.
So we need to figure out how to take it into account.
To see the invalidity of test procedures that are correct under the classical assumptions, when we
have nonspherical errors, consider the Octave script GLS/EffectsOLS.m. This script does a Monte
Carlo study, generating data that are either heteroscedastic or homoscedastic, and then computes
the empirical rejection frequency of a nominally 10% t-test. When the data are heteroscedastic,
we obtain something like what we see in Figure 9.1. This sort of heteroscedasticity causes us to
reject a true null hypothesis regarding the slope parameter much too often. You can experiment
with the script to look at the effects of other sorts of HET, and to vary the sample size.

Figure 9.1: Rejection frequency of 10% t-test, H0 is true.

Summary: OLS with heteroscedasticity and/or autocorrelation is:

unbiased with fixed or strongly exogenous regressors
biased with weakly exogenous regressors
has a different variance than before, so the previous test statistics arent valid
is consistent
is asymptotically normally distributed, but with a different limiting covariance matrix. Previous test statistics arent valid in this case for this reason.
is inefficient, as is shown below.

9.2

The GLS estimator

Suppose were known. Then one could form the Cholesky decomposition
P 0 P = 1
Here, P is an upper triangular matrix. We have
P 0 P = In
so
P 0 P P 0 = P 0 ,

which implies that

P P 0 = In
Lets take some time to play with the Cholesky decomposition. Try out the GLS/cholesky.m
Octave script to see that the above claims are true, and also to see how one can generate data from
a N (0, V ) distribition.
Consider the model
P y = P X + P ,
or, making the obvious definitions,
y = X + .
This variance of = P is
E(P 0 P 0 )

= P P 0
= In

Therefore, the model

y

= X +

E( )

V ( )

= In

satisfies the classical assumptions. The GLS estimator is simply OLS applied to the transformed
model:
GLS

(X 0 X )1 X 0 y

(X 0 P 0 P X)1 X 0 P 0 P y

(X 0 1 X)1 X 0 1 y

The GLS estimator is unbiased in the same circumstances under which the OLS estimator is
unbiased. For example, assuming X is nonstochastic
E(GLS )



= E (X 0 1 X)1 X 0 1 y


= E (X 0 1 X)1 X 0 1 (X +
= .

To get the variance of the estimator, we have

GLS

(X 0 X )1 X 0 y

(X 0 X )1 X 0 (X + )

= + (X 0 X )1 X 0
so
E



GLS


0 

GLS
=



E (X 0 X )1 X 0 0 X (X 0 X )1

(X 0 X )1 X 0 X (X 0 X )1

(X 0 X )1

(X 0 1 X)1

Either of these last formulas can be used.

All the previous results regarding the desirable properties of the least squares estimator hold,
when dealing with the transformed model, since the transformed model satisfies the classical
assumptions..
Tests are valid, using the previous formulas, as long as we substitute X in place of X.
Furthermore, any test that involves 2 can set it to 1. This is preferable to re-deriving the
appropriate formulas.
The GLS estimator is more efficient than the OLS estimator. This is a consequence of the
Gauss-Markov theorem, since the GLS estimator is based on a model that satisfies the classical
assumptions but the OLS estimator is not. To see this directly, note that
V ar(GLS )
V ar()

(X 0 X)1 X 0 X(X 0 X)1 (X 0 1 X)1

= AA

h
i
1
where A = (X 0 X) X 0 (X 0 1 X)1 X 0 1 . This may not seem obvious, but it is true,
0

as you can verify for yourself. Then noting that AA is a quadratic form in a positive definite
0

matrix, we conclude that AA is positive semi-definite, and that GLS is efficient relative to
OLS.
As one can verify by calculating first order conditions, the GLS estimator is the solution to
the minimization problem
GLS = arg min(y X)0 1 (y X)
so the metric 1 is used to weight the residuals.

9.3

Feasible GLS

The problem is that ordinarily isnt known, so this estimator isnt available.




Consider the dimension of : its an n n matrix with n2 n /2 + n = n2 + n /2 unique
elements (remember - it is symmetric, because its a covariance matrix).
The number of parameters to estimate is larger than n and increases faster than n. Theres
no way to devise an estimator that satisfies a LLN without adding restrictions.
The feasible GLS estimator is based upon making sufficient assumptions regarding the form
of so that a consistent estimator can be devised.
Suppose that we parameterize as a function of X and , where may include as well as other
parameters, so that
= (X, )
where is of fixed dimension. If we can consistently estimate , we can consistently estimate ,
as long as the elements of (X, ) are continuous functions of (by the Slutsky theorem). In this
case,
p

b = (X, )

(X, )

b we obtain the FGLS estimator. The

If we replace in the formulas for the GLS estimator with ,
FGLS estimator shares the same asymptotic properties as GLS. These are

1. Consistency
2. Asymptotic normality
3. Asymptotic efficiency if the errors are normally distributed. (Cramer-Rao).
4. Test procedures are asymptotically valid.
In practice, the usual way to proceed is
1. Define a consistent estimator of . This is a case-by-case proposition, depending on the parameterization (). Well see examples below.

b = (X, )
2. Form
1 ).
3. Calculate the Cholesky factorization Pb = Chol(
4. Transform the model using
P y = P X + P
5. Estimate using OLS on the transformed model.

9.4

Heteroscedasticity

Heteroscedasticity is the case where

E(0 ) =
is a diagonal matrix, so that the errors are uncorrelated, but have different variances. Heteroscedasticity is usually thought of as associated with cross sectional data, though there is absolutely no reason why time series data cannot also be heteroscedastic. Actually, the popular
ARCH (autoregressive conditionally heteroscedastic) models explicitly assume that a time series is
heteroscedastic.
Consider a supply function
qi = 1 + p Pi + s Si + i
where Pi is price and Si is some measure of size of the ith firm. One might suppose that unobservable factors (e.g., talent of managers, degree of coordination between production units, etc.)
account for the error term i . If there is more variability in these factors for large firms than for
small firms, then i may have a higher variance when Si is high than when it is low.
Another example, individual demand.
qi = 1 + p Pi + m Mi + i
where P is price and M is income. In this case, i can reflect variations in preferences. There are
more possibilities for expression of preferences when one is rich, so it is possible that the variance
of i could be higher when M is high.
Add example of group means.

OLS with heteroscedastic consistent varcov estimation

Eicker (1967) and White (1980) showed how to modify test statistics to account for heteroscedasticity of unknown form. The OLS estimator has asymptotic distribution




d
1
n N 0, Q1
X QX

as weve already seen. Recall that we defined


lim E

X 0 0 X
n


=

This matrix has dimension K K and can be consistently estimated, even if we cant estimate
consistently. The consistent estimator, under heteroscedasticity but no autocorrelation is
n

X
b= 1
xt x0t 2t

n t=1
One can then modify the previous test statistics to obtain tests that are valid when there is heteroscedasticity of unknown form. For example, the Wald test for H0 : R r = 0 would be


n R r

0


R

X 0X
n

1

X 0X
n

!1

1
R


a
R r 2 (q)

To see the effects of ignoring HET when doing OLS, and the good effect of using a HET consistent covariance estimator, consider the script bootstrap_example1.m. This script generates data
from a linear model with HET, then computes standard errors using the ordinary OLS formula, the
Eicker-White formula, and also bootstrap standard errors. Note that Eicker-White and bootstrap
pretty much agree, while the OLS formula gives standard errors that are quite different. Typical
output of this script follows:

octave:1> bootstrap_example1
Bootstrap standard errors
0.083376
0.090719
0.143284
*********************************************************
OLS estimation results
Observations 100
R-squared 0.014674
Sigma-squared 0.695267
Results (Ordinary var-cov estimator)
estimate
st.err.
t-stat.
p-value
1
-0.115
0.084
-1.369
0.174
2
-0.016
0.083
-0.197
0.845
3
-0.105
0.088
-1.189
0.237
*********************************************************
OLS estimation results
Observations 100
R-squared 0.014674
Sigma-squared 0.695267
Results (Het. consistent var-cov estimator)
estimate
st.err.
t-stat.
p-value
1
-0.115
0.084
-1.381
0.170
2
-0.016
0.090
-0.182
0.856
3
-0.105
0.140
-0.751
0.454
If you run this several times, you will notice that the OLS standard error for the last parameter
appears to be biased downward, at least comparing to the other two methods, which are
asymptotically valid.
The true coefficients are zero. With a standard error biased downward, the t-test for lack of
significance will reject more often than it should (the variables really are not significant, but
we will find that they seem to be more often than is due to Type-I error.
For example, you should see that the p-value for the last coefficient is smaller than 0.10 more
than 10% of the time. Run the script 20 times and youll see.

Detection
There exist many tests for the presence of heteroscedasticity. Well discuss three methods.
Goldfeld-Quandt

The sample is divided in to three parts, with n1 , n2 and n3 observations, where

n1 + n2 + n3 = n. The model is estimated using the first and third parts of the sample, separately,
so that 1 and 3 will be independent. Then we have
0

1 M 1 1 d 2
10 1
=
(n1 K)
2

2
and
0

30 3
3 M 3 3 d 2
=
(n3 K)
2

2
so

10 1 /(n1 K) d
F (n1 K, n3 K).
30 3 /(n3 K)

The distributional result is exact if the errors are normally distributed. This test is a two-tailed
test. Alternatively, and probably more conventionally, if one has prior ideas about the possible
magnitudes of the variances of the observations, one could order the observations accordingly,
from largest to smallest. In this case, one would use a conventional one-tailed F-test. Draw picture.
Ordering the observations is an important step if the test is to have any power.
The motive for dropping the middle observations is to increase the difference between the
average variance in the subsamples, supposing that there exists heteroscedasticity. This can
increase the power of the test. On the other hand, dropping too many observations will
substantially increase the variance of the statistics 10 1 and 30 3 . A rule of thumb, based on
Monte Carlo experiments is to drop around 25% of the observations.
If one doesnt have any ideas about the form of the het. the test will probably have low power
since a sensible data ordering isnt available.
Whites test When one has little idea if there exists heteroscedasticity, and no idea of its potential
form, the White test is a possibility. The idea is that if there is homoscedasticity, then
E(2t |xt ) = 2 , t
so that xt or functions of xt shouldnt help to explain E(2t ). The test works as follows:
1. Since t isnt available, use the consistent estimator t instead.
2. Regress
2t = 2 + zt0 + vt
where zt is a P -vector. zt may include some or all of the variables in xt , as well as other
variables. Whites original suggestion was to use xt , plus the set of all unique squares and
cross products of variables in xt .
3. Test the hypothesis that = 0. The qF statistic in this case is
qF =

P (ESSR ESSU ) /P
ESSU / (n P 1)

Note that ESSR = T SSU , so dividing both numerator and denominator by this we get
qF = (n P 1)

R2
1 R2

Note that this is the R2 of the artificial regression used to test for heteroscedasticity, not the
R2 of the original model.
An asymptotically equivalent statistic, under the null of no heteroscedasticity (so that R2 should
tend to zero), is
a

nR2 2 (P ).
This doesnt require normality of the errors, though it does assume that the fourth moment of t is
constant, under the null. Question: why is this necessary?
The White test has the disadvantage that it may not be very powerful unless the zt vector is
chosen well, and this is hard to do without knowledge of the form of heteroscedasticity.

 It also has the problem that specification errors other than heteroscedasticity may lead to
rejection.
Note: the null hypothesis of this test may be interpreted as = 0 for the variance model
V (2t ) = h( + zt0 ), where h() is an arbitrary function of unknown form. The test is more
general than is may appear from the regression that is used.
Plotting the residuals A very simple method is to simply plot the residuals (or their squares).
Draw pictures here. Like the Goldfeld-Quandt test, this will be more informative if the observations
are ordered according to the suspected form of the heteroscedasticity.

Correction
Correcting for heteroscedasticity requires that a parametric form for () be supplied, and that a
means for estimating consistently be determined. The estimation method will be specific to the
for supplied for (). Well consider two examples. Before this, lets consider the general nature of
GLS when there is heteroscedasticity.
When we have HET but no AUT, is a diagonal matrix:

12
.
..

0
..
.

0
n2

...

22
..

Likewise, 1 is diagonal

1
12

..
.
=

...

1
22

..

0
..
.

1
2
n

and so is the Cholesky decomposition P = chol(1 )

1
1

.
..

P =

0
1
2

...
..

0
..
.

1
n

We need to transform the model, just as before, in the general case:

P y = P X + P ,
or, making the obvious definitions,
y = X + .
Note that multiplying by P just divides the data for each observation (yi , xi ) by the corresponding
standard error of the error term, i . That is, yi = yi /i and xi = xi /i (note that xi is a K-vector:
we divided each element, including the 1 corresponding to the constant).
This makes sense. Consider Figure 9.2, which shows a true regression line with heteroscedastic
errors. Which sample is more informative about the location of the line? The ones with observations with smaller variances. So, the GLS solution is equivalent to OLS on the transformed data.

Figure 9.2: Motivation for GLS correction when there is HET

By the transformed data is the original data, weighted by the inverse of the standard error of the
observations error term. When the standard error is small, the weight is high, and vice versa. The
GLS correction for the case of HET is also known as weighted least squares, for this reason.
Multiplicative heteroscedasticity
Suppose the model is
yt

x0t + t

t2

E(2t ) = (zt0 )

but the other classical assumptions hold. In this case

2t = (zt0 ) + vt
and vt has mean zero. Nonlinear least squares could be used to estimate and consistently, were
t observable. The solution is to substitute the squared OLS residuals 2t in place of 2t , since it is
we can estimate 2 consistently using
consistent by the Slutsky theorem. Once we have and ,
t

t2 = (zt0 ) t2 .
In the second step, we transform the model by dividing by the standard deviation:
x0
t
yt
= t +

t
or

yt = x0
t + t .

Asymptotically, this model satisfies the classical assumptions.

This model is a bit complex in that NLS is required to estimate the model of the variance. A
simpler version would be
yt

x0t + t

t2

= E(2t ) = 2 zt

where zt is a single variable. There are still two parameters to be estimated, and the model
of the variance is still nonlinear in the parameters. However, the search method can be used
in this case to reduce the estimation problem to repeated applications of OLS.
First, we define an interval of reasonable values for , e.g., [0, 3].
Partition this interval into M equally spaced values, e.g., {0, .1, .2, ..., 2.9, 3}.
For each of these values, calculate the variable ztm .
The regression
2t = 2 ztm + vt
is linear in the parameters, conditional on m , so one can estimate 2 by OLS.
2
, m ), and the corresponding ESSm . Choose the pair with the minimum
Save the pairs (m

ESSm as the estimate.

Next, divide the model by the estimated standard deviations.
Can refine. Draw picture.
Works well when the parameter to be searched over is low dimensional, as in this case.
Groupwise heteroscedasticity
A common case is where we have repeated observations on each of a number of economic agents:
e.g., 10 years of macroeconomic data on each of a set of countries or regions, or daily observations
of transactions of 200 banks. This sort of data is a pooled cross-section time-series model. It may be
reasonable to presume that the variance is constant over time within the cross-sectional units, but
that it differs across them (e.g., firms or countries of different sizes...). The model is
yit
E(2it )

= x0it + it
=

i2 , t

where i = 1, 2, ..., G are the agents, and t = 1, 2, ..., n are the observations on each agent.
The other classical assumptions are presumed to hold.
In this case, the variance i2 is specific to each agent, but constant over the n observations
for that agent.
In this model, we assume that E(it is ) = 0. This is a strong assumption that well relax later.
To correct for heteroscedasticity, just estimate each i2 using the natural estimator:
n

i2 =

1X 2

n t=1 it

Figure 9.3: Residuals, Nerlove model, sorted by firm size

Regression residuals
1.5
Residuals

0.5

-0.5

-1

-1.5
0

20

40

60

80

100

120

140

160

Note that we use 1/n here since its possible that there are more than n regressors, so n K
could be negative. Asymptotically the difference is unimportant.
With each of these, transform the model as usual:
yit
x0
it
= it +

i
Do this for each cross-sectional group. This transformed model satisfies the classical assumptions, asymptotically.

Example: the Nerlove model (again!)

Remember the Nerlove data - see sections 3.8 and 5.8. Lets check the Nerlove data for evidence
of heteroscedasticity. In what follows, were going to use the model with the constant and output coefficient varying across 5 groups, but with the input price coefficients fixed (see Equation
5.5 for the rationale behind this). Figure 9.3, which is generated by the Octave program GLS/NerloveResiduals.m plots the residuals. We can see pretty clearly that the error variance is larger for
small firms than for larger firms.
Now lets try out some tests to formally check for heteroscedasticity. The Octave program
GLS/HetTests.m performs the White and Goldfeld-Quandt tests, using the above model. The results
are

Value
p-value
White's test
61.903
0.000
Value
p-value
GQ test
10.886
0.000
All in all, it is very clear that the data are heteroscedastic. That means that OLS estimation is
not efficient, and tests of restrictions that ignore heteroscedasticity are not valid. The previous
tests (CRTS, HOD1 and the Chow test) were calculated assuming homoscedasticity. The Octave

program GLS/NerloveRestrictions-Het.m uses the Wald test to check for CRTS and HOD1, but
using a heteroscedastic-consistent covariance estimator.1 The results are

Testing HOD1
Wald test

Value
6.161

p-value
0.013

Value
20.169

p-value
0.001

Testing CRTS
Wald test

We see that the previous conclusions are altered - both CRTS is and HOD1 are rejected at the 5%
level. Maybe the rejection of HOD1 is due to to Wald tests tendency to over-reject?
From the previous plot, it seems that the variance of  is a decreasing function of output.
Suppose that the 5 size groups have different error variances (heteroscedasticity by groups):
V ar(i ) = j2 ,
where j = 1 if i = 1, 2, ..., 29, etc., as before. The Octave script GLS/NerloveGLS.m estimates
the model using GLS (through a transformation of the model so that OLS can be applied). The
estimation results are i

*********************************************************
OLS estimation results
Observations 145
R-squared 0.958822
Sigma-squared 0.090800
Results (Het. consistent var-cov estimator)

constant1
constant2
constant3
constant4
constant5
output1
output2
output3
output4
output5
labor
fuel
capital

estimate
-1.046
-1.977
-3.616
-4.052
-5.308
0.391
0.649
0.897
0.962
1.101
0.007
0.498
-0.460

st.err.
1.276
1.364
1.656
1.462
1.586
0.090
0.090
0.134
0.112
0.090
0.208
0.081
0.253

t-stat.
-0.820
-1.450
-2.184
-2.771
-3.346
4.363
7.184
6.688
8.612
12.237
0.032
6.149
-1.818

p-value
0.414
0.149
0.031
0.006
0.001
0.000
0.000
0.000
0.000
0.000
0.975
0.000
0.071

*********************************************************
1 By the way, notice that GLS/NerloveResiduals.m and GLS/HetTests.m use the restricted LS estimator directly to restrict
the fully general model with all coefficients varying to the model with only the constant and the output coefficient varying.
But GLS/NerloveRestrictions-Het.m estimates the model by substituting the restrictions into the model. The methods are
equivalent, but the second is more convenient and easier to understand.

*********************************************************
OLS estimation results
Observations 145
R-squared 0.987429
Sigma-squared 1.092393
Results (Het. consistent var-cov estimator)

constant1
constant2
constant3
constant4
constant5
output1
output2
output3
output4
output5
labor
fuel
capital

estimate
-1.580
-2.497
-4.108
-4.494
-5.765
0.392
0.648
0.892
0.951
1.093
0.103
0.492
-0.366

st.err.
0.917
0.988
1.327
1.180
1.274
0.090
0.094
0.138
0.109
0.086
0.141
0.044
0.165

t-stat.
-1.723
-2.528
-3.097
-3.808
-4.525
4.346
6.917
6.474
8.755
12.684
0.733
11.294
-2.217

p-value
0.087
0.013
0.002
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.465
0.000
0.028

*********************************************************
Testing HOD1
Wald test

Value
9.312

p-value
0.002

The first panel of output are the OLS estimation results, which are used to consistently estimate
the j2 . The second panel of results are the GLS estimation results. Some comments:
The R2 measures are not comparable - the dependent variables are not the same. The measure for the GLS results uses the transformed dependent variable. One could calculate a
comparable R2 measure, but I have not done so.
The differences in estimated standard errors (smaller in general for GLS) can be interpreted
as evidence of improved efficiency of GLS, since the OLS standard errors are calculated using
the Huber-White estimator. They would not be comparable if the ordinary (inconsistent)
estimator had been used.
Note that the previously noted pattern in the output coefficients persists. The nonconstant
CRTS result is robust.
The coefficient on capital is now negative and significant at the 3% level. That seems to
indicate some kind of problem with the model or the data, or economic theory.
Note that HOD1 is now rejected. Problem of Wald test over-rejecting? Specification error in
model?

9.5

Autocorrelation

Autocorrelation, which is the serial correlation of the error term, is a problem that is usually
associated with time series data, but also can affect cross-sectional data. For example, a shock to
oil prices will simultaneously affect all countries, so one could expect contemporaneous correlation
of macroeconomic variables across countries.

Example
Consider the Keeling-Whorf data on atmospheric CO2 concentrations an Mauna Loa, Hawaii (see

http://en.wikipedia.org/wiki/Keeling_Curve and http://cdiac.ornl.gov/ftp/ndp001/maunaloa.

txt).
From the file maunaloa.txt: THE DATA FILE PRESENTED IN THIS SUBDIRECTORY CONTAINS MONTHLY AND ANNUAL ATMOSPHERIC CO2 CONCENTRATIONS DERIVED FROM THE
SCRIPPS INSTITUTION OF OCEANOGRAPHYS (SIOs) CONTINUOUS MONITORING PROGRAM
AT MAUNA LOA OBSERVATORY, HAWAII. THIS RECORD CONSTITUTES THE LONGEST CONTINUOUS RECORD OF ATMOSPHERIC CO2 CONCENTRATIONS AVAILABLE IN THE WORLD.
MONTHLY AND ANNUAL AVERAGE MOLE FRACTIONS OF CO2 IN WATER-VAPOR-FREE AIR
ARE GIVEN FROM MARCH 1958 THROUGH DECEMBER 2003, EXCEPT FOR A FEW INTERRUPTIONS.

Figure 9.4: Residuals from time trend for CO2 data

The data is available in Octave format at CO2.data .

If we fit the model CO2t = 1 + 2 t + t , we get the results

octave:8> CO2Example
warning: load: file found in load path
*********************************************************
OLS estimation results
Observations 468
R-squared 0.979239
Sigma-squared 5.696791
Results (Het. consistent var-cov estimator)

1
2

estimate
316.918
0.121

st.err.
0.227
0.001

t-stat.
1394.406
141.521

p-value
0.000
0.000

*********************************************************

It seems pretty clear that CO2 concentrations have been going up in the last 50 years, surprise,
surprise. Lets look at a residual plot for the last 3 years of the data, see Figure 9.4. Note that there
is a very predictable pattern. This is pretty strong evidence that the errors of the model are not
independent of one another, which means there seems to be autocorrelation.

Figure 9.5: Autocorrelation induced by misspecification

Causes
Autocorrelation is the existence of correlation across the error term:
E(t s ) 6= 0, t 6= s.
Why might this occur? Plausible explanations include
1. Lags in adjustment to shocks. In a model such as
yt = x0t + t ,
one could interpret x0t as the equilibrium value. Suppose xt is constant over a number of
observations. One can interpret t as a shock that moves the system away from equilibrium.
If the time needed to return to equilibrium is long with respect to the observation frequency,
one could expect t+1 to be positive, conditional on t positive, which induces a correlation.
2. Unobserved factors that are correlated over time. The error term is often assumed to correspond to unobservable factors. If these factors are correlated, there will be autocorrelation.
3. Misspecification of the model. Suppose that the DGP is
yt = 0 + 1 xt + 2 x2t + t
but we estimate
yt = 0 + 1 xt + t
The effects are illustrated in Figure 9.5.

Effects on the OLS estimator

The variance of the OLS estimator is the same as in the case of heteroscedasticity - the standard
formula does not apply. The correct formula is given in equation 9.1. Next we discuss two GLS corrections for OLS. These will potentially induce inconsistency when the regressors are nonstochastic
(see Chapter 6) and should either not be used in that case (which is usually the relevant case) or
used with caution. The more recommended procedure is discussed in section 9.5.

AR(1)
There are many types of autocorrelation. Well consider two examples. The first is the most
commonly encountered case: autoregressive order 1 (AR(1) errors. The model is
yt

= x0t + t

= t1 + ut

ut

iid(0, u2 )

E(t us )

0, t < s

We assume that the model satisfies the other classical assumptions.

We need a stationarity assumption: || < 1. Otherwise the variance of t explodes as t
increases, so standard asymptotics will not apply.
By recursive substitution we obtain
t

t1 + ut

(t2 + ut1 ) + ut

2 t2 + ut1 + ut

2 (t3 + ut2 ) + ut1 + ut

In the limit the lagged drops out, since m 0 as m , so we obtain

t =

m utm

m=0

With this, the variance of t is found as

E(2t )

= u2

2m

m=0

u2
1 2

If we had directly assumed that t were covariance stationary, we could obtain this using
V (t )

= 2 E(2t1 ) + 2E(t1 ut ) + E(u2t )

= 2 V (t ) + u2 ,

so
V (t ) =

u2
1 2

The variance is the 0th order autocovariance: 0 = V (t )

 Note that the variance does not depend on t

Likewise, the first order autocovariance 1 is
Cov(t , t1 )

E((t1 + ut ) t1 )

= s =
=

V (t )
u2
1 2

Using the same method, we find that for s < t

Cov(t , ts ) = s =

s u2
1 2

The autocovariances dont depend on t: the process {t } is covariance stationary

The correlation (in general, for r.v.s x and y) is defined as
corr(x, y) =

cov(x, y)
se(x)se(y)

but in this case, the two standard errors are the same, so the s-order autocorrelation s is
s = s
All this means that the overall matrix has the form

..
u2
.
=

1 2

| {z }

this is the variance

n1
|

n1

..
.

n2
..
.

..

{z

this is the correlation matrix

So we have homoscedasticity, but elements off the main diagonal are not zero. All of this
depends only on two parameters, and u2 . If we can estimate these consistently, we can
apply FGLS.
It turns out that its easy to estimate these consistently. The steps are
1. Estimate the model yt = x0t + t by OLS.
2. Take the residuals, and estimate the model
t =
t1 + ut
p

Since t t , this regression is asymptotically equivalent to the regression

t = t1 + ut
which satisfies the classical assumptions. Therefore, obtained by applying OLS to t =
p

t1 + ut is consistent. Also, since ut ut , the estimator

n

u2 =

1X 2 p 2
(
u ) u
n t=2 t

 = (
3. With the consistent estimators
u2 and , form
u2 , ) using the previous structure of
, and estimate by FGLS. Actually, one can omit the factor
u2 /(1 2 ), since it cancels out
in the formula


1
1 X
1 y).
F GLS = X 0
(X 0

One can iterate the process, by taking the first FGLS estimator of , re-estimating and u2 ,
etc. If one iterates to convergences its equivalent to MLE (supposing normal errors).
An asymptotically equivalent approach is to simply estimate the transformed model
yt yt1 = (xt xt1 )0 + ut
using n 1 observations (since y0 and x0 arent available). This is the method of Cochrane
and Orcutt. Dropping the first observation is asymptotically irrelevant, but it can be very
important in small samples. One can recuperate the first observation by putting
y1
x1

= y1

1 2

= x1

1 2

This somewhat odd-looking result is related to the Cholesky factorization of 1 . See Davidson and MacKinnon, pg. 348-49 for more discussion. Note that the variance of y1 is u2 ,
asymptotically, so we see that the transformed model will be homoscedastic (and nonautocorrelated, since the u0 s are uncorrelated with the y 0 s, in different time periods.

MA(1)
The linear regression model with moving average order 1 errors is
yt

= x0t + t

= ut + ut1

ut

iid(0, u2 )

E(t us )

0, t < s

In this case,
V (t )

h
i
2
= 0 = E (ut + ut1 )
=

u2 + 2 u2

u2 (1 + 2 )

Similarly
1

E [(ut + ut1 ) (ut1 + ut2 )]

u2

and
2

[(ut + ut1 ) (ut2 + ut3 )]

so in this case

1 + 2

1 + 2

0
..
.

..
.

2
= u

0
..
.

..

1 + 2

Note that the first order autocorrelation is

1

2
u
2 (1+2 )
u

1
0

(1 + 2 )

This achieves a maximum at = 1 and a minimum at = 1, and the maximal and minimal
autocorrelations are 1/2 and -1/2. Therefore, series that are more strongly autocorrelated
cant be MA(1) processes.
Again the covariance matrix has a simple structure that depends on only two parameters. The
problem in this case is that one cant estimate using OLS on
t = ut + ut1
because the ut are unobservable and they cant be estimated consistently. However, there is a
simple way to estimate the parameters.
Since the model is homoscedastic, we can estimate
V (t ) = 2 = u2 (1 + 2 )
using the typical estimator:
n

1X 2
c2 = 2 (1
\

+ 2 ) =

u
n t=1 t
By the Slutsky theorem, we can interpret this as defining an (unidentified) estimator of both
u2 and , e.g., use this as

X
c2 (1 + b2 ) = 1

2
u
n t=1 t
However, this isnt sufficient to define consistent estimators of the parameters, since its
unidentified - two unknowns, one equation.
To solve this problem, estimate the covariance of t and t1 using
n

X
d2 = 1
d t , t1 ) =
t t1
Cov(
u
n t=2
This is a consistent estimator, following a LLN (and given that the epsilon hats are consistent
for the epsilons). As above, this can be interpreted as defining an unidentified estimator of
the two parameters:
n

X
c2 = 1

t t1
u
n t=2
Now solve these two equations to obtain identified (and therefore consistent) estimators of

Figure 9.6: Efficiency of OLS and FGLS, AR1 errors

(a) OLS

(b) GLS

both and u2 . Define the consistent estimator

c2 )

= (,

u
following the form weve seen above, and transform the model using the Cholesky decomposition. The transformed model satisfies the classical assumptions asymptotically.
Note: there is no guarantee that estimated using the above method will be positive definite,
which may pose a problem. Another method would be to use ML estimation, if one is willing
to make distributional assumptions regarding the white noise errors.

Monte Carlo example: AR1

Lets look at a Monte Carlo study that compares OLS and GLS when we have AR1 errors. The
model is
y t = 1 + x t + t
t = t1 + ut
with = 0.9. The sample size is n = 30, and 1000 Monte Carlo replications are done. The Octave
script is GLS/AR1Errors.m. Figure 9.6 shows histograms of the estimated coefficient of x minus
the true value. We can see that the GLS histogram is much more concentrated about 0, which is
indicative of the efficiency of GLS relative to OLS.

Asymptotically valid inferences with autocorrelation of unknown form

See Hamilton Ch. 10, pp. 261-2 and 280-84.
When the form of autocorrelation is unknown, one may decide to use the OLS estimator, without correction. Weve seen that this estimator has the limiting distribution




d
1
n N 0, Q1
X QX

where, as before, is

= lim E
n

X 0 0 X
n

We need a consistent estimate of . Define mt = xt t (recall that xt is defined as a K 1 vector).

Note that

x1

x2

xn

2
..
.

n
=
=

n
X
t=1
n
X

xt t
mt

t=1

so that
1
= lim E
n n

"

n
X

n
X

mt

t=1

!#
m0t

t=1

We assume that mt is covariance stationary (so that the covariance between mt and mts does not
depend on t).
Define the v th autocovariance of mt as
v = E(mt m0tv ).
Note that E(mt m0t+v ) = 0v . (show this with an example). In general, we expect that:
mt will be autocorrelated, since t is potentially autocorrelated:
v = E(mt m0tv ) 6= 0
Note that this autocovariance does not depend on t, due to covariance stationarity.
contemporaneously correlated ( E(mit mjt ) 6= 0 ), since the regressors in xt will in general be
correlated (more on this later).
and heteroscedastic (E(m2it ) = i2 , which depends upon i ), again since the regressors will
have different variances.
While one could estimate parametrically, we in general have little information upon which to
base a parametric specification. Recent research has focused on consistent nonparametric estimators of .
Now define
1
n = E
n

"

n
X
t=1

!
mt

n
X

!#
m0t

t=1

We have (show that the following is true, by expanding sum and shifting rows to left)
n = 0 +

n1
n2
1
(1 + 01 ) +
(2 + 02 ) +
n1 + 0n1
n
n
n

The natural, consistent estimator of v is

n
X
cv = 1

m
tm
0tv .
n t=v+1

where
m
t = xt t
(note: one could put 1/(n v) instead of 1/n here). So, a natural, but inconsistent, estimator of
n would be
n







0
c0 + n 2
c0 + + 1 [
[
c0 + n 1
c1 +
c2 +
=
+

n1
1
2
n1
n
n
n
n1

X nv 
c0 +
cv +
c0 .
=

v
n
v=1

This estimator is inconsistent in general, since the number of parameters to estimate is more than
the number of observations, and increases more rapidly than n, so information does not build up
as n .
On the other hand, supposing that v tends to zero sufficiently rapidly as v tends to , a
modified estimator
n =
c0 +

q(n) 


cv +
c0 ,

v=1
p

where q(n) as n will be consistent, provided q(n) grows sufficiently slowly.

The assumption that autocorrelations die off is reasonable in many cases. For example, the
AR(1) model with || < 1 has autocorrelations that die off.
The term

nv
n

can be dropped because it tends to one for v < q(n), given that q(n) increases

slowly relative to n.
A disadvantage of this estimator is that is may not be positive definite. This could cause one
to calculate a negative 2 statistic, for example!
Newey and West proposed and estimator (Econometrica, 1987) that solves the problem of
possible nonpositive definiteness of the above estimator. Their estimator is
n =
c0 +

q(n) 

X
v=1



v
cv +
c0v .

1
q+1

This estimator is p.d. by construction. The condition for consistency is that n1/4 q(n) 0.
Note that this is a very slow rate of growth for q. This estimator is nonparametric - weve
placed no parametric restrictions on the form of . It is an example of a kernel estimator.
p
n
n and Q
d
Finally, since n has as its limit,
. We can now use
X =

1
0
nX X

to consistently

estimate the limiting distribution of the OLS estimator under heteroscedasticity and autocorrelation of unknown form. With this, asymptotically valid tests are constructed in the usual way.

Testing for autocorrelation

Durbin-Watson test
The Durbin-Watson test is not strictly valid in most situations where we would like to use it.
Nevertheless, it is encountered often enough so that one should know something about it. The
Durbin-Watson test statistic is
Pn

DW

=
=

2
(
t t1 )
t=2P
n
2t
t=1
Pn
2t 2
t t1
t=2
Pn
2t
t=1

+ 2t1

Figure 9.7: Durbin-Watson critical values

The null hypothesis is that the first order autocorrelation of the errors is zero: H0 : 1 = 0.
The alternative is of course HA : 1 6= 0. Note that the alternative is not that the errors are
AR(1), since many general patterns of autocorrelation will have the first order autocorrelation different than zero. For this reason the test is useful for detecting autocorrelation in
general. For the same reason, one shouldnt just assume that an AR(1) model is appropriate
when the DW test rejects the null.
p

Under the null, the middle term tends to zero, and the other two tend to one, so DW 2.
Supposing that we had an AR(1) error process with = 1. In this case the middle term tends
p

to 2, so DW 0
Supposing that we had an AR(1) error process with = 1. In this case the middle term
p

tends to 2, so DW 4
These are the extremes: DW always lies between 0 and 4.
The distribution of the test statistic depends on the matrix of regressors, X, so tables cant
give exact critical values. The give upper and lower bounds, which correspond to the extremes that are possible. See Figure 9.7. There are means of determining exact critical
values conditional on X.
Note that DW can be used to test for nonlinearity (add discussion).
The DW test is based upon the assumption that the matrix X is fixed in repeated samples.
This is often unreasonable in the context of economic time series, which is precisely the
context where the test would have application. It is possible to relate the DW test to other
test statistics which are valid without strict exogeneity.

Breusch-Godfrey test
This test uses an auxiliary regression, as does the White test for heteroscedasticity. The regression is
t = x0t + 1 t1 + 2 t2 + + P tP + vt
and the test statistic is the nR2 statistic, just as in the White test. There are P restrictions, so the
test statistic is asymptotically distributed as a 2 (P ).
The intuition is that the lagged errors shouldnt contribute to explaining the current error if
there is no autocorrelation.
xt is included as a regressor to account for the fact that the t are not independent even if
the t are. This is a technicality that we wont go into here.
This test is valid even if the regressors are stochastic and contain lagged dependent variables,
so it is considerably more useful than the DW test for typical time series data.
The alternative is not that the model is an AR(P), following the argument above. The alternative is simply that some or all of the first P autocorrelations are different from zero. This
is compatible with many specific forms of autocorrelation.

Lagged dependent variables and autocorrelation

0

Weve seen that the OLS estimator is consistent under autocorrelation, as long as plim Xn = 0.
This will be the case when E(X 0 ) = 0, following a LLN. An important exception is the case where
X contains lagged y 0 s and the errors are autocorrelated.
Example 22. Dynamic model with MA1 errors. Consider the model
yt

+ yt1 + xt + t

t

t + t1

We can easily see that a regressor is not weakly exogenous:

E(yt1 t )

= E {( + yt2 + xt1 + t1 + t2 )(t + t1 )}

6=

2
since one of the terms is E(t1
) which is clearly nonzero. In this case E(xt t ) 6= 0, and therefore
0

plim Xn 6= 0. Since
plim = + plim

X 0
n

the OLS estimator is inconsistent in this case. One needs to estimate by instrumental variables
(IV), which well get to later
The Octave script GLS/DynamicMA.m does a Monte Carlo study. The sample size is n = 100.
The true coefficients are = 1 = 0.9 and = 1. The MA parameter is = 0.95. Figure 9.8
gives the results. You can see that the constant and the autoregressive parameter have a lot of bias.
By re-running the script with = 0, you will see that much of the bias disappears (not all - why?).

Examples
Nerlove model, yet again The Nerlove model uses cross-sectional data, so one may not think
of performing tests for autocorrelation. However, specification error can induce autocorrelated
errors. Consider the simple Nerlove model
ln C = 1 + 2 ln Q + 3 ln PL + 4 ln PF + 5 ln PK + 
and the extended Nerlove model
ln C =

5
X

j Dj +

j=1

5
X

j Dj ln Q + L ln PL + F ln PF + K ln PK + 

j=1

discussed around equation 5.5. If you have done the exercises, you have seen evidence that the
extended model is preferred. So if it is in fact the proper model, the simple model is misspecified.
Lets check if this misspecification might induce autocorrelated errors.
The Octave program GLS/NerloveAR.m estimates the simple Nerlove model, and plots the
residuals as a function of ln Q, and it calculates a Breusch-Godfrey test statistic. The residual plot
is in Figure 9.9 , and the test results are:

Breusch-Godfrey test

Value
34.930

p-value
0.000

Clearly, there is a problem of autocorrelated residuals.

Repeat the autocorrelation tests using the extended Nerlove model (Equation 5.5) to see the
problem is solved.
Klein model

Kleins Model I is a simple macroeconometric model. One of the equations in the

model explains consumption (C) as a function of profits (P ), both current and lagged, as well as
the sum of wages in the private sector (W p ) and wages in the government sector (W g ). Have a
look at the README file for this data set. This gives the variable names and other information.
Consider the model
Ct = 0 + 1 Pt + 2 Pt1 + 3 (Wtp + Wtg ) + 1t
The Octave program GLS/Klein.m estimates this model by OLS, plots the residuals, and performs
the Breusch-Godfrey test, using 1 lag of the residuals. The estimation and test results are:

*********************************************************
OLS estimation results
Observations 21
R-squared 0.981008
Sigma-squared 1.051732
Results (Ordinary var-cov estimator)

Constant
Profits
Lagged Profits
Wages

estimate
16.237
0.193
0.090
0.796

st.err.
1.303
0.091
0.091
0.040

t-stat.
12.464
2.115
0.992
19.933

p-value
0.000
0.049
0.335
0.000

Figure 9.8: Dynamic model with MA(1) errors

(a)

(b)

(c)

Figure 9.9: Residuals of simple Nerlove model

2
Residuals
Quadratic fit to Residuals

1.5

0.5

-0.5

-1
0

Figure 9.10: OLS residuals, Klein consumption equation

Regression residuals
2
Residuals

-1

-2

-3
0

10

15

*********************************************************
Value
p-value
Breusch-Godfrey test
1.539
0.215

and the residual plot is in Figure 9.10. The test does not reject the null of nonautocorrelatetd
errors, but we should remember that we have only 21 observations, so power is likely to be fairly
low. The residual plot leads me to suspect that there may be autocorrelation - there are some
significant runs below and above the x-axis. Your opinion may differ.
Since it seems that there may be autocorrelation, letss try an AR(1) correction. The Octave
program GLS/KleinAR1.m estimates the Klein consumption equation assuming that the errors follow the AR(1) pattern. The results, with the Breusch-Godfrey test for remaining autocorrelation
are:

*********************************************************
OLS estimation results
Observations 21

20

R-squared 0.967090
Sigma-squared 0.983171
Results (Ordinary var-cov estimator)

Constant
Profits
Lagged Profits
Wages

estimate
16.992
0.215
0.076
0.774

st.err.
1.492
0.096
0.094
0.048

t-stat.
11.388
2.232
0.806
16.234

p-value
0.000
0.039
0.431
0.000

*********************************************************
Value
p-value
Breusch-Godfrey test
2.129
0.345

The test is farther away from the rejection region than before, and the residual plot is a bit
more favorable for the hypothesis of nonautocorrelated residuals, IMHO. For this reason, it
seems that the AR(1) correction might have improved the estimation.
Nevertheless, there has not been much of an effect on the estimated coefficients nor on their
estimated standard errors. This is probably because the estimated AR(1) coefficient is not
very large (around 0.2)
The existence or not of autocorrelation in this model will be important later, in the section
on simultaneous equations.

9.6

Exercises

1. Comparing the variances of the OLS and GLS estimators, I claimed that the following holds:
V ar(GLS )
V ar()

= AA

Verify that this is true.

2. Show that the GLS estimator can be defined as
GLS = arg min(y X)0 1 (y X)
3. The limiting distribution of the OLS estimator with heteroscedasticity of unknown form is




d
1
n N 0, Q1
X QX ,
where


lim E

X 0 0 X
n


=

Explain why
n

X
b= 1

xt x0t 2t
n t=1
is a consistent estimator of this matrix.

4. Define the v th autocovariance of a covariance stationary process mt , where E(mt ) = 0 as

v = E(mt m0tv ).
Show that E(mt m0t+v ) = 0v .
5. For the Nerlove model with dummies and interactions discussed above (see Section 9.4 and
equation 5.5)
ln C =

5
X

j Dj +

j=1

5
X

j Dj ln Q + L ln PL + F ln PF + K ln PK + 

j=1

above, we did a GLS correction based on the assumption that there is HET by groups (V (t |xt ) =
j2 ). Lets assume that this model is correctly specified, except that there may or may not be
HET, and if it is present it may be of the form assumed, or perhaps of some other form. What
happens if the assumed form of HET is incorrect?
(a) Is the FGLS based on the assumed form of HET consistent?
(b) Is it efficient? Is it likely to be efficient with respect to OLS?
(c) Are hypothesis tests using the FGLS estimator valid? If not, can they be made valid
following some procedure? Explain.
(d) Are the t-statistics reported in Section 9.4 valid?
(e) Which estimator do you prefer, the OLS estimator or the FGLS estimator? Discuss.
6. Perhaps we can be a little more parsimonious with the Nerlove data (nerlove.data ), rather
than using so many parameters to account for non-constant returns to scale, and to account
for heteroscedasticity. Consider the original model
ln C = + Q ln Q + L ln PL + F ln PF + K ln PK + 
(a) Estimate by OLS, plot the residuals, and test for autocorrelation and heteroscedasticity.
Explain your findings.
(b) Consider the model
2

ln C = + Q ln Q + Q (ln Q) + L ln PL + F ln PF + K ln PK + 
i. Explain how this model can account for non-constant returns to scale.
ii. estimate this model, and test for autocorrelation and heteroscedasticity. You should
find that there is HET, but no strong evidence of AUT. Why is this the case?
iii. Do a GLS correction where it is assumed that V (i ) =

2
.
(ln Qi )2

In GRETL, there is

a weighted least squares option that you can use. Why does this assumed form of
HET make sense?
iv. plot the weighted residuals versus output. Is there evidence of HET, or has the
correction eliminated the problem?
v. plot the fitted values for returns to scale, for all of the firms.
7. The hall.csv or hall.gdt dataset contains monthly observation on 3 variables: the consumption
ratio ct /ct1 ; the gross return of an equally weighted index of assets ewrt ; and the gross
return of the same index, but weighted by value, vwrt . The idea is that a representative
consumer may finance consumption by investing in assets. Present wealth is used for two

things: consumption and investment. The return on investment defines wealth in the next
period, and the process repeats. For the moment, explore the properties of the variables.
(a) Are the variances constant over time?
(b) Do the variables appear to be autocorrelated? Hint: regress a variable on its own lags.
(c) Do the variable seem to be normally distributed?
(d) Look at the properties of the growth rates of the variables: repeat a-c for growth rates.
The growth rate of a variable xt is given by log (xt /xt1 ).
8. Consider the model
yt = C + A1 yt1 + t
E(t 0t ) =
E(t 0s ) = 0, t 6= s
where yt and t are G 1 vectors, C is a G 1 of constants, and A1 is a G G matrix of
parameters. The matrix is a GG covariance matrix. Assume that we have n observations.
This is a vector autoregressive model, of order 1 - commonly referred to as a VAR(1) model.
(a) Show how the model can be written in the form Y = X + , where Y is a Gn 1
vector, is a (G + G2 )1 parameter vector, and the other items are conformable. What
is the structure of X? What is the structure of the covariance matrix of ?
(b) This model has HET and AUT. Verify this statement.
"
#
"
#
0.8 0.1
1 0.5
0
(c) Set G = 2,C = (0 0) A =
,=
. Simulate data from this
0.2 0.5
0.5 1
model, then estimate the model using OLS and feasible GLS. You should find that the
two estimators are identical, which might seem surprising, given that there is HET and
AUT.
(d) (optional, and advanced). Prove analytically that the OLS and GLS estimators are identical. Hint: this model is of the form of seemingly unrelated regressions.
9. Consider the model
yt = + 1 yt1 + 2 yt2 + t
where t is a N (0, 1) white noise error. This is an autogressive model of order 2 (AR2) model.
Suppose that data is generated from the AR2 model, but the econometrician mistakenly
decides to estimate an AR1 model (yt = + 1 yt1 + t ).
(a) simulate data from the AR2 model, setting 1 = 0.5 and 2 = 0.4, using a sample size of
n = 30.
(b) Estimate the AR1 model by OLS, using the simulated data
(c) test the hypothesis that 1 = 0.5
(d) test for autocorrelation using the test of your choice
(e) repeat the above steps 10000 times.
i. What percentage of the time does a t-test reject the hypothesis that 1 = 0.5?
ii. What percentage of the time is the hypothesis of no autocorrelation rejected?
(f) discuss your findings. Include a residual plot for a representative sample.

10. Modify the script given in Subsection 9.5 so that the first observation is dropped, rather than
given special treatment. This corresponds to using the Cochrane-Orcutt method, whereas the
script as provided implements the Prais-Winsten method. Check if there is an efficiency loss
when the first observation is dropped.

Chapter 10

Endogeneity and simultaneity

Several times weve encountered cases where correlation between regressors and the error term
lead to biasedness and inconsistency of the OLS estimator. Cases include autocorrelation with
lagged dependent variables (Exampe 22) and measurement error in the regressors (Example 19).
Another important case is that of simultaneous equations. The cause is different, but the effect is
the same.

10.1

Simultaneous equations

Up until now our model is

y = X +
where we assume weak exogeneity of the regressors, so that E(xt t ) = 0. With weak exogeneity,
the OLS estimator has desirable large sample properties (consistency, asymptotic normality).
Simultaneous equations is a different prospect. An example of a simultaneous equation system
is a simple supply-demand system:

"
E

1t

2t

Demand: qt

= 1 + 2 pt + 3 yt + 1t

Supply: qt
!
i
1t 2t

= 1 + 2 pt + 2t
"
#
11 12
=

22
, t

The presumption is that qt and pt are jointly determined at the same time by the intersection of
these equations. Well assume that yt is determined by some unrelated process. Its easy to see that
we have correlation between regressors and errors. Solving for pt :
1 + 2 pt + 3 yt + 1t
2 pt 2 pt
pt

= 1 + 2 pt + 2t
= 1 1 + 3 yt + 1t 2t
1 1
3 yt
1t 2t
=
+
+
2 2
2 2
2 2

Now consider whether pt is uncorrelated with 1t :



E(pt 1t )

=
=

3 yt
1t 2t
1 1
E
+
+
2 2
2 2
2 2
11 12
2 2

123


1t

Because of this correlation, weak exogeneity does not hold, and OLS estimation of the demand
equation will be biased and inconsistent. The same applies to the supply equation, for the same
reason.
In this model, qt and pt are the endogenous varibles (endogs), that are determined within the
system. yt is an exogenous variable (exogs). These concepts are a bit tricky, and well return to it
in a minute. First, some notation. Suppose we group together current endogs in the vector Yt . If
there are G endogs, Yt is G 1. Group current and lagged exogs, as well as lagged endogs in the
vector Xt , which is K 1. Stack the errors of the G equations into the error vector Et . The model,
with additional assumtions, can be written as
Yt0
Et
E(Et Es0 )

Xt0 B + Et0

N (0, ), t
=

0, t 6= s

There are G equations here, and the parameters that enter into each equation are contained in the
columns of the matrices and B. We can stack all n observations and write the model as
Y

XB + E

E(X 0 E)

0(KG)

vec(E) N (0, )
where
E10

,X = . 2
.

,E = . 2
.

Xn0

En0

Y10

Y =

Y20
..
.
Yn0

X10

Y is n G, X is n K, and E is n G.
This system is complete, in that there are as many equations as endogs.
There is a normality assumption. This isnt necessary, but allows us to consider the relationship between least squares and ML estimators.
Since there is no autocorrelation of the Et s, and since the columns of E are individually
homoscedastic, then

11 In

12 In

22 In
..

1G In
..
.
..
.

GG In

= In
X may contain lagged endogenous and exogenous variables. These variables are predetermined.
We need to define what is meant by endogenous and exogenous when classifying the
current period variables. Remember the definition of weak exogeneity Assumption 15, the
regressors are weakly exogenous if E(Et |Xt ) = 0. Endogenous regressors are those for which
this assumption does not hold. As long as there is no autocorrelation, lagged endogenous
variables are weakly exogenous.

10.2

Reduced form

Recall that the model is

Yt0

Xt0 B + Et0

V (Et )

This is the model in structural form.

Definition 23. [Structural form] An equation is in structural form when more than one current
period endogenous variable is included.
The solution for the current period endogs is easy to find. It is
Yt0

Xt0 B1 + Et0 1

Xt0 + Vt0

Now only one current period endog appears in each equation. This is the reduced form.
Definition 24. [Reduced form] An equation is in reduced form if only one current period endog is
included.
An example is our supply/demand system. The reduced form for quantity is obtained by solving
the supply equation for price and substituting into demand:


qt 1 2t
1 + 2
+ 3 yt + 1t
2
2 1 2 (1 + 2t ) + 2 3 yt + 2 1t
2 3 yt
2 1t 2 2t
2 1 2 1
+
+
2 2
2 2
2 2
11 + 21 yt + V1t


qt

2 qt 2 qt

qt

=
=

Similarly, the rf for price is

1 + 2 pt + 2t

1 + 2 pt + 3 yt + 1t

2 pt 2 pt

pt

1 1 + 3 yt + 1t 2t
3 yt
1t 2t
1 1
+
+
2 2
2 2
2 2
12 + 22 yt + V2t

The interesting thing about the rf is that the equations individually satisfy the classical assumptions,
since yt is uncorrelated with 1t and 2t by assumption, and therefore E(yt Vit ) = 0, i=1,2, t. The
errors of the rf are

"

V1t
V2t

"
=

2 1t 2 2t
2 2
1t 2t
2 2

The variance of V1t is



V (V1t )

=
=



2 1t 2 2t
2 1t 2 2t
2 2
2 2
2
2 11 22 2 12 + 2 22

(2 2 )

 This is constant over time, so the first rf equation is homoscedastic.

Likewise, since the t are independent over time, so are the Vt .
The variance of the second rf error is


V (V2t )



1t 2t
1t 2t
2 2
2 2
11 212 + 22

(2 2 )

and the contemporaneous covariance of the errors across equations is



E(V1t V2t )



2 1t 2 2t
1t 2t
2 2
2 2
2 11 (2 + 2 ) 12 + 22

= E
=

(2 2 )

In summary the rf equations individually satisfy the classical assumptions, under the assumtions weve made, but they are contemporaneously correlated.
The general form of the rf is
Yt0

Xt0 B1 + Et0 1

Xt0 + Vt0

so we have that



0

0
Vt = 1 Et N 0, 1 1 , t
and that the Vt are timewise independent (note that this wouldnt be the case if the Et were
autocorrelated).
From the reduced form, we can easily see that the endogenous variables are correlated with
the structural errors:
E(Et Yt0 ) = E Et Xt0 B1 + Et0 1

= E Et Xt0 B1 + Et Et0 1

= 1

10.3

(10.1)

Estimation of the reduced form equations

From above, the RF equations are

Yt0

= Xt0 B1 + Et0 1
= Xt0 + Vt0

and
Vt N (0, ) , t

0
is simply OLS applied to this
where we define 1 1 . The rf parameter estimator ,
model, equation by equation::
= (X 0 X)1 X 0 Y

which is simply
= (X 0 X)1 X 0

y1

y2

yG

that is, OLS equation by equation using all the exogs in the estimation of each column of .
It may seem odd that we use OLS on the reduced form, since the rf equations are correlated,

0
because 1 1 is a full matrix. Why dont we do GLS to improve efficiency of estimation
of the RF parameters?
OLS equation by equation to get the rf is equivalent to

y1

y2
..
.

yG

0
..
.

..

0
1
..

2
.

..
0 .
G
X

v1

v2
..
.

vG

where yi is the n 1 vector of observations of the ith endog, X is the entire n K matrix of exogs,
i is the ith column of , and vi is the ith column of V. Use the notation
y = X + v
to indicate the pooled model. Following this notation, the error covariance matrix is
V (v) = In
This is a special case of a type of model known as a set of seemingly unrelated equations (SUR)
since the parameter vector of each equation is different. The important feature of this special
case is that the regressors are the same in each equation. The equations are contemporanously
correlated, because of the non-zero off diagonal elements in .
Note that each equation of the system individually satisfies the classical assumptions.
Normally when doing SUR, one simply does GLS on the whole system y = X + v, where
V (v) = In , which is in general more efficient than OLS on each equation.
However, when the regressors are the same in all equations, as is true in the present case of
estimation of the RF parameters, SUR OLS. To show this note that in this case X = In X.
Using the rules
1. (A B)1 = (A1 B 1 )
2. (A B)0 = (A0 B 0 ) and
3. (A B)(C D) = (AC BD), we get

SU R

=
=
=
=

1
0
1
0
1
(In X) ( In ) (In X)
(In X) ( In ) y



1 1


1 X 0 (In X)
X0 y




(X 0 X)1 1 X 0 y


IG (X 0 X)1 X 0 y

2

.
.
.

G
Note that this provides the answer to the exercise 8d in the chapter on GLS.
So the unrestricted rf coefficients can be estimated efficiently (assuming normality) by OLS,
even if the equations are correlated.

 We have ignored any potential zeros in the matrix , which if they exist could potentially
increase the efficiency of estimation of the rf.
Another example where SUROLS is in estimation of vector autoregressions which is discussed in Section 15.3.

10.4

Bias and inconsistency of OLS estimation of a structural

equation

Considering the first equation (this is without loss of generality, since we can always reorder the
equations) we can partition the Y matrix as
Y =

Y1

Y2

y is the first column

Y1 are the other endogenous variables that enter the first equation
Y2 are endogs that are excluded from this equation
Similarly, partition X as
X=

X1

X2

X1 are the included exogs, and X2 are the excluded exogs.

Finally, partition the error matrix as
E=

E12

Assume that has ones on the main diagonal. These are normalization restrictions that simply
scale the remaining coefficients on each equation, and which scale the variances of the error terms.
Given this scaling and our partitioning, the coefficient matrices can be written as

12

1 22
0
32
"
#
1 B12
0

B22

With this, the first equation can be written as

y

(10.2)

= Y1 1 + X1 1 +
= Z +

The problem, as weve seen, is that the columns of Z corresponding to Y1 are correlated with ,
because these are endogenous variables, and as we saw in equation 10.1, the endogenous variables
are correlated with the structural errors, so they dont satisfy weak exogeneity. So, E(Z 0 ) 6=0.
What are the properties of the OLS estimator in this situation?
1

Z 0y

Z 0 Z 0 +

(Z 0 Z)

(Z 0 Z)

0 + (Z 0 Z)

Z 0

Its clear that the OLS estimator is biased in general. Also,

0 =

Z 0Z
n

1

Z 0
n

Say that lim Zn = A,a.s., and lim ZnZ = QZ , a.s. Then


lim 0 = Q1
Z A 6= 0, a.s.
So the OLS estimator of a structural equation is inconsistent. In general, correlation between
regressors and errors leads to this problem, whether due to measurement error, simultaneity, or
omitted regressors.

10.5

Identification by exclusion restrictions

The identification problem in simultaneous equations is in fact of the same nature as the identification problem in any estimation setting: does the limiting objective function have the proper
curvature so that there is a unique global minimum or maximum at the true parameter value? In
the context of IV estimation, this is the case if the limiting covariance of the IV estimator is positive
definite and plim n1 W 0 = 0. This matrix is
0
1 2
V (IV ) = (QXW Q1

W W QXW )

The necessary and sufficient condition for identification is simply that this matrix be positive
definite, and that the instruments be (asymptotically) uncorrelated with .
For this matrix to be positive definite, we need that the conditions noted above hold: QW W
must be positive definite and QXW must be of full rank ( K ).
These identification conditions are not that intuitive nor is it very obvious how to check them.

Necessary conditions
If we use IV estimation for a single equation of the system, the equation can be written as
y = Z +
where
Z=

Y1

X1

Notation:
Let K be the total numer of weakly exogenous variables.
Let K = cols(X1 ) be the number of included exogs, and let K = K K be the number
of excluded exogs (in this equation).
Let G = cols(Y1 ) + 1 be the total number of included endogs, and let G = G G be the
number of excluded endogs.
Using this notation, consider the selection of instruments.
Now the X1 are weakly exogenous and can serve as their own instruments.

 It turns out that X exhausts the set of possible instruments, in that if the variables in X dont
lead to an identified model then no other instruments will identify the model either. Assuming this is true (well prove it in a moment), then a necessary condition for identification is
that cols(X2 ) cols(Y1 ) since if not then at least one instrument must be used twice, so W
will not have full column rank:
(W ) < K + G 1 (QZW ) < K + G 1
This is the order condition for identification in a set of simultaneous equations. When the
only identifying information is exclusion restrictions on the variables that enter an equation,
then the number of excluded exogs must be greater than or equal to the number of included
endogs, minus 1 (the normalized lhs endog), e.g.,
K G 1
To show that this is in fact a necessary condition consider some arbitrary set of instruments
W. A necessary condition for identification is that


1
plim W 0 Z
n

= K + G 1

where
Z=

Y1

X1

Recall that weve partitioned the model

Y = XB + E
as
Y =
X=

X1

Y1
X2

Y2

Given the reduced form

Y = X + V
we can write the reduced form using the same partition
h

Y1

Y2

X1

X2

"

11

12

13

21

22

23

#
+

V1

V2

so we have
Y1 = X1 12 + X2 22 + V1
so

h
1 0
1
W Z = W 0 X1 12 + X2 22 + V1
n
n

X1

Because the W s are uncorrelated with the V1 s, by assumption, the cross between W and V1
converges in probability to zero, so
h
1
1
plim W 0 Z = plim W 0 X1 12 + X2 22
n
n

X1

Since the far rhs term is formed only of linear combinations of columns of X, the rank of this
matrix can never be greater than K, regardless of the choice of instruments. If Z has more than K

columns, then it is not of full column rank. When Z has more than K columns we have
G 1 + K > K
or noting that K = K K ,
G 1 > K
In this case, the limiting matrix is not of full column rank, and the identification condition fails.

Sufficient conditions
Identification essentially requires that the structural parameters be recoverable from the data.
This wont be the case, in general, unless the structural model is subject to some restrictions.
Weve already identified necessary conditions. Turning to sufficient conditions (again, were only
considering identification through zero restricitions on the parameters, for the moment).
The model is
Yt0

Xt0 B + Et

V (Et )

This leads to the reduced form

Yt0

= Xt0 B1 + Et 1

= Xt0 + Vt

0
V (Vt ) = 1 1
=

The reduced form parameters are consistently estimable, but none of them are known a priori,
and there are no restrictions on their values. The problem is that more than one structural form
has the same reduced form, so knowledge of the reduced form parameters alone isnt enough to
determine the structural parameters. To see this, consider the model
Yt0 F
V (Et F )

= Xt0 BF + Et F
= F 0 F

where F is some arbirary nonsingular G G matrix. The rf of this new model is

Yt0

= Xt0 BF (F )

+ Et F (F )

= Xt0 BF F 1 1 + Et F F 1 1
= Xt0 B1 + Et 1
= Xt0 + Vt
Likewise, the covariance of the rf of the transformed model is
1

V (Et F (F )

V (Et 1 )

Since the two structural forms lead to the same rf, and the rf is all that is directly estimable, the
models are said to be observationally equivalent. What we need for identification are restrictions
on and B such that the only admissible F is an identity matrix (if all of the equations are to be

identified). Take the coefficient matrices as partitioned before:

"

12

=
0

1
0

22

32

B12
B22

The coefficients of the first equation of the transformed model are simply these coefficients multiplied by the first column of F . This gives

"

#"

12

=
0

1
0

f11

F2

#
22 "
f11
32
F

2
B12
B22

For identification of the first equation we need that there be enough restrictions so that the only
admissible

"

f11
F2

be the leading column of an identity matrix, so that

12

1
0

#
22 "
1
f11

=
32
0
F2

B12
1
B22
0

Note that the third and fifth rows are

"

32

"
F2 =

B22

Supposing that the leading matrix is of full column rank, e.g.,

"

32

#!

"
= cols

B22

32

#!
=G1

B22

then the only way this can hold, without additional restrictions on the models parameters, is if F2
is a vector of zeros. Given that F2 is a vector of zeros, then the first equation
h

12

"

f11

#
= 1 f11 = 1

F2

Therefore, as long as
"

then

"

32

#!
=G1

B22
f11
F2

"
=

0G1

The first equation is identified in this case, so the condition is sufficient for identification. It is also

necessary, since the condition implies that this submatrix must have at least G 1 rows. Since this
matrix has
G + K = G G + K
rows, we obtain
G G + K G 1
or
K G 1
which is the previously derived necessary condition.
The above result is fairly intuitive (draw picture here). The necessary condition ensures that
there are enough variables not in the equation of interest to potentially move the other equations, so as to trace out the equation of interest. The sufficient condition ensures that those other
equations in fact do move around as the variables change their values. Some points:
When an equation has K = G 1, is is exactly identified, in that omission of an identifiying
restriction is not possible without loosing consistency.
When K > G 1, the equation is overidentified, since one could drop a restriction and still
retain consistency. Overidentifying restrictions are therefore testable. When an equation is
overidentified we have more instruments than are strictly necessary for consistent estimation.
Since estimation by IV with more instruments is more efficient asymptotically, one should
employ overidentifying restrictions if one is confident that theyre true.
We can repeat this partition for each equation in the system, to see which equations are
identified and which arent.
These results are valid assuming that the only identifying information comes from knowing
which variables appear in which equations, e.g., by exclusion restrictions, and through the
use of a normalization. There are other sorts of identifying information that can be used.
These include
1. Cross equation restrictions
2. Additional restrictions on parameters within equations (as in the Klein model discussed
below)
3. Restrictions on the covariance matrix of the errors
4. Nonlinearities in variables
When these sorts of information are available, the above conditions arent necessary for
identification, though they are of course still sufficient.
To give an example of how other information can be used, consider the model
Y = XB + E
where is an upper triangular matrix with 1s on the main diagonal. This is a triangular system of
equations. In this case, the first equation is
y1 = XB1 + E1
Since only exogs appear on the rhs, this equation is identified.
The second equation is

y2 = 21 y1 + XB2 + E2
This equation has K = 0 excluded exogs, and G = 2 included endogs, so it fails the order
(necessary) condition for identification.
However, suppose that we have the restriction 21 = 0, so that the first and second structural
errors are uncorrelated. In this case
E(y1t 2t ) = E {(Xt0 B1 + 1t )2t } = 0
so theres no problem of simultaneity. If the entire matrix is diagonal, then following the
same logic, all of the equations are identified. This is known as a fully recursive model.

Example: Kleins Model 1

To give an example of determining identification status, consider the following macro model (this
is the widely known Kleins Model 1)
= 0 + 1 Pt + 2 Pt1 + 3 (Wtp + Wtg ) + 1t

Consumption: Ct
Investment: It
Private Wages:

= 0 + 1 Pt + 2 Pt1 + 3 Kt1 + 2t

Wtp

= 0 + 1 Xt + 2 Xt1 + 3 At + 3t

Output: Xt

= Ct + It + Gt
= Xt Tt Wtp

Profits: Pt
Capital Stock: Kt

1t

2t
3t

= Kt1 + It

11

IID 0 ,
0

12

13

22

23
33

The other variables are the government wage bill, Wtg , taxes, Tt , government nonwage spending,
Gt ,and a time trend, At . The endogenous variables are the lhs variables,
Yt0 =

Ct

It

Wtp

Xt

Pt

Kt

and the predetermined variables are all others:

Xt0 =

Wtg

Gt

Tt

At

Pt1

Kt1

Xt1

The model assumes that the errors of the equations are contemporaneously correlated, by nonautocorrelated. The model written as Y = XB + E gives

B=

0
0

1 0

0
0

0
0

0
1

0
0

To check this identification of the consumption equation, we need to extract 32 and B22 , the
submatrices of coefficients of endogs and exogs that dont appear in this equation. These are the
rows that have zeros in the first column, and we need to drop the first column. We get

"

32
B22

We need to find a set of 5 rows of this matrix gives a full-rank 55 matrix. For example, selecting
rows 3,4,5,6, and 7 we obtain the matrix

A=
0

0
3

0
3
0

0 1 0

0 0
0
0 0
1

This matrix is of full rank, so the sufficient condition for identification is met. Counting included
endogs, G = 3, and counting excluded exogs, K = 5, so
K L = G 1
5L
L

=31
=3

The equation is over-identified by three restrictions, according to the counting rules, which
are correct when the only identifying information are the exclusion restrictions. However,
there is additional information in this case. Both Wtp and Wtg enter the consumption equation, and their coefficients are restricted to be the same. For this reason the consumption
equation is in fact overidentified by four restrictions.

10.6

Note about the rest of this chaper

In class, I will not teach the material in the rest of this chapter at this time, but instead we will go
on to GMM. The material that follows is easier to understand in the context of GMM, and we get a
nice unified theory.

10.7

2SLS

When we have no information regarding cross-equation restrictions or the structure of the error
covariance matrix, one can estimate the parameters of a single equation of the system without
regard to the other equations.
This isnt always efficient, as well see, but it has the advantage that misspecifications in other
equations will not affect the consistency of the estimator of the parameters of the equation
of interest.
Also, estimation of the equation wont be affected by identification problems in other equations.
The 2SLS estimator is very simple: it is the GIV estimator, using all of the weakly exogenous
variables as instruments. In the first stage, each column of Y1 is regressed on all the weakly
exogenous variables in the system, e.g., the entire X matrix. The fitted values are
Y1

= X(X 0 X)1 X 0 Y1
= PX Y1
1
= X

Since these fitted values are the projection of Y1 on the space spanned by X, and since any vector
in this space is uncorrelated with by assumption, Y1 is uncorrelated with . Since Y1 is simply
the reduced-form prediction, it is correlated with Y1 , The only other requirement is that the instruments be linearly independent. This should be the case when the order condition is satisfied, since
there are more columns in X2 than in Y1 in this case.
The second stage substitutes Y1 in place of Y1 , and estimates by OLS. This original model is
y

= Y1 1 + X1 1 +
= Z +

and the second stage model is

y = Y1 1 + X1 1 + .
Since X1 is in the space spanned by X, PX X1 = X1 , so we can write the second stage model as
y

= PX Y1 1 + PX X1 1 +
PX Z +

The OLS estimator applied to this model is

= (Z 0 PX Z)1 Z 0 PX y
which is exactly what we get if we estimate using IV, with the reduced form predictions of the
endogs used as instruments. Note that if we define
Z

=
=

PX Z
h
i
Y1 X1

so that Z are the instruments for Z, then we can write

= (Z 0 Z)1 Z 0 y

 Important note: OLS on the transformed model can be used to calculate the 2SLS estimate
of , since we see that its equivalent to IV using a particular set of instruments. However the
OLS covariance formula is not valid. We need to apply the IV covariance formula already seen
above.
Actually, there is also a simplification of the general IV variance formula. Define
Z

PX Z
h
i
Y X

The IV covariance estimator would ordinarily be


1 

1
2
= Z 0 Z
V ()
Z 0 Z Z 0 Z

IV
However, looking at the last term in brackets
Z 0 Z =

Y1

X1

i0 h

Y1

X1

"
=

Y10 (PX )Y1

X10 Y1

Y10 (PX )X1

X10 X1

but since PX is idempotent and since PX X = X, we can write

h

Y1

X1

i0 h

Y1

X1

"
=
=

Y10 PX PX Y1

Y10 PX X1

X10 PX Y1

X10 X1

Y1

X1

i0 h

Y1

X1

= Z 0 Z
Therefore, the second and last term in the variance formula cancel, so the 2SLS varcov estimator
simplifies to

1
2
= Z 0 Z
V ()

IV
which, following some algebra similar to the above, can also be written as

1
2
= Z 0 Z
V ()

IV

(10.3)

Finally, recall that though this is presented in terms of the first equation, it is general since any
equation can be placed first.
Properties of 2SLS:
1. Consistent
2. Asymptotically normal
3. Biased when the mean esists (the existence of moments is a technical issue we wont go into
here).
4. Asymptotically inefficient, except in special circumstances (more on this later).

10.8

Testing the overidentifying restrictions

The selection of which variables are endogs and which are exogs is part of the specification of the
model. As such, there is room for error here: one might erroneously classify a variable as exog
when it is in fact correlated with the error term. A general test for the specification on the model
can be formulated as follows:

The IV estimator can be calculated by applying OLS to the transformed model, so the IV objective function at the minimized value is

0


s(IV ) = y X IV PW y X IV ,
but
IV

= y X IV
= y X(X 0 PW X)1 X 0 PW y


= I X(X 0 PW X)1 X 0 PW y


= I X(X 0 PW X)1 X 0 PW (X + )
= A (X + )

where
A I X(X 0 PW X)1 X 0 PW
so
s(IV ) = (0 + 0 X 0 ) A0 PW A (X + )
Moreover, A0 PW A is idempotent, as can be verified by multiplication:
A0 P W A

=
=
=

I PW X(X 0 PW X)1 X 0 PW I X(X 0 PW X)1 X 0 PW




PW PW X(X 0 PW X)1 X 0 PW PW PW X(X 0 PW X)1 X 0 PW


I PW X(X 0 PW X)1 X 0 PW .

Furthermore, A is orthogonal to X
AX

I X(X 0 PW X)1 X 0 PW X

= X X
=

so
s(IV ) = 0 A0 PW A
Supposing the are normally distributed, with variance 2 , then the random variable
0 A0 PW A
s(IV )
=
2
2
is a quadratic form of a N (0, 1) random variable with an idempotent matrix in the middle, so
s(IV )
2 ((A0 PW A))
2
This isnt available, since we need to estimate 2 . Substituting a consistent estimator,
s(IV ) a 2
((A0 PW A))
c
2

Even if the arent normally distributed, the asymptotic result still holds. The last thing we
need to determine is the rank of the idempotent matrix. We have
A0 PW A = PW PW X(X 0 PW X)1 X 0 PW

so
(A0 PW A)

= T r PW PW X(X 0 PW X)1 X 0 PW

= T rPW T rX 0 PW PW X(X 0 PW X)1

= T rW (W 0 W )1 W 0 KX
= T rW 0 W (W 0 W )1 KX
= KW KX
where KW is the number of columns of W and KX is the number of columns of X. The degrees of freedom of the test is simply the number of overidentifying restrictions: the number
of instruments we have beyond the number that is strictly necessary for consistent estimation.
This test is an overall specification test: the joint null hypothesis is that the model is correctly
specified and that the W form valid instruments (e.g., that the variables classified as exogs
really are uncorrelated with . Rejection can mean that either the model y = Z + is
misspecified, or that there is correlation between X and .
This is a particular case of the GMM criterion test, which is covered in the second half of the
course. See Section 14.9.
Note that since
IV = A
and
s(IV ) = 0 A0 PW A
we can write
s(IV )
c2

0 W (W 0 W )1 W 0 W (W 0 W )1 W 0
=
0 /n
= n(RSSIV |W /T SSIV )
= nRu2

where Ru2 is the uncentered R2 from a regression of the IV residuals on all of the instruments
W . This is a convenient way to calculate the test statistic.
On an aside, consider IV estimation of a just-identified model, using the standard notation
y = X +
and W is the matrix of instruments. If we have exact identification then cols(W ) = cols(X), so
0

W X is a square matrix. The transformed model is

PW y = PW X + PW
and the fonc are
X 0 PW (y X IV ) = 0
The IV estimator is
1
IV = (X 0 PW X) X 0 PW y

Considering the inverse here

(X 0 PW X)

X 0 W (W 0 W )1 W 0 X

1

(W 0 X)1 X 0 W (W 0 W )1

(W 0 X)1 (W 0 W ) (X 0 W )

1

Now multiplying this by X 0 PW y, we obtain

IV

X 0 PW y

X 0 W (W 0 W )1 W 0 y

(W 0 X)1 (W 0 W ) (X 0 W )

(W 0 X)1 (W 0 W ) (X 0 W )

(W 0 X)1 W 0 y

The objective function for the generalized IV estimator is

s(IV )

0


y X IV PW y X IV




0
y 0 PW y X IV IV
X 0 PW y X IV


0
0
y 0 PW y X IV IV
X 0 PW y + IV
X 0 PW X IV




0
y 0 PW y X IV IV
X 0 PW y + X 0 PW X IV


y 0 PW y X IV


=
=
=
=
=

by the fonc for generalized IV. However, when were in the just indentified case, this is
s(IV )

= y 0 PW y X(W 0 X)1 W 0 y


= y 0 PW I X(W 0 X)1 W 0 y


= y 0 W (W 0 W )1 W 0 W (W 0 W )1 W 0 X(W 0 X)1 W 0 y
=

The value of the objective function of the IV estimator is zero in the just identified case. This makes
sense, since weve already shown that the objective function after dividing by 2 is asymptotically
2 with degrees of freedom equal to the number of overidentifying restrictions. In the present case,
there are no overidentifying restrictions, so we have a 2 (0) rv, which has mean 0 and variance 0,
e.g., its simply 0. This means were not able to test the identifying restrictions in the case of exact
identification.

10.9

System methods of estimation

2SLS is a single equation method of estimation, as noted above. The advantage of a single equation
method is that its unaffected by the other equations of the system, so they dont need to be
specified (except for defining what are the exogs, so 2SLS can use the complete set of instruments).
The disadvantage of 2SLS is that its inefficient, in general.
Recall that overidentification improves efficiency of estimation, since an overidentified equation can use more instruments than are necessary for consistent estimation.
Secondly, the assumption is that

XB + E

E(X 0 E)

0(KG)

vec(E)

N (0, )

Since there is no autocorrelation of the Et s, and since the columns of E are individually
homoscedastic, then

11 In

12 In

22 In
..

1G In
..
.
..
.

GG In

In

This means that the structural equations are heteroscedastic and correlated with one another
In general, ignoring this will lead to inefficient estimation, following the section on GLS.
When equations are correlated with one another estimation should account for the correlation in order to obtain efficiency.
Also, since the equations are correlated, information about one equation is implicitly information about all equations. Therefore, overidentification restrictions in any equation improve
efficiency for all equations, even the just identified equations.
Single equation methods cant use these types of information, and are therefore inefficient
(in general).

3SLS
Note: It is easier and more practical to treat the 3SLS estimator as a generalized method of moments estimator (see Chapter 14). I no longer teach the following section, but it is retained for its
possible historical interest. Another alternative is to use FIML (Subsection 10.9), if you are willing
to make distributional assumptions on the errors. This is computationally feasible with modern
computers.
Following our above notation, each structural equation can be written as
yi

= Yi 1 + Xi 1 + i
= Zi i + i

Grouping the G equations together we get

y1

y2
..
.

yG

Z1

0
..
.

Z2

..
0

0
..
.
0
ZG

or
y = Z +

2
..
.

2
..
.

where we already have that

E(0 )

In

The 3SLS estimator is just 2SLS combined with a GLS correction that takes advantage of the
structure of . Define Z as

X(X 0 X)1 X 0 Z1

0
..
.

X(X 0 X)1 X 0 Z2

Y1

..

Y2

0
..
.

X1

X2
..

0
..
.

0
X(X 0 X)1 X 0 ZG

0
YG

0
..
.

XG

These instruments are simply the unrestricted rf predicitions of the endogs, combined with the
exogs. The distinction is that if the model is overidentified, then
= B1
does
may be subject to some zero restrictions, depending on the restrictions on and B, and
is calculated using OLS equation by equation, as
not impose these restrictions. Also, note that
was discussed in Section 10.3.
The 2SLS estimator would be
= (Z 0 Z)1 Z 0 y
as can be verified by simple multiplication, and noting that the inverse of a block-diagonal matrix is
just the matrix with the inverses of the blocks on the main diagonal. This IV estimator still ignores
the covariance information. The natural extension is to add the GLS transformation, putting the
inverse of the error covariance into the formula, which gives the 3SLS estimator
3SLS

=
=

1

1
1
Z 0 ( In ) y
Z 0 ( In ) Z


1 0 1


Z 0 1 In Z
Z In y

This estimator requires knowledge of . The solution is to define a feasible estimator using a
consistent estimator of . The obvious solution is to use an estimator based on the 2SLS residuals:
i = yi Zi i,2SLS
(IMPORTANT NOTE: this is calculated using Zi , not Zi ). Then the element i, j of is estimated
by

ij =

0i j
n

into the formula above to get the feasible 3SLS estimator.

Substitute
Analogously to what we did in the case of 2SLS, the asymptotic distribution of the 3SLS esti-

mator can be shown to be

!
Z 0 ( I )1 Z 1


a
n

n 3SLS N 0, lim E
n

n
A formula for estimating the variance of the 3SLS estimator in finite samples (cancelling out the
powers of n) is

  
 1
1 In Z
V 3SLS = Z 0
This is analogous to the 2SLS formula in equation (10.3), combined with the GLS correction.
In the case that all equations are just identified, 3SLS is numerically equivalent to 2SLS.
Proving this is easiest if we use a GMM interpretation of 2SLS and 3SLS. GMM is presented
in the next econometrics course. For now, take it on faith.

FIML
Full information maximum likelihood is an alternative estimation method. FIML will be asymptotically efficient, since ML estimators based on a given information set are asymptotically efficient
w.r.t. all other estimators that use the same information set, and in the case of the full-information
ML estimator we use the entire information set. The 2SLS and 3SLS estimators dont require
distributional assumptions, while FIML of course does. Our model is, recall
Yt0
Et
E(Et Es0 )

Xt0 B + Et0

N (0, ), t
=

0, t 6= s

The joint normality of Et means that the density for Et is the multivariate normal, which is
g/2

(2)

1 1/2

det



1 0 1
exp Et Et
2

The transformation from Et to Yt requires the Jacobian

| det

dEt
| = | det |
dYt0

so the density for Yt is

(2)G/2 | det | det 1

1/2



1
0
exp (Yt0 Xt0 B) 1 (Yt0 Xt0 B)
2

Given the assumption of independence over time, the joint log-likelihood function is
n

ln L(B, , ) =

n
1X 0
nG
0
ln(2) + n ln(| det |) ln det 1
(Y Xt0 B) 1 (Yt0 Xt0 B)
2
2
2 t=1 t

This is a nonlinear in the parameters objective function. Maximixation of this can be done
using iterative numeric methods. Well see how to do this in the next section.
It turns out that the asymptotic distribution of 3SLS and FIML are the same, assuming normality of the errors.
One can calculate the FIML estimator by iterating the 3SLS estimator, thus avoiding the use
of a nonlinear optimizer. The steps are

 3SLS and B
3SLS as normal.
1. Calculate
=B
3SLS
1 . This is new, we didnt estimate in this way before. This
2. Calculate
3SLS
estimator may have some zeros in it. When Greene says iterated 3SLS doesnt lead to
but only updates
and B

FIML, he means this for a procedure that doesnt update ,

If you update
you do converge to FIML.
and .
and calculate
using
and B
to get the estimated
3. Calculate the instruments Y = X
errors, applying the usual estimator.
4. Apply 3SLS using these new instruments and the estimate of .
5. Repeat steps 2-4 until there is no change in the parameters.
FIML is fully efficient, since its an ML estimator that uses all information. This implies
that 3SLS is fully efficient when the errors are normally distributed. Also, if each equation is
just identified and the errors are normal, then 2SLS will be fully efficient, since in this case
2SLS3SLS.
When the errors arent normally distributed, the likelihood function is of course different
than whats written above.

10.10

Example: 2SLS and Kleins Model 1

The Octave program Simeq/Klein.m performs 2SLS estimation for the 3 equations of Kleins model
1, assuming nonautocorrelated errors, so that lagged endogenous variables can be used as instruments. The results are:

CONSUMPTION EQUATION
*******************************************************
2SLS estimation results
Observations 21
R-squared 0.976711
Sigma-squared 1.044059

Constant
Profits
Lagged Profits
Wages

estimate
16.555
0.017
0.216
0.810

st.err.
1.321
0.118
0.107
0.040

t-stat.
12.534
0.147
2.016
20.129

p-value
0.000
0.885
0.060
0.000

*******************************************************
INVESTMENT EQUATION
*******************************************************
2SLS estimation results
Observations 21
R-squared 0.884884
Sigma-squared 1.383184

Constant

estimate
20.278

st.err.
7.543

t-stat.
2.688

p-value
0.016

Profits
Lagged Profits
Lagged Capital

0.150
0.616
-0.158

0.173
0.163
0.036

0.867
3.784
-4.368

0.398
0.001
0.000

*******************************************************
WAGES EQUATION
*******************************************************
2SLS estimation results
Observations 21
R-squared 0.987414
Sigma-squared 0.476427

Constant
Output
Lagged Output
Trend

estimate
1.500
0.439
0.147
0.130

st.err.
1.148
0.036
0.039
0.029

t-stat.
1.307
12.316
3.777
4.475

p-value
0.209
0.000
0.002
0.000

*******************************************************

The above results are not valid (specifically, they are inconsistent) if the errors are autocorrelated, since lagged endogenous variables will not be valid instruments in that case. You might
consider eliminating the lagged endogenous variables as instruments, and re-estimating by 2SLS,
to obtain consistent parameter estimates in this more complex case. Standard errors will still be
estimated inconsistently, unless use a Newey-West type covariance estimator. Food for thought...

Chapter 11

Numeric optimization methods

Readings: Hamilton, ch. 5, section 7 (pp. 133-139) ; Gourieroux and Monfort, Vol. 1, ch. 13, pp.
443-60 ; Goffe, et. al. (1994).
The next chapter introduces extremum estimators, which are minimizers or maximizers of
objective functions. If were going to be applying extremum estimators, well need to know how
to find an extremum. This section gives a very brief introduction to what is a large literature on
numeric optimization methods. Well consider a few well-known techniques, and one fairly new
technique that may allow one to solve difficult problems. The main objective is to become familiar
with the issues, and to learn how to use the BFGS algorithm at the practical level.
The general problem we consider is how to find the maximizing element (a K -vector) of a
function s(). This function may not be continuous, and it may not be differentiable. Even if it is
twice continuously differentiable, it may not be globally concave, so local maxima, minima and
saddlepoints may all exist. Supposing s() were a quadratic function of , e.g.,
1
s() = a + b0 + 0 C,
2
the first order conditions would be linear:
D s() = b + C
so the maximizing (minimizing) element would be = C 1 b. This is the sort of problem we have
with linear models estimated by OLS. Its also the case for feasible GLS, since conditional on the
estimate of the varcov matrix, we have a quadratic objective function in the remaining parameters.
More general problems will not have linear f.o.c., and we will not be able to solve for the
maximizer analytically. This is when we need a numeric optimization method.

11.1

Search

The idea is to create a grid over the parameter space and evaluate the function at each point on
the grid. Select the best point. Then refine the grid in the neighborhood of the best point, and
continue until the accuracy is good enough. See Figure 11.1. One has to be careful that the grid
is fine enough in relationship to the irregularity of the function to ensure that sharp peaks are not
missed entirely.
To check q values in each dimension of a K dimensional parameter space, we need to check q K
points. For example, if q = 100 and K = 10, there would be 10010 points to check. If 1000 points
can be checked in a second, it would take 3. 171 109 years to perform the calculations, which
is approximately 2/3 the age of the earth. The search method is a very reasonable choice if K is
146

Figure 11.1: Search method

small, but it quickly becomes infeasible if K is moderate or large.

11.2

Derivative-based methods

Introduction
Derivative-based methods are defined by
1. the method for choosing the initial value, 1
2. the iteration method for choosing k+1 given k (based upon derivatives)
3. the stopping criterion.
The iteration method can be broken into two problems: choosing the stepsize ak (a scalar) and
choosing the direction of movement, dk , which is of the same dimension of , so that
(k+1) = (k) + ak dk .
A locally increasing direction of search d is a direction such that
s( + ad)
>0
a
for a positive but small. That is, if we go in direction d, we will improve on the objective function,
at least if we dont go too far in that direction.
As long as the gradient at is not zero there exist increasing directions, and they can all be
represented as Qk g(k ) where Qk is a symmetric pd matrix and g () = D s() is the gradient

Figure 11.2: Increasing directions of search

at . To see this, take a T.S. expansion around a0 = 0

s( + ad)

= s( + 0d) + (a 0) g( + 0d)0 d + o(1)

= s() + ag()0 d + o(1)

For small enough a the o(1) term can be ignored. If d is to be an increasing direction, we
need g()0 d > 0. Defining d = Qg(), where Q is positive definite, we guarantee that
g()0 d = g()0 Qg() > 0
unless g() = 0. Every increasing direction can be represented in this way (p.d. matrices are
those such that the angle between g and Qg() is less that 90 degrees). See Figure 11.2.
With this, the iteration rule becomes
(k+1) = (k) + ak Qk g(k )
and we keep going until the gradient becomes zero, so that there is no increasing direction. The
problem is how to choose a and Q.
Conditional on Q, choosing a is fairly straightforward. A simple line search is an attractive
possibility, since a is a scalar.
The remaining problem is how to choose Q.
Note also that this gives no guarantees to find a global maximum.

Steepest descent
Steepest descent (ascent if were maximizing) just sets Q to and identity matrix, since the gradient
provides the direction of maximum rate of change of the objective function.
Advantages: fast - doesnt require anything more than first derivatives.
Disadvantages: This doesnt always work too well however: see the Rosenbrock, or banana
function: http://en.wikipedia.org/wiki/Rosenbrock_function.

Newton s method
Newtons method uses information about the slope and curvature of the objective function to
determine which direction and how far to move from an initial point. Supposing were trying to
maximize sn (). Take a second order Taylors series approximation of sn () about k (an initial
guess).



0


sn () sn (k ) + g(k )0 k + 1/2 k H(k ) k
To attempt to maximize sn (), we can maximize the portion of the right-hand side that depends
on , i.e., we can maximize

0


s() = g(k )0 + 1/2 k H(k ) k
with respect to . This is a much easier problem, since it is a quadratic function in , so it has linear
first order conditions. These are
D s() = g(k ) + H(k ) k

So the solution for the next round estimate is

k+1 = k H(k )1 g(k )
See http://en.wikipedia.org/wiki/Newton%27s_method_in_optimization for more information. This is illustrated in Figure 11.3.
However, its good to include a stepsize, since the approximation to sn () may be bad far away
so the actual iteration formula is
from the maximizer ,
k+1 = k ak H(k )1 g(k )
A potential problem is that the Hessian may not be negative definite when were far from the
maximizing point. So H(k )1 may not be positive definite, and H(k )1 g(k ) may not
define an increasing direction of search. This can happen when the objective function has
flat regions, in which case the Hessian matrix is very ill-conditioned (e.g., is nearly singular),
or when were in the vicinity of a local minimum, H(k ) is positive definite, and our direction
is a decreasing direction of search. Matrix inverses by computers are subject to large errors
when the matrix is ill-conditioned. Also, we certainly dont want to go in the direction of
a minimum when were maximizing. To solve this problem, Quasi-Newton methods simply
add a positive definite component to H() to ensure that the resulting matrix is positive
definite, e.g., Q = H() + bI, where b is chosen large enough so that Q is well-conditioned
and positive definite. This has the benefit that improvement in the objective function is
guaranteed. See http://en.wikipedia.org/wiki/Quasi-Newton_method.
Another variation of quasi-Newton methods is to approximate the Hessian by using successive

Figure 11.3: Newton iteration

gradient evaluations. This avoids actual calculation of the Hessian, which is an order of
magnitude (in the dimension of the parameter vector) more costly than calculation of the
gradient. They can be done to ensure that the approximation is p.d. DFP and BFGS are two
well-known examples.
Stopping criteria
The last thing we need is to decide when to stop. A digital computer is subject to limited machine precision and round-off errors. For these reasons, it is unreasonable to hope that a program
can exactly find the point that maximizes a function. We need to define acceptable tolerances.
Some stopping criteria are:
Negligable change in parameters:
|jk jk1 | < 1 , j
Negligable relative change:
|

jk jk1
jk1

| < 2 , j

Negligable change of function:

|s(k ) s(k1 )| < 3
Gradient negligibly different from zero:
|gj (k )| < 4 , j
Or, even better, check all of these.

 Also, if were maximizing, its good to check that the last round (real, not approximate)
Hessian is negative definite.
Starting values
The Newton-Raphson and related algorithms work well if the objective function is concave
(when maximizing), but not so well if there are convex regions and local minima or multiple local
maxima. The algorithm may converge to a local minimum or to a local maximum that is not
optimal. The algorithm may also have difficulties converging at all.
The usual way to ensure that a global maximum has been found is to use many different
starting values, and choose the solution that returns the highest objective function value.
THIS IS IMPORTANT in practice. More on this later.
Calculating derivatives
The Newton-Raphson algorithm requires first and second derivatives. It is often difficult to
calculate derivatives (especially the Hessian) analytically if the function sn () is complicated. Possible solutions are to calculate derivatives numerically, or to use programs such as MuPAD or
Mathematica to calculate analytic derivatives. For example, Figure 11.4 shows Sage

calculating

a couple of derivatives. The KAIST Sage cell server will let you try Sage online, its address is

http://aleph.sagemath.org/.
Numeric derivatives are less accurate than analytic derivatives, and are usually more costly
to evaluate. Both factors usually cause optimization programs to be less successful when
numeric derivatives are used.
One advantage of numeric derivatives is that you dont have to worry about having made
an error in calculating the analytic derivative. When programming analytic derivatives its
a good idea to check that they are correct by using numeric derivatives. This is a lesson I
learned the hard way when writing my thesis.
Numeric second derivatives are much more accurate if the data are scaled so that the elements of the gradient are of the same order of magnitude. Example: if the model is yt =
h(xt +zt )+t , and estimation is by NLS, suppose that D sn () = 1000 and D sn () = 0.001.
One could define = /1000; xt = 1000xt ; = 1000; zt = zt /1000. In this case, the gradients D sn () and D sn () will both be 1.
In general, estimation programs always work better if data is scaled in this way, since roundoff errors are less likely to become important. This is important in practice.
There are algorithms (such as BFGS and DFP) that use the sequential gradient evaluations
to build up an approximation to the Hessian. The iterations are faster because the actual
Hessian isnt calculated, but more iterations usually are required for convergence. Versions
of BFGS are probably the most widely used optimizers in econometrics.
Switching between algorithms during iterations is sometimes useful.

11.3

Simulated Annealing

Simulated annealing is an algorithm which can find an optimum in the presence of nonconcavities,
discontinuities and multiple local minima/maxima. Basically, the algorithm randomly selects evaluation points, accepts all points that yield an increase in the objective function, but also accepts
some points that decrease the objective function. This allows the algorithm to escape from local
1 Sage

is free software that has both symbolic and numeric computational capabilities. See http://www.sagemath.org/

Figure 11.4: Using Sage to get analytic derivatives

minima. As more and more points are tried, periodically the algorithm focuses on the best point
so far, and reduces the range over which random points are generated. Also, the probability that
a negative move is accepted reduces. The algorithm relies on many evaluations, as in the search
method, but focuses in on promising areas, which reduces function evaluations with respect to the
search method. It does not require derivatives to be evaluated. I have a program to do this if youre
interested.

11.4

Examples of nonlinear optimization

This section gives a few examples of how some nonlinear statistical models may be estimated using
maximum likelihood. The intention at the moment is simply illustrative, to get the general idea
of the form of a nonlinear statistical model and how optimization is used. Later, more specific
examples will be provided for learning how to optimize, and even later the specifics of ML will be
explained.

Discrete Choice: The logit model

In this section we will consider maximum likelihood estimation of the logit model for binary 0/1
dependent variables. We will use the BFGS algotithm to find the MLE.
A binary response is a variable that takes on only two values, customarily 0 and 1, which can
be thought of as codes for whether or not a condisiton is satisfied. For example, 0=drive to work,
1=take the bus. Often the observed binary variable, say y, is related to an unobserved (latent)
continuous varable, say y . We would like to know the effect of covariates, x, on y. The model can
be represented as
y

g(x)

1(y > 0)

P r(y = 1)

= F [g(x)]

p(x, )

The log-likelihood function is

n

1X
sn () =
(yi ln p(xi , ) + (1 yi ) ln [1 p(xi , )])
n i=1
For the logit model, the probability has the specific form
p(x, ) =

1
1 + exp(x0 )

You should download and examine LogitDGP.m , which generates data according to the logit
model, logit.m , which calculates the loglikelihood, and EstimateLogit.m , which sets things up and
calls the estimation routine, which uses the BFGS algorithm.
Here are some estimation results with n = 100, and the true = (0, 1)0 .

***********************************************
Trial of MLE estimation of Logit model
MLE Estimation Results
BFGS convergence: Normal convergence
Average Log-L: 0.607063

Observations: 100
constant
slope

estimate
0.5400
0.7566

st. err
0.2229
0.2374

t-stat
2.4224
3.1863

p-value
0.0154
0.0014

Information Criteria
CAIC : 132.6230
BIC : 130.6230
AIC : 125.4127
***********************************************

The estimation program is calling mle_results(), which in turn calls a number of other routines.

Count Data: The MEPS data and the Poisson model

Demand for health care is usually thought of a a derived demand: health care is an input to a
home production function that produces health, and health is an argument of the utility function.
Grossman (1972), for example, models health as a capital stock that is subject to depreciation (e.g.,
the effects of ageing). Health care visits restore the stock. Under the home production framework,
individuals decide when to make health care visits to maintain their health stock, or to deal with
negative shocks to the stock in the form of accidents or illnesses. As such, individual demand will
be a function of the parameters of the individuals utility functions.
The MEPS health data file , meps1996.data, contains 4564 observations on six measures of
health care usage. The data is from the 1996 Medical Expenditure Panel Survey (MEPS). You can
get more information at http://www.meps.ahrq.gov/. The six measures of use are are officebased visits (OBDV), outpatient visits (OPV), inpatient visits (IPV), emergency room visits (ERV),
dental visits (VDV), and number of prescription drugs taken (PRESCR). These form columns 1 6 of meps1996.data. The conditioning variables are public insurance (PUBLIC), private insurance
(PRIV), sex (SEX), age (AGE), years of education (EDUC), and income (INCOME). These form
columns 7 - 12 of the file, in the order given here. PRIV and PUBLIC are 0/1 binary variables,
where a 1 indicates that the person has access to public or private insurance coverage. SEX is
also 0/1, where 1 indicates that the person is female. This data will be used in examples fairly
extensively in what follows.
The program ExploreMEPS.m shows how the data may be read in, and gives some descriptive
information about variables, which follows:
All of the measures of use are count data, which means that they take on the values 0, 1, 2, .... It
might be reasonable to try to use this information by specifying the density as a count data density.
One of the simplest count data densities is the Poisson density, which is
fY (y)

exp()y
.
y!

The Poisson average log-likelihood function is

n

sn () =

1X
(i + yi ln i ln yi !)
n i=1

We will parameterize the model as

i

exp(x0i )

xi

[1 P U BLIC P RIV SEX AGE EDU C IN C]0

(11.1)

This ensures that the mean is positive, as is required for the Poisson model. Note that for this
parameterization
j =

/j

so
j xj = xj ,
the elasticity of the conditional mean of y with respect to the j th conditioning variable.
The program EstimatePoisson.m estimates a Poisson model using the full data set. The results
of the estimation, using OBDV as the dependent variable are here:

MPITB extensions found

OBDV

******************************************************
Poisson model, MEPS 1996 full data set
MLE Estimation Results
BFGS convergence: Normal convergence
Average Log-L: -3.671090
Observations: 4564

constant
pub. ins.
priv. ins.
sex
age
edu
inc

estimate
-0.791
0.848
0.294
0.487
0.024
0.029
-0.000

st. err
0.149
0.076
0.071
0.055
0.002
0.010
0.000

t-stat
-5.290
11.093
4.137
8.797
11.471
3.061
-0.978

p-value
0.000
0.000
0.000
0.000
0.000
0.002
0.328

Information Criteria
CAIC : 33575.6881
Avg. CAIC:
7.3566
BIC : 33568.6881
Avg. BIC:
7.3551
AIC : 33523.7064
Avg. AIC:
7.3452
******************************************************

Duration data and the Weibull model

In some cases the dependent variable may be the time that passes between the occurence of two
events. For example, it may be the duration of a strike, or the time needed to find a job once
one is unemployed. Such variables take on values on the positive real line, and are referred to as
duration data.
A spell is the period of time between the occurence of initial event and the concluding event.
For example, the initial event could be the loss of a job, and the final event is the finding of a new
job. The spell is the period of unemployment.
Let t0 be the time the initial event occurs, and t1 be the time the concluding event occurs.
For simplicity, assume that time is measured in years. The random variable D is the duration
of the spell, D = t1 t0 . Define the density function of D, fD (t), with distribution function
FD (t) = Pr(D < t).
Several questions may be of interest. For example, one might wish to know the expected time
one has to wait to find a job given that one has already waited s years. The probability that a spell
lasts more than s years is
Pr(D > s) = 1 Pr(D s) = 1 FD (s).
The density of D conditional on the spell being longer than s years is
fD (t)
.
1 FD (s)

fD (t|D > s) =

The expectanced additional time required for the spell to end given that is has already lasted s
years is the expectation of D with respect to this density, minus s.

Z
E = E(D|D > s) s =
t

fD (z)
dz
z
1 FD (s)


s

To estimate this function, one needs to specify the density fD (t) as a parametric density, then
estimate by maximum likelihood. There are a number of possibilities including the exponential
density, the lognormal, etc. A reasonably flexible model that is a generalization of the exponential
density is the Weibull density

fD (t|) = e(t) (t)1 .

According to this model, E(D) = . The log-likelihood is just the product of the log densities.
To illustrate application of this model, 402 observations on the lifespan of dwarf mongooses
(see Figure 11.5) in Serengeti National Park (Tanzania) were used to fit a Weibull model. The
spell in this case is the lifetime of an individual mongoose. The parameter estimates and standard
= 0.559 (0.034) and = 0.867 (0.033) and the log-likelihood value is -659.3. Figure
errors are
11.6 presents fitted life expectancy (expected additional years of life) as a function of age, with
95% confidence bands. The plot is accompanied by a nonparametric Kaplan-Meier estimate of
life-expectancy. This nonparametric estimator simply averages all spell lengths greater than age,
and then subtracts age. This is consistent by the LLN.
In the figure one can see that the model doesnt fit the data well, in that it predicts life expectancy quite differently than does the nonparametric model. For ages 4-6, the nonparametric
estimate is outside the confidence interval that results from the parametric model, which casts
doubt upon the parametric model. Mongooses that are between 2-6 years old seem to have a
lower life expectancy than is predicted by the Weibull model, whereas young mongooses that survive beyond infancy have a higher life expectancy, up to a bit beyond 2 years. Due to the dramatic

Figure 11.5: Dwarf mongooses

Figure 11.6: Life expectancy of mongooses, Weibull model

change in the death rate as a function of t, one might specify fD (t) as a mixture of two Weibull
densities,




1
2
fD (t|) = e(1 t) 1 1 (1 t)1 1 + (1 ) e(2 t) 2 2 (2 t)2 1 .
The parameters i and i , i = 1, 2 are the parameters of the two Weibull densities, and is the
parameter that mixes the two.
With the same data, can be estimated using the mixed model. The results are a log-likelihood
= -623.17. Note that a standard likelihood ratio test cannot be used to chose between the two
models, since under the null that = 1 (single density), the two parameters 2 and 2 are not
identified. It is possible to take this into account, but this topic is out of the scope of this course.
Nevertheless, the improvement in the likelihood function is considerable. The parameter estimates
are
Parameter

Estimate

St. Error

0.233

0.016

1.722

0.166

1.731

0.101

1.522

0.096

0.428

0.035

Note that the mixture parameter is highly significant. This model leads to the fit in Figure 11.7.
Note that the parametric and nonparametric fits are quite close to one another, up to around 6
years. The disagreement after this point is not too important, since less than 5% of mongooses
live more than 6 years, which implies that the Kaplan-Meier nonparametric estimate has a high
variance (since its an average of a small number of observations).
Mixture models are often an effective way to model complex responses, though they can suffer
from overparameterization. Alternatives will be discussed later.

11.5

Numeric optimization: pitfalls

In this section well examine two common problems that can be encountered when doing numeric
optimization of nonlinear models, and some solutions.

Poor scaling of the data

When the data is scaled so that the magnitudes of the first and second derivatives are of different
orders, problems can easily result. If we uncomment the appropriate line in EstimatePoisson.m,
the data will not be scaled, and the estimation program will have difficulty converging (it seems
to take an infinite amount of time). With unscaled data, the elements of the score vector have
very different magnitudes at the initial value of (all zeros). To see this run CheckScore.m. With
unscaled data, one element of the gradient is very large, and the maximum and minimum elements
are 5 orders of magnitude apart. This causes convergence problems due to serious numerical
inaccuracy when doing inversions to calculate the BFGS direction of search. With scaled data,
none of the elements of the gradient are very large, and the maximum difference in orders of
magnitude is 3. Convergence is quick.

Multiple optima
Multiple optima (one global, others local) can complicate life, since we have limited means of
determining if there is a higher maximum the the one were at. Think of climbing a mountain in

Figure 11.7: Life expectancy of mongooses, mixed Weibull model

an unknown range, in a very foggy place (Figure 11.8). You can go up until theres nowhere else
to go up, but since youre in the fog you dont know if the true summit is across the gap thats at
your feet. Do you claim victory and go home, or do you trudge down the gap and explore the other
side?
The best way to avoid stopping at a local maximum is to use many starting values, for example
on a grid, or randomly generated. Or perhaps one might have priors about possible values for the
parameters (e.g., from previous studies of similar data).
Lets try to find the true minimizer of minus 1 times the foggy mountain function (since the
algorithms are set up to minimize). From the picture, you can see its close to (0, 0), but lets
pretend there is fog, and that we dont know that. The program FoggyMountain.m shows that
poor start values can lead to problems. It uses SA, which finds the true global minimum, and it
shows that BFGS using a battery of random start values can also find the global minimum help.
The output of one run is here:

MPITB extensions found

======================================================
BFGSMIN final results
Used numeric gradient
-----------------------------------------------------STRONG CONVERGENCE
Function conv 1 Param conv 1 Gradient conv 1

Figure 11.8: A foggy mountain

-----------------------------------------------------Objective function value -0.0130329

Stepsize 0.102833
43 iterations
-----------------------------------------------------param
gradient change
15.9999 -0.0000
0.0000
-28.8119
0.0000
0.0000
The result with poor start values
ans =
16.000

-28.812

================================================
SAMIN final results
NORMAL CONVERGENCE
Func. tol. 1.000000e-10 Param. tol. 1.000000e-03
Obj. fn. value -0.100023
parameter

search width
0.037419

0.000018

-0.000000
0.000051
================================================
Now try a battery of random start values and
a short BFGS on each, then iterate to convergence
The result using 20 randoms start values
ans =
3.7417e-02

2.7628e-07

The true maximizer is near (0.037,0)

In that run, the single BFGS run with bad start values converged to a point far from the true
minimizer, which simulated annealing and BFGS using a battery of random start values both found
the true maximizer. Using a battery of random start values, we managed to find the global max.
The moral of the story is to be cautious and dont publish your results too quickly.

11.6

Examples

11.7

Exercises

1. In octave, type help bfgsmin_example, to find out the location of the file. Edit the file to
examine it and learn how to call bfgsmin. Run it, and examine the output.
2. In octave, type help samin_example, to find out the location of the file. Edit the file to
examine it and learn how to call samin. Run it, and examine the output.
2

3. Numerically minimize the function sin(x) + 0.01 (x a) , setting a = 0, using the software of
your choice. Plot the function over the interval (2, 2). Does the software find the global
minimum? Does this depend on the starting value you use? Outline a strategy that would
allow you to find the minimum reliably, when a can take on any given value in the interval
(, ).
4. Numerically compute the OLS estimator of the Nerlove model by using an interative minimization algorithm to minimize the sum of squared residuals. Verify that the results coincide
with those given in subsection 3.8. The important part of this problem is to learn how to
minimize a function that depends on both parameters and data. Try to write your function
so that it is easy to use it with an arbitrary data set.

Chapter 12

Asymptotic properties of extremum

estimators
Readings: Hayashi (2000), Ch. 7; Gourieroux and Monfort (1995), Vol. 2, Ch. 24; Amemiya,
Ch. 4 section 4.1; Davidson and MacKinnon, pp. 591-96; Gallant, Ch. 3; Newey and McFadden
(1994), Large Sample Estimation and Hypothesis Testing, in Handbook of Econometrics, Vol. 4,
Ch. 36.

12.1

Extremum estimators

Well begin with study of extremum estimators in general. Let Zn = {z1 , z2 , ..., zn } be the available
data, arranged in a n p matrix, based on a sample of size n (there are p variables). Our paradigm
is that data are generated as a draw from the joint density fZn (z). This density may not be known,
but it exists in principle. The draw from the density may be thought of as the outcome of a
random experiment that is characterized by the probability space {, F, P }. When the experiment
is performed, is the result, and Zn () = {Z1 (), Z2 (), ..., Zn ()} = {z1 , z2 , ..., zn } is the
realized data. The probability space is rich enough to allow us to consider events defined in terms
of an infinite sequence of data Z = {z1 , z2 , ..., }.
Definition 25. [Extremum estimator] An extremum estimator is the optimizing element of an
objective function sn (Zn , ) over a set .
Because the data Zn () depends on , we can emphasize this by writing sn (, ). Ill be loose
with notation and interchange when convenient.
Example 26. OLS. Let the d.g.p. be yt = x0t 0 + t , t = 1, 2, ..., n, 0 . Stacking observations

0
vertically, yn = Xn 0 + n , where Xn =
. Let Zn = [yn Xn ]. The least
x1 x2 xn
squares estimator is defined as
arg min sn (Zn , )

where
sn (Zn , ) = 1/n

n
X

(yt x0t )

t=1

As you already know, = (X X)

0

X y.

.
Example 27. Maximum likelihood. Suppose that the continuous random variables Yt IIN (0 , 02 ), t =

164

1, 2, ..., n. If  is a standard normal random variable, its density is

1/2

f (z; ) = (2)

z2
exp
2



.

We have that t = (Yt 0 )/0 is standard normal, and the Jacobian |t /yt | = 1/0 . Thus, doing
a change of variable, the density of a single observation on Y is
1/2

fY (yt ; , ) = (2)

1
(1/) exp
2

yt

2 !
.

The maximum likelihood estimator is maximizes the joint density of the sample. Because the data
are i.i.d., the joint density of the sample {y1 , y2 , ..., yn } is the product of the densities of each
observation, and the ML estimator is
arg max Ln () =

n
Y

(2)

t=1

1/2

(yt )
(1/) exp
2

Because the natural logarithm is strictly increasing on (0, ), maximization of the average logarithmic likelihood function is achieved at the same as for the likelihood function. So, the ML
estimator arg max sn () where
n
2
X

(yt )
sn () = (1/n) ln Ln () = ln 2 log (1/n)
2
t=1

. Well come back to this in more detail

Solution of the f.o.c. leads to the familiar result that = y
later.

Example 28. Bayesian estimator

Bayesian point estimators such as the posterior mode, median or mean can be expressed as
extremum estimators. For example, the posterior mean E(|Zn ) is the minimizer (with respect to
) of the function
Z

( ) f (Zn ; )()/f (Zn )d

sn () =

where f (Zn ; ) is the likelihood function, () is a prior density, and f (Zn ) is the marginal
likelihood of the data. These concepts are explained later, for now the point is that Bayesian
estimators can be thought of as extremum estimators, and the theory for extremum estimators will
apply.
Note that the objective function sn (Zn , ) is a random function, because it depends on Zn () =
{Z1 (), Z2 (), ..., Zn ()} = {z1 , z2 , ..., zn }. We need to consider what happens as different outcomes occur. These different outcomes lead to different data being generated, and the
different data causes the objective function to change. Note, however, that for a fixed , the
data Zn () = {Z1 (), Z2 (), ..., Zn ()} = {z1 , z2 , ..., zn } are a fixed realization, and the objective
function sn (Zn , ) becomes a non-random function of . When actually computing an extremum
estimator, we treat the data as fixed, and employ algorithms for optimization of nonstochastic
functions. When analyzing the properties of an extremum estimator, we need to investigate what
happens throughout : we do not focus only on the that generated the observed data. This is
because we would like to find estimators that work well on average for any data set that can result
from .
Well often write the objective function suppressing the dependence on Zn , as sn (, ) or simply
sn (), depending on context. The first of these emphasizes the fact that the objective function is

random, and the second is more compact. However, the data is still in there, and because the data
is randomly sampled, the objective function is random, too.

12.2

Existence

If sn () is continuous in and is compact, then a maximizer exists, by the Weierstrass maximum

theorem (Debreu, 1959). In some cases of interest, sn () may not be continuous. Nevertheless, it
may still converge to a continous function, in which case existence will not be a problem, at least
asymptotically. Henceforth in this course, we assume that sn () is continuous.

12.3

Consistency

The following theorem is patterned on a proof in Gallant (1987) (the article, ref. later), which
well see in its original form later in the course. It is interesting to compare the following proof
with Amemiyas Theorem 4.1.1, which is done in terms of convergence in probability.
Theorem 29. [Consistency of e.e.] Suppose that n is obtained by maximizing sn () over .
Assume
(a) Compactness: The parameter space is an open bounded subset of Euclidean space <K . So the
closure of , , is compact.
(b) Uniform Convergence: There is a nonstochastic function s () that is continuous in on
such that
lim sup |sn (, ) s ()| = 0, a.s.

(c) Identification: s () has a unique global maximum at 0 , i.e., s (0 ) > s (), 6=

0 ,
a.s.
Then n 0 .

Proof: Select a and hold it fixed. Then {sn (, )} is a fixed sequence of functions.
Suppose that is such that sn (, ) converges to s (). This happens with probability one by
assumption (b). The sequence {n } lies in the compact set , by assumption (a) and the fact that
maximixation is over . Since every sequence from a compact set has at least one limit point
(Bolzano-Weierstrass), say that is a limit point of {n }. There is a subsequence {nm } ({nm } is
By uniform convergence and
simply a sequence of increasing integers) with limm n = .
m

continuity,

lim snm (nm ) = s ().

n
o
To see this, first of all, select an element t from the sequence nm . Then uniform convergence
implies
lim snm (t ) = s (t )

Continuity of s () implies that

lim s (t ) = s ()

n o
So the above claim is true.
since the limit as t of t is .
Next, by maximization
snm (nm ) snm (0 )
which holds in the limit, so
lim snm (nm ) lim snm (0 ).

However,

lim snm (nm ) = s (),

as seen above, and

lim snm (0 ) = s (0 )

by uniform convergence, so
s (0 ).
s ()
=
But by assumption (3), there is a unique global maximum of s () at 0 , so we must have s ()
0
0

s ( ), and = in the limit. Finally, all of the above limits hold almost surely, since so far we
have held fixed, but now we need to consider all . Therefore {n } has only one limit point,
0 , except on a set C with P (C) = 0.
Discussion of the proof:
This proof relies on the identification assumption of a unique global maximum at 0 . An
equivalent way to state this is
(c) Identification: Any point in with s () s (0 ) must be such that k 0 k= 0, which
matches the way we will write the assumption in the section on nonparametric inference.
We assume that n is in fact a global maximum of sn () . It is not required to be unique
for n finite, though the identification assumption requires that the limiting objective function
have a unique maximizing argument. The previous section on numeric optimization methods
showed that actually finding the global maximum of sn () may be a non-trivial problem.
See Amemiyas Example 4.1.4 for a case where discontinuity leads to breakdown of consistency.
The assumption that 0 is in the interior of (part of the identification assumption) has not
been used to prove consistency, so we could directly assume that 0 is simply an element of a
compact set . The reason that we assume its in the interior here is that this is necessary for
subsequent proof of asymptotic normality, and Id like to maintain a minimal set of simple
assumptions, for clarity. Parameters on the boundary of the parameter set cause theoretical
difficulties that we will not deal with in this course. Just note that conventional hypothesis
testing methods do not apply in this case.
Note that sn () is not required to be continuous, though s () is.
The following figures illustrate why uniform convergence is important. In the second figure,
if the function is not converging quickly enough around the lower of the two maxima. If
the pointwise convergence in this region is slow enough, there is no guarantee that the
maximizer will be in the neighborhood of the global maximizer of s (), even when n is
very large. Uniform comvergence means that we are in the situation of the top graphic. As
long as n is large enough, the maximum will be in the neighborhood of the global maximum
of s ().

With uniform convergence, the maximum of the sample

objective function eventually must be in the neighborhood
of the maximum of the limiting objective function

With pointwise convergence, the sample objective function

may have its maximum far away from that of the limiting
objective function

Sufficient conditions for assumption (b)

We need a uniform strong law of large numbers in order to verify assumption (2) of Theorem 29.
To verify the uniform convergence assumption, it is often feasible to employ the following set of
stronger assumptions:
the parameter space is compact, which is given by assumption (b)
the objective function sn () is continuous and bounded with probability one on the entire
parameter space
a standard SLLN can be shown to apply to some point in the parameter space. That is, we
a.s.

can show that sn () s () for some . Note that in most cases, the objective function will

be an average of terms, such as

n

sn () =

1X
st ()
n t=1

As long as the st () are not too strongly dependent, and have finite variances, we can usually
find a SLLN that will apply.
With these assumptions, it can be shown that pointwise convergence holds throughout the parameter space, so we obtain the needed uniform convergence.
These are reasonable conditions in many cases, and henceforth when dealing with specific
estimators well simply assume that pointwise almost sure convergence can be extended to uniform
almost sure convergence in this way.
More on the limiting objective function
The limiting objective function in assumption (b) is s (). What is the nature of this function and
where does it come from?
Remember our paradigm - data is presumed to be generated as a draw from fZn (z), and the
objective function is sn (Zn , ).
Usually, sn (Zn , ) is an average of terms.
The limiting objective function is found by applying a strong (weak) law of large numbers to
sn (Zn , ).
A strong (weak) LLN says that an average of terms converges almost surely (in probability)
to the limit of the expectation of the average.
Supposing one holds,
Z
s () = lim Esn (Zn , ) = lim
n

sn (z, )fZn (z)dz

Zn

Now suppose that the density fZn (z) that characterizes the DGP is parametric: fZn (z; ), %,
and the data is generated by 0 %. Now we have two parameters to worry about, and . We
are probably interested in learning about the true DGP, which means that 0 is the item of interest.
When the DGP is parametric, the limiting objective function is
Z
s () = lim Esn (Zn , ) = lim
n

sn (z, )fZn (z; 0 )dz

Zn

and we can write the limiting objective function as s (, 0 ) to emphasize the dependence on the
a.s.
parameter of the DGP. From the theorem, we know that n 0 What is the relationship between
0 and 0 ?
and may have different dimensions. Often, the statistical model (with parameter ) only
partially describes the DGP. For example, the case of OLS with errors of unknown distribution.
In some cases, the dimension of may be greater than that of . For example, fitting a
polynomial to an unknown nonlinear function.
If knowledge of 0 is sufficient for knowledge of 0 , we have a correctly and fully specified
model. 0 is referred to as the true parameter value.
If knowledge of 0 is sufficient for knowledge of some but not all elements of 0 , we have
a correctly specified semiparametric model. 0 is referred to as the true parameter value,
understanding that not all parameters of the DGP are estimated.

 If knowledge of 0 is not sufficient for knowledge of any elements of 0 , or if it causes

us to draw false conclusions regarding at least some of the elements of 0 , our model is
misspecified. 0 is referred to as the pseudo-true parameter value.
Summary
The theorem for consistency is really quite intuitive. It says that with probability one, an extremum
estimator converges to the value that maximizes the limit of the expectation of the objective function. Because the objective function may or may not make sense, depending on how good or poor
is the model, we may or may not be estimating parameters of the DGP.

12.4

Example: Consistency of Least Squares

We suppose that data is generated by random sampling of (Y, X), where yt = 0 xt +t . (X, ) has
the common distribution function FZ = x (x and are independent) with support Z = X E.
2
Suppose that the variances X
and 2 are finite. The sample objective function for a sample size n

is
sn ()

1/n

n
X

(yt xt ) = 1/n

n
X

t=1

1/n

n
X

(0 xt + t xt )

i=1
2

(xt (0 )) + 2/n

t=1

n
X

xt (0 ) t + 1/n

t=1

n
X

2t

t=1

Considering the last term, by the SLLN,

1/n

n
X

a.s.
2t

Z Z
X

t=1

2 dX dE = 2 .

Considering the second term, since E() = 0 and X and are independent, the SLLN implies
that it converges to zero.
Finally, for the first term, for a given , we assume that a SLLN applies so that
1/n

n
X

(xt (0 ))

a.s.

(x (0 )) dX

(12.1)

t=1

2

x2 dX

2

E X2

Finally, the objective function is clearly continuous, and the parameter space is assumed to be
compact, so the convergence is also uniform. Thus,
s () = 0

2

E X 2 + 2

A minimizer of this is clearly = 0 .

Exercise 30. Show that in order for the above solution to be unique it is necessary that E(X 2 ) 6= 0.
Interpret this condition.
This example shows that Theorem 29 can be used to prove strong consistency of the OLS
estimator. There are easier ways to show this, of course - this is only an example of application of
the theorem.

12.5

Example: Inconsistency of Misspecified Least Squares

You already know that the OLS estimator is inconsistent when relevant variables are omitted. Lets
verify this result in the context of extremum estimators. We suppose that data is generated by
random sampling of (Y, X), where yt = 0 xt +t . (X, ) has the common distribution function
2
FZ = x (x and are independent) with support Z = X E. Suppose that the variances X
and

2 are finite. However, the econometrician is unaware of the true DGP, and instead proposes the
misspecified model yt = 0 wt +t . Suppose that E(W ) = 0 but that E(W X) 6= 0.
The sample objective function for a sample size n is
sn ()

1/n

n
X

(yt wt ) = 1/n

t=1

1/n

n
X
t=1

n
X

(0 xt + t wt )

i=1
2

(0 xt ) + 1/n

n
X

(wt ) + 1/n

t=1

n
X

2t

+ 2/n

t=1

n
X

0 xt t 2/n

t=1

n
X

0 xt wt 2/n

t=1

Using arguments similar to above,

s () = 2 E W 2 20 E(W X) + C
So, 0 =

0 E(W X)
E(W 2 ) ,

which is the true parameter of the DGP, multiplied by the pseudo-true value of

a regression of X on W. The OLS estimator is not consistent for the true parameter, 0

12.6

Example: Linearization of a nonlinear model

Ref. Gourieroux and Monfort, section 8.3.4. White, Intnl Econ. Rev. 1980 is an earlier reference.
Suppose we have a nonlinear model
yi = h(xi , 0 ) + i
where
i iid(0, 2 )
The nonlinear least squares estimator solves
n

1X
2
n = arg min
(yi h(xi , ))
n i=1
Well study this more later, but for now it is clear that the foc for minimization will require solving
a set of nonlinear equations. A common approach to the problem seeks to avoid this difficulty by
linearizing the model. A first order Taylors series expansion about the point x0 with remainder
gives
0

yi = h(x0 , 0 ) + (xi x0 )

h(x0 , 0 )
+ i
x

where i encompasses both i and the Taylors series remainder. Note that i is no longer a classical
error - its mean is not zero. We should expect problems.
Define

= h(x0 , 0 ) x00

h(x0 , 0 )
x

h(x0 , 0 )
x

n
X
t=1

t xt wt

Given this, one might try to estimate and by applying OLS to

yi = + xi + i
Question, will
and be consistent for and ?
The answer is no, as one can see by interpreting
and as extremum estimators. Let
= (, 0 )0 .
n

= arg min sn () =

1X
2
(yi xi )
n i=1

The objective function converges to its expectation

u.a.s.

sn () s () = EX EY |X (y x)
and converges a.s. to the 0 that minimizes s ():
2

0 = arg min EX EY |X (y x)
Noting that

2
= EX EY |X h(x, 0 ) + x

2
= 2 + EX h(x, 0 ) x

EX EY |X (y x0 )

since cross products involving drop out. 0 and 0 correspond to the hyperplane that is closest to
the true regression function h(x, 0 ) according to the mean squared error criterion. This depends
on both the shape of h() and the density function of the conditioning variables.

Inconsistency of the linear approximation, even at

the approximation point
x
h(x,)
x

Tangent line

x
x

x
Fitted line

x
x

x_0

It is clear that the tangent line does not minimize MSE, since, for example, if h(x, 0 ) is
concave, all errors between the tangent line and the true function are negative.
Note that the true underlying parameter 0 is not estimated consistently, either (it may be
of a different dimension than the dimension of the parameter of the approximating model,
which is 2 in this example).

 Second order and higher-order approximations suffer from exactly the same problem, though
to a less severe degree, of course. For this reason, translog, Generalized Leontiev and other
flexible functional forms based upon second-order approximations in general suffer from
bias and inconsistency. The bias may not be too important for analysis of conditional means,
but it can be very important for analyzing first and second derivatives. In production and
consumer analysis, first and second derivatives (e.g., elasticities of substitution) are often
of interest, so in this case, one should be cautious of unthinking application of models that
impose stong restrictions on second derivatives.
This sort of linearization about a long run equilibrium is a common practice in dynamic
macroeconomic models. It is justified for the purposes of theoretical analysis of a model
given the models parameters, but it is not justifiable for the estimation of the parameters of
the model using data. The section on simulation-based methods offers a means of obtaining
consistent estimators of the parameters of dynamic macro models that are too complex for
standard methods of analysis.

12.7

Asymptotic Normality

A consistent estimator is oftentimes not very useful unless we know how fast it is likely to be converging to the true value, and the probability that it is far away from the true value. Establishment
of asymptotic normality with a known scaling factor solves these two problems. The following
theorem is similar to Amemiyas Theorem 4.1.3 (pg. 111).
Theorem 31. [Asymptotic normality of e.e.] In addition to the assumptions of Theorem 29, assume
(a) Jn () D2 sn () exists and is continuous in an open, convex neighborhood of 0 .
a.s.

(b) {Jn (n )} J (0 ), a finite negative definite matrix, for any sequence {n } that converges
almost surely to 0 .



d
(c) nDsn (0 ) 
N 0, I (0 ) , where I (0 ) = limn V ar nD sn (0 )



d
Then n 0 N 0, J (0 )1 I (0 )J (0 )1
Proof: By Taylor expansion:


D sn (n ) = D sn (0 ) + D2 sn ( ) 0
where = + (1 )0 , 0 1.
Note that will be in the neighborhood where D2 sn () exists with probability one as n
becomes large, by consistency.
Now the l.h.s. of this equation is zero, at least asymptotically, since n is a maximizer and
the f.o.c. must hold exactly since the limiting objective function is strictly concave in a
neighborhood of 0 .
a.s.

Also, since is between n and 0 , and since n 0 , assumption (b) gives

a.s.

D2 sn ( ) J (0 )
So



0 = D sn (0 ) + J (0 ) + os (1) 0
And
0=



 
nD sn (0 ) + J (0 ) + os (1) n 0

Now



d
nD sn (0 ) N 0, I (0 ) by assumption c, so



 

d
J (0 ) + os (1) n 0 N 0, I (0 )


 a.s.
Also, J (0 ) + os (1) J (0 ), so




d
n 0 N 0, J (0 )1 I (0 )J (0 )1
by the Slutsky Theorem (see Gallant, Theorem 4.6).
Skip this in lecture. A note on the order of these matrices: Supposing that sn () is representable as an average of n terms, which is the case for all estimators we consider, D2 sn () is
also an average of n matrices, the elements of which are not centered (they do not have zero
expectation). Supposing a SLLN applies, the almost sure limit of D2 sn (0 ), J (0 ) = O(1),



d
as we saw in Example 86. On the other hand, assumption (c): nD sn (0 ) N 0, I (0 )
means that

nD sn (0 ) = Op ()

where we use the result of Example 84. If we were to omit the

D sn (0 )

n, wed have

= n 2 Op (1)
 1
= Op n 2

where we use the fact that Op (nr )Op (nq ) = Op (nr+q ). The sequence D sn (0 ) is centered,

so we need to scale by n to avoid convergence to zero.

12.8

Example: Classical linear model

Lets use the results to get the asymptotic distribution of the OLS estimator applied to the classical
model, to verify that we obtain the results seen before. The OLS criterion is

sn ()

=
=
=

1
0
(y X) (y X)
n

0


1
X 0 +  X X 0 +  X
n
i

0


1h 0
X 0 X 0 20 X + 0 
n

The first derivative is

D sn () =


1
2X 0 X 0 2X 0 
n

so, evaluating at 0 ,
D sn ( 0 ) = 2

X 0
n

This has expectation 0, so the variance is the expectation of the outer product:

V ar nD sn ( )

"
= E
=
=

X 0
n2
n

X 0 0 X
n
0
2X X
4
n
E4


#
X 0 0
n2
n

Therefore
I ( 0 )

=
=

lim V ar nD sn ( 0 )

n
42 QX

The second derivative is

Jn () = D2 sn ( 0 ) =

1
[2X 0 X] .
n

A SLLN tells us that this converges almost surely to the limit of its expectation:
J ( 0 ) = 2QX
Theres no parameter in that last expression, so uniformity is not an issue.
The asymptotic normality theorem (31) tells us that


n 0



d
N 0, J ( 0 )1 I ( 0 )J ( 0 )1

which is, given the above,

  1 




Q1

QX
d
2
0
X

4 QX
n N 0,
2
2
or





d
2
n 0 N 0, Q1
X  .

This is the same thing we saw in equation 4.1, of course. So, the theory seems to work :-)

12.9

Exercises

1. Suppose that xi uniform(0,1), and yi = 1 x2i + i , where i is iid(0, 2 ). Suppose we

estimate the misspecified model yi = + xi + i by OLS. Find the numeric values of 0 and
0 that are the probability limits of
and
2. Verify your results using Octave by generating data that follows the above model, and calculating the OLS estimator. When the sample size is very large the estimator should be very
close to the analytical results you obtained in question 1.
3. Use the asymptotic normality theorem to find the asymptotic distribution of the ML estimator
of 0 for the model y = x 0 + , where N (0, 1) and is independent of x. This means
sn () 
2
0
0
finding
0 sn (), J ( ),
 , and I( ). The expressions may involve the unspecified
density of x.

Chapter 13

Maximum likelihood estimation

The maximum likelihood estimator is important because it uses all of the information in a fully
specified statistical model. Its use of all of the information causes it to have a number of attractive
properties, foremost of which is asymptotic efficiency. For this reason, the ML estimator can serve
as a benchmark against which other estimators may be measured. The ML estimator requires that
the statistical model be fully specified, which essentially means that there is enough information
to draw data from the DGP, given the parameter. This is a fairly strong requirement, and for this
reason we need to be concerned about the possible misspecification of the statistical model. If this
is the case, the ML estimator will not have the nice properties that it has under correct specification.

13.1

The likelihood function

Suppose
of size
vectors y and z. Suppose the joint density of
 we have a sample

 n of the random

Y = y1 . . . yn and Z = z1 . . . zn is characterized by a parameter vector 0 :
fY Z (Y, Z, 0 ).
This is the joint density of the sample. This density can be factored as
fY Z (Y, Z, 0 ) = fY |Z (Y |Z, 0 )fZ (Z, 0 )
The likelihood function is just this density evaluated at other values
L(Y, Z, ) = f (Y, Z, ), ,
where is a parameter space.
The maximum likelihood estimator of 0 is the value of that maximizes the likelihood function.
Note that if 0 and 0 share no elements, then the maximizer of the conditional likelihood
function fY |Z (Y |Z, ) with respect to is the same as the maximizer of the overall likelihood
function fY Z (Y, Z, ) = fY |Z (Y |Z, )fZ (Z, ), for the elements of that correspond to . In this
case, the variables Z are said to be exogenous for estimation of , and we may more conveniently
work with the conditional likelihood function fY |Z (Y |Z, ) for the purposes of estimating 0 .
The maximum likelihood estimator of 0 = arg max fY |Z (Y |Z, )
If the n observations are independent, the likelihood function can be written as
L(Y |Z, ) =

n
Y
t=1

177

f (yt |zt , )

where the ft are possibly of different form.

If this is not possible, we can always factor the likelihood into contributions of observations,
by using the fact that a joint density can be factored into the product of a marginal and
conditional (doing this iteratively)
L(Y, ) = f (y1 |z1 , )f (y2 |y1 , z2 , )f (y3 |y1 , y2 , z3 , ) f (yn |y1, y2 , . . . ytn , zn , )
To simplify notation, define
xt

= {y1 , y2 , ..., yt1 , zt }

so x1 = z1 , x2 = {y1 , z2 }, etc. - it contains exogenous and predetermined endogeous variables.

Now the likelihood function can be written as
L(Y, ) =

n
Y

f (yt |xt , )

t=1

The criterion function can be defined as the average log-likelihood function:

n

sn () =

1
1X
ln L(Y, ) =
ln f (yt |xt , )
n
n t=1

The maximum likelihood estimator may thus be defined equivalently as

= arg max sn (),
where the set maximized over is defined below. Since ln() is a monotonic increasing function, ln L

and L maximize at the same value of . Dividing by n has no effect on .

Example 32. Example: Bernoulli trial
Suppose that we are flipping a coin that may be biased, so that the probability of a heads may
not be 0.5. Maybe were interested in estimating the probability of a heads. Let Y = 1(heads) be
a binary variable that indicates whether or not a heads is observed. The outcome of a toss is a
Bernoulli random variable:
fY (y, p0 )

1y

= py0 (1 p0 )
=

, y {0, 1}

0, y
/ {0, 1}

So a representative term that enters the likelihood function is

1y

fY (y, p) = py (1 p)
and

ln fY (y, p) = y ln p + (1 y) ln (1 p)
The derivative of this is
ln fY (y, p)
p

=
=

y (1 y)

p (1 p)
yp
p (1 p)

Averaging this over a sample of size n gives

n

sn (p)
1 X yi p
=
p
n i=1 p (1 p)
Setting to zero and solving gives
(13.1)

p = y

So its easy to calculate the MLE of p0 in this case. For future reference, note that E(Y ) =
PY =1 y
1y
2
= p0 and V ar(Y ) = E(Y 2 ) [E(Y )] = p0 p20 .
Y =0 yp0 (1 p0 )
Now imagine that we had a bag full of bent coins, each bent around a sphere of a different
radius (with the head pointing to the outside of the sphere). We might suspect that the probability
of a heads could depend upon the radius. Suppose that pi p(xi , ) = (1 + exp(x0i ))
h
i0
xi = 1 ri , so that is a 21 vector. Now

where

pi ()
= pi (1 pi ) xi

so
ln fY (y, )

y pi
pi (1 pi ) xi
pi (1 pi )
(yi p(xi , )) xi

=
=

So the derivative of the average log lihelihood function is now

sn ()
=

Pn

i=1

(yi p(xi , )) xi
n

This is a set of 2 nonlinear equations in the two unknown elements in . There is no explicit
solution for the two elements that set the equations to zero. This is commonly the case with
ML estimators: they are often nonlinear, and finding the value of the estimate often requires
use of numeric methods to find solutions to the first order conditions. See Chapter 11 for more
information on how to do this.
Example 33. Example: Likelihood function of classical linear regression model
The classical linear regression model with normality is outlined in Section 3.6. The likelihood
function for this model is presented in Section 4.3. A Octave/Matlab example that shows how to
compute the maximum likelihood estimator for data that follows the CLRM with normality is in
NormalExample.m , which makes use of NormalLF.m .

13.2

Consistency of MLE

The MLE is an extremum estimator, given basic assumptions it is consistent for the value that
maximizes the limiting objective function, following Theorem 29. The question is: what is the
value that maximizes s ()?
Remember that sn () =

1
n

ln L(Y, ), and L(Y, 0 ) is the true density of the sample data. For

any 6= 0
 

 

L()
L()
E ln
ln E
L(0 )
L(0 )
by Jensens inequality ( ln () is a concave function).

Now, the expectation on the RHS is


E

L()
L(0 )

Z
=

L()
L(0 )dy = 1,
L(0 )

since L(0 ) is the density function of the observations, and since the integral of any density is 1.
Therefore, since ln(1) = 0,
 

L()
E ln
0,
L(0 )
or
E (sn ()) E (sn (0 )) 0.
a.s.

A SLLN tells us that sn () s (, 0 ) = lim E (sn ()), and with continuity and a compact
parameter space, this is uniform, so
s (, 0 ) s (0 , 0 ) 0
except on a set of zero probability. Note: the 0 appears because the expectation is taken with
respect to the true density L(0 ).
By the identification assumption there is a unique maximizer, so the inequality is strict if 6= 0 :
s (, 0 ) s (0 , 0 ) < 0, 6= 0 , a.s.
Therefore, 0 is the unique maximizer of s (, 0 ), and thus, Theorem 29 tells us that
lim = 0 , a.s.

So, the ML estimator is consistent for the true parameter value.

13.3

The score function

Assumption: (Differentiability) Assume that sn () is twice continuously differentiable in a

neighborhood N (0 ) of 0 , at least when n is large enough.
To maximize the log-likelihood function, take derivatives:
gn (Y, )

= D sn ()
n
1X
D ln f (yt |xx , )
=
n t=1
n

1X
gt ().
n t=1

This is the score vector (with dim K 1). Note that the score function has Y as an argument, which
implies that it is a random function. Y (and any exogeneous variables) will often be suppressed
for clarity, but one should not forget that they are still there.
The ML estimator sets the derivatives to zero:
n

X
= 1
0.
gn ()
gt ()
n t=1
We will show that E [gt ()] = 0, t. This is the expectation taken with respect to the density f (),

not necessarily f (0 ) .
Z
E [gt ()]

[D ln f (yt |xt , )]f (yt |x, )dyt

=
Z
=

1
[D f (yt |xt , )] f (yt |xt , )dyt
f (yt |xt , )

Z
=

D f (yt |xt , )dyt .

Given some regularity conditions on boundedness of D f, we can switch the order of integration
and differentiation, by the dominated convergence theorem. This gives
Z
E [gt ()]

f (yt |xt , )dyt

= D
= D 1
=

where we use the fact that the integral of the density is 1.

So E (gt () = 0 : the expectation of the score vector is zero.
This hold for all t, so it implies that E gn (Y, ) = 0.

13.4

Asymptotic normality of MLE

Recall that we assume that the log-likelihood function sn () is twice continuously differentiable.
about the true value 0 :
Take a first order Taylors series expansion of g(Y, )
0



= g(0 ) + (D0 g( )) 0
g()

or with appropriate definitions



J ( ) 0 = g(0 ),
where = + (1 )0 , 0 < < 1. Assume J ( ) is invertible (well justify this in a minute).
So


n 0 = J ( )1 ng(0 )
Now consider J ( ), the matrix of second derivatives of the average log likelihood function.

This is
J ( )

D0 g( )

D2 sn ( )
n
1X 2
D ln ft ( )
n t=1

=
where the notation

D2 sn ()

2 sn ()
.
0

Given that this is an average of terms, it should usually be the case that this satisfies a strong law
of large numbers (SLLN). Regularity conditions are a set of assumptions that guarantee that this
will happen. There are different sets of assumptions that can be used to justify appeal to different
SLLNs. For example, the D2 ln ft ( ) must not be too strongly dependent over time, and their
variances must not become infinite. We dont assume any particular set here, since the appropriate

assumptions will depend upon the particularities of a given model. However, we assume that a
SLLN applies.
a.s.
Also, since we know that is consistent, and since = + (1 )0 , we have that 0 .

Also, by the above differentiability assumption, J () is continuous in . Given this, J ( ) converges to the limit of its expectation:


a.s.
J ( ) lim E D2 sn (0 ) = J (0 ) <
n

This matrix converges to a finite limit.

Re-arranging orders of limits and differentiation, which is legitimate given certain regularity
conditions related to the boundedness of the log-likelihood function, we get
= D2 lim E (sn (0 ))

J (0 )

= D2 s (0 , 0 )
Weve already seen that
s (, 0 ) < s (0 , 0 )
i.e., 0 maximizes the limiting objective function. Since there is a unique maximizer, and by the
assumption that sn () is twice continuously differentiable (which holds in the limit), then J (0 )
must be negative definite, and therefore of full rank. Therefore the previous inversion is justified,
asymptotically, and we have



n 0 = J ( )1 ng(0 ).
Now consider

(13.2)

ng(0 ). This is

ngn (0 )

=
=

nD sn ()
X
n
n
D ln ft (yt |xt , 0 )
n t=1
n

1 X

gt (0 )
n t=1

Weve already seen that E [gt ()] = 0. As such, it is reasonable to assume that a CLT applies.
a.s.

Note that gn (0 ) 0, by consistency. To avoid this collapse to a degenerate r.v. (a constant

vector) we need to scale by n. A generic CLT states that, for Xn a random vector that satisfies
certain conditions,
d

Xn E(Xn ) N (0, lim V (Xn ))

The certain conditions that Xn must satisfy depend on the case at hand. Usually, Xn will be of

the form of an average, scaled by n:

Xn =
This is the case for

Pn
n

t=1

Xt

ng(0 ) for example. Then the properties of Xn depend on the properties of

the Xt . For example, if the Xt have finite variances and are not too strongly dependent, then a CLT

for dependent processes will apply. Supposing that a CLT applies, and noting that E( ngn (0 ) = 0,
we get

ngn (0 ) N [0, I (0 )]

(13.3)

where
I (0 )

0

lim E0 n [gn (0 )] [gn (0 )]

n

= lim V0
ngn (0 )
=

This can also be written as

I (0 ) is known as the information matrix.
a.s.

Combining [13.2] and [13.3], and noting that J ( ) J (0 ), we get





a
n 0 N 0, J (0 )1 I (0 )J (0 )1 .
The MLE estimator is asymptotically normally distributed.
Definition 34. Consistent and asymptotically normal (CAN).An estimator
of a parameter 0 is


d
n-consistent and asymptotically normally distributed if n 0 N (0, V ) where V is a
finite positive definite matrix.
 p

n 0 0. These are

known as superconsistent estimators, since in ordinary circumstances with stationary data, n is

There do exist, in special cases, estimators that are consistent such that

the highest factor that we can multiply by and still get convergence to a stable limiting distribution.
Definition 35. Asymptotically unbiased. An estimator of a parameter 0 is asymptotically unbiased if
= .
limn E ()
Estimators that are CAN are asymptotically unbiased, though not all consistent estimators are
asymptotically unbiased. Such cases are unusual, though.

13.5

The information matrix equality

We will show that J () = I (). Let ft () be short for f (yt |xt , )

Z
1

ft ()dy, so

=
Z

D ft ()dy
Z

(D ln ft ()) ft ()dy

Now differentiate again:

Z
0

Z
 2

D ln ft () ft ()dy + [D ln ft ()] D0 ft ()dy
Z
 2

E D ln ft () + [D ln ft ()] [D0 ln ft ()] ft ()dy


E D2 ln ft () + E [D ln ft ()] [D0 ln ft ()]

E [Jt ()] + E [gt ()] [gt ()]

=
=

Now sum over n and multiply by

(13.4)

1
n

" n
#
n
1X
1X
0
E
[Jt ()] = E
[gt ()] [gt ()]
n t=1
n t=1

(13.5)

The scores gt and gs are uncorrelated for t 6= s, since for t > s, ft (yt |y1 , ..., yt1 , ) has conditioned
on prior information, so what was random in s is fixed in t. (This forms the basis for a specification
test proposed by White: if the scores appear to be correlated one may question the specification of
the model). This allows us to write
0

E [Jn ()] = E n [g()] [g()]

since all cross products between different periods expect to zero. Finally take limits, we get
J () = I ().

(13.6)

This holds for all , in particular, for 0 . Using this,





a.s.
n 0 N 0, J (0 )1 I (0 )J (0 )1
simplifies to

or





a.s.
n 0 N 0, I (0 )1

(13.7)





a.s.
n 0 N 0, J (0 )1

(13.8)

To estimate the asymptotic variance, we need estimators of J (0 ) and I (0 ). We can use

n

1X
0
gt ()gt ()
n t=1

I\
(0 )

J\
(0 )

= Jn ().

as is intuitive if one considers equation 13.5. Note, one cant use

h
ih
i0
gn ()

I\
(0 ) = n gn ()
to estimate the information matrix. Why not?
From this we see that there are alternative ways to estimate V (0 ) that are all valid. These
include
V\
(0 )

J\
(0 )

V\
(0 )

I\
(0 )

V\
(0 )

J\
(0 )

1
1

\
I\
(0 )J (0 )

These are known as the inverse Hessian, outer product of the gradient (OPG) and sandwich estimators, respectively. The sandwich form is the most robust, since it coincides with the covariance
estimator of the quasi-ML estimator.

Example, Coin flipping, again

In section 32 we saw that the MLE for the parameter of a Bernoulli trial, with i.i.d. data, is the

p p0 ). We can
sample mean: p = y (equation 13.1). Now lets find the limiting variance of n (

do this in a simple way:

lim V ar n (
p p0 )

lim nV ar (
p p0 )

lim nV ar (
p)

lim nV ar (
y)
P 
yt
= lim nV ar
n
1X
= lim
V ar(yt ) (by independence of obs.)
n
1
= lim nV ar(y) (by identically distributed obs.)
n
= V ar(y)
=

p0 (1 p0 )

While that is simple, lets verify this using the methods of Chapter 12 give the same answer. The
log-likelihood function is
n

sn (p)

1X
{yt ln p + (1 yt ) ln (1 p)}
n t=1

so


Esn (p) = p0 ln p + 1 p0 ln (1 p)


by the fact that the observations are i.i.d. Thus, s (p) = p0 ln p + 1 p0 ln (1 p). A bit of
calculation shows that

D2 sn (p)p=p0 Jn () =

p0

1
,
(1 p0 )

1 0
(p ).
which doesnt depend upon n. By results weve seen on MLE, lim V ar n p p0 = J


0
0
1 0
And in this case, J (p ) = p 1 p . So, we get the same limiting variance using both
methods.

13.6

The Cramr-Rao lower bound

Theorem 36. [Cramer-Rao Lower Bound] The limiting variance of a CAN estimator of 0 , say ,
minus the inverse of the information matrix is a positive semidefinite matrix.
Proof: Since the estimator is CAN, it is asymptotically unbiased, so
lim E ( ) = 0

Differentiate wrt 0 :
D0 lim E ( )
n

Z
=
=

lim

h

i
D0 f (Y, ) dy

0 (this is a K K matrix of zeros).

Noting that D0 f (Y, ) = f ()D0 ln f (), we can write

lim

Z 

Z




0
f ()D ln f ()dy + lim
f (Y, )D0 dy = 0.
n



R
Now note that D0 = IK , and f (Y, )(IK )dy = IK . With this we have
lim

Z 


f ()D0 ln f ()dy = IK .

Playing with powers of n we get

Z
lim

 1

n
n [D0 ln f ()] f ()dy
{z
}
|n

IK

Note that the bracketed part is just the transpose of the score vector, g(), so we can write
lim E

i
h 

n
ng()0 = IK



This means that the covariance of the score function with n , for any CAN estimator, is


Therefore,
an identity matrix. Using this, suppose the variance of n tends to V ().
 # "
#
" 

n
V ()
IK
=
.
V

IK
I ()
ng()

(13.9)

Since this is a covariance matrix, it is positive semi-definite. Therefore, for any K -vector ,
h

1
()
0 I

"

V ()

IK

IK

I ()

#"

I ()1

0.

This simplifies to
h
i
I 1 () 0.
0 V ()

I 1 () is positive semidefinite. This conludes the proof.

Since is arbitrary, V ()

1
() is a lower bound for the asymptotic variance of a CAN estimator.
This means that I
is asymptot(Asymptotic efficiency) Given two CAN estimators of a parameter 0 , say and ,

V ()
is a positive semidefinite matrix.
ically efficient with respect to if V ()
A direct proof of asymptotic efficiency of an estimator is infeasible, but if one can show that
the asymptotic variance is equal to the inverse of the information matrix, then the estimator is
asymptotically efficient. In particular, the MLE is asymptotically efficient with respect to any other
CAN estimator.

13.7

Likelihood ratio-type tests

Suppose we would like to test a set of q possibly nonlinear restrictions r() = 0, where the q k
matrix D0 r() has rank q. The Wald test can be calculated using the unrestricted model. The score
test can be calculated using only the restricted model. The likelihood ratio test, on the other hand,
uses both the restricted and the unrestricted estimators. The test statistic is


ln L()

LR = 2 ln L()
where is the unrestricted estimate and is the restricted estimate. To show that it is asymptoti about :
cally 2 , take a second order Taylors series expansion of ln L()
' ln L()
+
ln L()



n  0

J ()
2

 0 by the fonc and we need to multiply the

(note, the first order term drops out since D ln L()
second-order term by n since J () is defined in terms of

1
n

ln L()) so


0



LR ' n J ()
J (0 ) = I(0 ), by the information matrix equality. So
As n , J ()

0


a
LR = n I (0 )

(13.10)

We also have that, from the theory on the asymptotic normality of the MLE and the information
matrix equality



a
n 0 = I (0 )1 n1/2 g(0 ).

An analogous result for the restricted estimator is (this is unproven here, to prove this set up the
Lagrangean for MLE subject to R = r, and manipulate the first order conditions) :




1

a
n 0 = I (0 )1 In R0 RI (0 )1 R0
RI (0 )1 n1/2 g(0 ).
Combining the last two equations


1

a
n = n1/2 I (0 )1 R0 RI (0 )1 R0
RI (0 )1 g(0 )
so, substituting into [13.10]
h
i
i
1 h
a
LR = n1/2 g(0 )0 I (0 )1 R0 RI (0 )1 R0
RI (0 )1 n1/2 g(0 )
But since
d

n1/2 g(0 ) N (0, I (0 ))

the linear function
d

RI (0 )1 n1/2 g(0 ) N (0, RI (0 )1 R0 ).

We can see that LR is a quadratic form of this rv, with the inverse of its variance in the middle, so
d

LR 2 (q).

Summary of MLE
Consistent
Asymptotically normal (CAN)
Asymptotically efficient
Asymptotically unbiased
LR test is available for testing hypothesis
The presentation is for general MLE: we havent specified the distribution or the linearity/nonlinearity of the estimator

13.8

Example: ML of Nerlove model, assuming normality

As we saw in Section 4.3, the ML and OLS estimators of in the linear model y = X +  coincide
when  is assumed to be i.i.d. normally distributed. The Octave script NerloveMLE.m verifies this
result, for the basic Nerlove model (eqn. 3.10). The output of the script follows:

******************************************************
check MLE with normality, compare to OLS
MLE Estimation Results
BFGS convergence: Normal convergence
Average Log-L: -0.465806
Observations: 145
estimate
constant
output
labor
fuel
capital
sig

st. err
-3.527
0.720
0.436
0.427
-0.220
0.386

t-stat
1.689
0.032
0.241
0.074
0.318
0.041

p-value
-2.088
22.491
1.808
5.751
-0.691
9.290

0.037
0.000
0.071
0.000
0.490
0.000

Information Criteria
CAIC : 170.9442
Avg. CAIC:
1.1789
BIC : 164.9442
Avg. BIC: 1.1375
AIC : 147.0838
Avg. AIC: 1.0144
******************************************************
Compare the output to that of Nerlove.m , which does OLS. The script also provides a basic example
of how to use the MLE estimation routing mle_results.m

13.9

Example: Binary response models

This section extends the Bernoulli trial model to binary response models with conditioning variables, as such models arise in a variety of contexts.
Assume that
y

= x0

1(y > 0)

N (0, 1)
Here, y is an unobserved (latent) continuous variable, and y is a binary variable that indicates
whether y is negative or positive. Then the probit model results, where P r(y = 1|x) = P r( <
x0 ) = (x0 ), where
Z

() =

is the standard normal distribution function.

(2)1/2 exp(

2
)d
2

The logit model results if the errors  are not normal, but rather have a logistic distribution.
This distribution is similar to the standard normal, but has fatter tails. The probability has the
following parameterization
P r(y = 1|x) = (x0 ) = (1 + exp(x0 ))

In general, a binary response model will require that the choice probability be parameterized
in some form which could be logit, probit, or something else. For a vector of explanatory variables
x, the response probability will be parameterized in some manner
P r(y = 1|x) = p(x, )
Again, if p(x, ) = (x0 ), we have a logit model. If p(x, ) = (x0 ), where () is the standard
normal distribution function, then we have a probit model.
Regardless of the parameterization, we are dealing with a Bernoulli density,
fYi (yi |xi ) = p(xi , )yi (1 p(x, ))1yi
is the
so as long as the observations are independent, the maximum likelihood (ML) estimator, ,
maximizer of
n

sn ()

1X
(yi ln p(xi , ) + (1 yi ) ln [1 p(xi , )])
n i=1

1X
s(yi , xi , ).
n i=1

(13.11)

Following the above theoretical results, tends in probability to the 0 that maximizes the uniform
almost sure limit of sn (). Noting that Eyi = p(xi , 0 ), and following a SLLN for i.i.d. processes,
sn () converges almost surely to the expectation of a representative term s(y, x, ). First one can
take the expectation conditional on x to get
Ey|x {y ln p(x, ) + (1 y) ln [1 p(x, )]} = p(x, 0 ) ln p(x, ) + [1 p(x, 0 )] ln [1 p(x, )] .
Next taking expectation over x we get the limiting objective function
Z
{p(x, 0 ) ln p(x, ) + [1 p(x, 0 )] ln [1 p(x, )]} (x)dx,

s () =

(13.12)

where (x) is the (joint - the integral is understood to be multiple, and X is the support of x)
density function of the explanatory variables x. This is clearly continuous in , as long as p(x, ) is
continuous, and if the parameter space is compact we therefore have uniform almost sure convergence. Note that p(x, ) is continous for the logit and probit models, for example. The maximizing
element of s (), , solves the first order conditions
Z 
X


p(x, 0 )
1 p(x, 0 )

p(x, )
p(x, ) (x)dx = 0
p(x, )
1 p(x, )

This is clearly solved by = 0 . Provided the solution is unique, is consistent. Question: whats
needed to ensure that the solution is unique?
The asymptotic normality theorem tells us that




d
n 0 N 0, J (0 )1 I (0 )J (0 )1 .


In the case of i.i.d. observations I (0 ) = limn V ar nD sn (0 ) is simply the expectation of a
typical element of the outer product of the gradient.
Theres no need to subtract the mean, since its zero, following the f.o.c. in the consistency
proof above and the fact that observations are i.i.d.
The terms in n also drop out by the same argument:

lim V ar nD sn (0 )

1X
lim V ar nD
s(0 )
n
n t
X
1
= lim V ar D
s(0 )
n
n
t
X
1
D s(0 )
= lim V ar
n n
t
=

=
=

lim V arD s(0 )

V arD s(0 )

So we get

I (0 ) = E

s(y, x, 0 ) 0 s(y, x, 0 ) .

Likewise,
J (0 ) = E

2
s(y, x, 0 ).
0

Expectations are jointly over y and x, or equivalently, first over y conditional on x, then over x.
From above, a typical element of the objective function is
s(y, x, 0 ) = y ln p(x, 0 ) + (1 y) ln [1 p(x, 0 )] .
Now suppose that we are dealing with a correctly specified logit model:
p(x, ) = (1 + exp(x0 ))

We can simplify the above results in this case. We have that

p(x, )

(1 + exp(x0 ))

(1 + exp(x0 ))

=
=

exp(x0 )x

exp(x0 )
x
1 + exp(x0 )
p(x, ) (1 p(x, )) x


p(x, ) p(x, )2 x.
1

So

s(y, x, 0 ) =

2
s(0 ) =
0

[y p(x, 0 )] x

(13.13)



p(x, 0 ) p(x, 0 )2 xx0 .

Taking expectations over y then x gives

Z
I (0 )

=
Z
=



EY y 2 2p(x, 0 )p(x, 0 ) + p(x, 0 )2 xx0 (x)dx

(13.14)



p(x, 0 ) p(x, 0 )2 xx0 (x)dx.

(13.15)

where we use the fact that EY (y) = EY (y 2 ) = p(x, 0 ). Likewise,

Z
J (0 ) =



p(x, 0 ) p(x, 0 )2 xx0 (x)dx.

(13.16)

Note that we arrive at the expected result: the information matrix equality holds (that is, J (0 ) =
I (0 )). With this,




d
n 0 N 0, J (0 )1 I (0 )J (0 )1
simplifies to





d
n 0 N 0, J (0 )1

which can also be expressed as





d
n 0 N 0, I (0 )1 .
On a final note, the logit and standard normal CDFs are very similar - the logit distribution
is a bit more fat-tailed. While coefficients will vary slightly between the two models, functions of
will be virtually identical for the two models.
interest such as estimated probabilities p(x, )

13.10

Examples

For examples of MLE using logit and Poisson model applied to data, see Section 11.4 in the chapter
on Numerical Optimization. You should examine the scripts and run them to see how MLE is
actually done.

Estimation of a simple DSGE model

Dynamic stochastic general equilibrium model are widely used tools in macroeconomics. These
are models in which current decisions depend upon expectations of future events. An example is
the simple real business cycle model discussed in the file rbc.pdf, by Fernndez-Villaverde, which
is available on the Dynare web page www.dynare.org. The file EstimateRBC_ML.mod shows how
this model may be estimated, using maximum likelihood methods. The estimation process involves
forming a linear approximation to the true model, which means that the estimator is not actually
the true maximum likelihood estimator, it is actually a quasi-ML estimator. The quasi-likelihood
is computed by putting the linearized model in state-space form, and then computing the likelihood iteratively using Kalman filtering. State space models and Kalman filtering are introduced in
Section 15.2. Once the likelihood function is available, the methods studied in this Chapter may
be applied. The intention at the moment is simple to show that ML is an estimation method that
may be applied to complicated and more or less realistic economic models.

13.11

Exercises

1. Consider coin tossing with a single possibly biased coin. The density function for the random
variable y = 1(heads) is
fY (y, p0 )

1y

= py0 (1 p0 )
=

0, y
/ {0, 1}

, y {0, 1}

Suppose that we have a sample of size n. We know from above that the ML estimator is
pb0 = y. We also know from the theory above that



a
n (
y p0 ) N 0, J (p0 )1 I (p0 )J (p0 )1

a) find the analytic expression for gt () and show that E [gt ()] = 0
b) find the analytical expressions for J (p0 ) and I (p0 ) for this problem

c) verify that the result for lim V ar n (

p p) found in section 13.5 is equal to J (p0 )1 I (p0 )J (p0 )1

d) Write an Octave program that does a Monte Carlo study that shows that n (
y p0 ) is
approximately normally distributed when n is large. Please give me histograms that show

the sampling frequency of n (

y p0 ) for several values of n.
2. The exponential density is
(
fX (x) =

ex , x > 0
0, x < 0

Suppose we have an independently and identically distributed sample of size n, {xi } , i =

1, 2, ..., n, where each xi follows this exponential distribution.
(a) write the log likelihood function
(b) compute the maximum likelihood estimator of the parameter .
3. Consider the model yt = x0t + t where the errors follow the Cauchy (Student-t with 1
degree of freedom) density. So
f (t ) =

1
, < t <
(1 + 2t )

The Cauchy density has a shape similar to a normal density, but with much thicker tails.
Thus, extremely small and large errors occur much more frequently with this density than
would happen if the errors were normally distributed. Find the score function gn () where

0
= 0 .
4. Consider the model classical linear regression model yt = x0t + t where t IIN (0, 2 ).

0
Find the score function gn () where = 0 .
5. Compare the first order conditions that define the ML estimators of problems 2 and 3 and
interpret the differences. Why are the first order conditions that define an efficient estimator
different in the two cases?
6. Assume a d.g.p. follows the logit model: Pr(y = 1|x) = 1 + exp( 0 x)

1

(a) Assume that x uniform(-a,a). Find the asymptotic distribution of the ML estimator of
0 (this is a scalar parameter).
(b) Now assume that x uniform(-2a,2a). Again find the asymptotic distribution of the
ML estimator of 0 .
(c) Comment on the results
7. There is an ML estimation routine in the provided software that accompanies these notes.
Edit (to see what it does) then run the script mle_example.m. Interpret the output.
8. Estimate the simple Nerlove model discussed in section 3.8 by ML, assuming that the errors
are i.i.d. N (0, 2 ) and compare to the results you get from running Nerlove.m .

9. Using logit.m and EstimateLogit.m as templates, write a function to calculate the probit log
likelihood, and a script to estimate a probit model. Run it using data that actually follows a
logit model (you can generate it in the same way that is done in the logit example).
10. Study mle_results.m to see what it does. Examine the functions that mle_results.m calls,
and in turn the functions that those functions call. Write a complete description of how the
whole chain works.
11. In Subsection 11.4 a model is presented for data on health care usage, along with some
Octave scripts. Look at the Poisson estimation results for the OBDV measure of health care
use and give an economic interpretation. Estimate Poisson models for the other 5 measures
of health care usage, using the provided scripts.

Chapter 14

Generalized method of moments

Readings: Hamilton Ch. 14 ; Davidson and MacKinnon, Ch. 17 (see pg. 587 for refs. to applications); Newey and McFadden (1994), "Large Sample Estimation and Hypothesis Testing", in
Handbook of Econometrics, Vol. 4, Ch. 36.

14.1

Motivation

Sampling from 2 (0 )
Example 37. (Method of moments, v1) Suppose we draw a random sample of yt from the 2 (0 )
distribution. Here, 0 is the parameter of interest. The first moment (expectation), 1 , of a random
variable will in general be a function of the parameters of the distribution:1 = 1 (0 ) .
In this example, if Y 2 (0 ), then E(Y ) = 0 , so the relationship is the identity function
1 (0 ) = 0 , though in general the relationship may be more complicated. The sample first moment
is

c1 =

n
X

yt /n.

t=1

Define the single observation contribution to the moment condition as the true moment minus
the tth observations contribution to the sample moment:
m1t () = 1 () yt
The corresponding average moment condition is
m1 () = 1 ()
c1
where the sample moment
c1 = y =

Pn

t=1

yt /n.

The method of moments principle is to choose the estimator of the parameter to set the estimate
0. Then the equation is solved
of the population moment equal to the sample moment, i.e., m1 ()
for the estimator. In this case,
=
m1 ()

n
X

yt /n = 0

t=1

is solved by = y. Since y =

Pn

t=1

yt /n 0 by the LLN, the estimator is consistent.

Example 38. (Method of moments, v2) The variance of a 2 (0 ) r.v. is

V (yt ) = E yt 0
194

2

= 20 .

The sample variance is V (yt ) =

Pn

y )2
t=1 (yt
n

. Define the average moment condition as the

population moment minus the sample moment:

m2 () = V (yt ) V (yt )
Pn
2
(yt y)
= 2 t=1
n
We can see that the average moment condition is the average of the contributions
2

m2t () = V (yt ) (yt y)

The MM estimator using the variance would set
= 2
m2 ()

Pn

t=1

(yt y)
0.
n

Again, by the LLN, the sample variance is consistent for the true variance, that is,
Pn

t=1

(yt y) p
20 .
n

So, the estimator is half the sample variance:

1
=
2

Pn

t=1

(yt y)
,
n

This estimator is also consistent for 0 .

Example 39. Try some MM estimation yourself: heres an Octave script that implements the two
MM estimators discussed above: GMM/chi2mm.m
Note that when you run the script, the two estimators give different results. Each of the two
estimators is consistent.
With two moment-parameter equations and only one parameter, we have overidentification,
which means that we have more information than is strictly necessary for consistent estimation of the parameter.
The idea behind GMM is to combine information from the two moment-parameter equations
to form a new estimator which will be more efficient, in general (proof of this below).

Sampling from t(0 )

Heres another example based upon the t-distribution. The density function of a t-distributed r.v.
Yt is




 0
0 + 1 /2 
2 0 ( +1)/2
0
1
+
y
/
fYt (yt , ) =
t
1/2
(0 ) (0 /2)
Given an iid sample of size n, one could estimate 0 by maximizing the log-likelihood function
arg max ln Ln () =

n
X

ln fYt (yt , )

t=1

This approach is attractive since ML estimators are asymptotically efficient. This is because
the ML estimator uses all of the available information (e.g., the distribution is fully specified
up to a parameter). Recalling that a distribution is completely characterized by its moments,

the ML estimator is interpretable as a GMM estimator that uses all of the moments. The
method of moments estimator uses only K moments to estimate a Kdimensional parameter. Since information is discarded, in general, by the MM estimator, efficiency is lost relative
to the ML estimator.
Example 40. (Method of moments). A t-distributed r.v. with density fYt (yt , 0 ) has mean zero and


variance V (yt ) = 0 / 0 2 (for 0 > 2).
Using the notation introduced previously, define a moment contribution m1t () = / ( 2)yt2
Pn
Pn
and the average moment condition m1 () = 1/n t=1 m1t () = / ( 2) 1/n t=1 yt2 . As




before, when evaluated at the true parameter value 0 , both E0 m1t (0 ) = 0 and E0 m1 (0 ) =
0.
0 yields a MM estimator:
Choosing to set m1 ()
=

2
1

(14.1)

Pn 2
i yi

This estimator is based on only one moment of the distribution - it uses less information than the
ML estimator, so it is intuitively clear that the MM estimator will be inefficient relative to the ML
estimator.
Example 41. (Method of moments). An alternative MM estimator could be based upon the fourth
moment of the t-distribution. The fourth moment of a t-distributed r.v. is
4

E(yt4 )

2
3 0
= 0
,
( 2) (0 4)

provided that 0 > 4. We can define a second moment condition

2

3 ()
1X 4
m2 () =
y

( 2) ( 4) n t=1 t
0. If you solve this youll see that
A second, different MM estimator chooses to set m2 ()
the estimate is different from that in equation 14.1.
This estimator isnt efficient either, since it uses only one moment. A GMM estimator would
use the two moment conditions together to estimate the single parameter. The GMM estimator
is overidentified, which leads to an estimator which is efficient relative to the just identified MM
estimators (more on efficiency later).

14.2

Definition of GMM estimator

For the purposes of this course, the following definition of the GMM estimator is sufficiently general:
Definition 42. The GMM estimator of the K -dimensional parameter vector 0 , arg min sn ()
Pn
mn ()0 Wn mn (), where mn () = n1 t=1 mt () is a g-vector, g K, with E m() = 0, and Wn
converges almost surely to a finite g g symmetric positive definite matrix W .
Whats the reason for using GMM if MLE is asymptotically efficient?
Robustness: GMM is based upon a limited set of moment conditions. For consistency, only
these moment conditions need to be correctly specified, whereas MLE in effect requires correct specification of every conceivable moment condition. GMM is robust with respect to distributional misspecification. The price for robustness is loss of efficiency with respect to the MLE

estimator. Keep in mind that the true distribution is not known so if we erroneously specify a
distribution and estimate by MLE, the estimator will be inconsistent in general (not always).
Feasibility: in some cases the MLE estimator is not available, because we are not able to
deduce the likelihood function. More on this in the section on simulation-based estimation.
The GMM estimator may still be feasible even though MLE is not available.
Example 43. The Octave script GMM/chi2gmm.m implements GMM using the same 2 data as
was using in Example 39, above. The two moment conditions, based on the sample mean and
sample variance are combined. The weight matrix is an identity matrix, I2 . In Octave, type help
gmm_estimate to get more information on how the GMM estimation routine works.

14.3

Consistency

We simply assume that the assumptions of Theorem 29 hold, so the GMM estimator is strongly
consistent. The only assumption that warrants additional comments is that of identification. In
Theorem 29, the third assumption reads: (c) Identification: s () has a unique global maximum
at 0 , i.e., s (0 ) > s (), 6= 0 . Taking the case of a quadratic objective function sn () =
mn ()0 Wn mn (), first consider mn ().
a.s.

Applying a uniform law of large numbers, we get mn () m ().

Since E0 mn (0 ) = 0 by assumption, m (0 ) = 0.
Since s (0 ) = m (0 )0 W m (0 ) = 0, in order for asymptotic identification, we need that
m () 6= 0 for 6= 0 , for at least some element of the vector. This and the assumption that
a.s.

Wn W , a finite positive g g definite g g matrix guarantee that 0 is asymptotically

identified.
Note that asymptotic identification does not rule out the possibility of lack of identification
for a given data set - there may be multiple minimizing solutions in finite samples.
Example 44. Increase n in the Octave script GMM/chi2gmm.m to see evidence of the consistency
of the GMM estimator.

14.4

Asymptotic normality

We also simply assume that the conditions of Theorem 31 hold, so we will have asymptotic normality. However, we do need to find the structure of the asymptotic variance-covariance matrix of
the estimator. From Theorem 31, we have




d
n 0 N 0, J (0 )1 I (0 )J (0 )1
where J (0 ) is the almost sure limit of

2
0 sn ()

when evaluated at 0 and

I (0 ) = lim V ar n sn (0 ).
n

We need to determine the form of these matrices given the objective function sn () = mn ()0 Wn mn ().
Now using the product rule from the introduction,



0
sn () = 2
mn () Wn mn ()

(this is analogous to

0 0
X X

= 2X 0 X which appears when computing the first order condi-

tions for the OLS estimator)

Define the K g matrix
Dn ()
so:

0
m () ,
n

s() = 2D()W m () .

(14.2)

(Note that sn (), Dn (), Wn and mn () all depend on the sample size n, but it is omitted to
unclutter the notation).
To take second derivatives, let Di be the i th row of D(). Using the product rule,
2
s()
0 i

2Di ()W m ()
0

0
D
2Di W D + 2m W
0 i
0

When evaluating the term



2m() W
D()0i
0
0

at 0 , assume that

0
0 D()i

satisfies a LLN, so that it converges almost surely to a finite limit. In

this case, we have

2m(0 )0 W

0 0 a.s.
D(
)
i 0,
0

a.s.

since m(0 ) = op (1) and W W .

Stacking these results over the K rows of D, we get
lim

2
0
sn (0 ) = J (0 ) = 2D W D
, a.s.,
0

where we define lim D = D , a.s., and lim W = W , a.s. (we assume a LLN holds).
With regard to I (0 ), following equation 14.2, and noting that the scores have mean zero at
0 (since Em(0 ) = 0 by assumption), we have
I (0 )

lim V ar n sn (0 )
n

= lim E4nDW m(0 )m(0 )0 W D0

n



= lim E4DW
nm(0 )
nm(0 )0 W D0
=

Now, given that m(0 ) is an average of centered (mean-zero) quantities, it is reasonable to expect

a CLT to apply, after multiplication by n. Assuming this,

nm(0 ) N (0, ),

where


= lim E nm(0 )m(0 )0 .
n

Using this, and the last equation, we get

0
I (0 ) = 4D W W D

Figure 14.1: Asymptotic Normality of GMM estimator, 2 example

(a) n = 30

(b) n = 1000

Using these results, the asymptotic normality theorem (31) gives us


h
i

d
0 1
0
0 1
n 0 N 0, (D W D
) D W W D
(D W D
)
,
the asymptotic distribution of the GMM estimator for arbitrary weighting matrix Wn . Note that for
J to be positive definite, D must have full row rank, (D ) = k. This is related to identification.
If the rows of mn () were not linearly independent of one another, then neither Dn nor D would
have full row rank. Identification plus two times differentiability of the objective function lead to
J being positive definite.
Example 45. The Octave script GMM/AsymptoticNormalityGMM.m does a Monte
 Carlo of the

0
2
GMM estimator for the data. Histograms for 1000 replications of n are given in
Figure 14.1. On the left are results for n = 30, on the right are results for n = 1000. Note that
the two distributions are fairly similar. In both cases the distribution is approximately normal. The
distribution for the small sample size is somewhat asymmetric. This has mostly disappeared for
the larger sample size.

14.5

Choosing the weighting matrix

W is a weighting matrix, which determines the relative importance of violations of the individual
moment conditions. For example, if we are much more sure of the first moment condition, which
is based upon the variance, than of the second, which is based upon the fourth moment, we could
set

"
W =

with a much larger than b. In this case, errors in the second moment condition have less weight in
the objective function.
Since moments are not independent, in general, we should expect that there be a correlation
between the moment conditions, so it may not be desirable to set the off-diagonal elements
to 0. W may be a random, data dependent matrix.

 We have already seen that the choice of W will influence the asymptotic distribution of the
GMM estimator. Since the GMM estimator is already inefficient w.r.t. MLE, we might like to
choose the W matrix to make the GMM estimator efficient within the class of GMM estimators
defined by mn ().
To provide a little intuition, consider the linear model y = x0 + , where N (0, ). That
is, he have heteroscedasticity and autocorrelation.
Let P be the Cholesky factorization of 1 , e.g, P 0 P = 1 .
Then the model P y = P X + P satisfies the classical assumptions of homoscedasticity and
1

nonautocorrelation, since V (P ) = P V ()P 0 = P P 0 = P (P 0 P )1 P 0 = P P 1 (P 0 )

In . (Note: we use (AB)

P0 =

= B 1 A1 for A, B both nonsingular). This means that the

transformed model is efficient.

The OLS estimator of the model P y = P X + P minimizes the objective function (y
X)0 1 (y X). Interpreting (y X) = () as moment conditions (note that they do
have zero expectation when evaluated at 0 ), the optimal weighting matrix is seen to be the
inverse of the covariance matrix of the moment conditions. This result carries over to GMM
estimation. (Note: this presentation of GLS is not a GMM estimator, because the number of
moment conditions here is equal to the sample size, n. Later well see that GLS can be put
into the GMM framework defined above).
Theorem 46. If is a GMM estimator that minimizes mn ()0 Wn mn (), the asymptotic variance of


a.s
0
0 0
will be minimized by choosing Wn so that Wn W = 1
, where = limn E nm( )m( ) .
Proof: For W = 1
, the asymptotic variance
0
(D W D
)
0
simplifies to D 1
D

1

0
0
D W W D
(D W D
)

. Now, let A be the difference between the general form and the simplified

form:
0
A = (D W D
)
1

0
Set B = (D W D
)

0
0
D W W D
(D W D
)

0
D W D 1
D

1

0
D 1
D

1
0

D 1
. You can show that A = B B . This

is a quadratic form in a p.d. matrix, so it is p.s.d., which concludes the proof.

The result

h

i

d
0 1
n 0 N 0, D 1
D

allows us to treat
N

0
D 1
D
,
n
0

(14.3)

1 !
,

where the means approximately distributed as. To operationalize this we need estimators of
D and .
 

d
The obvious estimator of D
is simply mn , which is consistent by the consistency of
assuming that m0 is continuous in . Stochastic equicontinuity results can give us this
,
result even if

0
m
n is not

continuous.

Example 47. To see the effect of using an efficient weight matrix, consider the Octave script
GMM/EfficientGMM.m. This modifies the previous Monte Carlo for the 2 data. This new Monte
Carlo computes the GMM estimator in two ways:

Figure 14.2: Inefficient and Efficient GMM estimators, 2 data

(a) inefficient

(b) efficient

1) based on an identity weight matrix

2) using an estimated optimal weight matrix. The estimated efficient weight matrix is computed as
the inverse of the estimated covariance of the moment conditions, using the inefficient estimator
of the first step. See the next section for more on how to do this.
Figure 14.2 shows the results, plotting histograms for 1000 replications of



n 0 . Note that

the use of the estimated efficient weight matrix leads to much better results in this case. This is a
simple case where it is possible to get a good estimate of the efficient weight matrix. This is not
always so. See the next section.

14.6

Estimation of the variance-covariance matrix

(See Hamilton Ch. 10, pp. 261-2 and 280-84) .

In the case that we wish to use the optimal weighting matrix, we need an estimate of , the

limiting variance-covariance matrix of nmn (0 ). While one could estimate parametrically,

we in general have little information upon which to base a parametric specification. In general, we
expect that:
mt will be autocorrelated (ts = E(mt m0ts ) 6= 0). Note that this autocovariance will not
depend on t if the moment conditions are covariance stationary.
contemporaneously correlated, since the individual moment conditions will not in general
be independent of one another (E(mit mjt ) 6= 0).
2
and have different variances (E(m2it ) = it
).

Since we need to estimate so many components if we are to take the parametric approach, it is
unlikely that we would arrive at a correct parametric specification. For this reason, research has
focused on consistent nonparametric estimators of .
Henceforth we assume that mt is covariance stationary (the covariance between mt and mts
does not depend on t). Define the vth autocovariance of the moment conditions v = E(mt m0ts ).
Note that E(mt m0t+s ) = 0v . Recall that mt and m are functions of , so for now assume that we

 Now
have some consistent estimator of 0 , so that m
t = mt ().
n

"
!
!#
n
n
X
X


0
0 0
0
= E nm( )m( ) = E n 1/n
mt
1/n
mt
t=1

"
= E 1/n

n
X

!
mt

t=1

t=1

!#

n
X

m0t

t=1

n2
1
n1
(1 + 01 ) +
(2 + 02 ) +
n1 + 0n1
0 +
n
n
n

A natural, consistent estimator of v is

cv = 1/n

n
X

m
tm
0tv .

t=v+1

(you might use n v in the denominator instead). So, a natural, but inconsistent, estimator of
would be







0
c0 + n 2
c0 + + [
[
c1 +
c2 +
c0 + n 1
=
n1 + n1
1
2
n
n
n1

X nv 
c0 +
cv +
c0v .
=

n
v=1

This estimator is inconsistent in general, since the number of parameters to estimate is more than
the number of observations, and increases more rapidly than n, so information does not build up
as n .
On the other hand, supposing that v tends to zero sufficiently rapidly as v tends to , a
modified estimator
=
c0 +

q(n) 


cv +
c0 ,

v=1
p

where q(n) as n will be consistent, provided q(n) grows sufficiently slowly. The term
nv
n

can be dropped because q(n) must be op (n). This allows information to accumulate at a rate

that satisfies a LLN. A disadvantage of this estimator is that it may not be positive definite. This
could cause one to calculate a negative 2 statistic, for example!
requires an estimate of m(0 ), which in turn requires an estimate
Note: the formula for
of , which is based upon an estimate of ! The solution to this circularity is to set the
weighting matrix W arbitrarily (for example to an identity matrix), obtain a first consistent
then re-estimate 0 . The
but inefficient estimate of 0 , then use this estimate to form ,
nor change appreciably between iterations.
process can be iterated until neither

Newey-West covariance estimator

The Newey-West estimator (Econometrica, 1987) solves the problem of possible nonpositive definiteness of the above estimator. Their estimator is
=
c0 +

q(n) 

X
v=1



v
cv +
c0v .

1
q+1

This estimator is p.d. by construction. The condition for consistency is that n1/4 q 0. Note
that this is a very slow rate of growth for q. This estimator is nonparametric - weve placed no
parametric restrictions on the form of . It is an example of a kernel estimator.
In a more recent paper, Newey and West (Review of Economic Studies, 1994) use pre-whitening

before applying the kernel estimator. The idea is to fit a VAR model to the moment conditions.
It is expected that the residuals of the VAR model will be more nearly white noise, so that the
Newey-West covariance estimator might perform better with short lag lengths..
The VAR model is
m
t = 1 m
t1 + + p m
tp + ut
This is estimated, giving the residuals u
t . Then the Newey-West covariance estimator is applied to
these pre-whitened residuals, and the covariance is estimated combining the fitted VAR
c
c1 m
cp m
m
t =
t1 + +
tp
with the kernel estimate of the covariance of the ut . See Newey-West for details.
I have a program that does this if youre interested.

14.7

Estimation using conditional moments

So far, the moment conditions have been presented as unconditional expectations. One common
way of defining unconditional moment conditions is based upon conditional moment conditions.
Suppose that a random variable Y has zero expectation conditional on the random variable X
Z
EY |X Y =

Y f (Y |X)dY = 0

Then the unconditional expectation of the product of Y and a function g(X) of X is also zero. The
unconditional expectation is
Z Z
EY g(X) =


Y g(X)f (Y, X)dY

dX.

This can be factored into a conditional expectation and an expectation w.r.t. the marginal density
of X :

Z Z

EY g(X) =

Y g(X)f (Y |X)dY
X

f (X)dX.

Since g(X) doesnt depend on Y it can be pulled out of the integral

Z Z

EY g(X) =

Y f (Y |X)dY
X

g(X)f (X)dX.

But the term in parentheses on the rhs is zero by assumption, so

EY g(X) = 0
as claimed.
This is important econometrically, since models often imply restrictions on conditional moments. Suppose a model tells us that the function K(yt , xt ) has expectation, conditional on the
information set It , equal to k(xt , ),
E K(yt , xt )|It = k(xt , ).
For example, in the context of the classical linear model yt = x0t + t , we can set K(yt , xt ) =
yt so that k(xt , ) = x0t .

With this, the error function

t () = K(yt , xt ) k(xt , )
has conditional expectation equal to zero
E t ()|It = 0.
This is a scalar moment condition, which isnt sufficient to identify a K -dimensional parameter
(K > 1). However, the above result allows us to form various unconditional expectations
mt () = Z(wt )t ()
where Z(wt ) is a g 1-vector valued function of wt and wt is a set of variables drawn from the
information set It . The Z(wt ) are instrumental variables. We now have g moment conditions, so as
long as g > K the necessary condition for identification holds.
One can form the n g matrix

Zn

Z1 (w1 )

Z2 (w1 )

Z1 (w2 )
=
..
.

Z2 (w2 )

Z1 (wn ) Z2 (wn )

Z10

Z20

0
Zn

Zg (w1 )

Zg (w2 )
..
.

Zg (wn )

With this we can form the g moment conditions

mn ()

1 ()

1 0
Zn

2 ()
..
.

n ()
Define the vector of error functions

1 ()

hn () =

2 ()
..
.

n ()
With this, we can write

1 0
Z hn ()
n n
n
1X
=
Zt ht ()
n t=1

mn () =

1X
mt ()
n t=1

where Z(t,) is the tth row of Zn . This fits the previous treatment.

14.8

Estimation using dynamic moment conditions

Note that dynamic moment conditions simplify the var-cov matrix, but are often harder to formulate. The will be added in future editions. For now, the Hansen application below is enough.

14.9

A specification test

The first order conditions for minimization, using the an estimate of the optimal weighting matrix,
are



 

0   1
s() = 2
mn mn 0

or

0
1 mn ()
D()

Consider a Taylor expansion of m():



= mn (0 ) + D0 ( ) 0
m()
n

(14.4)

1 we obtain
where is between and 0 . Multiplying by D()



= D()

1 m()
1 mn (0 ) + D()
1 D( )0 0
D()
The lhs is zero, so
h
i


1 mn (0 ) = D()
1 D( )0 0
D()
or




1

1 D( )0
1 mn (0 )
0 = D()
D()

With this, and taking into account the original expansion (equation 14.4), we get

=
nm()

nmn (0 )


1

1 D( )0
1 mn (0 ).
nDn0 ( ) D()
D()

With some factoring, this last can be written as

=
nm()



1

1/2

1/2
1/2 Dn0 ( ) D()
1 D( )0

Dn0 ( )
n
mn (0 )

1/2 to get
and then multiply be

=
1/2 m()
n

Now



1

1/2

1/2 Dn0 ( ) D()

1 D( )0
1/2

Ig
Dn0 ( )
n
mn (0 )

1/2 mn (0 ) N (0, Ig )
n

1

1/2 Dn0 ( ) D()

1 D( )0
1/2 converges in probability to
and the big matrix Ig
Dn0 ( )


1
1/2 0
1/2
0
P = Ig D
D 1
D . However, one can easily verify that P is idempotent
D
and has rank g K, (recall that the rank of an idempotent matrix is equal to its trace). We know
that N (0, Ig )0 P N (0, Ig ) 2 (g K). So, a quadratic form on the r.h.s. has an asymptotic chisquare distribution. The quadratic form on the l.h.s. must also have the same distribution, so we

finally get

0 

d

= nm()
0

1/2 m()
1/2 m()
1 m()
n
n
2 (g K)

or
d

2 (g K)
n sn ()
supposing the model is correctly specified. This is a convenient test since we just multiply the
optimized value of the objective function by n, and compare with a 2 (g K) critical value. The
test is a general test of whether or not the moments used to estimate are correctly specified.
This wont work when the estimator is just identified. The f.o.c. are
0.
1 m()
D sn () = D
are square and invertible (at least asymptotically,
But with exact identification, both D and
assuming that asymptotic normality hold), so
0.
m()
So the moment conditions are zero regardless of the weighting matrix used. As such, we
= 0, so the test breaks
might as well use an identity matrix and save trouble. Also sn ()
down.
A note: this sort of test often over-rejects in finite samples. One should be cautious in rejecting
a model when this test rejects.

14.10

Example: Generalized instrumental variables estimator

The IV estimator may appear a bit unusual at first, but it will grow on you over time. We have
in fact already seen the IV estimator above, in the discussion of conditional moments. Lets look
at the special case of a linear model with iid errors, but with correlation between regressors and
errors:
yt

E(x0t t ) 6=

x0t + t
0

Lets assume, just to keep things simple, that the errors are iid
The model in matrix form is y = X + 
Let K = dim(xt ). Consider some vector zt of dimension G 1, where G K. Assume that
E(zt t ) = 0. The variables zt are instrumental variables. Consider the moment conditions
mt () = zt t
= zt (yt x0t )
We can arrange the instruments in the n G matrix
z10

0
z2
=
..
.

zn0

The average moment conditions are

1 0
Z
n
1
= (Z 0 y Z 0 X)
n

mn () =

The generalized instrumental variables estimator is just the GMM estimator based upon these moment conditions. When G = K, we have exact identification, and it is referred to as the instrumental variables estimator.
= 0, which imply that
The first order conditions for GMM are Dn Wn mn ()
Dn Wn Z 0 X IV = Dn Wn Z 0 y
0

Exercise 48. Verify that Dn = XnZ . Remember that (assuming differentiability) identification
of the GMM estimator requires that this matrix must converge to a matrix with full row rank. Can
just any variable that is uncorrelated with the error be used as an instrument, or is there some
other condition?

Exercise 49. Verify that the efficient weight matrix is Wn =

Z0Z
n

1

(up to a constant).

If we accept what is stated in these two exercises, then

X 0Z
n

Z 0Z
n

1

Z 0 X IV =

X 0Z
n

Z 0Z
n

1

Z 0y

Noting that the powers of n cancel, we get

1

X 0 Z (Z 0 Z)
or

1
Z 0 X IV = X 0 Z (Z 0 Z) Z 0 y


1
1
1
IV = X 0 Z (Z 0 Z) Z 0 X
X 0 Z (Z 0 Z) Z 0 y

(14.5)

Another way of arriving to the same point is to define the projection matrix PZ
PZ = Z(Z 0 Z)1 Z 0
Anything that is projected onto the space spanned by Z will be uncorrelated with , by the definition of Z. Transforming the model with this projection matrix we get
PZ y = PZ X + PZ
or
y = X +
Now we have that and X are uncorrelated, since this is simply
E(X 0 )

= E(X 0 PZ0 PZ )
=

E(X 0 PZ )

and
PZ X = Z(Z 0 Z)1 Z 0 X

is the fitted value from a regression of X on Z. This is a linear combination of the columns of Z,
so it must be uncorrelated with . This implies that applying OLS to the model
y = X +
will lead to a consistent estimator, given a few more assumptions.
Exercise 50. Verify algebraically that applying OLS to the above model gives the IV estimator of
equation 14.5.
With the definition of PZ , we can write

IV = (X 0 PZ X)1 X 0 PZ y
from which we obtain
IV

(X 0 PZ X)1 X 0 PZ (X0 + )

= 0 + (X 0 PZ X)1 X 0 PZ
so
IV 0

(X 0 PZ X)1 X 0 PZ

1 0
= X 0 Z(Z 0 Z)1 Z 0 X
X Z(Z 0 Z)1 Z 0

Now we can introduce factors of n to get

IV 0 =

X 0Z
n

Z 0Z
n

!

Z 0X
n

!1 

X 0Z
n



Z 0Z
n

1 

Z 0
n

Assuming that each of the terms with a n in the denominator satisfies a LLN, so that
p

Z0Z
n

QZZ , a finite pd matrix

X0Z
n

QXZ , a finite matrix with rank K (= cols(X) ). That is to say, the instruments must

be correlated with the regressors.

Z0 p
n

then the plim of the rhs is zero. This last term has plim 0 since we assume that Z and are
uncorrelated, e.g.,
E(zt0 t ) = 0,
Given these assumtions the IV estimator is consistent
p
IV 0 .

Furthermore, scaling by

n, we have



n IV 0 =

X 0Z
n



Z 0Z
n

1 

Z 0X
n

!1 

Assuming that the far right term satifies a CLT, so that

0
d
Z

n

N (0, QZZ 2 )

X 0Z
n



Z 0Z
n

1 

Z 0

then we get





d
0
1 2
n IV 0 N 0, (QXZ Q1

ZZ QXZ )

The estimators for QXZ and QZZ are the obvious ones. An estimator for 2 is
0 

1
2
d
y X IV
y X IV .
IV =
n
This estimator is consistent following the proof of consistency of the OLS estimator of 2 , when
the classical assumptions hold.
The formula used to estimate the variance of IV is

1
1
2
d
V (IV ) = (X 0 Z) (Z 0 Z) (Z 0 X)
IV

The GIV estimator is

1. Consistent
2. Asymptotically normally distributed
3. Biased in general, since even though E(X 0 PZ ) = 0, E(X 0 PZ X)1 X 0 PZ may not be zero,
since (X 0 PZ X)1 and X 0 PZ are not independent.
An important point is that the asymptotic distribution of IV depends upon QXZ and QZZ , and
these depend upon the choice of Z. The choice of instruments influences the efficiency of the estimator.
This point was made above, when optimal instruments were discussed.
When we have two sets of instruments, Z1 and Z 2 such that Z1 Z2 , then the IV estimator using Z2 is at least as efficiently asymptotically as the estimator that used Z1 . More
instruments leads to more asymptotically efficient estimation, in general.
The penalty for indiscriminant use of instruments is that the small sample bias of the IV
estimator rises as the number of instruments increases. The reason for this is that PZ X
becomes closer and closer to X itself as the number of instruments increases.
Exercise 51. How would one adapt the GIV estimator presented here to deal with the case of HET
and AUT?

Example 52. Recall Example 19 which deals with a dynamic model with measurement error. The
model is
yt

= + yt1
+ xt + t

yt

= yt + t

where t and t are independent Gaussian white noise errors. Suppose that yt is not observed, and
instead we observe yt . If we estimate the equation
yt = + yt1 + xt + t
by OLS, we have seen in Example 19 that the estimator is biased an inconsistent. What about
using the GIV estimator? Consider using as instruments Z = [1 xt xt1 xt2 ]. The lags of xt are
correlated with yt1 as long as and are different from zero, and by assumption xt and its
lags are uncorrelated with t and t (and thus theyre also uncorrelated with t ). Thus, these
are legitimate instruments. As we have 4 instruments and 3 parameters, this is an overidentified
situation. The Octave script GMM/MeasurementErrorIV.m does a Monte Carlo study using 1000
replications, with a sample size of 100. The results are comparable with those in Example 19.
Using the GIV estimator, descriptive statistics for 1000 replications are

octave:3> MeasurementErrorIV
rm: cannot remove `meas_error.out': No such file or directory
mean st. dev.
min
max
0.000
0.241
-1.250
1.541
-0.016
0.149
-0.868
0.827
-0.001
0.177
-0.757
0.876
octave:4>

Figure 14.3: GIV estimation results for , dynamic model with measurement error

If you compare these with the results for the OLS estimator, you will see that the bias of the GIV
estimator is much less for estimation of . If you increase the sample size, you will see that the GIV
estimator is consistent, but that the OLS estimator is not.
A histogram for is in Figure 14.3. You can compare with the similar figure for the OLS
estimator, Figure 7.4.

2SLS
In the general discussion of GIV above, we havent considered from where we get the instruments.
Two stage least squares is an example of a particular GIV estimator, where the instruments are
obtained in a particular way. Consider a single equation from a system of simultaneous equations.
Refer back to equation 10.2 for context. The model is
y

= Y1 1 + X1 1 +
= Z +

where Y1 are current period endogenous variables that are correlated with the error term. X1 are
exogenous and predetermined variables that are assumed not to be correlated with the error term.

Let X be all of the weakly exogenous variables (please refer back for context). The problem, recall,
is that the variables in Y1 are correlated with .
h
i
Define Z = Y1 X1 as the vector of predictions of Z when regressed upon X:
1
Z = X (X 0 X) X 0 Z

Remember that X are all of the exogenous variables from all equations. The fitted values of
a regression of X1 on X are just X1 , because X contains X1 . So, Y1 are the reduced form
predictions of Y1 .
Since Z is a linear combination of the weakly exogenous variables X, it must be uncorrelated
t (yt z0t ) and so
with . This suggests the K-dimensional moment condition mt () = z
m() = 1/n

t (yt z0t ) .
z

Since we have K parameters and K moment conditions, the GMM estimator will set m
identically equal to zero, regardless of W, so we have
!1
=

t z0t
z

1

0y
0Z
Z
(
zt yt ) = Z

This is the standard formula for 2SLS. We use the exogenous variables and the reduced form
predictions of the endogenous variables as instruments, and apply IV estimation. See Hamilton
pp. 420-21 for the varcov formula (which is the standard formula for 2SLS), and for how to deal
with t heterogeneous and dependent (basically, just use the Newey-West or some other consistent
estimator of , and apply the usual formula).
Note that autocorrelation of t causes lagged endogenous variables to loose their status as
legitimate instruments. Some caution is warranted if this suspicion arises.

14.11

Nonlinear simultaneous equations

GMM provides a convenient way to estimate nonlinear systems of simultaneous equations. We

have a system of equations of the form
y1t

= f1 (zt , 10 ) + 1t

y2t

= f2 (zt , 20 ) + 2t
..
.

yGt

0
= fG (zt , G
) + Gt ,

or in compact notation
yt = f (zt , 0 ) + t ,
00 0
where f () is a G -vector valued function, and 0 = (100 , 200 , , G
).

We need to find an Ai 1 vector of instruments xit , for each equation, that are uncorrelated
with it . Typical instruments would be low order monomials in the exogenous variables in zt , with

their lagged values. Then we can define the

P

G
i=1


Ai 1 orthogonality conditions

(y1t f1 (zt , 1 )) x1t

(y2t f2 (zt , 2 )) x2t

mt () =
..

(yGt fG (zt , G )) xGt

A note on identification: selection of instruments that ensure identification is a non-trivial
problem.
A note on efficiency: the selected set of instruments has important effects on the efficiency of
estimation. Unfortunately there is little theory offering guidance on what is the optimal set.
More on this later.

14.12

Maximum likelihood

In the introduction we argued that ML will in general be more efficient than GMM since ML implicitly uses all of the moments of the distribution while GMM uses a limited number of moments.
Actually, a distribution with P parameters can be uniquely characterized by P moment conditions.
However, some sets of P moment conditions may contain more information than others, since the
moment conditions could be highly correlated. A GMM estimator that chose an optimal set of P
moment conditions would be fully efficient. Here well see that the optimal moment conditions are
simply the scores of the ML estimator.
Let yt be a G -vector of variables, and let Yt = (y10 , y20 , ..., yt0 )0 . Then at time t, Yt1 has been
observed (refer to it as the information set, since we assume the conditioning variables have been
selected to take advantage of all useful information). The likelihood function is the joint density
of the sample:
L() = f (y1 , y2 , ..., yn , )
which can be factored as
L() = f (yn |Yn1 , ) f (Yn1 , )
and we can repeat this to get
L() = f (yn |Yn1 , ) f (yn1 |Yn2 , ) ... f (y1 ).
The log-likelihood function is therefore
ln L() =

n
X

ln f (yt |Yt1 , ).

t=1

Define
mt (Yt , ) D ln f (yt |Yt1 , )
as the score of the tth observation. It can be shown that, under the regularity conditions, that the
scores have conditional mean zero when evaluated at 0 (see notes to Introduction to Econometrics):
E{mt (Yt , 0 )|Yt1 } = 0
so one could interpret these as moment conditions to use to define a just-identified GMM estimator

( if there are K parameters there are K score equations). The GMM estimator sets
1/n

n
X

= 1/n
mt (Yt , )

t=1

n
X

= 0,
D ln f (yt |Yt1 , )

t=1

which are precisely the first order conditions of MLE. Therefore, MLE can be interpreted as a GMM


0 1
estimator. The GMM varcov formula is V = D 1 D
.
Consistent estimates of variance components are as follows
D

n
X

d
m(Y
,
)
=
1/n
D2 ln f (yt |Yt1 , )
D
=
t

0
t=1

It is important to note that mt and mts , s > 0 are both conditionally and unconditionally
uncorrelated. Conditional uncorrelation follows from the fact that mts is a function of Yts ,
which is in the information set at time t. Unconditional uncorrelation follows from the fact
that conditional uncorrelation hold regardless of the realization of Yt1 , so marginalizing
with respect to Yt1 preserves uncorrelation (see the section on ML estimation, above). The
fact that the scores are serially uncorrelated implies that can be estimated by the estimator
of the 0th autocovariance of the moment conditions:
b = 1/n

n
X

t (Yt , )
0 = 1/n
mt (Yt , )m

n h
ih
i0
X
D ln f (yt |Yt1 , )

D ln f (yt |Yt1 , )
t=1

t=1

Recall from study of ML estimation that the information matrix equality (equation 13.4) states that
E

n

0 o


D ln f (yt |Yt1 , 0 ) D ln f (yt |Yt1 , 0 )
= E D2 ln f (yt |Yt1 , 0 ) .

This result implies the well known (and already seeen) result that we can estimate V in any of
three ways:
The sandwich version:

Vc
=n

nP
n

o
2

D
ln
f
(y
|Y
,
)
t
t1
t=1
ih
i0 1
Pn h

D ln f (yt |Yt1 , )

t=1 D ln f (yt |Yt1 , )

nP
o
n
2

t=1 D ln f (yt |Yt1 , )

or the inverse of the negative of the Hessian (since the middle and last term cancel, except
for a minus sign):
"
#1
n
X
2

c
V = 1/n
D ln f (yt |Yt1 , )
,
t=1

or the inverse of the outer product of the gradient (since the middle and last cancel except
for a minus sign, and the first term converges to minus the inverse of the middle term, which
is still inside the overall inverse)
(
)
n h
ih
i0 1
X

c
V = 1/n
D ln f (yt |Yt1 , ) D ln f (yt |Yt1 , )
.
t=1

This simplification is a special result for the MLE estimator - it doesnt apply to GMM estimators in
general.

Asymptotically, if the model is correctly specified, all of these forms converge to the same limit.
In small samples they will differ. In particular, there is evidence that the outer product of the
gradient formula does not perform very well in small samples (see Davidson and MacKinnon, pg.
477). Whites Information matrix test (Econometrica, 1982) is based upon comparing the two ways
to estimate the information matrix: outer product of gradient or negative of the Hessian. If they
differ by too much, this is evidence of misspecification of the model.

14.13

Example: OLS as a GMM estimator - the Nerlove model

again

The simple Nerlove model can be estimated using GMM. The Octave script NerloveGMM.m estimates the model by GMM and by OLS. It also illustrates that the weight matrix does not matter
when the moments just identify the parameter. You are encouraged to examine the script and run
it.

14.14

Example: The MEPS data

The MEPS data on health care usage discussed in section 11.4 estimated a Poisson model by maximum likelihood (probably misspecified). Perhaps the same latent factors (e.g., chronic illness)
that induce one to make doctor visits also influence the decision of whether or not to purchase
insurance. If this is the case, the PRIV variable could well be endogenous, in which case, the Poisson ML estimator would be inconsistent, even if the conditional mean were correctly specified.
The Octave script meps.m estimates the parameters of the model presented in equation 11.1, using
Poisson ML (better thought of as quasi-ML), and IV estimation1 . Both estimation methods are
implemented using a GMM form. Running that script gives the output

OBDV

******************************************************
IV
GMM Estimation Results
BFGS convergence: Normal convergence
Objective function value: 0.004273
Observations: 4564
No moment covariance supplied, assuming efficient weight matrix

X^2 test

Value
19.502

df
3.000

constant
pub. ins.
priv. ins.
sex

estimate
-0.441
-0.127
-1.429
0.537

st. err
0.213
0.149
0.254
0.053

1 The

p-value
0.000
t-stat
-2.072
-0.851
-5.624
10.133

p-value
0.038
0.395
0.000
0.000

validity of the instruments used may be debatable, but real data sets often dont contain ideal instruments.

age
0.031
0.002
13.431
0.000
edu
0.072
0.011
6.535
0.000
inc
0.000
0.000
4.500
0.000
******************************************************

******************************************************
Poisson QML
GMM Estimation Results
BFGS convergence: Normal convergence
Objective function value: 0.000000
Observations: 4564
No moment covariance supplied, assuming efficient weight matrix
Exactly identified, no spec. test
estimate
st. err
t-stat
p-value
constant
-0.791
0.149
-5.289
0.000
pub. ins.
0.848
0.076
11.092
0.000
priv. ins.
0.294
0.071
4.136
0.000
sex
0.487
0.055
8.796
0.000
age
0.024
0.002
11.469
0.000
edu
0.029
0.010
3.060
0.002
inc
-0.000
0.000
-0.978
0.328
******************************************************

Note how the Poisson QML results, estimated here using a GMM routine, are the same as were
obtained using the ML estimation routine (see subsection 11.4). This is an example of how (Q)ML
may be represented as a GMM estimator. Also note that the IV and QML results are considerably
different. Treating PRIV as potentially endogenous causes the sign of its coefficient to change.
Perhaps it is logical that people who own private insurance make fewer visits, if they have to
make a co-payment. Note that income becomes positive and significant when PRIV is treated as
endogenous.
Perhaps the difference in the results depending upon whether or not PRIV is treated as endogenous can suggest a method for testing exogeneity. Onward to the Hausman test!

14.15

Example: The Hausman Test

This section discusses the Hausman test, which was originally presented in Hausman, J.A. (1978),
Specification tests in econometrics, Econometrica, 46, 1251-71.
Consider the simple linear regression model yt = x0t + t . We assume that the functional form
and the choice of regressors is correct, but that the some of the regressors may be correlated with
For example, this will be a
the error term, which as you know will produce inconsistency of .
problem if
if some regressors are endogeneous

Figure 14.4: OLS

OLS estimates

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0
2.28

2.3

2.32

2.34

2.36

2.38

some regressors are measured with error

lagged values of the dependent variable are used as regressors and t is autocorrelated.
To illustrate, the Octave program OLSvsIV.m performs a Monte Carlo experiment where errors are
correlated with regressors, and estimation is by OLS and IV. The true value of the slope coefficient
used to generate the data is = 2. Figure 14.4 shows that the OLS estimator is quite biased,
while Figure 14.5 shows that the IV estimator is on average much closer to the true value. If you
play with the program, increasing the sample size, you can see evidence that the OLS estimator is
asymptotically biased, while the IV estimator is consistent.
We have seen that inconsistent and the consistent estimators converge to different probability
limits. This is the idea behind the Hausman test - a pair of consistent estimators converge to
the same probability limit, while if one is consistent and the other is not they converge to different
limits. If we accept that one is consistent (e.g., the IV estimator), but we are doubting if the other is
consistent (e.g., the OLS estimator), we might try to check if the difference between the estimators
is significantly different from zero.
If were doubting about the consistency of OLS (or QML, etc.), why should we be interested in
testing - why not just use the IV estimator? Because the OLS estimator is more efficient when
the regressors are exogenous and the other classical assumptions (including normality of the
errors) hold. When we have a more efficient estimator that relies on stronger assumptions
(such as exogeneity) than the IV estimator, we might prefer to use it, unless we have evidence
that the assumptions are false.
So, lets consider the covariance between the MLE estimator (or any other fully efficient estima Now, lets recall some results from MLE. Equation 13.2
tor) and some other CAN estimator, say .
is:



a.s.
n 0 J (0 )1 ng(0 ).
Equation 13.6 is
J () = I ().

Figure 14.5: IV
IV estimates

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0
1.9

1.92

1.94

1.96

1.98

2.02

2.04

2.06

2.08

Combining these two equations, we get




a.s.
n 0 I (0 )1 ng(0 ).
Also, equation 13.9 tells us that the asymptotic covariance between any CAN estimator and the
MLE score vector is
 # "
" 
#

n
V ()
IK
V
=
.

IK
I ()
ng()
Now, consider
"

IK
0K



 #
#" 

n

n
a.s.
 .


I ()1
n
ng()
0K

The asymptotic covariance of this is



n
 =
V 
n
=

"

IK

0K

#"

V ()

0K I ()1
IK
"
#

V ()
I ()1
I ()1

I ()1

IK

#"

I ()

IK

0K

0K

I ()1

which, for clarity in what follows, we might write as


 "

n
V ()
 =
V 
I ()1
n

I ()1

V ()

#
.

So, the asymptotic covariance between the MLE and any other CAN estimator is equal to the MLE
asymptotic variance (the inverse of the information matrix).
Now, suppose we with to test whether the the two estimators are in fact both converging to 0 ,
versus the alternative hypothesis that the MLE estimator is not in fact consistent (the consistency

of is a maintained hypothesis). Under the null hypothesis that they are, we have
h
IK

IK




n 0


 = n ,
n 0

will be asymptotically normally distributed as





d
V ()
.
n N 0, V ()
So,


0 
1 

d
V ()

n
V ()
2 (),

where is the rank of the difference of the asymptotic variances. A statistic that has the same
asymptotic distribution is


1 

0 
d
V ()

V ()
2 ().

This is the Hausman test statistic, in its original form. The reason that this test has power under
the alternative hypothesis is that in that case the MLE estimator will not be consistent, and will
converge to A , say, where A 6= 0 . Then the mean of the asymptotic distribution of vector
 
n will be 0 A , a non-zero vector, so the test statistic will eventually reject, regardless
of how small a significance level is used.
Note: if the test is based on a sub-vector of the entire parameter vector of the MLE, it is
possible that the inconsistency of the MLE will not show up in the portion of the vector that
has been used. If this is the case, the test may not have power to detect the inconsistency.
This may occur, for example, when the consistent but inefficient estimator is not identified
for all the parameters of the model.
Some things to note:
The rank, , of the difference of the asymptotic variances is often less than the dimension
of the matrices, and it may be difficult to determine what the true rank is. If the true rank
is lower than what is taken to be true, the test will be biased against rejection of the null
hypothesis. The contrary holds if we underestimate the rank.
A solution to this problem is to use a rank 1 test, by comparing only a single coefficient. For
example, if a variable is suspected of possibly being endogenous, that variables coefficients
may be compared.
This simple formula only holds when the estimator that is being tested for consistency is
fully efficient under the null hypothesis. This means that it must be a ML estimator or a
fully efficient estimator that has the same asymptotic distribution as the ML estimator. This
is quite restrictive since modern estimators such as GMM and QML are not in general fully
efficient.
Following up on this last point, lets think of two not necessarily efficient estimators, 1 and 2 ,
where one is assumed to be consistent, but the other may not be. We assume for expositional
simplicity that both 1 and 2 belong to the same parameter space, and that they can be expressed
as generalized method of moments (GMM) estimators. The estimators are defined (suppressing
the dependence upon data) by
i

arg min mi (i )0 Wi mi (i )
i

Figure 14.6: Incorrect rank and the Hausman test

where mi (i ) is a gi 1 vector of moment conditions, and Wi is a gi gi positive definite weighting

matrix, i = 1, 2. Consider the omnibus GMM estimator


1 , 2 = arg min

m1 (1 )0

m2 (2 )0

"

W1

0(g1 g2 )

0(g2 g1 )

W2

#"

m1 (1 )
m2 (2 )

#
.

(14.6)

Suppose that the asymptotic covariance of the omnibus moment vector is

(

lim V ar

12

"
n

m1 (1 )
m2 (2 )

#)
(14.7)

!
.

The standard Hausman test is equivalent to a Wald test of the equality of 1 and 2 (or subvectors of
the two) applied to the omnibus GMM estimator, but with the covariance of the moment conditions
estimated as
!
c1

0(g1 g2 )
b
=
.
c2
0(g g )

While this is clearly an inconsistent estimator in general, the omitted 12 term cancels out of the
test statistic when one of the estimators is asymptotically efficient, as we have seen above, and
thus it need not be estimated.
The general solution when neither of the estimators is efficient is clear: the entire matrix
must be estimated consistently, since the 12 term will not cancel out. Methods for consistently
estimating the asymptotic covariance of a vector of moment conditions are well-known, e.g., the
Newey-West estimator discussed previously. The Hausman test using a proper estimator of the
overall covariance matrix will now have an asymptotic 2 distribution when neither estimator is

efficient. This is
However, the test suffers from a loss of power due to the fact that the omnibus GMM estimator
of equation 14.6 is defined using an inefficient weight matrix. A new test can be defined by using
an alternative omnibus GMM estimator



h
1 , 2 = arg min m1 (1 )0

i  1
e

m2 (2 )

"

m1 (1 )
m2 (2 )

#
,

(14.8)

e is a consistent estimator of the overall covariance matrix of equation 14.7. By standard

where
arguments, this is a more efficient estimator than that defined by equation 14.6, so the Wald test
using this alternative is more powerful. See my article in Applied Economics, 2004, for more details,
including simulation results. The Octave script hausman.m calculates the Wald test corresponding
to the efficient joint GMM estimator (the H2 test in my paper), for a simple linear model.

14.16

Application: Nonlinear rational expectations

Readings: Hansen and Singleton, 1982 ; Tauchen, 1986

Though GMM estimation has many applications, application to rational expectations models is
elegant, since theory directly suggests the moment conditions. Hansen and Singletons 1982 paper
is also a classic worth studying in itself. Though I strongly recommend reading the paper, Ill use a
simplified model with similar notation to Hamiltons. The literature on estimation of these models
has grown a lot since these early papers. After work like the cited papers, people moved to ML
estimation of linearized models, using Kalman filtering. Current methods are usually Bayesian, and
involve sophisticated filtering methods to compute the likelihood function for nonlinear models
with non-normal shocks. There is a lot of interesting stuff that is beyond the scope of this course.
I have done some work using simulation-based estimation methods applied to such models. The
methods explained in this section are intended to provide an example of GMM estimation. They
are not the state of the art for estimation of such models.
We assume a representative consumer maximizes expected discounted utility over an infinite
horizon. Utility is temporally additive, and the expected utility hypothesis holds. The future

consumption stream is the stochastic sequence {ct }t=0 . The objective function at time t is the
discounted expected utility

s E (u(ct+s )|It ) .

(14.9)

s=0

The parameter is between 0 and 1, and reflects discounting.

It is the information set at time t, and includes the all realizations of random variables indexed t and earlier.
The choice variable is ct - current consumption, which is constained to be less than or equal
to current wealth wt .
Suppose the consumer can invest in a risky asset. A dollar invested in the asset yields a gross
return
(1 + rt+1 ) =

pt+1 + dt+1
pt

where pt is the price and dt is the dividend in period t. The price of ct is normalized to 1.
Current wealth wt = (1 + rt )it1 , where it1 is investment in period t 1. So the problem
is to allocate current wealth between current consumption and investment to finance future
consumption: wt = ct + it .

 Future net rates of return rt+s , s > 0 are not known in period t: the asset is risky.
A partial set of necessary conditions for utility maximization have the form:
u0 (ct ) = E {(1 + rt+1 ) u0 (ct+1 )|It } .

(14.10)

To see that the condition is necessary, suppose that the lhs < rhs. Then by reducing current consumption marginally would cause equation 14.9 to drop by u0 (ct ), since there is no discounting of
the current period. At the same time, the marginal reduction in consumption finances investment,
which has gross return (1 + rt+1 ) , which could finance consumption in period t+1. This increase in
consumption would cause the objective function to increase by E {(1 + rt+1 ) u0 (ct+1 )|It } . Therefore, unless the condition holds, the expected discounted utility function is not maximized.
To use this we need to choose the functional form of utility. A constant relative risk aversion
form is
u(ct ) =

c1
1
t
1

where is the coefficient of relative risk aversion. With this form,

u0 (ct ) = c
t
so the foc are


c
= E (1 + rt+1 ) c
t
t+1 |It
While it is true that



E c
(1 + rt+1 ) c
|It = 0
t
t+1
so that we could use this to define moment conditions, it is unlikely that ct is stationary, even
though it is in real terms, and our theory requires stationarity. To solve this, divide though by c
t
(
E

1-


(1 + rt+1 )

ct+1
ct

 )!
|It = 0

(note that ct can be passed though the conditional expectation since ct is chosen based only upon
information available in time t).
Now

(
1-

(1 + rt+1 )

ct+1
ct

 )

is analogous to ht () defined above: its a scalar moment condition. To get a vector of moment
conditions we need some instruments. Suppose that zt is a vector of variables drawn from the
information set It . We can use the necessary conditions to form the expressions


 
ct+1
1 (1 + rt+1 ) ct
zt mt ()
represents and .
Therefore, the above expression may be interpreted as a moment condition which can be
used for GMM estimation of the parameters 0 .
Note that at time t, mts has been observed, and is therefore an element of the information set. By
rational expectations, the autocovariances of the moment conditions other than 0 should be zero.
The optimal weighting matrix is therefore the inverse of the variance of the moment conditions:



= lim E nm(0 )m(0 )0
which can be consistently estimated by
= 1/n

n
X

t ()
0
mt ()m

t=1

As before, this estimate depends on an initial consistent estimate of , which can be obtained by

setting the weighting matrix W arbitrarily (to an identity matrix, for example). After obtaining ,
we then minimize
1 m().
s() = m()0
This process can be iterated, e.g., use the new estimate to re-estimate , use this to estimate 0 ,
and repeat until the estimates dont change.
In principle, we could use a very large number of moment conditions in estimation, since
any current or lagged variable could be used in xt . Since use of more moment conditions
will lead to a more (asymptotically) efficient estimator, one might be tempted to use many
instrumental variables. We will do a computer lab that will show that this may not be a good
idea with finite samples. This issue has been studied using Monte Carlos (Tauchen, JBES,
1986). The reason for poor performance when using many instruments is that the estimate
of becomes very imprecise.
Empirical papers that use this approach often have serious problems in obtaining precise
estimates of the parameters, and identification can be problematic. Note that we are basing everything on a single partial first order condition. Probably this f.o.c. is simply not
informative enough.

14.17

Empirical example: a portfolio model

The Octave program portfolio.m performs GMM estimation of a portfolio model, using the data file
tauchen.data. The columns of this data file are c, p, and d in that order. There are 95 observations
(source: Tauchen, JBES, 1986). As instruments we use lags of c and r, as well as a constant. For
a single lag the estimation results are

MPITB extensions found

******************************************************
Example of GMM estimation of rational expectations model
GMM Estimation Results
BFGS convergence: Normal convergence
Objective function value: 0.000014
Observations: 94

X^2 test

Value
0.001
estimate

df
1.000
st. err

p-value
0.971
t-stat

p-value

beta
0.915
0.009
97.271
0.000
gamma
0.569
0.319
1.783
0.075
******************************************************

For two lags the estimation results are

MPITB extensions found

******************************************************
Example of GMM estimation of rational expectations model
GMM Estimation Results
BFGS convergence: Normal convergence
Objective function value: 0.037882
Observations: 93

X^2 test

Value
3.523

df
3.000

p-value
0.318

estimate
st. err
t-stat
p-value
beta
0.857
0.024
35.636
0.000
gamma
-2.351
0.315
-7.462
0.000
******************************************************

Pretty clearly, the results are sensitive to the choice of instruments. Maybe there is some problem
here: poor instruments, or possibly a conditional moment that is not very informative. Moment
conditions formed from Euler conditions sometimes do not identify the parameter of a model. See
Hansen, Heaton and Yarron, (1996) JBES V14, N3. I believe that this is the case here, though I
havent checked it carefully.

14.18

Exercises

1. Do the exercises in section 14.10.

2. Show how the GIV estimator presented in section 14.10 can be adapted to account for an
error term with HET and/or AUT.
3. For the GIV estimator presented in section 14.10, find the form of the expressions I (0 ) and
J (0 ) that appear in the asymptotic distribution of the estimator, assuming that an efficient
weight matrix is used.
4. The Octave script meps.m estimates a model for office-based doctpr visits (OBDV) using two
different moment conditions, a Poisson QML approach and an IV approach. If all conditioning variables are exogenous, both approaches should be consistent. If the PRIV variable is
endogenous, only the IV approach should be consistent. Neither of the two estimators is efficient in any case, since we already know that this data exhibits variability that exceeds what
is implied by the Poisson model (e.g., negative binomial and other models fit much better).
Test the exogeneity of the variable PRIV with a GMM-based Hausman-type test, using the
Octave script hausman.m for hints about how to set up the test.
5. Using Octave, generate data from the logit dgp. The script EstimateLogit.m should prove
quite helpful.
(a) Recall that E(yt |xt ) = p(xt , ) = [1 + exp(xt 0)]1 . Consider the moment condtions
(exactly identified) mt () = [yt p(xt , )]xt Estimate by GMM (using gmm_results),
using these moments.
(b) Estimate by ML (using mle_results).
(c) The two estimators should coincide. Prove analytically that the estimators coicide.
has a 2 (g K) distribution.
0
1 m()
6. Verify the missing steps needed to show that n m()
That is, show that the monster matrix is idempotent and has trace equal to g K.
7. For the portfolio example, experiment with the program using lags of 3 and 4 periods to
define instruments
(a) Iterate the estimation of = (, ) and to convergence.
(b) Comment on the results. Are the results sensitive to the set of instruments used? Look
Are these good instruments? Are the instruments highly correlated
as well as .
at
with one another? Is there something analogous to collinearity going on here?
8. Run the Octave script GMM/chi2gmm.m with several sample sizes. Do the results you obtain
seem to agree with the consistency of the GMM estimator? Explain.
9. The GMM estimator with an arbitrary weight matrix has the asymptotic distribution

h
i

d
0 1
0
0 1
n 0 N 0, (D W D
) D W W D
(D W D
)
Supposing that you compute a GMM estimator using an arbitrary weight matrix, so that this
result applies. Carefully explain how you could test the hypothesis H0 : R0 = r versus
HA : R0 6= r, where R is a given q k matrix, and r is a given q 1 vector. I suggest that
you use a Wald test. Explain exactly what is the test statistic, and how to compute every
quantity that appears in the statistic.

10. (proof that the GMM optimal weight matrix is one such that W = 1
) Consider the
difference of the asymptotic variance using an arbitrary weight matrix, minus the asymptotic
variance using the optimal weight matrix:
1

0
A = (D W D
)
0
Set B = (D W D
)

0
0
D W W D
(D W D
)

0
D W D 1
D

1

0
D 1
D

1
0

D 1
. Verify that A = B B . What

is the implication of this? Explain.

11. Recall the dynamic model with measurement error that was discussed in class:
yt

= + yt1
+ xt + t

yt

= yt + t

where t and t are independent Gaussian white noise errors. Suppose that yt is not observed, and instead we observe yt . If we estimate the equation
yt = + yt1 + xt + t
The Octave script GMM/SpecTest.m performs a Monte Carlo study of the performance of the
GMM criterion test,
d

n sn ()
2 (g K)

Examine the script and describe what it does. Run this script to verify that the test overrejects. Increase the sample size, to determine if the over-rejection problem becomes less
severe. Discuss your findings.

Chapter 15

Models for time series data

Hamilton, Time Series Analysis is a good reference for this section.
Up to now weve considered the behavior of the dependent variable yt as a function of other
variables xt . These variables can of course contain lagged dependent variables, e.g., xt = (wt , yt1 , ..., ytj ).
Pure time series methods consider the behavior of yt as a function only of its own lagged values,
unconditional on other observable variables. One can think of this as modeling the behavior of yt
after marginalizing out all other variables. While its not immediately clear why a model that has
other explanatory variables should marginalize to a linear in the parameters time series model,
most time series work is done with linear models, though nonlinear time series is also a large and
growing field. Well stick with linear time series models.

Basic concepts
Definition 53. [Stochastic process]A stochastic process is a sequence of random variables, indexed
by time: {Yt }
t=

Definition 54. [Time series] A time series is one observation of a stochastic process, over a specific
interval: {yt }nt=1 .
So a time series is a sample of size n from a stochastic process. Its important to keep in mind
that conceptually, one could draw another sample, and that the values would be different.
Definition 55. [Autocovariance] The j th autocovariance of a stochastic process is jt = E(yt
t )(ytj tj ) where t = E (yt ) .

Definition 56. [Covariance (weak) stationarity] A stochastic process is covariance stationary if it

has time constant mean and autocovariances of all orders:

= ,

jt

= j , t

As weve seen, this implies that j = j : the autocovariances depend only one the interval
between observations, but not the time of the observations.
Definition 57. [Strong stationarity]A stochastic process is strongly stationary if the joint distribution of an arbitrary collection of the {Yt } doesnt depend on t.
227

Since moments are determined by the distribution, strong stationarityweak stationarity.

What is the mean of Yt ? The time series is one sample from the stochastic process. One could
think of M repeated samples from the stoch. proc., e.g., {ytm } By a LLN, we would expect that
M
1 X
p
ytm E(Yt )
M M
m=1

lim

The problem is, we have only one sample to work with, since we cant go back in time and collect
another. How can E(Yt ) be estimated then? It turns out that ergodicity is the needed property.
Definition 58. [Ergodicity]. A stationary stochastic process is ergodic (for the mean) if the time
average converges to the mean
n

1X
p
yt
n t=1

(15.1)

A sufficient condition for ergodicity is that the autocovariances be absolutely summable:

|j | <

j=0

This implies that the autocovariances die off, so that the yt are not so strongly dependent that they
dont satisfy a LLN.
Definition 59. [Autocorrelation] The j th autocorrelation, j is just the j th autocovariance divided
by the variance:
j =

j
0

(15.2)

Definition 60. [White noise] White noise is just the time series literature term for a classical error.
t is white noise if i) E(t ) = 0, t, ii) V (t ) = 2 , t and iii) t and s are independent, t 6= s.
Gaussian white noise just adds a normality assumption.

15.1

ARMA models

With these concepts, we can discuss ARMA models. These are closely related to the AR and MA error processes that weve already discussed. The main difference is that the lhs variable is observed
directly now.

MA(q) processes
A q th order moving average (MA) process is
yt = + t + 1 t1 + 2 t2 + + q tq
where t is white noise. The variance is
0

= E (yt )

= E (t + 1 t1 + 2 t2 + + q tq )


= 2 1 + 12 + 22 + + q2

Similarly, the autocovariances are

j

= j + j+1 1 + j+2 2 + + q qj , j q
=

0, j > q

Therefore an MA(q) process is necessarily covariance stationary and ergodic, as long as 2 and all
of the j are finite.

AR(p) processes
An AR(p) process can be represented as
yt = c + 1 yt1 + 2 yt2 + + p ytp + t
The dynamic behavior of an AR(p) process can be studied by writing this pth order difference
equation as a vector first order difference equation:

yt

yt1
..
.

0
..
.

ytp+1

0
..
.

1
..

0
..

..

..

yt1

yt2
..
.

0
..
.

ytp

or
Yt = C + F Yt1 + Et
With this, we can recursively work forward in time:
Yt+1

= C + F Yt + Et+1
= C + F (C + F Yt1 + Et ) + Et+1
= C + F C + F 2 Yt1 + F Et + Et+1

and
Yt+2

= C + F Yt+1 + Et+2


= C + F C + F C + F 2 Yt1 + F Et + Et+1 + Et+2
= C + F C + F 2 C + F 3 Yt1 + F 2 Et + F Et+1 + Et+2

or in general
Yt+j = C + F C + + F j C + F j+1 Yt1 + F j Et + F j1 Et+1 + + F Et+j1 + Et+j
Consider the impact of a shock in period t on yt+j . This is simply
Yt+j
j
= F(1,1)
Et0 (1,1)
If the system is to be stationary, then as we move forward in time this impact must die off. Otherwise a shock causes a permanent change in the mean of yt . Therefore, stationarity requires that
j
lim F(1,1)
=0

 Save this result, well need it in a minute.

Consider the eigenvalues of the matrix F. These are the such that
|F IP | = 0
The determinant here can be expressed as a polynomial. For example, for p = 1, the matrix F is
simply
F = 1
so
|1 | = 0
can be written as
1 = 0
When p = 2, the matrix F is
"
F =
so

"
F IP =

and
|F IP | = 2 1 2
So the eigenvalues are the roots of the polynomial
2 1 2
which can be found using the quadratic equation. This generalizes. For a pth order AR process, the
eigenvalues are the roots of
p p1 1 p2 2 p1 p = 0
Supposing that all of the roots of this polynomial are distinct, then the matrix F can be factored as
F = T T 1
where T is the matrix which has as its columns the eigenvectors of F, and is a diagonal matrix
with the eigenvalues on the main diagonal. Using this decomposition, we can write
F j = T T 1

T T 1 T T 1

where T T 1 is repeated j times. This gives

F j = T j T 1
and

j1

0
j =

j2
..

.
jp

Supposing that the i i = 1, 2, ..., p are all real valued, it is clear that
j
lim F(1,1)
=0

requires that
|i | < 1, i = 1, 2, ..., p
e.g., the eigenvalues must be less than one in absolute value.
It may be the case that some eigenvalues are complex-valued. The previous result generalizes
to the requirement that the eigenvalues be less than one in modulus, where the modulus of a
complex number a + bi is
mod(a + bi) =

p
a2 + b2

This leads to the famous statement that stationarity requires the roots of the determinantal
polynomial to lie inside the complex unit circle. draw picture here.
When there are roots on the unit circle (unit roots) or outside the unit circle, we leave the
world of stationary processes.
j
is a dynamic multiplier or an impulse-response func Dynamic multipliers: yt+j /t = F(1,1)

tion. Real eigenvalues lead to steady movements, whereas complex eigenvalues lead to
ocillatory behavior. Of course, when there are multiple eigenvalues the overall effect can be
a mixture. pictures
Invertibility of AR process
To begin with, define the lag operator L
Lyt = yt1
The lag operator is defined to behave just as an algebraic quantity, e.g.,
L2 yt

= L(Lyt )
= Lyt1
= yt2

or
(1 L)(1 + L)yt

1 Lyt + Lyt L2 yt

1 yt2

A mean-zero AR(p) process can be written as

yt 1 yt1 2 yt2 p ytp = t
or
yt (1 1 L 2 L2 p Lp ) = t
Factor this polynomial as
1 1 L 2 L2 p Lp = (1 1 L)(1 2 L) (1 p L)
For the moment, just assume that the i are coefficients to be determined. Since L is defined to

operate as an algebraic quantitiy, determination of the i is the same as determination of the i

such that the following two expressions are the same for all z :
1 1 z 2 z 2 p z p = (1 1 z)(1 2 z) (1 p z)
Multiply both sides by z p
z p 1 z 1p 2 z 2p p1 z 1 p = (z 1 1 )(z 1 2 ) (z 1 p )
and now define = z 1 so we get
p 1 p1 2 p2 p1 p = ( 1 )( 2 ) ( p )
The LHS is precisely the determinantal polynomial that gives the eigenvalues of F. Therefore, the
i that are the coefficients of the factorization are simply the eigenvalues of the matrix F.
Now consider a different stationary process
(1 L)yt = t
Stationarity, as above, implies that || < 1.
Multiply both sides by 1 + L + 2 L2 + ... + j Lj to get




1 + L + 2 L2 + ... + j Lj (1 L)yt = 1 + L + 2 L2 + ... + j Lj t
or, multiplying the polynomials on the LHS, we get




1 + L + 2 L2 + ... + j Lj L 2 L2 ... j Lj j+1 Lj+1 yt = 1 + L + 2 L2 + ... + j Lj t
and with cancellations we have




1 j+1 Lj+1 yt = 1 + L + 2 L2 + ... + j Lj t
so


yt = j+1 Lj+1 yt + 1 + L + 2 L2 + ... + j Lj t
Now as j , j+1 Lj+1 yt 0, since || < 1, so


yt
= 1 + L + 2 L2 + ... + j Lj t
and the approximation becomes better and better as j increases. However, we started with
(1 L)yt = t
Substituting this into the above equation we have


yt
= 1 + L + 2 L2 + ... + j Lj (1 L)yt
so


1 + L + 2 L2 + ... + j Lj (1 L)
=1
and the approximation becomes arbitrarily good as j increases arbitrarily. Therefore, for || < 1,

define
1

(1 L)

j Lj

j=0

Recall that our mean zero AR(p) process

yt (1 1 L 2 L2 p Lp ) = t
can be written using the factorization
yt (1 1 L)(1 2 L) (1 p L) = t
where the are the eigenvalues of F, and given stationarity, all the |i | < 1. Therefore, we can
invert each first order polynomial on the LHS to get

X
X
X
yt =
j1 Lj
j2 Lj
jp Lj t
j=0

j=0

j=0

The RHS is a product of infinite-order polynomials in L, which can be represented as

yt = (1 + 1 L + 2 L2 + )t
where the i are real-valued and absolutely summable.
The i are formed of products of powers of the i , which are in turn functions of the i .
The i are real-valued because any complex-valued i always occur in conjugate pairs. This
means that if a + bi is an eigenvalue of F, then so is a bi. In multiplication
(a + bi) (a bi)

a2 abi + abi b2 i2

a2 + b2

which is real-valued.
This shows that an AR(p) process is representable as an infinite-order MA(q) process.
Recall before that by recursive substitution, an AR(p) process can be written as
Yt+j = C + F C + + F j C + F j+1 Yt1 + F j Et + F j1 Et+1 + + F Et+j1 + Et+j
If the process is mean zero, then everything with a C drops out. Take this and lag it by j
periods to get
Yt = F j+1 Ytj1 + F j Etj + F j1 Etj+1 + + F Et1 + Et
As j , the lagged Y on the RHS drops out. The Ets are vectors of zeros except for their
first element, so we see that the first equation here, in the limit, is just
yt =

Fj

1,1 tj

j=0

which makes explicit the relationship between the i and the i (and the i as well, recalling
the previous factorization of F j ).

Moments of AR(p) process

The AR(p) process is
yt = c + 1 yt1 + 2 yt2 + + p ytp + t
Assuming stationarity, E(yt ) = , t, so
= c + 1 + 2 + ... + p
so
=

c
1 1 2 ... p

and
c = 1 ... p
so
yt = 1 ... p + 1 yt1 + 2 yt2 + + p ytp + t
= 1 (yt1 ) + 2 (yt2 ) + ... + p (ytp ) + t
With this, the second moments are easy to find: The variance is
0 = 1 1 + 2 2 + ... + p p + 2
The autocovariances of orders j 1 follow the rule
j

= E [(yt ) (ytj ))]

= E [(1 (yt1 ) + 2 (yt2 ) + ... + p (ytp ) + t ) (ytj )]
= 1 j1 + 2 j2 + ... + p jp

Using the fact that j = j , one can take the p + 1 equations for j = 0, 1, ..., p, which have p + 1
unknowns ( 2 , 0 , 1 , ..., p ) and solve for the unknowns. With these, the j for j > p can be solved
for recursively.

Invertibility of MA(q) process

An MA(q) can be written as
yt = (1 + 1 L + ... + q Lq )t
As before, the polynomial on the RHS can be factored as
(1 + 1 L + ... + q Lq ) = (1 1 L)(1 2 L)...(1 q L)
and each of the (1 i L) can be inverted as long as each of the |i | < 1. If this is the case, then we
can write
(1 + 1 L + ... + q Lq )1 (yt ) = t
where
(1 + 1 L + ... + q Lq )1
will be an infinite-order polynomial in L, so we get

X
j=0

j Lj (ytj ) = t

with 0 = 1, or
(yt ) 1 (yt1 ) 2 (yt2 ) + ... = t
or
yt = c + 1 yt1 + 2 yt2 + ... + t
where
c = + 1 + 2 + ...
So we see that an MA(q) has an infinite AR representation, as long as the |i | < 1, i = 1, 2, ..., q.
It turns out that one can always manipulate the parameters of an MA(q) process to find an
invertible representation. For example, the two MA(1) processes
yt = (1 L)t
and
yt = (1 1 L)t
have exactly the same moments if
2 = 2 2
For example, weve seen that
0 = 2 (1 + 2 ).
Given the above relationships amongst the parameters,
0 = 2 2 (1 + 2 ) = 2 (1 + 2 )
so the variances are the same. It turns out that all the autocovariances will be the same,
as is easily checked. This means that the two MA processes are observationally equivalent.
As before, its impossible to distinguish between observationally equivalent processes on the
basis of data.
For a given MA(q) process, its always possible to manipulate the parameters to find an
invertible representation (which is unique).
Its important to find an invertible representation, since its the only representation that allows one to represent t as a function of past y 0 s. The other representations express t as a
function of future y 0 s
Why is invertibility important? The most important reason is that it provides a justification
for the use of parsimonious models. Since an AR(1) process has an MA() representation,
one can reverse the argument and note that at least some MA() processes have an AR(1)
representation. Likewise, some AR() processes have an MA(1) representation. At the time
of estimation, its a lot easier to estimate the single AR(1) or MA(1) coefficient rather than
the infinite number of coefficients associated with the MA() or AR() representation.
This is the reason that ARMA models are popular. Combining low-order AR and MA models
can usually offer a satisfactory representation of univariate time series data using a reasonable number of parameters.
Stationarity and invertibility of ARMA models is similar to what weve seen - we wont go
into the details. Likewise, calculating moments is similar.
Exercise 61. Calculate the autocovariances of an ARMA(1,1) model:(1 + L)yt = c + (1 + L)t

15.2

State space models

Ref. Gouriroux and Monfort (1997) Time Series and Dynamic Models, chapter 15
A linear Gaussian state space model has the form
zt+1 =At zt + Bt t
yt =Ct zt + Dt t
0
is Gaussian white noise. The first equation is the state equation, and the second
0t t0
is the measurement equation. The variables yt are observable. Some or all of the state variables zt
where

may not be observable.

Kalman filter
Example: Estimation of ARMA models
mle with normality, conditional ML

Example: Estimation of a linearized RBC model

15.3

VAR models

1. Consider the model

yt = C + A1 yt1 + t
E(t 0t ) =
E(t 0s ) = 0, t 6= s
where yt and t are G 1 vectors, C is a G 1 of constants, and A1 is a G G matrix of
parameters. The matrix is a GG covariance matrix. Assume that we have n observations.
This is a vector autoregressive model, of order 1 - commonly referred to as a VAR(1) model.
As shown in Section 10.3, it is efficient to estimate a VAR model using OLS equation by equation,
there is no need to use GLS, in spite of the cross equation correlations.

15.4

ARCH and GARCH

ARCH (autoregressive conditionally heteoscedastic) models appeared in the literature in 1982, in

Engle, Robert F. (1982). "Autoregressive Conditional Heteroscedasticity with Estimates of Variance
of United Kingdom Inflation", Econometrica 50:987-1008. This paper stimulated a very large
growth in the literature for a number of years afterward. The related GARCH (generalized ARCH)
model is now one of the most widely used models for financial time series.
Financial time series often exhibit several type of behavior:
volatility clustering: periods of low variation can be followed by periods of high variation
fat tails, or excess kurtosis: the marginal density of a series is more strongly peaked and has
fatter tails than does a normal distribution with the same mean and variance.
other features, such as leverage (correlation between returns and volatility) and perhaps
slight autocorrelation within the bounds allowed by arbitrage.

The data set nysewk.gdt, which is provided with Gretl, provides an example. If we compute 100
times the growth rate of the series, using log differences, we can obtain the plots in Figure 15.1. In
the first we clearly see volatility clusters, and in the second, we see excess kurtosis and tails fatter
than the normal distribution. The skewness suggests that leverage may be present.
The presence of volatility clusters indicates that the variance of the series is not constant over
time, conditional on past events. Engles ARCH paper was the first to model this feature.

ARCH
A basic ARCH specification is
yt = + yt1 + t
t = t ut
t2 = +

q
X

i 2ti

i=1

where the ut are Gaussian white noise shocks. The ARCH variance is a moving average process.
Previous large shocks to the series cause the conditional variance of the series to increase. There is
no leverage: negative shocks have the same impact on the future variance as do positive shocks..
for t2 to be positive for all realizations of {t }, we need > 0, i 0, i.
to ensure that the model is covariance stationary, we need

i < 1. Otherwise, the vari-

ances will explode off to infinity.

Given that t is a linear combination of normal random variables, it is also normally distributed.
To find the likelihood, first note that the series ut = (yt gt ) /t is iid Gaussian, so the likelihood
is simply the product of standard normal densities.
u N (0, I), so
 2
n
Y
1
u
exp t
f (u) =
2
2
t=1
The joint density for y can be constructed using a change of variables. We have ut = (yt yt1 ) /t ,
Qn 1
1
u
t
so u
t=1 t , so doing a change of variables,
yt = t and | y 0 | =
n
Y

1 1
1

f (y; ) =
exp

2
2
t
t=1

yt yt1
t

2 !

where includes the parameters in gt and the alpha parameters of the ARCH specification. Taking
logs,
n
n
X

1X
ln t
ln L() = n ln 2
2 t=1
t=1

yt yt1
t

2
.

In principle, this is easy to maximize. Some complications can arise when the restrictions for
positivity and stationarity are imposed. Consider a fairly short data series with low volatility in
the initial part, and high volatility at the end. This data appears to have a nonstationary variance
sequence. If one attempts to estimate and ARCH model with stationarity imposed, the data and
the restrictions are saying two different things, which can make maximization of the likelihood
function difficult.
The Octave script GarchExample.m illustrates estimation of an ARCH(1) model, using the NYSE
closing price data.

Figure 15.1: NYSE weekly close price, 100 log differences

(a) Time series plot

(b) Frequency distribution

GARCH
Note that an ARCH model specifies the variance process as a moving average. For the same reason
that an ARMA model may be used to parsimoniously model a series instead of a high order AR
or MA, one can do the same thing for the variance series. A basic GARCH(p,q) (Bollerslev, Tim
(1986). "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics,
31:307-327) specification is
yt = + yt1 + t
t = t ut
t2 = +

q
X
i=1

i 2ti +

p
X

2
i ti

i=1

The idea is that a GARCH model with low values of p and q may fit the data as well or better than
an ARCH model with large q.
the model also requires restrictions for positive variance and stationarity, which are:
>0
i 0, i = 1, ..., q
i 0, i = 1, ..., p
Pq
Pp

i=1 i +
i=1 i < 1.
to estimate a GARCH model, you need to initialize 02 at some value. The sample unconditional variance is one possibility. Another choice could be the sample variance of the initial
elements of the sequence. One can also backcast the conditional variance.
The GARCH model also requires restrictions on the parameters to ensure stationarity and positivity
of the variance. A useful modification is the EGARCH model (exponential GARCH, Nelson, D. B.
(1991). "Conditional heteroskedasticity in asset returns: A new approach", Econometrica 59: 347370). This model treats the logarithm of the variance as an ARMA process, so the variance will
be positive without restrictions on the parameters. It is also possible to introduce asymmetry
(leverage) and non-normality.
The Octave script GarchExample.m illustrates estimation of a GARCH(1,1) model, using the
NYSE closing price data. You can get the same results more quickly using Gretl, which takes
advantage of C code for the model. If you play with the example, you can see that the results
are sensitive to start values. The likelihood function does not appear to have a nice well-defined
global maximum. Thus, one needs to use care when estimating this sort of model, or rely on some
software that is known to work well.
Note that the test of homoscedasticity against ARCH or GARCH involves parameters being
on the boundary of the parameter space. Also, the reducton of GARCH to ARCH has the same
problem. Testing needs to be done taking this into account. See Demos and Sentana (1998)
Journal of Econometrics.

15.5

Nonstationarity and cointegration

15.6

Exercises

1. Use Matlab to estimate the same GARCH(1,1) model as in the GarchExample.m script provided above. Also, estimate an ARCH(4) model for the same data. If unconstrained estimation does not satisfy stationarity restrictions, then do contrained estimation. Compare

likelihood values. Which of the two models do you prefer? But do the models have the same
number of parameters? Find out what is the consistent Akaike information criterion or the
Bayes information criterion and what they are used for. Compute one or the other, or both,
and discuss what they tell you about selecting between the two models.

Chapter 16

Bayesian methods
References I have used to prepare these notes: Cameron and Trivedi, Microeconometrics: Methods
and Applications, Chapter 13; Chernozhukov and Hong (2003), An MCMC approach to classical
estimation, Journal of Econometrics; Gallant and Tauchen, EMM: A program for efficient method
of moments estimation; Hoogerheide, van Dijk and van Oest (2007) Simulation Based Bayesian
Econometric Inference: Principles and Some Recent Computational Advances.
This chaper provides a brief introduction to Bayesian methods, which form a large part of
econometric research, especially in the last two decades. Advances in computational methods,
combined with practical advantages of Bayesian methods have contributed to the popularity of
this approach.

16.1

Definitions

The Bayesian approach treats the parameter of a model as a random vector. The parameter has a
density, (), which is known as the prior. It is assumed that the econometrician can provide this
density, which reflects current beliefs about the parameter.
We also have sample information, y={y1 , y2 , ...yn }. Were already familiar with the likelihood
function, f (y|), which is the density of the sample given a parameter value.
Given these two pieces, we can write the joint density of the sample and the parameter:
f (y, ) = f (y|)()
We can get the marginal likelihood by integrating out the parameter, integrating over its support
:

Z
f (y) =

f (y, )d

The last step is to get the posterior of the parameter. This is simply the density of the parameter
conditional on the sample, and we get it in the normal way we get a conditional density, using
Bayes theorem
f (|y) =

f (y, )
f (y|)()
=
f (y)
f (y)

The posterior reflects the learning that occurs about the parameter when one receives the sample
information. The sources of information used to make the posterior are the prior and the likelihood
function. Once we have the posterior, one can provide a complete probabilistic description about
our updated beliefs about the parameter, using quantiles or moments of the posterior. The posterior
mean or median provide the Bayesian analogue of the frequentist point estimator in the form of
the ML estimator. We can define regions analogous to confidence intervals by using quantiles of
241

the posterior, or the marginal posterior.

So far, this is pretty straightforward. The complications are often computational. To illustrate,
the posterior mean is
R

Z
E(|y) =

f (|y)d =

f (y|)()d
f (y, )d

One can see that a means of integrating will be needed. Only in very special cases will the integrals
have analytic solutions. Otherwise, computational methods will be needed.

16.2

Philosophy, etc.

So, the classical paradigm views the data as generated by a data generating process, which is a
perhaps unknown model characterized by a parameter vector, and the data is generated from the
model at a particular value of the parameter vector. Bayesians view data as given, and update
beliefs about a random parameter using the information about the parameter contained in the
data.
Bayesians and frequentists have a long tradition of arguing about the meaning and interpretation of their respective procedures. Heres my take on the debate. Fundamentally, I take the
frequentist view: I find it pleasing to think about a model with a fixed non-random parameter
about which we would like to learn. I like the idea of a point estimator that gives a best guess
about the true parameter. However, we shouldnt reinvent the wheel each time we get a new sample: previous samples have information about the parameter, and we should use all of the available
information. A pure frequentist approach would require writing the joint likelihood of all samples,
which would almost certainly constitute an impossible task. The Bayesian approach concentrates
all of the information coming from previous work in the form of a prior. A fairly simple, easy to use
prior may not exactly capture all previous information, but it could offer a handy and reasonably
accurate summary. So, the idea of a prior as a summary of what we have learned may simply be
viewed as a practical solution to the problem of using all the available information. Given that its
a summary, one may as well use a convenient form, as long as its plausible and the results dont
depend too exaggerately on the prior.
About the likelihood function, fortunately, Bayesians and frequentists are in agreement, so
theres no need for further comment.
When we get to how to generate and interpret results, there is some divergence. Frequentists
maximize the likelihood function, and compute standard errors, etc., using the methods already
explained in these notes. A frequentist could test the hypothesis that 0 = by seeing if the data
are sufficiently likely conditional on the parameter value . A Bayesian would check if is a
plausible value conditional on the observed data.
I have criticized the frequentist practice of using only the current sample, ignoring what previous work has told us about the parameter, simply because its too hard to write the overall joint
likelihood for all samples. So to be fair, heres a criticism of the Bayesian approach. If were doing
Bayesian learning, what is it were learning about? If its not a fixed parameter value then what
is it? What is the process that generated the sample data? If the parameter is random, was the
sample generated at a single realization, or at many realizations? If we had an infinite sample,
then the Bayesian estimators (e.g., posterior mean or median) converge to a point. What is that
point if its not the same true parameter value that the frequentists are trying to estimate? Why
would one use noninformative priors for ones whole career - dont we believe what we learned
from the last paper we wrote? These questions often receive no answer, or obscure answers.
It turns out that one can analyze Bayesian estimators from a classical (frequentist) perspective.
It also turns out that Bayesian estimators may be easier to compute reliably than analogous classical

Figure 16.1: Bayesian estimation, exponential likelihood, lognormal prior

(a) N=10

(b) N=50

estimators. These computational advantages, combined with the ability to use information from
previous work in an intelligent way, make the study of Bayesian methods attractive for frequentists.
If a Bayesian takes the view that there is a fixed data generating process, and Bayesian learning
leads in the limit to the same fixed true value that frequentists posit, then the study of frequentist
theory will be useful to a Bayesian practitioner.
For the rest of this, I will adopt the classical, frequentist perspective, and study the behavior of
Bayesian estimators in this context.

16.3

Example

Suppose data is generated by i.i.d. sampling from an exponential distribution with mean . An
exponential random variable takes values on the positive real numbers. Waiting times are often
modeled using the exponential distribution.
The density of a typical sample element is f (y|) =

1 y/
.
e

The likelihood is simply the

product of the sample contributions.

Suppose the prior for is lognormal(1,1). This means that the logarithm of is standard
normal. We use a lognormal prior because it enforces the requirement that the parameter of
the exponential density be positive.
The Octave script BayesExample1.m implements Bayesian estimation for this setup.
with a sample of 10 observations, we obtain the results in panel (a) of Figure 16.1, while
with a sample of size 50 we obtain the results in panel (b). Note how the posterior is more
concentrated around the true parameter value in panel (b). Also note how the posterior
mean is closer to the prior mean when the sample is small. When the sample is small,
the likelihood function has less weight, and more of the information comes from the prior.
When the sample is larger, the likelihood function will have more weight, and its effect will
dominate the priors.

16.4

Theory

Chernozhukov and Hong (2003) An MCMC Approach to Classical Estimation http://www.sciencedirect.

com/science/article/pii/S0304407603001003 is a very interesting article that shows how Bayesian

Figure 16.2: Chernozhukov and Hong, Theorem 2

methods may be used with criterion functions that are associated with classical estimation techniques. For example, it is possible to compute a posterior mean version of a GMM estimator.
Chernozhukov and Hong provide their Theorem 2, which proves consistency and asymptotic normality for a general class of such estimators. When the criterion function Ln () in their paper is
set to the log-likelihood function, the pseudo-prior () is a real Bayesian prior, and the penalty
function n is the squared loss function (see the paper), then the class of estimators discussed by
CH reduces to the ordinary Bayesian posterior mean. As such, their Theorem 2, in Figure 16.2
tells us that this estimator is consistent and asymptotically normally distributed. In particular,
the Bayesian posterior mean has the same asymptotic distribution as does the ordinary maximum
likelihood estimator.
the intuition is clear: as the amount of information coming from the sample increases, the
likelihood function brings an increasing amount of information relative to the prior. Eventually, the prior is no longer important for determining the shape of the posterior.
when the sample is large, the shape of the posterior depends on the likelihood function. The
likelihood function collapses around 0 when the sample is generated at 0 . The same is true
of the posterior, it narrows around 0 . This causes the posterior mean to converge to the true
parameter value. In fact, all quantiles of the posterior converge to 0 . Chernozhukov and
Hong discuss estimators defined using quantiles.
For an econometrician coming from the frequentist perspective, this is attractive. The Bayesian
estimator has the same asymptotic behavior as the MLE. There may be computational advantages to using the Bayesian approach (this is the main motivation of CHs paper) and if an
informed prior is used, the small sample performance of the Bayesian estimator may be better
than that of a similar classical estimator.

16.5

Computational methods

To compute the posterior mean, we need to evaluate E(|y) =

f (|y)d =

f (y|)()d/

Note that both of the integrals are multiple integrals, with the dimension given by that of the
parameter, .
Under some special circumstances, the integrals may have analytic solutions: e.g., Gaussian
likelihood with a Gaussian prior leads to a Gaussian posterior.
When the dimension of the parameter is low, quadrature methods may be used. What was
done in as was done in BayesExample1.m is an unsofisticated example of this. More sophisticated methods use an intelligently chosen grid to reduce the number of function evaluations.
Still, these methods only work for dimensions up to 3 or so.

f (y, )d.

 Otherwise, some form of simulation-based Monte Carlo integration must be used. The
PS
basic idea is that E(|y) can be approximated by (1/S) s=1 s , where s is a random draw
from the posterior distribution f (|y). The trick is how to make draws from the posterior when
in general we cant compute the posterior.
the law of large numbers tells us that this average will converge to the desired expectation as S gets large
convergence will be more rapid if the random draws are independent of one another,
but insisting on independence may have computational drawbacks.
Monte Carlo methods include importance sampling, Markov chain Monte Carlo (MCMC) and
sequential Monte Carlo (SMC, also known as particle filtering). The great expansion of these
methods over the years has caused Bayesian econometrics to become much more widely used
than it was in the not so distant (for some of us) past. There is much literature - here we will
only look at a basic example that captures the main ideas.

MCMC
Variants of Markov chain Monte Carlo have become a very widely used means of computing
Bayesian estimates. See Tierney (1994) Markov Chains for Exploring Posterior Distributions
Annals of Statistics and Chib and Greenberg (1995) Understanding the Metropolis-Hastings algorithm The American Statistician.
Lets consider the basic Metropolis-Hastings MCMC algorithm. We will generate a long realization of a Markov chain process for , as follows:
The prior density is (), as above. Let g( ; s ) be a proposal density, which generates a new
trial parameter value given the most recently accepted parameter value s . A proposal will be
accepted if

f ( |y) g(s ; )
>
f (s |y) g( ; s )

where is a U (0, 1) random variate.

There are two parts to the numerator and denominator: the posterior, and the proposal density.
Focusing on the numerator, when the trial value of the proposal has a higher posterior, acceptance
is favored. The other factor is the density associated with returning to s when starting at ,
which has to do with the reversability of the Markov chain. If this is too low, acceptance is not
favored. We dont want to jump to a new region if we will never get back. The two together mean
that we will jump to a new area only if we are able to eventually jump back with a reasonably
high probability. The probability of jumping is higher when the new area has a higher posterior
density, but lower if its hard to get back. The idea is to sample from all regions of the posterior,
those with high and low density, sampling more heavily from regions of high density. We want
to go occasionally to regions of low density, but it is important not to get stuck there. Consider
a bimodal density: we want to explore the area around both modes. To be able to do that, it is
important that the proposal density allows us to be able to jump between modes. Understanding
in detail why this makes sense is the tricky and elegant part of the theory, see the references for
more information.
Note that the ratio of posteriors is equal to the ratio of likelihoods times the ratio of priors:
f (y| ) ( )
f ( |y)
=
s
f ( |y)
f (y|s ) (s )
because the marginal likelihood f (y) is the same in both cases. We dont need to compute
that integral! We dont need to know the posterior, either. The acceptance criterion can be

written as: accept if

f (y| ) ( ) g(s ; )
>
f (y|s ) (s ) g( ; s )

otherwise, reject
From this, we see that the information needed to determine if a proposal is accepted or
rejected is the prior, the proposal density, and the likelihood function f (y|).
the steps are:
1. the algorithm is initialized at some 1
2. for s = 2, ..., S,
(a) draw from g( ; s )
(b) according to the acceptance/rejection criterion, if the result is acceptance, set s+1 = ,
otherwise set s+1 = s
(c) iterate
Once the chain is considered to have stabilized, say at iteration r, the values of s for s > r
are taken to be draws from the posterior. The posterior mean is computed as the simple
average of the value. Quantiles, etc., can be computed in the appropriate fashion.
the art of applying these methods consists of providing a good candidate density so that the
acceptance rate is reasonably high. Otherwise, the chain will be highly autocorrelated, with
long intervals where the same value of appears. There is a vast literature on this, and the
vastness of the literature should serve as a warning that getting this to work in practice is not
necessarily a simple matter. If it were, there would be fewer papers on the topic.

16.6

Examples

Simple example
The simple exponential example with lognormal prior can be implemented using MH MCMC, and
this is done in the Octave script BayesExample2.m . Play around with the sample size and the
tuning parameter, and note the effects on the computed posterior mean and on the acceptance
rate. An example of output is given in Figure 16.3. In that Figure, the chain shows relatively long
periods of rejection, meaning that the tuning parameter needs to be lowered, to cause the random
walk to be a little less random.

Bayesian estimation of DSGE model

In Section 13.10, a simple DSGE model was estimated by ML. EstimateRBC_Bayesian.mod is a
Dynare .mod file that lets you do the same thing using Bayesian methods, with MCMC.

Figure 16.3: Metropolis-Hastings MCMC, exponential likelihood, lognormal prior

16.7

Exercises

1. In the MH MCMC algorithm, discuss the need for tuning the proposal density to achieve a
reasonable acceptance rate. Whats the effect of too low an acceptance rate? And whats the
problem with a high acceptance rate?

Chapter 17

Introduction to panel data

Reference: Cameron and Trivedi, 2005, Microeconometrics: Methods and Applications, Part V, Chapters 21 and 22 (plus 23 if you have special interest in the topic).
In this chapter well look at panel data. Panel data is an important area in applied econometrics,
simply because much of the available data has this structure. Also, it provides an example where
things weve already studied (GLS, endogeneity, GMM, Hausman test) come into play. There has
been much work in this area, and the intention is not to give a complete overview, but rather to
highlight the issues and see how the tools we have studied can be applied.

17.1

Generalities

Panel data combines cross sectional and time series data: we have a time series for each of the
agents observed in a cross section. The addition of temporal information can in principle allow
us to investigate issues such as persistence, habit formation, and dynamics. Starting from the
perspective of a single time series, the addition of cross-sectional information allows investigation
of heterogeneity. In both cases, if parameters are common across units or over time, the additional
data allows for more precise estimation.
The basic idea is to allow variables to have two indices, i = 1, 2, ..., n and t = 1, 2, ..., T . The
simple linear model
yi = + xi + i
becomes
yit = + xit + it
We could think of allowing the parameters to change over time and over cross sectional units. This
would give
yit = it + xit it + it
The problem here is that there are more parameters than observations, to the model is not identified. We need some restraint! The proper restrictions to use of course depend on the problem at
hand, and a single model is unlikely to be appropriate for all situations. For example, one could
have time and cross-sectional dummies, and slopes that vary by time:
yit = i + t + xit t + it
There is a lot of room for playing around here. We also need to consider whether or not n and T
are fixed or growing. Well need at least one of them to be growing in order to do asymptotics.
To provide some focus, well consider common slope parameters, but agent-specific intercepts,

248

which:
yit = i + xit + it

(17.1)

I will refer to this as the simple linear panel model. This is the model most often encountered
in the applied literature. It is like the original cross-sectional model, in that the 0 s are constant
over time for all i. However were now allowing for the constant to vary across i (some individual
heterogeneity). The 0 s are fixed over time, which is a testable restriction, of course. We can
consider what happens as n but T is fixed. This would be relevant for microeconometric
panels, (e.g., the PSID data) where a survey of a large number of individuals may be done for a
limited number of time periods. Macroeconometric applications might look at longer time series for
a small number of cross-sectional units (e.g., 40 years of quarterly data for 15 European countries).
For that case, we could keep n fixed (seems appropriate when dealing with the EU countries), and
do asymptotics as T increases, as is normal for time series. The asymptotic results depend on how
we do this, of course.
Why bother using panel data, what are the benefits? The model
yit = i + xit + it
is a restricted version of
yit = i + xit i + it
which could be estimated for each i in turn. Why use the panel approach?
Because the restrictions that i = j = ... = , if true, lead to more efficient estimation.
Estimation for each i in turn will be very uninformative if T is small.
Another reason is that panel data allows us to estimate parameters that are not identified by
cross sectional (time series) data. For example, if the model is
yit = i + t + xit t + it
and we have only cross sectional data, we cannot estimate the i . If we have only time
series data on a single cross sectional unit i = 1, we cannot estimate the t . Cross-sectional
variation allows us to estimate parameters indexed by time, and time series variation allows
us to estimate parameters indexed by cross-sectional unit. Parameters indexed by both i and
t will require other forms of restrictions in order to be estimable.
The main issues are:
can be estimated consistently? This is almost always a goal.
can the i be estimated consistently? This is often of secondary interest.
sometimes, were interested in estimating the distribution of i across i.
are the i correlated with xit ?
does the presence of i complicate estimation of ?
what about the covariance stucture? Were likely to have HET and AUT, so GLS issue will
probably be relevant. Potential for efficiency gains.

17.2

Static issues and panel data

To begin with, assume that the xit are weakly exogenous variables (uncorrelated with it ), and
that the model is static: xit does not contain lags of yit . The basic problem we have in the panel

data model yit = i + xit + it is the presence of the i . These are individual-specific parameters.
Or, possibly more accurately, they can be thought of as individual-specific variables that are not
observed (latent variables). The reason for thinking of them as variables is because the agent may
choose their values following some process.
Define = E(i ), so E(i ) = 0. Our model yit = i + xit + it may be written
yit = i + xit + it
= + xit + (i + it )
= + xit + it
Note that E(it ) = 0. A way of thinking about the data generating process is this: First, i is drawn,
either in turn from the set of n fixed values, or randomly, and then x is drawn from fX (z|i ). In
either case, the important point is that the distribution of x may vary depending on the realization,
i . Thus, there may be correlation between i and xit , which means that E(xit it ) 6=0 in the above
equation. This means that OLS estimation of the model would lead to biased and inconsistent
estimates. However, it is possible (but unlikely for economic data) that xit and it are independent
or at least uncorrelated, if the distribution of xit is constant with respect to the realization of i .
In this case OLS estimation would be consistent.
Fixed effects: when E(xit it ) 6=0, the model is called the fixed effects model
Random effects: when E(xit it ) = 0, the model is called the random effects model.
I find this to be pretty poor nomenclature, because the issue is not whether effects are fixed
or random (they are always random, unconditional on i). The issue is whether or not the effects
are correlated with the other regressors. In economics, it seems likely that the unobserved variable
is probably correlated with the observed regressors, x (this is simply the presence of collinearity
between observed and unobserved variables, and collinearity is usually the rule rather than the
exception). So, we expect that the fixed effects model is probably the relevant one unless special
circumstances mean that the i are uncorrelated with the xit .

17.3

Estimation of the simple linear panel model

Fixed effects: The within estimator

How can we estimate the parameters of the simple linear panel model (equation 17.1) and what
properties do the estimators have? First, we assume that the i are correlated with the xit (fixed
effects model ). The model can be written as yit = + xit + it , and we have that E(xit it ) 6=0.
As such, OLS estimation of this model will give biased an inconsistent estimated of the parameters
and . The within estimator is a solution - this involves subtracting the time series average
from each cross sectional unit.
xi =

T
1X
xit
T t=1

i =

T
1X
it
T t=1

yi =

T
T
T
1X
1X
1X
yit = i +
xit +
it
T t=1
T t=1
T t=1

y i = i + xi + i

(17.2)

The transformed model is

yit y i = i + xit + it i xi i

yit

xit

(17.3)

it

where xit = xit xi and it = it i . In this model, it is clear that xit and it are uncorrelated,
as long as the original regressors xit are strongly exogenous with respect to the original error it
(E(xit is ) = 0, t, s). Thus OLS will give consistent estimates of the parameters of this model, .
What about the i ? Can they be estimated? An estimator is

i =

T

1 X
yit xit
T t=1

Its fairly obvious that this is a consistent estimator if T . For a short panel with fixed T, this
estimator is not consistent. Nevertheless, the variation in the
i can be fairly informative about
the heterogeneity. A couple of notes:
an equivalent approach is to estimate the model
yit =

n
X

dj,it i + xit + it

j=1

by OLS. The dj , j = 1, 2, ..., n are n dummy variables that take on the value 1 if j = 1, zero
otherwise. They are indicators of the cross sectional unit of the observation. (Write out form
of regressor matrix on blackboard). Estimating this model by OLS gives numerically exactly
the same results as the within estimator, and you get the
i automatically. See Cameron
and Trivedi, section 21.6.4 for details. An interesting and important result known as the
Frisch-Waugh-Lovell Theorem can be used to show that the two means of estimation give
identical results.
This last expression makes it clear why the within estimator cannot estimate slope coefficients corresponding to variables that have no time variation. Such variables are perfectly
collinear with the cross sectional dummies dj . The corresponding coefficients are not identified.
OLS estimation of the within model is consistent, but probably not efficient, because it is
highly probable that the it are not iid. There is very likely heteroscedasticity across the i and
autocorrelation between the T observations corresponding to a given i. One needs to estimate
the covariance matrix of the parameter estimates taking this into account. It is possible to use
GLS corrections if you make assumptions regarding the het. and autocor. Quasi-GLS, using a
possibly misspecified model of the error covariance, can lead to more efficient estimates than
simple OLS. One can then combine it with subsequent panel-robust covariance estimation
to deal with the misspecification of the error covariance, which would invalidate inferences
if ignored. The White heteroscedasticity consistent covariance estimator is easily extended
to panel data with independence across i, but with heteroscedasticity and autocorrelation
within i, and heteroscedasticity between i. See Cameron and Trivedi, Section 21.2.3.

Estimation with random effects

The original model is
yit = i + xit + it

This can be written as

yit

= + xit + (i + it )

yit

= + xit + it

(17.4)

where E(it ) = 0, and E(xit it ) = 0. As such, the OLS estimator of this model is consistent. We
can recover estimates of the i as discussed above. It is to be noted that the error it is almost
certainly heteroscedastic and autocorrelated, so OLS will not be efficient, and inferences based
on OLS need to be done taking this into account. One could attempt to use GLS, or panel-robust
covariance matrix estimation, or both, as above.
There are other estimators when we have random effects, a well-known example being the
between estimator, which operates on the time averages of the cross sectional units. There is
no advantage to doing this, as the overall estimator is already consistent, and averaging looses
information (efficiency loss). One would still need to deal with cross sectional heteroscedasticity
when using the between estimator, so there is no gain in simplicity, either.
It is to be emphasized that random effects is not a plausible assumption with most economic
data, so use of this estimator is discouraged, even if your statistical package offers it as an option. Think carefully about whether the assumption is warranted before trusting the results of this
estimator.

Hausman test
Suppose youre doubting about whether fixed or random effects are present. If we have fixed effects, then the within estimator will be consistent, but the estimator of the previous section will
not. Evidence that the two estimators are converging to different limits is evidence in favor of
fixed effects, not random effects. A Hausman test statistic can be computed, using the difference
between the two estimators. The null hypothesis is random effects so that both estimators are
consistent. When the test rejects, we conclude that fixed effects are present, so the within estimator should be used. Now, what happens if the test does not reject? One could optimistically turn
to the random effects model, but its probably more realistic to conclude that the test may have
low power. Failure to reject does not mean that the null hypothesis is true. After all, estimation of
the covariance matrices needed to compute the Hausman test is a non-trivial issue, and is a source
of considerable noise in the test statistic (noise=low power). Finally, the simple version of the
Hausman test requires that the estimator under the null be fully efficient. Achieving this goal is
probably a utopian prospect. A conservative approach would acknowledge that neither estimator
is likely to be efficient, and to operate accordingly. I have a little paper on this topic, Creel, Applied
Economics, 2004. See also Cameron and Trivedi, section 21.4.3.

17.4

Dynamic panel data

When we have panel data, we have information on both yit as well as yi,t1 . One may naturally
think of including yi,t1 as a regressor, to capture dynamic effects that cant be analyed with only
cross-sectional data. Excluding dynamic effects is often the reason for detection of spurious AUT of
the errors. With dynamics, there is likely to be less of a problem of autocorrelation, but one should
still be concerned that some might still be present. The model becomes
yit

= i + yi,t1 + xit + it

yit

= + yi,t1 + xit + (i + it )

yit

= + yi,t1 + xit + it

We assume that the xit are uncorrelated with it . Note that i is a component that determines
both yit and its lag, yi,t1 . Thus, i and yi,t1 are correlated, even if the i are pure random
effects (uncorrelated with xit ). So, yi,t1 is correlated with it . For this reason, OLS estimation
is inconsistent even for the random effects model, and its also of course still inconsistent for the
fixed effects model. When regressors are correlated with the errors, the natural thing to do is start
thinking of instrumental variables estimation, or GMM.
To illustrate, consider a simple linear dynamic panel model
yit = i + 0 yit1 + it

(17.5)

where it N (0, 1), i N (0, 1), 0 = 0, 0.3, 0.6, 0.9 and i and i are independently distributed.
Tables 17.1 and 17.2 present bias and RMSE for the within estimator (labeled as ML) and some
simulation-based estimators. Note that the within estimator is very biased, and has a large RMSE.
The overidentified SBIL estimator has the lowest RMSE. Simulation-based estimators are discussed
in a later Chapter. Perhaps these results will stimulate your interest.
Table 17.1: Dynamic panel data model. Bias. Source for ML and II is Gouriroux, Phillips and Yu,
2010, Table 2. SBIL, SMIL and II are exactly identified, using the ML auxiliary statistic. SBIL(OI)
and SMIL(OI) are overidentified, using both the naive and ML auxiliary statistics.
T
5
5
5
5
5
5
5
5

N
100
100
100
100
200
200
200
200

0.0
0.3
0.6
0.9
0.0
0.3
0.6
0.9

ML
-0.199
-0.274
-0.362
-0.464
-0.200
-0.275
-0.363
-0.465

II
0.001
-0.001
0.000
0.000
0.000
-0.010
-0.000
-0.003

SBIL
0.004
0.003
0.004
-0.022
0.001
0.001
0.001
-0.010

SBIL(OI)
-0.000
-0.001
-0.001
-0.000
0.000
-0.001
-0.001
0.001

Table 17.2: Dynamic panel data model. RMSE. Source for ML and II is Gouriroux, Phillips and Yu,
2010, Table 2. SBIL, SMIL and II are exactly identified, using the ML auxiliary statistic. SBIL(OI)
and SMIL(OI) are overidentified, using both the naive and ML auxiliary statistics.
T
5
5
5
5
5
5
5
5

N
100
100
100
100
200
200
200
200

0.0
0.3
0.6
0.9
0.0
0.3
0.6
0.9

ML
0.204
0.278
0.365
0.467
0.203
0.277
0.365
0.467

II
0.057
0.081
0.070
0.076
0.041
0.074
0.050
0.054

SBIL
0.059
0.065
0.071
0.059
0.041
0.046
0.050
0.046

SBIL(OI)
0.044
0.041
0.036
0.033
0.031
0.029
0.025
0.027

Arellano-Bond estimator
The first thing is to realize that the i that are a component of the error are correlated with all
regressors in the general case of fixed effects. Getting rid of the i is a step in the direction of
solving the problem. We could subtract the time averages, as above for the within estimator,
but this would give us problems later when we need to define instruments. Instead, consider the

model in first differences

yit yi,t1

= i + yi,t1 + xit + it i yi,t2 xi,t1 i,t1

yit yi,t1

= (yi,t1 yi,t2 ) + (xit xi,t1 ) + it i,t1

or
yit = yi,t1 + xit + it
Now the pesky i are no longer in the picture. Note that we loose one observation when doing
first differencing. OLS estimation of this model will still be inconsistent, because yi,t1 is clearly
correlated with i,t1 . Note also that the error it is serially correlated even if the it are not.
There is no problem of correlation between xit and it . Thus, to do GMM, we need to find
instruments for yi,t1 , but the variables in xit can serve as their own instruments.
How about using yi.t2 as an instrument? It is clearly correlated with yi,t1 = (yi,t1 yi,t2 ),
and as long as the it are not serially correlated, then yi.t2 is not correlated with it = it i,t1 .
We can also use additional lags yi.ts , s 2 to increase efficiency, because GMM with additional
instruments is asymptotically more efficient than with less instruments. This sort of estimator is
widely known in the literature as an Arellano-Bond estimator, due to the influential 1991 paper of
Arellano and Bond (1991).
Note that this sort of estimators requires T = 3 at a minimum. Suppose T = 4. Then for
t = 1 and t = 2, we cannot compute the moment conditions. For t = 3, we can compute the
moment conditions using a single lag yi,1 as an instrument. When t = 4, we can use both
yi,1 and yi,2 as instruments. This sort of unbalancedness in the instruments requires a bit of
care when programming. Also, additional instruments increase asymptotic efficiency but can
lead to increased small sample bias, so one should be a little careful with using too many
instruments. Some robustness checks, looking at the stability of the estimates are a way to
proceed.
One should note that serial correlation of the it will cause this estimator to be inconsistent.
Serial correlation of the errors may be due to dynamic misspecification, and this can be solved
by including additional lags of the dependent variable. However, serial correlation may also
be due to factors not captured in lags of the dependent variable. If this is a possibility, then
the validity of the Arellano-Bond type instruments is in question.
A final note is that the error it is serially correlated, and very likely heteroscedastic across
i. One needs to take this into account when computing the covariance of the GMM estimator.
One can also attempt to use GLS style weighting to improve efficiency. There are many
possibilities.

17.5

Exercises

1. In the context of a dynamic model with fixed effects, why is the differencing used in the
within estimation approach (equation 17.3) problematic? That is, why does the ArellanoBond estimator operate on the model in first differences instead of using the within approach?
2. Consider the simple linear panel data model with random effects (equation 17.4). Suppose
that the it are independent across cross sectional units, so that E(it js ) = 0, i 6= j, t, s.
With a cross sectional unit, the errors are independently and identically distributed, so

E(2it ) = i2 , but E(it is ) = 0, t 6= s. More compactly, let i =

i1
0
2
0
Then the assumptions are that E(i i ) = i IT , and E(i j ) = 0, i 6= j.

i2

iT

i0

(a) write out the form of the entire covariance matrix (nT nT ) of all errors, = E(0 ),
h
i0
where  = 01 02 0T
is the column vector of nT errors.
(b) suppose that n is fixed, and consider asymptotics as T grows. Is it possible to estimate
the i consistently? If so, how?
(c) suppose that T is fixed, and consider asymptotics an n grows. Is it possible to estimate
the i consistently? If so, how?
(d) For one of the two preceeding parts (b) and (c), consistent estimation is possible. For
that case, outline how to do within estimation using a GLS correction.

Chapter 18

Quasi-ML
Quasi-ML is the estimator one obtains when a misspecified probability model is used to calculate
an ML estimator.
Given a sample of size n of a random vector
y and a vector of 
conditioning variables
x, suppose



the joint density of Y =

y1 . . . yn conditional on X = x1 . . . xn is a member of
the parametric family pY (Y|X, ), . The true joint density is associated with the vector 0 :
pY (Y|X, 0 ).

As long as the marginal density of X doesnt depend on 0 , this conditional density fully characterizes the random characteristics of samples: i.e., it fully describes the probabilistically important
features of the d.g.p. The likelihood function is just this density evaluated at other values
L(Y|X, ) = pY (Y|X, ), .





The likelihood
y1 . . . yt1 , Y0 = 0, and let Xt =
x1 . . . xt
function, taking into account possible dependence of observations, can be written as

Let Yt1 =

L(Y|X, )

n
Y
t=1
n
Y

pt (yt |Yt1 , Xt , )
pt ()

t=1

The average log-likelihood function is:

n

sn () =

1
1X
ln L(Y|X, ) =
ln pt ()
n
n t=1

Suppose that we do not have knowledge of the family of densities pt (). Mistakenly, we may
assume that the conditional density of yt is a member of the family ft (yt |Yt1 , Xt , ), ,
where there is no 0 such that ft (yt |Yt1 , Xt , 0 ) = pt (yt |Yt1 , Xt , 0 ), t (this is what we
mean by misspecified).
This setup allows for heterogeneous time series data, with dynamic misspecification.
The QML estimator is the argument that maximizes the misspecified average log likelihood, which

256

we refer to as the quasi-log likelihood function. This objective function is

n

sn ()

1X
ln ft (yt |Yt1 , Xt , 0 )
n t=1

1X
ln ft ()
n t=1

and the QML is

n = arg max sn ()

A SLLN for dependent sequences applies (we assume), so that

n

a.s.

sn () lim E
n

1X
ln ft () s ()
n t=1

We assume that this can be strengthened to uniform convergence, a.s., following the previous
arguments. The pseudo-true value of is the value that maximizes s():
0 = arg max s ()

Given assumptions so that theorem 29 is applicable, we obtain

lim n = 0 , a.s.

Applying the asymptotic normality theorem,





d
n 0 N 0, J (0 )1 I (0 )J (0 )1
where
J (0 ) = lim ED2 sn (0 )
n

and

I (0 ) = lim V ar nD sn (0 ).
n

Note that asymptotic normality only requires that the additional assumptions regarding J
and I hold in a neighborhood of 0 for J and at 0 , for I, not throughout . In this sense,
asymptotic normality is a local property.

18.1

Consistent Estimation of Variance Components

Consistent estimation of J (0 ) is straightforward. Assumption (b) of Theorem 31 implies that

n

1X 2
1X 2
a.s.
Jn (n ) =
D ln ft (n ) lim E
D ln ft (0 ) = J (0 ).
n
n t=1
n t=1
That is, just calculate the Hessian using the estimate n in place of 0 .
Consistent estimation of I (0 ) is more difficult, and may be impossible.
Notation: Let gt D ft (0 )

We need to estimate
I (0 )

=
=

lim V ar nD sn (0 )

n
1X
D ln ft (0 )
lim V ar n
n
n t=1

n
X
1
V ar
gt
n n
t=1
( n
! n
!0 )
X
X
1
(gt Egt )
(gt Egt )
= lim E
n n
t=1
t=1

lim

This is going to contain a term

n

1X
0
(Egt ) (Egt )
n n
t=1
lim

which will not tend to zero, in general. This term is not consistently estimable in general, since it
requires calculating an expectation using the true density under the d.g.p., which is unknown.
There are important cases where I (0 ) is consistently estimable. For example, suppose that
the data come from a random sample (i.e., they are iid). This would be the case with cross
sectional data, for example. (Note: under i.i.d. sampling, the joint distribution of (yt , xt ) is
identical. This does not imply that the conditional density f (yt |xt ) is identical).
With random sampling, the limiting objective function is simply
s (0 ) = EX E0 ln f (y|x, 0 )
where E0 means expectation of y|x and EX means expectation respect to the marginal density
of x.
By the requirement that the limiting objective function be maximized at 0 we have
D EX E0 ln f (y|x, 0 ) = D s (0 ) = 0
The dominated convergence theorem allows switching the order of expectation and differentiation, so
D EX E0 ln f (y|x, 0 ) = EX E0 D ln f (y|x, 0 ) = 0
The CLT implies that
n

1 X
d

D ln f (y|x, 0 ) N (0, I (0 )).

n t=1
That is, its not necessary to subtract the individual means, since they are zero. Given this,
and due to independent observations, a consistent estimator is
n

1X
0 ln ft ()

Ib =
D ln ft ()D
n t=1
This is an important case where consistent estimation of the covariance matrix is possible. Other
cases exist, even for dynamically misspecified time series models.

18.2

Example: the MEPS Data

To check the plausibility of the Poisson model for the MEPS data, we can compare the sample
unconditional
variance with the estimated unconditional variance according to the Poisson model:
Pn

V[
(y) = t=1 t . Using the program PoissonVariance.m, for OBDV and ERV, we get We see that
n

Table 18.1: Marginal Variances, Sample and Estimated (Poisson)

OBDV
38.09
3.28

Sample
Estimated

ERV
0.151
0.086

even after conditioning, the overdispersion is not captured in either case. There is huge problem
with OBDV, and a significant problem with ERV. In both cases the Poisson model does not appear
to be plausible. You can check this for the other use measures if you like.

Infinite mixture models: the negative binomial model

Reference: Cameron and Trivedi (1998) Regression analysis of count data, chapter 4.
The two measures seem to exhibit extra-Poisson variation. To capture unobserved heterogeneity, a possibility is the random parameters approach. Consider the possibility that the constant term
in a Poisson model were random:
fY (y|x, )

exp()y
y!
exp(x0 + )

exp(x0) exp()

=
where = exp(x0) and = exp(). Now captures the randomness in the constant. The problem
is that we dont observe , so we will need to marginalize it to get a usable density
Z

fY (y|x) =

exp[]y
fv (z)dz
y!

This density can be used directly, perhaps using numerical integration to evaluate the likelihood
function. In some cases, though, the integral will have an analytic solution. For example, if
follows a certain one parameter gamma density, then
fY (y|x, ) =

(y + )
(y + 1)()

 

y
(18.1)

where = (, ). appears since it is the parameter of the gamma density.

For this density, E(y|x) = , which we have parameterized = exp(x0 )
The variance depends upon how is parameterized.
If = /, where > 0, then V (y|x) = + . Note that is a function of x, so that
the variance is too. This is referred to as the NB-I model.
If = 1/, where > 0, then V (y|x) = + 2 . This is referred to as the NB-II model.
So both forms of the NB model allow for overdispersion, with the NB-II model allowing for a more
radical form.

Testing reduction of a NB model to a Poisson model cannot be done by testing = 0 using

standard Wald or LR procedures. The critical values need to be adjusted to account for the fact
that = 0 is on the boundary of the parameter space. Without getting into details, suppose that
the data were in fact Poisson, so there is equidispersion and the true = 0. Then about half
the time the sample data will be underdispersed, and about half the time overdispersed. When the
data is underdispersed, the MLE of will be
= 0. Thus, under the null, there will be a probability
p

spike in the asymptotic distribution of n(

) = n
at 0, so standard testing methods will
not be valid.
This program will do estimation using the NB model. Note how modelargs is used to select a
NB-I or NB-II density. Here are NB-I estimation results for OBDV:

MPITB extensions found

OBDV
======================================================
BFGSMIN final results
Used analytic gradient
-----------------------------------------------------STRONG CONVERGENCE
Function conv 1 Param conv 1 Gradient conv 1
-----------------------------------------------------Objective function value 2.18573
Stepsize 0.0007
17 iterations
-----------------------------------------------------param
1.0965
0.2551
0.2024
0.2289
0.1969
0.0769
0.0000
1.7146

gradient
0.0000
-0.0000
-0.0000
0.0000
0.0000
0.0000
-0.0000
-0.0000

change
-0.0000
0.0000
0.0000
-0.0000
-0.0000
-0.0000
0.0000
0.0000

******************************************************
Negative Binomial model, MEPS 1996 full data set
MLE Estimation Results
BFGS convergence: Normal convergence
Average Log-L: -2.185730
Observations: 4564
constant
pub. ins.
priv. ins.
sex
age
edu
inc
alpha

estimate
-0.523
0.765
0.451
0.458
0.016
0.027
0.000
5.555

st. err
0.104
0.054
0.049
0.034
0.001
0.007
0.000
0.296

t-stat
-5.005
14.198
9.196
13.512
11.869
3.979
0.000
18.752

p-value
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000

Information Criteria
CAIC : 20026.7513
Avg. CAIC:
4.3880
BIC : 20018.7513
Avg. BIC:
4.3862
AIC : 19967.3437
Avg. AIC:
4.3750
******************************************************
Note that the parameter values of the last BFGS iteration are different that those reported
in the final results. This reflects two things - first, the data were scaled before doing the BFGS
minimization, but the mle_results script takes this into account and reports the results using the
original scaling. But also, the parameterization = exp( ) is used to enforce the restriction
that > 0. The unrestricted parameter = log is used to define the log-likelihood function,
since the BFGS minimization algorithm does not do contrained minimization. To get the standard
error and t-statistic of the estimate of , we need to use the delta method. This is done inside

mle_results, making use of the function parameterize.m .

Likewise, here are NB-II results:
MPITB extensions found
OBDV
======================================================
BFGSMIN final results
Used analytic gradient
-----------------------------------------------------STRONG CONVERGENCE
Function conv 1 Param conv 1 Gradient conv 1
-----------------------------------------------------Objective function value 2.18496
Stepsize 0.0104394
13 iterations
-----------------------------------------------------param
1.0375
0.3673
0.2136
0.2816
0.3027
0.0843
-0.0048
0.4780

gradient
0.0000
-0.0000
0.0000
0.0000
0.0000
-0.0000
0.0000
-0.0000

change
-0.0000
0.0000
-0.0000
-0.0000
0.0000
0.0000
-0.0000
0.0000

******************************************************
Negative Binomial model, MEPS 1996 full data set
MLE Estimation Results
BFGS convergence: Normal convergence
Average Log-L: -2.184962
Observations: 4564
constant

estimate
-1.068

st. err
0.161

t-stat
-6.622

p-value
0.000

pub. ins.
priv. ins.
sex
age
edu
inc
alpha

1.101
0.476
0.564
0.025
0.029
-0.000
1.613

0.095
0.081
0.050
0.002
0.009
0.000
0.055

11.611
5.880
11.166
12.240
3.106
-0.176
29.099

0.000
0.000
0.000
0.000
0.002
0.861
0.000

Information Criteria
CAIC : 20019.7439
Avg. CAIC:
4.3864
BIC : 20011.7439
Avg. BIC:
4.3847
AIC : 19960.3362
Avg. AIC:
4.3734
******************************************************
For the OBDV usage measurel, the NB-II model does a slightly better job than the NB-I
model, in terms of the average log-likelihood and the information criteria (more on this last
in a moment).
Note that both versions of the NB model fit much better than does the Poisson model (see
11.4).
The estimated is highly significant.
To check the plausibility of the NB-II model, we can compare the sample unconditional variance
Pn
t )2
t +
(
.
with the estimated unconditional variance according to the NB-II model: V[
(y) = t=1
n

For OBDV and ERV (estimation results not reported), we get For OBDV, the overdispersion problem
Table 18.2: Marginal Variances, Sample and Estimated (NB-II)

Sample
Estimated

OBDV
38.09
30.58

ERV
0.151
0.182

is significantly better than in the Poisson case, but there is still some that is not captured. For ERV,
the negative binomial model seems to capture the overdispersion adequately.

Finite mixture models: the mixed negative binomial model

The finite mixture approach to fitting health care demand was introduced by Deb and Trivedi
(1997). The mixture approach has the intuitive appeal of allowing for subgroups of the population
with different health status. If individuals are classified as healthy or unhealthy then two subgroups
are defined. A finer classification scheme would lead to more subgroups. Many studies have
incorporated objective and/or subjective indicators of health status in an effort to capture this
heterogeneity. The available objective measures, such as limitations on activity, are not necessarily
very informative about a persons overall health status. Subjective, self-reported measures may
suffer from the same problem, and may also not be exogenous
Finite mixture models are conceptually simple. The density is
fY (y, 1 , ..., p , 1 , ..., p1 ) =

p1
X

(i)

i fY (y, i ) + p fYp (y, p ),

i=1

where i > 0, i = 1, 2, ..., p, p = 1

Pp1
i=1

i , and

Pp

i=1

i = 1. Identification requires that the

i are ordered in some way, for example, 1 2 p and i 6= j , i 6= j. This is simple to

accomplish post-estimation by rearrangement and possible elimination of redundant component
densities.

 The properties of the mixture density follow in a straightforward way from those of the components. In particular, the moment generating function is the same mixture of the moment
Pp
generating functions of the component densities, so, for example, E(Y |x) = i=1 i i (x),
where i (x) is the mean of the ith component density.
Mixture densities may suffer from overparameterization, since the total number of parameters grows rapidly with the number of component densities. It is possible to constrained
parameters across the mixtures.
Testing for the number of component densities is a tricky issue. For example, testing for
p = 1 (a single component, which is to say, no mixture) versus p = 2 (a mixture of two
components) involves the restriction 1 = 1, which is on the boundary of the parameter
space. Not that when 1 = 1, the parameters of the second component can take on any
value without affecting the density. Usual methods such as the likelihood ratio test are not
applicable when parameters are on the boundary under the null hypothesis. Information
criteria means of choosing the model (see below) are valid.
The following results are for a mixture of 2 NB-II models, for the OBDV data, which you can
replicate using this program .

OBDV
******************************************************
Mixed Negative Binomial model, MEPS 1996 full data set
MLE Estimation Results
BFGS convergence: Normal convergence
Average Log-L: -2.164783
Observations: 4564
constant
pub. ins.
priv. ins.
sex
age
edu
inc
alpha
constant
pub. ins.
priv. ins.
sex
age
edu
inc
alpha
Mix

estimate
0.127
0.861
0.146
0.346
0.024
0.025
-0.000
1.351
0.525
0.422
0.377
0.400
0.296
0.111
0.014
1.034
0.257

st. err
0.512
0.174
0.193
0.115
0.004
0.016
0.000
0.168
0.196
0.048
0.087
0.059
0.036
0.042
0.051
0.187
0.162

t-stat
0.247
4.962
0.755
3.017
6.117
1.590
-0.214
8.061
2.678
8.752
4.349
6.773
8.178
2.634
0.274
5.518
1.582

p-value
0.805
0.000
0.450
0.003
0.000
0.112
0.831
0.000
0.007
0.000
0.000
0.000
0.000
0.008
0.784
0.000
0.114

Information Criteria
CAIC : 19920.3807
Avg. CAIC:
4.3647
BIC : 19903.3807
Avg. BIC:
4.3610
AIC : 19794.1395
Avg. AIC:
4.3370
******************************************************

It is worth noting that the mixture parameter is not significantly different from zero, but also
not that the coefficients of public insurance and age, for example, differ quite a bit between the
two latent classes.

Information criteria
As seen above, a Poisson model cant be tested (using standard methods) as a restriction of a
negative binomial model. But it seems, based upon the values of the likelihood functions and the
fact that the NB model fits the variance much better, that the NB model is more appropriate. How
can we determine which of a set of competing models is the best?
The information criteria approach is one possibility. Information criteria are functions of the
log-likelihood, with a penalty for the number of parameters used. Three popular information
criteria are the Akaike (AIC), Bayes (BIC) and consistent Akaike (CAIC). The formulae are
CAIC

+ k(ln n + 1)
2 ln L()

BIC

+ k ln n
2 ln L()

AIC

+ 2k
2 ln L()

It can be shown that the CAIC and BIC will select the correctly specified model from a group of
models, asymptotically. This doesnt mean, of course, that the correct model is necesarily in the
group. The AIC is not consistent, and will asymptotically favor an over-parameterized model over
the correctly specified model. Here are information criteria values for the models weve seen, for
OBDV. Pretty clearly, the NB models are better than the Poisson. The one additional parameter
Table 18.3: Information Criteria, OBDV
Model
Poisson
NB-I
NB-II
MNB-II

AIC
7.345
4.375
4.373
4.337

BIC
7.355
4.386
4.385
4.361

CAIC
7.357
4.388
4.386
4.365

gives a very significant improvement in the likelihood function value. Between the NB-I and NB-II
models, the NB-II is slightly favored. But one should remember that information criteria values
are statistics, with variances. With another sample, it may well be that the NB-I model would be
favored, since the differences are so small. The MNB-II model is favored over the others, by all 3
information criteria.
Why is all of this in the chapter on QML? Lets suppose that the correct model for OBDV is in fact
the NB-II model. It turns out in this case that the Poisson model will give consistent estimates of the
slope parameters (if a model is a member of the linear-exponential family and the conditional mean
is correctly specified, then the parameters of the conditional mean will be consistently estimated).
So the Poisson estimator would be a QML estimator that is consistent for some parameters of the
true model. The ordinary OPG or inverse Hessian ML covariance estimators are however biased
and inconsistent, since the information matrix equality does not hold for QML estimators. But for
i.i.d. data (which is the case for the MEPS data) the QML asymptotic covariance can be consistently
estimated, as discussed above, using the sandwich form for the ML estimator. mle_results in fact
reports sandwich results, so the Poisson estimation results would be reliable for inference even
if the true model is the NB-I or NB-II. Not that they are in fact similar to the results for the NB
models.
However, if we assume that the correct model is the MNB-II model, as is favored by the information criteria, then both the Poisson and NB-x models will have misspecified mean functions, so

the parameters that influence the means would be estimated with bias and inconsistently.

18.3

Exercises

1. Considering the MEPS data (the description is in Section 11.4), for the OBDV (y) measure,
let be a latent index of health status that has expectation equal to unity.1 We suspect that
and P RIV may be correlated, but we assume that is uncorrelated with the other regressors.
We assume that
E(y|P U B, P RIV, AGE, EDU C, IN C, )
= exp(1 + 2 P U B + 3 P RIV + 4 AGE + 5 EDU C + 6 IN C).
We use the Poisson QML estimator of the model
y

Poisson()

exp(1 + 2 P U B + 3 P RIV +

(18.2)

4 AGE + 5 EDU C + 6 IN C).

Since much previous evidence indicates that health care services usage is overdispersed2 , this
is almost certainly not an ML estimator, and thus is not efficient. However, when and P RIV
are uncorrelated, this estimator is consistent for the i parameters, since the conditional
mean is correctly specified in that case. When and P RIV are correlated, Mullahys (1997)
NLIV estimator that uses the residual function
=

y
1,

where is defined in equation 18.2, with appropriate instruments, is consistent. As instruments we use all the exogenous regressors, as well as the cross products of P U B with the
variables in Z = {AGE, EDU C, IN C}. That is, the full set of instruments is
W = {1

PUB

P U B Z }.

(a) Calculate the Poisson QML estimates.

(b) Calculate the generalized IV estimates (do it using a GMM formulation - see the portfolio
example for hints how to do this).
(c) Calculate the Hausman test statistic to test the exogeneity of PRIV.
(d) comment on the results

1A

restriction of this sort is necessary for identification.

exists when the conditional variance is greater than the conditional mean. If this is the case, the Poisson
specification is not correct.
2 Overdispersion

Chapter 19

Nonlinear least squares (NLS)

Readings: Davidson and MacKinnon, Ch. 2 and 5 ; Gallant, Ch. 1

19.1

Introduction and definition

Nonlinear least squares (NLS) is a means of estimating the parameter of the model
yt = f (xt , 0 ) + t .
In general, t will be heteroscedastic and autocorrelated, and possibly nonnormally distributed. However, dealing with this is exactly as in the case of linear models, so well just
treat the iid case here,
t iid(0, 2 )
If we stack the observations vertically, defining
y = (y1 , y2 , ..., yn )0

f = (f (x1 , ), f (x1 , ), ..., f (x1 , ))0

and
= (1 , 2 , ..., n )0
we can write the n observations as
y = f () +
Using this notation, the NLS estimator can be defined as
1
1
0
arg min sn () = [y f ()] [y f ()] = k y f () k2

n
n
The estimator minimizes the weighted sum of squared errors, which is the same as minimizing the Euclidean distance between y and f ().
The objective function can be written as
sn () =

1 0
[y y 2y0 f () + f ()0 f ()] ,
n

which gives the first order conditions

267




0
0
0.

f () y +
f () f ()

Define the n K matrix

D0 f ().

F()

(19.1)

Using this, the first order conditions can be written as

in place of F().
In shorthand, use F
0,
0y + F
0 f ()
F
or
h
i
0.
0 y f ()
F

(19.2)

This bears a good deal of similarity to the f.o.c. for the linear model - the derivative of the prediction
is simply X, so the f.o.c. (with spherical
is orthogonal to the prediction error. If f () = X, then F
errors) simplify to
X0 y X0 X = 0,
the usual 0LS f.o.c.
We can interpret this geometrically: INSERT drawings of geometrical depiction of OLS and NLS
(see Davidson and MacKinnon, pgs. 8,13 and 46).
Note that the nonlinearity of the manifold leads to potential multiple local maxima, minima
and saddlepoints: the objective function sn () is not necessarily well-behaved and may be
difficult to minimize.

19.2

Identification

As before, identification can be considered conditional on the sample, and asymptotically. The
condition for asymptotic identification is that sn () tend to a limiting function s () such that
s (0 ) < s (), 6= 0 . This will be the case if s (0 ) is strictly convex at 0 , which requires
that D2 s (0 ) be positive definite. Consider the objective function:
n

sn ()

1X
2
[yt f (xt , )]
n t=1

n
2
1 X
f (xt , 0 ) + t ft (xt , )
n t=1

n
n
2 1 X
1 X
2
ft (0 ) ft () +
(t )
n t=1
n t=1

n

2 X
ft (0 ) ft () t
n t=1

As in example 12.4, which illustrated the consistency of extremum estimators using OLS, we
conclude that the second term will converge to a constant which does not depend upon .
A LLN can be applied to the third term to conclude that it converges pointwise to 0, as long
as f () and are uncorrelated.
Next, pointwise convergence needs to be stregnthened to uniform almost sure convergence.
There are a number of possible assumptions one could use. Here, well just assume it holds.

 Turning to the first term, well assume a pointwise law of large numbers applies, so
n
2 a.s.
1 X
ft (0 ) ft ()
n t=1


2
f (z, 0 ) f (z, ) d(z),

(19.3)

where (x) is the distribution function of x. In many cases, f (x, ) will be bounded and
continuous, for all , so strengthening to uniform almost sure convergence is immediate.
1

For example if f (x, ) = [1 + exp(x)]

, f : <K (0, 1) , a bounded range, and the

function is continuous in .
Given these results, it is clear that a minimizer is 0 . When considering identification (asymptotic), the question is whether or not there may be some other minimizer. A local condition for
identification is that
2
2
s
()
=

0
0


2
f (x, 0 ) f (x, ) d(x)

be positive definite at 0 . Evaluating this derivative, we obtain (after a little work)

2
0




2
0
f (x, ) f (x, ) d(x)

Z
=2



0
D f (z, 0 )0 D0 f (z, 0 ) d(z)

the expectation of the outer product of the gradient of the regression function evaluated at 0 .
(Note: the uniform boundedness we have already assumed allows passing the derivative through
the integral, by the dominated convergence theorem.) This matrix will be positive definite (wp1) as
long as the gradient vector is of full rank (wp1). The tangent space to the regression manifold must
span a K -dimensional space if we are to consistently estimate a K -dimensional parameter vector.
This is analogous to the requirement that there be no perfect colinearity in a linear model. This
is a necessary condition for identification. Note that the LLN implies that the above expectation is
equal to
J (0 ) = 2 lim E

19.3

F0 F
n

Consistency

We simply assume that the conditions of Theorem 29 hold, so the estimator is consistent. Given
that the strong stochastic equicontinuity conditions hold, as discussed above, and given the above
identification conditions an a compact estimation space (the closure of the parameter space ),
the consistency proofs assumptions are satisfied.

19.4

Asymptotic normality

As in the case of GMM, we also simply assume that the conditions for asymptotic normality as
in Theorem 31 hold. The only remaining problem is to determine the form of the asymptotic
variance-covariance matrix. Recall that the result of the asymptotic normality theorem is




d
n 0 N 0, J (0 )1 I (0 )J (0 )1 ,
where J (0 ) is the almost sure limit of

2
0 sn ()

evaluated at 0 , and

I (0 ) = lim V ar nD sn (0 )

The objective function is

n

sn () =

1X
2
[yt f (xt , )]
n t=1

So

D sn () =

2X
[yt f (xt , )] D f (xt , ).
n t=1

Evaluating at 0 ,
n

2X
D sn ( ) =
t D f (xt , 0 ).
n t=1
0

Note that the expectation of this is zero, since t and xt are assumed to be uncorrelated. So to
calculate the variance, we can simply calculate the second moment about zero. Also note that
n
X

t D f (xt , 0 )

 0 0
f ( )

F0

t=1

With this we obtain

I (0 )

lim V ar nD sn (0 )
4
= lim nE 2 F0 F
n
F0 F
= 4 2 lim E
n
=

Weve already seen that

J (0 ) = 2 lim E

F0 F
,
n

where the expectation is with respect to the joint density of x and . Combining these expressions
for J (0 ) and I (0 ), and the result of the asymptotic normality theorem, we get


d
n 0 N

1 !

F0 F
2 .
0, lim E
n

We can consistently estimate the variance covariance matrix using

0F

F
n

!1

2,

(19.4)

is defined as in equation 19.1 and

where F
h

2 =

i0 h
i

y f ()
y f ()
n

the obvious estimator. Note the close correspondence to the results for the linear model.

19.5

Example: The Poisson model for count data

Suppose that yt conditional on xt is independently distributed Poisson. A Poisson random variable

is a count data variable, which means it can take the values {0,1,2,...}. This sort of model has been
used to study visits to doctors per year, number of patents registered by businesses per year, etc.

The Poisson density is

f (yt ) =

exp(t )yt t
, yt {0, 1, 2, ...}.
yt !

The mean of yt is t , as is the variance. Note that t must be positive. Suppose that the true mean
is
0t = exp(x0t 0 ),
which enforces the positivity of t . Suppose we estimate 0 by nonlinear least squares:
n
1X
2
(yt exp(x0t ))
= arg min sn () =
T t=1

We can write
sn ()

n

2
1X
exp(x0t 0 + t exp(x0t )
T t=1

n
n
n

2


1X
1X 2
1X
t + 2
t exp(x0t 0 exp(x0t )
exp(x0t 0 exp(x0t ) +
T t=1
T t=1
T t=1

The last term has expectation zero since the assumption that E(yt |xt ) = exp(x0t 0 ) implies that
E (t |xt ) = 0, which in turn implies that functions of xt are uncorrelated with t . Applying a strong
LLN, and noting that the objective function is continuous on a compact parameter space, we get
s () = Ex exp(x0 0 exp(x0 )

2

+ Ex exp(x0 0 )

where the last term comes from the fact that the conditional variance of is the same as the
variance of y. This function is clearly minimized at = 0 , so the NLS estimator is consistent as
long as identification holds.


Exercise 62. Determine the limiting distribution of n 0 . This means finding the the

sn () 
2
0
0
specific forms of
0 sn (), J ( ),
 , and I( ). Again, use a CLT as needed, no need to
verify that it can be applied.

19.6

The Gauss-Newton algorithm

Readings: Davidson and MacKinnon, Chapter 6, pgs. 201-207 .

The Gauss-Newton optimization technique is specifically designed for nonlinear least squares.
The idea is to linearize the nonlinear model, rather than the objective function. The model is
y = f (0 ) + .
At some in the parameter space, not equal to 0 , we have
y = f () +
where is a combination of the fundamental error term and the error due to evaluating the regression function at rather than the true value 0 . Take a first order Taylors series approximation
around a point 1 :





y = f (1 ) + D0 f 1
1 + + approximation error.

Define z y f (1 ) and b ( 1 ). Then the last equation can be written as

z = F(1 )b + ,
where, as above, F(1 ) D0 f (1 ) is the n K matrix of derivatives of the regression function,
evaluated at 1 , and is plus approximation error from the truncated Taylors series.
Note that F is known, given 1 .
Note that one could estimate b simply by performing OLS on the above equation.
Given b, we calculate a new round estimate of 0 as 2 = b + 1 . With this, take a new Taylors
series expansion around 2 and repeat the process. Stop when b = 0 (to within a specified
tolerance).
To see why this might work, consider the above approximation, but evaluated at the NLS estimator:


+ F()
+
y = f ()
The OLS estimate of b is
1 h
i

.
b = F
0 y f ()
0F

F
This must be zero, since
 h
i
0
0 y f ()
F
by definition of the NLS estimator (these are the normal equations as in equation 19.2, Since b 0
updating would stop.
when we evaluate at ,
The Gauss-Newton method doesnt require second derivatives, as does the Newton-Raphson
method, so its faster.
as a by The varcov estimator, as in equation 19.4 is simple to calculate, since we have F
product of the estimation process (i.e., its just the last round regressor matrix). In fact, a
normal OLS program will give the NLS varcov estimator directly, since its just the OLS varcov
estimator from the last iteration.
The method can suffer from convergence problems since F()0 F(), may be very nearly sin Consider
gular, even with an asymptotically identified model, especially if is very far from .
the example
y = 1 + 2 xt 3 + t
When evaluated at 2 0, 3 has virtually no effect on the NLS objective function, so F will
have rank that is essentially 2, rather than 3. In this case, F0 F will be nearly singular, so
(F0 F)1 will be subject to large roundoff errors.

19.7

Application: Limited dependent variables and sample selection

Readings: Davidson and MacKinnon, Ch. 15 (a quick reading is sufficient), J. Heckman, Sample
Selection Bias as a Specification Error, Econometrica, 1979 (This is a classic article, not required
for reading, and which is a bit out-dated. Nevertheless its a good place to start if you encounter
sample selection problems in your research).

Sample selection is a common problem in applied research. The problem occurs when observations used in estimation are sampled non-randomly, according to some selection scheme.

Example: Labor Supply

Labor supply of a person is a positive number of hours per unit time supposing the offer wage is
higher than the reservation wage, which is the wage at which the person prefers not to work. The
model (very simple, with t subscripts suppressed):
Characteristics of individual: x
Latent labor supply: s = x0 +
Offer wage: wo = z0 +
Reservation wage: wr = q0 +
Write the wage differential as
w

(z0 + ) (q0 + )

r0 +

We have the set of equations

Assume that

"

= x0 +

= r0 + .

"
N

0
0

# "
,

#!
.

We assume that the offer wage and the reservation wage, as well as the latent variable s are
unobservable. What is observed is
w

s =

1 [w > 0]
ws .

In other words, we observe whether or not a person is working. If the person is working, we
observe labor supply, which is equal to latent labor supply, s . Otherwise, s = 0 6= s . Note that we
are using a simplifying assumption that individuals can freely choose their weekly hours of work.
Suppose we estimated the model
s = x0 + residual
using only observations for which s > 0. The problem is that these observations are those for which
w > 0, or equivalently, < r0 and
E [| < r0 ] 6= 0,
since and are dependent. Furthermore, this expectation will in general depend on x since
elements of x can enter in r. Because of these two facts, least squares estimation is biased and
inconsistent.

Consider more carefully E [| < r0 ] . Given the joint normality of and , we can write
(see for example Spanos Statistical Foundations of Econometric Modelling, pg. 122)
= + ,
where has mean zero and is independent of . With this we can write
s = x0 + + .
If we condition this equation on < r0 we get
s = x0 + E(| < r0 ) +
which may be written as
s = x0 + E(| > r0 ) +
A useful result is that for
z N (0, 1)

E(z|z > z ) =

(z )
,
(z )

where () and () are the standard normal density and distribution function, respectively.
The quantity on the RHS above is known as the inverse Mills ratio:
IM R(z ) =

(z )
(z )

With this we can write (making use of the fact that the standard normal density is symmetric
about zero, so that (a) = (a)):
(r0 )
+
(r0 )
"
#
i
(r0 )
+ .
(r0 )

s = x0 +

(19.5)

(19.6)

x0

where = . The error term has conditional mean zero, and is uncorrelated with the regressors
(r0 )
x0 (r0 ) . At this point, we can estimate the equation by NLS.
Heckman showed how one can estimate this in a two step procedure where first is estimated, then equation 19.6 is estimated by least squares using the estimated value of to
form the regressors. This is inefficient and estimation of the covariance is a tricky issue. It is
probably easier (and more efficient) just to do MLE.
The model presented above depends strongly on joint normality. There exist many alternative
models which weaken the maintained assumptions. It is possible to estimate consistently
without distributional assumptions. See Ahn and Powell, Journal of Econometrics, 1994.

Chapter 20

Nonparametric inference
20.1

Possible pitfalls of parametric inference: estimation

Readings: H. White (1980) Using Least Squares to Approximate Unknown Regression Functions,
International Economic Review, pp. 149-70.
In this section we consider a simple example, which illustrates both why nonparametric methods may in some cases be preferred to parametric methods.
We suppose that data is generated by random sampling of (y, x), where y = f (x) +, x is
uniformly distributed on (0, 2), and is a classical error. Suppose that
f (x) = 1 +

3x  x 2

2
2

The problem of interest is to estimate the elasticity of f (x) with respect to x, throughout the range
of x.
In general, the functional form of f (x) is unknown. One idea is to take a Taylors series approximation to f (x) about some point x0 . Flexible functional forms such as the transcendental
logarithmic (usually know as the translog) can be interpreted as second order Taylors series approximations. Well work with a first order approximation, for simplicity. Approximating about
x0 :
h(x) = f (x0 ) + Dx f (x0 ) (x x0 )
If the approximation point is x0 = 0, we can write
h(x) = a + bx
The coefficient a is the value of the function at x = 0, and the slope is the value of the derivative
at x = 0. These are of course not known. One might try estimation by ordinary least squares. The
objective function is
s(a, b) = 1/n

n
X

(yt h(xt )) .

t=1

The limiting objective function, following the argument we used to get equations 12.1 and 19.3 is
Z

(f (x) h(x)) dx.

s (a, b) =
0

The theorem regarding the consistency of extremum estimators (Theorem 29) tells us that a
and
b will converge almost surely to the values that minimize the limiting objective function. Solving

275

Figure 20.1: True and simple approximating functions

3.5

approx
true
3.0

2.5

2.0

1.5

1.0
0

the first order conditions1 reveals that s (a, b) obtains its minimum at

estimated approximating function h(x)

therefore tends almost surely to

 0
a = 76 , b0 =

. The

h (x) = 7/6 + x/
In Figure 20.1 we see the true function and the limit of the approximation to see the asymptotic
bias as a function of x.
(The approximating model is the straight line, the true model has curvature.) Note that the
approximating model is in general inconsistent, even at the approximation point. This shows that
flexible functional forms based upon Taylors series approximations do not in general lead to
consistent estimation of functions.
The approximating model seems to fit the true model fairly well, asymptotically. However, we
are interested in the elasticity of the function. Recall that an elasticity is the marginal function
divided by the average function:
(x) = x0 (x)/(x)
Good approximation of the elasticity over the range of x will require a good approximation of both
f (x) and f 0 (x) over the range of x. The approximating elasticity is
(x) = xh0 (x)/h(x)
In Figure 20.2 we see the true elasticity and the elasticity obtained from the limiting approximating
model.
The true elasticity is the line that has negative slope for large x. Visually we see that the elasticity is not approximated so well. Root mean squared error in the approximation of the elasticity
is
Z

1/2
((x) (x)) dx
= . 31546
2

Now suppose we use the leading terms of a trigonometric series as the approximating model.
The reason for using a trigonometric series as an approximating model is motivated by the asymptotic properties of the Fourier flexible functional form (Gallant, 1981, 1982), which we will study
1 The following results were obtained using the free computer algebra system (CAS) Maxima. Unfortunately, I have lost
the source code to get the results :-(

Figure 20.2: True and approximating elasticities

0.7

approx
true
0.6

0.5

0.4

0.3

0.2

0.1

0.0
0

in more detail below. Normally with this type of model the number of basis functions is an increasing function of the sample size. Here we hold the set of basis function fixed. We will consider the
asymptotic behavior of a fixed model, which we interpret as an approximation to the estimators
behavior in finite samples. Consider the set of basis functions:
Z(x) =

x cos(x)

sin(x)

cos(2x)

sin(2x)

The approximating model is

gK (x) = Z(x).
Maintaining these basis functions as the sample size increases, we find that the limiting objective
function is minimized at


1
1
1
7
a1 = , a2 = , a3 = 2 , a4 = 0, a5 = 2 , a6 = 0 .
6

4
Substituting these values into gK (x) we obtain the almost sure limit of the approximation




1
1
g (x) = 7/6 + x/ + (cos x) 2 + (sin x) 0 + (cos 2x) 2 + (sin 2x) 0

(20.1)

In Figure 20.3 we have the approximation and the true function: Clearly the truncated trigonometric series model offers a better approximation, asymptotically, than does the linear model. In
Figure 20.4 we have the more flexible approximations elasticity and that of the true function: On
average, the fit is better, though there is some implausible wavyness in the estimate. Root mean
squared error in the approximation of the elasticity is
Z
0

g 0 (x)x
(x)
g (x)

2

!1/2
dx

= . 16213,

about half that of the RMSE when the first order approximation is used. If the trigonometric series
contained infinite terms, this error measure would be driven to zero, as we shall see.

Figure 20.3: True function and more flexible approximation

3.5

approx
true
3.0

2.5

2.0

1.5

1.0
0

Figure 20.4: True elasticity and more flexible approximation

0.7

approx
true
0.6

0.5

0.4

0.3

0.2

0.1

0.0
0

4
x

20.2

Possible pitfalls of parametric inference: hypothesis testing

What do we mean by the term nonparametric inference? Simply, this means inferences that are
possible without restricting the functions of interest to belong to a parametric family.
Consider means of testing for the hypothesis that consumers maximize utility. A consequence
of utility maximization is that the Slutsky matrix Dp2 h(p, U ), where h(p, U ) are the a set of
compensated demand functions, must be negative semi-definite. One approach to testing for
utility maximization would estimate a set of normal demand functions x(p, m).
Estimation of these functions by normal parametric methods requires specification of the
functional form of demand, for example
x(p, m) = x(p, m, 0 ) + , 0 0 ,
where x(p, m, 0 ) is a function of known form and 0 is a finite dimensional parameter.
to calculate (by solving the integrability prob After estimation, we could use x
= x(p, m, )
b p2 h(p, U ). If we can statistically reject that the matrix is negative
lem, which is non-trivial) D
semi-definite, we might conclude that consumers dont maximize utility.
The problem with this is that the reason for rejection of the theoretical proposition may be
that our choice of functional form is incorrect. In the introductory section we saw that functional form misspecification leads to inconsistent estimation of the function and its derivatives.
Testing using parametric models always means we are testing a compound hypothesis. The
hypothesis that is tested is 1) the economic proposition we wish to test, and 2) the model is
correctly specified. Failure of either 1) or 2) can lead to rejection (as can a Type-I error, even
when 2) holds). This is known as the model-induced augmenting hypothesis.
Varians WARP allows one to test for utility maximization without specifying the form of the
demand functions. The only assumptions used in the test are those directly implied by theory,
so rejection of the hypothesis calls into question the theory.
Nonparametric inference also allows direct testing of economic propositions, avoiding the
model-induced augmenting hypothesis. The cost of nonparametric methods is usually an
increase in complexity, and a loss of power, compared to what one would get using a wellspecified parametric model. The benefit is robustness against possible misspecification.

20.3

Estimation of regression functions

The Fourier functional form

Readings: Gallant, 1987, Identification and consistency in semi-nonparametric regression, in
Advances in Econometrics, Fifth World Congress, V. 1, Truman Bewley, ed., Cambridge.
Suppose we have a multivariate model
y = f (x) + ,

where f (x) is of unknown form and x is a P dimensional vector. For simplicity, assume that is a
classical error. Let us take the estimation of the vector of elasticities with typical element
xi =

xi f (x)
,
f (x) xi f (x)

at an arbitrary point xi .
The Fourier form, following Gallant (1982), but with a somewhat different parameterization,
may be written as

A X
J
X

gK (x | K ) = + x0 + 1/2x0 Cx +

(uj cos(jk0 x) vj sin(jk0 x)) .

(20.2)

=1 j=1

where the K-dimensional parameter vector

K = {, 0 , vec (C)0 , u11 , v11 , . . . , uJA , vJA }0 .

(20.3)

We assume that the conditioning variables x have each been transformed to lie in an interval
that is shorter than 2. This is required to avoid periodic behavior of the approximation,
which is desirable since economic functions arent periodic. For example, subtract sample
means, divide by the maxima of the conditioning variables, and multiply by 2 eps, where
eps is some positive number less than 2 in value.
The k are elementary multi-indices which are simply P vectors formed of integers (negative, positive and zero). The k , = 1, 2, ..., A are required to be linearly independent, and
we follow the convention that the first non-zero element be positive. For example
h

i0

is a potential multi-index to be used, but

h

i0

is not since its first nonzero element is negative. Nor is

h

i0

a multi-index we would use, since it is a scalar multiple of the original multi-index.

We parameterize the matrix C differently than does Gallant because it simplifies things in
practice. The cost of this is that we are no longer able to test a quadratic specification using
nested testing.
The vector of first partial derivatives is
Dx gK (x | K ) = + Cx +

A X
J
X

[(uj sin(jk0 x) vj cos(jk0 x)) jk ]

(20.4)

=1 j=1

and the matrix of second partial derivatives is

Dx2 gK (x|K ) = C +

A X
J
X

=1 j=1

(uj cos(jk0 x) + vj sin(jk0 x)) j 2 k k0

(20.5)

To define a compact notation for partial derivatives, let be an N -dimensional multi-index

with no negative elements. Define | | as the sum of the elements of . If we have N arguments
x of the (arbitrary) function h(x), use D h(x) to indicate a certain partial derivative:

||
D h(x)
h(x)
1
x1 x2 2 xNN

When is the zero vector, D h(x) h(x). Taking this definition and the last few equations into
account, we see that it is possible to define (1 K) vector Z (x) so that
D gK (x|K ) = z (x)0 K .

(20.6)

Both the approximating model and the derivatives of the approximating model are linear in
the parameters.
For the approximating model to the function (not derivatives), write gK (x|K ) = z0 K for
simplicity.
The following theorem can be used to prove the consistency of the Fourier form.
n is obtained by maximizing a sample
Theorem 63. [Gallant and Nychka, 1987] Suppose that h
objective function sn (h) over HKn where HK is a subset of some function space H on which is defined
a norm k h k. Consider the following conditions:
(a) Compactness: The closure of H with respect to k h k is compact in the relative topology defined
by k h k.
(b) Denseness: K HK , K = 1, 2, 3, ... is a dense subset of the closure of H with respect to k h k
and HK HK+1 .
(c) Uniform convergence: There is a point h in H and there is a function s (h, h ) that is
continuous in h with respect to k h k such that
lim sup | sn (h) s (h, h ) |= 0

almost surely.
(d) Identification: Any point h in the closure of H with s (h, h ) s (h , h ) must have k
h h k= 0.
n k= 0 almost surely, provided that limn Kn =
Under these conditions limn k h h
almost surely.
The modification of the original statement of the theorem that has been made is to set the
parameter space in Gallant and Nychkas (1987) Theorem 0 to a single point and to state the
theorem in terms of maximization rather than minimization.
This theorem is very similar in form to Theorem 29. The main differences are:
1. A generic norm k h k is used in place of the Euclidean norm. This norm may be stronger
than the Euclidean norm, so that convergence with respect to k h k implies convergence w.r.t
the Euclidean norm. Typically we will want to make sure that the norm is strong enough to
imply convergence of all functions of interest.
2. The estimation space H is a function space. It plays the role of the parameter space in
our discussion of parametric estimators. There is no restriction to a parametric family, only
a restriction to a space of functions that satisfy certain conditions. This formulation is much
less restrictive than the restriction to a parametric family.

3. There is a denseness assumption that was not present in the other theorem.
We will not prove this theorem (the proof is quite similar to the proof of theorem [29], see Gallant,
1987) but we will discuss its assumptions, in relation to the Fourier form as the approximating
model.
Sobolev norm Since all of the assumptions involve the norm k h k , we need to make explicit
what norm we wish to use. We need a norm that guarantees that the errors in approximation
of the functions we are interested in are accounted for. Since we are interested in first-order
elasticities in the present case, we need close approximation of both the function f (x) and its first
derivative f 0 (x), throughout the range of x. Let X be an open set that contains all values of x that
were interested in. The Sobolev norm is appropriate in this case. It is defined, making use of our
notation for partial derivatives, as:


k h km,X = max
sup D h(x)

| |m X

To see whether or not the function f (x) is well approximated by an approximating model gK (x |
K ), we would evaluate
k f (x) gK (x | K ) km,X .
We see that this norm takes into account errors in approximating the function and partial derivatives up to order m. If we want to estimate first order elasticities, as is the case in this example, the
relevant m would be m = 1. Furthermore, since we examine the sup over X , convergence w.r.t.
the Sobolev means uniform convergence, so that we obtain consistent estimates for all values of x.
Compactness

Verifying compactness with respect to this norm is quite technical and unenlight-

ening. It is proven by Elbadawi, Gallant and Souza, Econometrica, 1983. The basic requirement is
that if we need consistency w.r.t. k h km,X , then the functions of interest must belong to a Sobolev
space which takes into account derivatives of order m + 1. A Sobolev space is the set of functions
Wm,X (D) = {h(x) :k h(x) km,X < D},
where D is a finite constant. In plain words, the functions must have bounded partial derivatives
of one order higher than the derivatives we seek to estimate.
The estimation space and the estimation subspace Since in our case were interested in consistent estimation of first-order elasticities, well define the estimation space as follows:
Definition 64. [Estimation space] The estimation space H = W2,X (D). The estimation space is an
open set, and we presume that h H.
So we are assuming that the function to be estimated has bounded second derivatives throughout X .
With seminonparametric estimators, we dont actually optimize over the estimation space.
Rather, we optimize over a subspace, HKn , defined as:
Definition 65. [Estimation subspace] The estimation subspace HK is defined as
HK = {gK (x|K ) : gK (x|K ) W2,Z (D), K <K },
where gK (x, K ) is the Fourier form approximation as defined in Equation 20.2.

Denseness

The important point here is that HK is a space of functions that is indexed by a finite

dimensional parameter (K has K elements, as in equation 20.3). With n observations, n > K,

this parameter is estimable. Note that the true function h is not necessarily an element of HK , so
optimization over HK may not lead to a consistent estimator. In order for optimization over HK
to be equivalent to optimization over H, at least asymptotically, we need that:
1. The dimension of the parameter vector, dim Kn as n . This is achieved by making
A and J in equation 20.2 increasing functions of n, the sample size. It is clear that K will
have to grow more slowly than n. The second requirement is:
2. We need that the HK be dense subsets of H.
The estimation subspace HK , defined above, is a subset of the closure of the estimation space, H .
A set of subsets Aa of a set A is dense if the closure of the countable union of the subsets is equal
to the closure of A:

a=1 Aa = A
Use a picture here. The rest of the discussion of denseness is provided just for completeness: theres no
need to study it in detail. To show that HK is a dense subset of H with respect to k h k1,X , it is
useful to apply Theorem 1 of Gallant (1982), who in turn cites Edmunds and Moscatelli (1977). We
reproduce the theorem as presented by Gallant, with minor notational changes, for convenience of
reference:
Theorem 66. [Edmunds and Moscatelli, 1977] Let the real-valued function h (x) be continuously
differentiable up to order m on an open set containing the closure of X . Then it is possible to choose
a triangular array of coefficients 1 , 2 , . . . K , . . . , such that for every q with 0 q < m, and every
> 0, k h (x) hK (x|K ) kq,X = o(K m+q+ ) as K .
In the present application, q = 1, and m = 2. By definition of the estimation space, the elements
of H are once continuously differentiable on X , which is open and contains the closure of X , so
the theorem is applicable. Closely following Gallant and Nychka (1987), HK is the countable
union of the HK . The implication of Theorem 66 is that there is a sequence of {hK } from HK
such that
lim k h hK k1,X = 0,

for all h H. Therefore,

H HK .
However,
HK H,
so
HK H.
Therefore
H = HK ,
so HK is a dense subset of H, with respect to the norm k h k1,X .
Uniform convergence We now turn to the limiting objective function. We estimate by OLS. The
sample objective function stated in terms of maximization is
n

sn (K ) =

1X
2
(yt gK (xt | K ))
n t=1

With random sampling, as in the case of Equations 12.1 and 19.3, the limiting objective function
is
Z
s (g, f ) =

(f (x) g(x)) dx 2 .

(20.7)

where the true function f (x) takes the place of the generic function h in the presentation of the
theorem. Both g(x) and f (x) are elements of HK .
The pointwise convergence of the objective function needs to be strengthened to uniform convergence. We will simply assume that this holds, since the way to verify this depends upon the
specific application. We also have continuity of the objective function in g, with respect to the
norm k h k1,X since





s g 1 , f ) s g 0 , f )
1,X 0
Z h

2

2 i
1
0
lim

g
dx.
g
(x)

f
(x)
(x)

f
(x)
0

lim

kg 1 g 0 k

kg 1 g k1,X 0

By the dominated convergence theorem (which applies since the finite bound D used to define
W2,Z (D) is dominated by an integrable function), the limit and the integral can be interchanged,
so by inspection, the limit is zero.
Identification

The identification condition requires that for any point (g, f ) in H H, s (g, f )

s (f, f ) k g f k1,X = 0. This condition is clearly satisfied given that g and f are once
continuously differentiable (by the assumption that defines the estimation space).
Review of concepts

For the example of estimation of first-order elasticities, the relevant concepts

are:
Estimation space H = W2,X (D): the function space in the closure of which the true function
must lie.
Consistency norm k h k1,X . The closure of H is compact with respect to this norm.
Estimation subspace HK . The estimation subspace is the subset of H that is representable by
a Fourier form with parameter K . These are dense subsets of H.
Sample objective function sn (K ), the negative of the sum of squares. By standard arguments
this converges uniformly to the
Limiting objective function s ( g, f ), which is continuous in g and has a global maximum in
its first argument, over the closure of the infinite union of the estimation subpaces, at g = f.
As a result of this, first order elasticities
xi f (x)
f (x) xi f (x)
are consistently estimated for all x X .
Discussion

Consistency requires that the number of parameters used in the expansion increase

with the sample size, tending to infinity. If parameters are added at a high rate, the bias tends
relatively rapidly to zero. A basic problem is that a high rate of inclusion of additional parameters
causes the variance to tend more slowly to zero. The issue of how to chose the rate at which
parameters are added and which to add first is fairly complex. A problem is that the allowable

rates for asymptotic normality to obtain (Andrews 1991; Gallant and Souza, 1991) are very strict.
Supposing we stick to these rates, our approximating model is:
gK (x|K ) = z0 K .
Define ZK as the n K matrix of regressors obtained by stacking observations. The LS
estimator is
+
K = (Z0K ZK ) Z0K y,
+

where () is the Moore-Penrose generalized inverse.

This is used since Z0K ZK may be singular, as would be the case for K(n) large enough
when some dummy variables are included.
. The prediction, z0 K , of the unknown function f (x) is asymptotically normally distributed:

 0
d
n z K f (x) N (0, AV ),
where

#
" 
+
Z0K ZK
2
0
z
.
AV = lim E z
n
n

Formally, this is exactly the same as if we were dealing with a parametric linear model. I
emphasize, though, that this is only valid if K grows very slowly as n grows. If we cant
stick to acceptable rates, we should probably use some other method of approximating the
small sample distribution. Bootstrapping is a possibility. Well discuss this in the section on
simulation.

Kernel regression estimators

Readings: Bierens, 1987, Kernel estimators of regression functions, in Advances in Econometrics,
Fifth World Congress, V. 1, Truman Bewley, ed., Cambridge.
An alternative method to the semi-nonparametric method is a fully nonparametric method of
estimation. Kernel regression estimation is an example (others are splines, nearest neighbor, etc.).
Well consider the Nadaraya-Watson kernel regression estimator in a simple case.
Suppose we have an iid sample from the joint density f (x, y), where x is k -dimensional. The
model is
yt = g(xt ) + t ,
where
E(t |xt ) = 0.
The conditional expectation of y given x is g(x). By definition of the conditional expectation,
we have
Z
g(x)

=
=

f (x, y)
dy
h(x)
Z
1
yf (x, y)dy,
h(x)
y

where h(x) is the marginal density of x :

Z
h(x) =

f (x, y)dy.

 This suggests that we could estimate g(x) by estimating h(x) and

yf (x, y)dy.

Estimation of the denominator

A kernel estimator for h(x) has the form
n

1 X K [(x xt ) /n ]

,
h(x)
=
n t=1
nk
where n is the sample size and k is the dimension of x.
The function K() (the kernel) is absolutely integrable:
Z
|K(x)|dx < ,
and K() integrates to 1 :
Z
K(x)dx = 1.
In this respect, K() is like a density function, but we do not necessarily restrict K() to be
nonnegative.
The window width parameter, n is a sequence of positive numbers that satisfies
lim n

lim nnk

So, the window width must tend to zero, but not too quickly.

To show pointwise consistency of h(x)

for h(x), first consider the expectation of the estimator
(since the estimator is an average of iid terms we only need to consider the expectation of a
representative term):
h
i Z

E h(x)
= nk K [(x z) /n ] h(z)dz.
Change variables as z = (x z)/n , so z = x n z and | dzdz0 | = nk , we obtain
h
i

E h(x)

nk K (z ) h(x n z )nk dz

K (z ) h(x n z )dz .

=
=

Now, asymptotically,
h
i

lim E h(x)

lim
K (z ) h(x n z )dz
Z
lim K (z ) h(x n z )dz
n
Z
K (z ) h(x)dz
Z
h(x) K (z ) dz

h(x),

=
=
=

since n 0 and

K (z ) dz = 1 by assumption. (Note: that we can pass the limit through

the integral is a result of the dominated convergence theorem.. For this to hold we need that

h() be dominated by an absolutely integrable function.

Next, considering the variance of h(x),

we have, due to the iid assumption
h

nnk V

h(x)

nnk



n
K [(x xt ) /n ]
1 X
V
n2 t=1
nk
n

= nk

1X
V {K [(x xt ) /n ]}
n t=1

By the representative term argument, this is

h
i

nnk V h(x)
= nk V {K [(x z) /n ]}
Also, since V (x) = E(x2 ) E(x)2 we have
h
i

nnk V h(x)

n
o
2
2
nk E (K [(x z) /n ]) nk {E (K [(x z) /n ])}
Z
2
Z
2
=
nk K [(x z) /n ] h(z)dz nk
nk K [(x z) /n ] h(z)dz
Z
h
i2
2
=
nk K [(x z) /n ] h(z)dz nk E b
h(x)

The second term converges to zero:

h
i2
nk E b
h(x) 0,
by the previous result regarding the expectation and the fact that n 0. Therefore,
lim

nnk V

Z
i
2

h(x) = lim
nk K [(x z) /n ] h(z)dz.
n

Using exactly the same change of variables as before, this can be shown to be
Z
h
i
2

lim nnk V h(x)

= h(x) [K(z )] dz .

Since both

[K(z )] dz and h(x) are bounded, this is bounded, and since nnk by

assumption, we have that

h
i

V h(x)
0.
Since the bias and the variance both go to zero, we have pointwise consistency (convergence
in quadratic mean implies convergence in probability).
Estimation of the numerator
R
To estimate yf (x, y)dy, we need an estimator of f (x, y). The estimator has the same form as the
estimator for h(x), only with one dimension more:
n

1 X K [(y yt ) /n , (x xt ) /n ]
f(x, y) =
n t=1
nk+1
The kernel K () is required to have mean zero:
Z
yK (y, x) dy = 0

and to marginalize to the previous kernel for h(x) :

Z
K (y, x) dy = K(x).
With this kernel, we have
Z

y f(y, x)dy =

1 X K [(x xt ) /n ]
yt
n t=1
nk

by marginalization of the kernel, so we obtain

g(x)

=
=

h(x)
Pn
1

y f(y, x)dy

K[(xxt )/n ]
k
t=1 yt
n
Pn K[(xxt )/n ]
1
k
t=1
n
n
Pn
yt K [(x xt ) /n ]
Pt=1
.
n
t=1 K [(x xt ) /n ]
n

This is the Nadaraya-Watson kernel regression estimator.

Discussion
The kernel regression estimator for g(xt ) is a weighted average of the yj , j = 1, 2, ..., n,
where higher weights are associated with points that are closer to xt . The weights sum to 1.
The window width parameter n imposes smoothness. The estimator is increasingly flat as
n , since in this case each weight tends to 1/n.
A large window width reduces the variance (strong imposition of flatness), but increases the
bias.
A small window width reduces the bias, but makes very little use of information except points
that are in a small neighborhood of xt . Since relatively little information is used, the variance
is large when the window width is small.
The standard normal density is a popular choice for K(.) and K (y, x), though there are
possibly better alternatives.
Choice of the window width: Cross-validation
The selection of an appropriate window width is important. One popular method is cross validation. This consists of splitting the sample into two parts (e.g., 50%-50%). The first part is the in
sample data, which is used for estimation, and the second part is the out of sample data, used
for evaluation of the fit though RMSE or some other criterion. The steps are:
1. Split the data. The out of sample data is y out and xout .
2. Choose a window width .
3. With the in sample data, fit ytout corresponding to each xout
t . This fitted value is a function of
out
the in sample data, as well as the evaluation point xout
t , but it does not involve yt .

4. Repeat for all out of sample points.

5. Calculate RMSE()

6. Go to step 2, or to the next step if enough window widths have been tried.
7. Select the that minimizes RMSE() (Verify that a minimum has been found, for example
by plotting RMSE as a function of ).
8. Re-estimate using the best and all of the data.
This same principle can be used to choose A and J in a Fourier form model.

20.4

Density function estimation

Kernel density estimation

The previous discussion suggests that a kernel density estimator may easily be constructed. We
have already seen how joint densities may be estimated. If were interested in a conditional density,
for example of y conditional on x, then the kernel estimate of the conditional density is simply
fby|x

=
=

f(x, y)

h(x)
Pn
1

K [(yyt )/n ,(xxt )/n ]

k+1
n
Pn K[(xxt )/n ]
1
k
t=1
n
n
Pn
1
[(y yt ) /n , (x xt ) /n ]
t=1 K
P
n
n
t=1 K [(x xt ) /n ]
n

t=1

where we obtain the expressions for the joint and marginal densities from the section on kernel
regression.

Semi-nonparametric maximum likelihood

Readings: Gallant and Nychka, Econometrica, 1987. For a Fortran program to do this and a useful
discussion in the users guide, see this link. See also Cameron and Johansson, Journal of Applied
Econometrics, V. 12, 1997.
MLE is the estimation method of choice when we are confident about specifying the density. Is
is possible to obtain the benefits of MLE when were not so confident about the specification? In
part, yes.
Suppose were interested in the density of y conditional on x (both may be vectors). Suppose
that the density f (y|x, ) is a reasonable starting approximation to the true density. This density
can be reshaped by multiplying it by a squared polynomial. The new density is
gp (y|x, , ) =

h2p (y|)f (y|x, )

p (x, , )

where
hp (y|) =

p
X

k y k

k=0

and p (x, , ) is a normalizing factor to make the density integrate (sum) to one. Because
h2p (y|)/p (x, , ) is a homogenous function of it is necessary to impose a normalization: 0
is set to 1. The normalization factor p (, ) is calculated (following Cameron and Johansson)

using
E(Y r )

y r fY (y|, )

y=0

X
y=0

yr

[hp (y|)]
fY (y|)
p (, )

p
p X
X
X

y r fY (y|)k l y k y l /p (, )

y=0 k=0 l=0

p
p X
X
k=0 l=0
p X
p
X

k l

(
X

)
y

r+k+l

fY (y|) /p (, )

y=0

k l mk+l+r /p (, ).

k=0 l=0

By setting r = 0 we get that the normalizing factor is

20.8
p (, ) =

p X
p
X

(20.8)

k l mk+l

k=0 l=0

Recall that 0 is set to 1 to achieve identification. The mr in equation 20.8 are the raw moments
of the baseline density. Gallant and Nychka (1987) give conditions under which such a density
may be treated as correctly specified, asymptotically. Basically, the order of the polynomial must
increase as the sample size increases. However, there are technicalities.
Similarly to Cameron and Johannson (1997), we may develop a negative binomial polynomial
(NBP) density for count data. The negative binomial baseline density may be written (see equation
as
(y + )
fY (y|) =
(y + 1)()

 

y

where = {, }, > 0 and > 0. The usual means of incorporating conditioning variables
0

x is the parameterization = ex . When = / we have the negative binomial-I model

(NB-I). When = 1/ we have the negative binomial-II (NP-II) model. For the NB-I density,
V (Y ) = + . In the case of the NB-II model, we have V (Y ) = + 2 . For both forms,
E(Y ) = .
The reshaped density, with normalization to sum to one, is
2

[hp (y|)]
(y + )
fY (y|, ) =
p (, ) (y + 1)()

 

y
.

(20.9)

To get the normalization factor, we need the moment generating function:

MY (t) = et +

(20.10)

To illustrate, Figure 20.5 shows calculation of the first four raw moments of the NB density, calculated using MuPAD, which is a Computer Algebra System that (used to be?) free for personal
use. These are the moments you would need to use a second order polynomial (p = 2). MuPAD
will output these results in the form of C code, which is relatively easy to edit to write the likelihood function for the model. This has been done in NegBinSNP.cc, which is a C++ version of this
model that can be compiled to use with octave using the mkoctfile command. Note the impressive length of the expressions when the degree of the expansion is 4 or 5! This is an example of a
model that would be difficult to formulate without the help of a program like MuPAD.
It is possible that there is conditional heterogeneity such that the appropriate reshaping should

Figure 20.5: Negative binomial raw moments

be more local. This can be accomodated by allowing the k parameters to depend upon the conditioning variables, for example using polynomials.
Gallant and Nychka, Econometrica, 1987 prove that this sort of density can approximate a wide
variety of densities arbitrarily well as the degree of the polynomial increases with the sample size.
This approach is not without its drawbacks: the sample objective function can have an extremely
large number of local maxima that can lead to numeric difficulties. If someone could figure out
how to do in a way such that the sample objective function was nice and smooth, they would
probably get the paper published in a good journal. Any ideas?
Heres a plot of true and the limiting SNP approximations (with the order of the polynomial
fixed) to four different count data densities, which variously exhibit over and underdispersion, as
well as excess zeros. The baseline model is a negative binomial density.

Case 1

Case 2

.5
.4

.1

.3
.2

.05

.1

10

15

20

Case 3

10

15

20

25

Case 4

.25

.2

.2

.15

.15
.1
.1
.05
.05
1

20.5

2.5

7.5

10

12.5

15

Examples

MEPS health care usage data

Well use the MEPS OBDV data to illustrate kernel regression and semi-nonparametric maximum
likelihood.
Kernel regression estimation
Lets try a kernel regression fit for the OBDV data. The program OBDVkernel.m loads the MEPS
OBDV data, scans over a range of window widths and calculates leave-one-out CV scores, and
plots the fitted OBDV usage versus AGE, using the best window width. The plot is in Figure 20.6.
Note that usage increases with age, just as weve seen with the parametric models. Once could use
bootstrapping to generate a confidence interval to the fit.
Seminonparametric ML estimation and the MEPS data
Now lets estimate a seminonparametric density for the OBDV data. Well reshape a negative
binomial density, as discussed above. The program EstimateNBSNP.m loads the MEPS OBDV data
and estimates the model, using a NB-I baseline density and a 2nd order polynomial expansion.
The output is:

OBDV
======================================================
BFGSMIN final results
Used numeric gradient

Figure 20.6: Kernel fitted OBDV usage versus AGE

Kernel fit, OBDV visits versus AGE

3.29

3.285

3.28

3.275

3.27

3.265

3.26

3.255

20

25

30

35

40

45

50

55

Age

-----------------------------------------------------STRONG CONVERGENCE
Function conv 1 Param conv 1 Gradient conv 1
-----------------------------------------------------Objective function value 2.17061
Stepsize 0.0065
24 iterations
-----------------------------------------------------param
1.3826
0.2317
0.1839
0.2214
0.1898
0.0722
-0.0002
1.7853
-0.4358
0.1129

gradient
0.0000
-0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
-0.0000
0.0000
0.0000

change
-0.0000
0.0000
0.0000
-0.0000
-0.0000
-0.0000
-0.0000
-0.0000
-0.0000
0.0000

******************************************************
NegBin SNP model, MEPS full data set
MLE Estimation Results
BFGS convergence: Normal convergence
Average Log-L: -2.170614
Observations: 4564
constant
pub. ins.
priv. ins.

estimate
-0.147
0.695
0.409

st. err
0.126
0.050
0.046

t-stat
-1.173
13.936
8.833

p-value
0.241
0.000
0.000

60

65

Figure 20.7: Dollar-Euro

sex
age
edu
inc
gam1
gam2
lnalpha

0.443
0.016
0.025
-0.000
1.785
-0.436
0.113

0.034
0.001
0.006
0.000
0.141
0.029
0.027

13.148
11.880
3.903
-0.011
12.629
-14.786
4.166

0.000
0.000
0.000
0.991
0.000
0.000
0.000

Information Criteria
CAIC : 19907.6244
Avg. CAIC:
4.3619
BIC : 19897.6244
Avg. BIC:
4.3597
AIC : 19833.3649
Avg. AIC:
4.3456
******************************************************
Note that the CAIC and BIC are lower for this model than for the models presented in Table 18.3.
This model fits well, still being parsimonious. You can play around trying other use measures, using
a NP-II baseline density, and using other orders of expansions. Density functions formed in this way
may have MANY local maxima, so you need to be careful before accepting the results of a casual
run. To guard against having converged to a local maximum, one can try using multiple starting
values, or one could try simulated annealing as an optimization method. If you uncomment the
relevant lines in the program, you can use SA to do the minimization. This will take a lot of
time, compared to the default BFGS minimization. The chapter on parallel computations might be
interesting to read before trying this.

Financial data and volatility

The data set rates contains the growth rate (100log difference) of the daily spot $/euro and
$/yen exchange rates at New York, noon, from January 04, 1999 to February 12, 2008. There are
2291 observations. See the README file for details. Figures ?? and ?? show the data and their
histograms.
at the center of the histograms, the bars extend above the normal density that best fits the
data, and the tails are fatter than those of the best fit normal density. This feature of the data
is known as leptokurtosis.
in the series plots, we can see that the variance of the growth rates is not constant over time.
Volatility clusters are apparent, alternating between periods of stability and periods of more
wild swings. This is known as conditional heteroscedasticity. ARCH and GARCH well-known
models that are often applied to this sort of data.
Many structural economic models often cannot generate data that exhibits conditional heteroscedasticity without directly assuming shocks that are conditionally heteroscedastic. It

Figure 20.8: Dollar-Yen

Figure 20.9: Kernel regression fitted conditional second moments, Yen/Dollar and Euro/Dollar
(a) Yen/Dollar

(b) Euro/Dollar

would be nice to have an economic explanation for how conditional heteroscedasticity, leptokurtosis, and other (leverage, etc.) features of financial data result from the behavior of
economic agents, rather than from a black box that provides shocks.
2
2
The Octave script kernelfit.m performs kernel regression to fit E(yt2 |yt1,
yt2
), and generates the

plots in Figure 20.9.

From the point of view of learning the practical aspects of kernel regression, note how the
data is compactified in the example script.
In the Figure, note how current volatility depends on lags of the squared return rate - it is
high when both of the lags are high, but drops off quickly when either of the lags is low.
The fact that the plots are not flat suggests that this conditional moment contain information
about the process that generates the data. Perhaps attempting to match this moment might
be a means of estimating the parameters of the dgp. Well come back to this later.

20.6

Exercises

1. In Octave, type edit kernel_example.

(a) Look this script over, and describe in words what it does.
(b) Run the script and interpret the output.
(c) Experiment with different bandwidths, and comment on the effects of choosing small
and large values.
2. In Octave, type help kernel_regression.
(a) How can a kernel fit be done without supplying a bandwidth?
(b) How is the bandwidth chosen if a value is not provided?
(c) What is the default kernel used?
3. Using the Octave script OBDVkernel.m as a model, plot kernel regression fits for OBDV visits
as a function of income and education.

Chapter 21

Simulation-based estimation
Readings: Gourieroux and Monfort (1996) Simulation-Based Econometric Methods (Oxford University Press). There are many articles. Some of the seminal papers are Gallant and Tauchen (1996),
Which Moments to Match?, ECONOMETRIC THEORY, Vol. 12, 1996, pages 657-681; Gourieroux, Monfort and Renault (1993), Indirect Inference, J. Apl. Econometrics; Pakes and Pollard
(1989) Econometrica; McFadden (1989) Econometrica.

21.1

Motivation

Simulation methods are of interest when the DGP is fully characterized by a parameter vector, so
that simulated data can be generated, but the likelihood function and moments of the observable
varables are not calculable, so that MLE or GMM estimation is not possible. Many moderately
complex models result in intractible likelihoods or moments, as we will see. Simulation-based
estimation methods open up the possibility to estimate truly complex models. The desirability
introducing a great deal of complexity may be an issue1 , but it least it becomes a possibility.

Example: Multinomial and/or dynamic discrete response models

Let yi be a latent random vector of dimension m. Suppose that
yi = Xi + i
where Xi is m K. Suppose that
i N (0, )

(21.1)

Henceforth drop the i subscript when it is not needed for clarity.

y is not observed. Rather, we observe a many-to-one mapping
y = (y )
This mapping is such that each element of y is either zero or one (in some cases only one
element will be one).
Define
Ai = A(yi ) = {y |yi = (y )}
1 Remember that a model is an abstraction from reality, and abstraction helps us to isolate the important features of a
phenomenon.

297

Suppose random sampling of (yi , Xi ). In this case the elements of yi may not be independent
of one another (and clearly are not if is not diagonal). However, yi is independent of yj ,
i 6= j.
Let = ( 0 , (vec )0 )0 be the vector of parameters of the model. The contribution of the ith
observation to the likelihood function is
Z
pi () =

n(yi Xi , )dyi

Ai

where
M/2

n(, ) = (2)

||

1/2

0 1
exp
2


is the multivariate normal density of an M -dimensional random vector. The log-likelihood

function is

ln L() =

1X
ln pi ()
n i=1

and the MLE solves the score equations

n
n

1 X D pi ()
1X
gi () =
0.

n i=1
n i=1 pi ()

The problem is that evaluation of Li () and its derivative w.r.t. by standard methods of
numeric integration such as quadrature is computationally infeasible when m (the dimension
of y) is higher than 3 or 4 (as long as there are no restrictions on ).
The mapping (y ) has not been made specific so far. This setup is quite general: for different
choices of (y ) it nests the case of dynamic binary discrete choice models as well as the case
of multinomial discrete choice (the choice of one out of a finite set of alternatives).
Multinomial discrete choice is illustrated by a (very simple) job search model. We have
cross sectional data on individuals matching to a set of m jobs that are available (one
of which is unemployment). The utility of alternative j is
uj = Xj + j
Utilities of jobs, stacked in the vector ui are not observed. Rather, we observe the vector
formed of elements
yj = 1 [uj > uk , k m, k 6= j]
Only one of these elements is different than zero.
Dynamic discrete choice is illustrated by repeated choices over time between two alternatives. Let alternative j have utility
ujt

= Wjt jt ,

{1, 2}

{1, 2, ..., m}

Then
y

u2 u1

(W2 W1 ) + 2 1

X +

Now the mapping is (element-by-element)

y = 1 [y > 0] ,
that is yit = 1 if individual i chooses the second alternative in period t, zero otherwise.

Example: Marginalization of latent variables

Economic data often presents substantial heterogeneity that may be difficult to model. A possibility
is to introduce latent random variables. This can cause the problem that there may be no known
closed form for the distribution of observable variables after marginalizing out the unobservable
latent variables. For example, count data (that takes values 0, 1, 2, 3, ...) is often modeled using the
Poisson distribution
Pr(y = i) =

exp()i
i!

The mean and variance of the Poisson distribution are both equal to :
E(y) = V (y) = .
Often, one parameterizes the conditional mean as
i = exp(Xi ).
This ensures that the mean is positive (as it must be). Estimation by ML is straightforward.
Often, count data exhibits overdispersion which simply means that
V (y) > E(y).
If this is the case, a solution is to use the negative binomial distribution rather than the Poisson.
An alternative is to introduce a latent variable that reflects heterogeneity into the specification:
i = exp(Xi + i )
where i has some specified density with support S (this density may depend on additional parameters). Let d(i ) be the density of i . In some cases, the marginal density of y
Z
Pr(y = yi ) =
S

exp [ exp(Xi + i )] [exp(Xi + i )] i

d(i )
yi !

will have a closed-form solution (one can derive the negative binomial distribution in the way if
has an exponential distribution - see equation 18.1), but often this will not be possible. In this
case, simulation is a means of calculating Pr(y = i), which is then used to do ML estimation. This
would be an example of the Simulated Maximum Likelihood (SML) estimation.
In this case, since there is only one latent variable, quadrature is probably a better choice.
However, a more flexible model with heterogeneity would allow all parameters (not just the

constant) to vary. For example

Z
Pr(y = yi ) =
S

exp [ exp(Xi i )] [exp(Xi i )] i

d(i )
yi !

entails a K = dim i -dimensional integral, which will not be evaluable by quadrature when
K gets large.

Estimation of models specified in terms of stochastic differential equations

It is often convenient to formulate models in terms of continuous time using differential equations.
A realistic model should account for exogenous shocks to the system, which can be done by assuming a random component. This leads to a model that is expressed as a system of stochastic
differential equations. Consider the process
dyt = g(, yt )dt + h(, yt )dWt
which is assumed to be stationary. {Wt } is a standard Brownian motion (Weiner process), such
that
Z

dWt N (0, T )

W (T ) =
0

Brownian motion is a continuous-time stochastic process such that

W (0) = 0
[W (s) W (t)] N (0, s t)
[W (s) W (t)] and [W (j) W (k)] are independent for s > t > j > k. That is, nonoverlapping segments are independent.
One can think of Brownian motion the accumulation of independent normally distributed shocks
with infinitesimal variance.
The function g(, yt ) is the deterministic part.
h(, yt ) determines the variance of the shocks.
To estimate a model of this sort, we typically have data that are assumed to be observations of yt
in discrete points y1 , y2 , ...yT . That is, though yt is a continuous process it is observed in discrete
time.
To perform inference on , direct ML or GMM estimation is not usually feasible, because one
cannot, in general, deduce the transition density f (yt |yt1 , ). This density is necessary to evaluate
the likelihood function or to evaluate moment conditions (which are based upon expectations with
respect to this density).
A typical solution is to discretize the model, by which we mean to find a discrete time
approximation to the model. The discretized version of the model is
yt yt1
t

= g(, yt1 ) + h(, yt1 )t

N (0, 1)

The discretization induces a new parameter, (that is, the 0 which defines the best approximation of the discretization to the actual (unknown) discrete time version of the model is
not equal to 0 which is the true parameter value). This is an approximation, and as such
ML estimation of (which is actually quasi-maximum likelihood, QML) based upon this

equation is in general biased and inconsistent for the original parameter, . Nevertheless, the
approximation shouldnt be too bad, which will be useful, as we will see.
The important point about these three examples is that computational difficulties prevent
direct application of ML, GMM, etc. Nevertheless the model is fully specified in probabilistic
terms up to a parameter vector. This means that the model is simulable, conditional on the
parameter vector.

21.2

Simulated maximum likelihood (SML)

For simplicity, consider cross-sectional data. An ML estimator solves

n

1X
M L = arg max sn () =
ln p(yt |Xt , )
n t=1
where p(yt |Xt , ) is the density function of the tth observation. When p(yt |Xt , ) does not have a
known closed form, M L is an infeasible estimator. However, it may be possible to define a random
function such that
E f (, yt , Xt , ) = p(yt |Xt , )
where the density of is known. If this is the case, the simulator
p (yt , Xt , ) =

H
1 X
f (ts , yt , Xt , )
H s=1

is unbiased for p(yt |Xt , ).

The SML simply substitutes p (yt , Xt , ) in place of p(yt |Xt , ) in the log-likelihood function,
that is

1X
ln p (yt , Xt , )
SM L = arg max sn () =
n i=1

Example: multinomial probit

Recall that the utility of alternative j is
uj = Xj + j
and the vector y is formed of elements
yj = 1 [uj > uk , k m, k 6= j]
The problem is that Pr(yj = 1|) cant be calculated when m is larger than 4 or 5. However, it is
easy to simulate this probability.
Draw i from the distribution N (0, )
Calculate u
i = Xi + i (where Xi is the matrix formed by stacking the Xij )
Define yij = 1 [uij > uik , k m, k 6= j]
Repeat this H times and define
PH

eij =

h=1

yijh

 Define
ei as the m-vector formed of the
eij . Each element of
ei is between 0 and 1, and the
elements sum to one.
ei
Now p (yi , Xi , ) = yi0
The SML multinomial probit log-likelihood function is
n

ln L(, ) =

1X 0
y ln p (yi , Xi , )
n i=1 i

This is to be maximized w.r.t. and .

Notes:
The H draws of i are draw only once and are used repeatedly during the iterations used to
The draws are different for each i. If the i are re-drawn at every iteration the
find and .
estimator will not converge.
The log-likelihood function with this simulator is a discontinuous function of and . This
does not cause problems from a theoretical point of view since it can be shown that ln L(, )
is stochastically equicontinuous. However, it does cause problems if one attempts to use a
gradient-based optimization method such as Newton-Raphson.
It may be the case, particularly if few simulations, H, are used, that some elements of
ei are
zero. If the corresponding element of yi is equal to 1, there will be a log(0) problem.
Solutions to discontinuity:
1) use an estimation method that doesnt require a continuous and differentiable objective function, for example, simulated annealing. This is computationally costly.
2) Smooth the simulated probabilities so that they are continuous functions of the parameters. For example, apply a kernel transformation such as





m
m
yij = A uij max uik
+ .5 1 uij = max uik
k=1

k=1

where A is a large positive number. This approximates a step function such that yij
is very close to zero if uij is not the maximum, and yij is very close to 1 if uij is the
maximum. This makes yij a continuous function of and , so that pij and therefore
p

ln L(, ) will be continuous and differentiable. Consistency requires that A(n) ,

so that the approximation to a step function becomes arbitrarily close as the sample size
increases. There are alternative methods (e.g., Gibbs sampling) that may work better,
but this is too technical to discuss here.
To solve to log(0) problem, one possibility is to search the web for the slog function. Also,
increase H if this is a serious problem.

Properties
The properties of the SML estimator depend on how H is set. The following is taken from Lee
(1995) Asymptotic Bias in Simulated Maximum Likelihood Estimation of Discrete Choice Models,
Econometric Theory, 11, pp. 437-83.
Theorem 67. [Lee] 1) if limn n1/2 /H = 0, then


d
n SM L 0 N (0, I 1 (0 ))

2) if limn n1/2 /H = , a finite constant, then



d
n SM L 0 N (B, I 1 (0 ))
where B is a finite vector of constants.
This means that the SML estimator is asymptotically biased if H doesnt grow faster than
n1/2 .
The varcov is the typical inverse of the information matrix, so that as long as H grows fast
enough the estimator is consistent and fully asymptotically efficient.

21.3

Method of simulated moments (MSM)

Suppose we have a DGP(y|x, ) which is simulable given , but is such that the density of y is not
calculable.
Once could, in principle, base a GMM estimator upon the moment conditions
mt () = [K(yt , xt ) k(xt , )] zt
where

Z
k(xt , ) =

K(yt , xt )p(y|xt , )dy,

zt is a vector of instruments in the information set and p(y|xt , ) is the density of y conditional on
xt . The problem is that this density is not available.
However k(xt , ) is readily simulated using
H
1 X
e
K(e
yth , xt )
k (xt , ) =
H
h=1

a.s.
By the law of large numbers, e
k (xt , ) k (xt , ) , as H , which provides a clear intuitive

basis for the estimator, though in fact we obtain consistency even for H finite, since a law of
large numbers is also operating across the n observations of real data, so errors introduced
by simulation cancel themselves out.
This allows us to form the moment conditions
h
i
m
ft () = K(yt , xt ) e
k (xt , ) zt

(21.2)

where zt is drawn from the information set. As before, form

n

m()
e

1X
m
ft ()
n i=1
"
#
n
H
1X
1 X
h
K(yt , xt )
k(e
yt , xt ) zt
n i=1
H

(21.3)

h=1

with which we form the GMM criterion and estimate as usual. Note that the unbiased simulator k(e
yth , xt ) appears linearly within the sums.

Properties
Suppose that the optimal weighting matrix is used. McFadden (ref. above) and Pakes and Pollard
(refs. above) show that the asymptotic distribution of the MSM estimator is very similar to that of
the infeasible GMM estimator. In particular, assuming that the optimal weighting matrix is used,
and for H finite,


 





1
d
0 1
D 1 D
n M SM 0 N 0, 1 +
H

0
where D 1 D

1

(21.4)

is the asymptotic variance of the infeasible GMM estimator.

That is, the asymptotic variance is inflated by a factor 1 + 1/H. For this reason the MSM
estimator is not fully asymptotically efficient relative to the infeasible GMM estimator, for H
finite, but the efficiency loss is small and controllable, by setting H reasonably large.
The estimator is asymptotically unbiased even for H = 1. This is an advantage relative to
SML.
If one doesnt use the optimal weighting matrix, the asymptotic varcov is just the ordinary
GMM varcov, inflated by 1 + 1/H.
The above presentation is in terms of a specific moment condition based upon the conditional
mean. Simulated GMM can be applied to moment conditions of any form.

Comments
Why is SML inconsistent if H is finite, while MSM is? The reason is that SML is based upon an
average of logarithms of an unbiased simulator (the densities of the observations). To use the
multinomial probit model as an example, the log-likelihood function is
n

ln L(, ) =

1X 0
y ln pi (, )
n i=1 i

ln L(, ) =

1X 0
y ln pi (, )
n i=1 i

The SML version is

The problem is that

E ln(
pi (, )) 6= ln(E pi (, ))
in spite of the fact that
E pi (, ) = pi (, )
due to the fact that ln() is a nonlinear transformation. The only way for the two to be equal (in
the limit) is if H tends to infinite so that p () tends to p ().
The reason that MSM does not suffer from this problem is that in this case the unbiased simulator appears linearly within every sum of terms, and it appears within a sum over n (see equation
[21.3]). Therefore the SLLN applies to cancel out simulation errors, from which we get consistency.
That is, using simple notation for the random sampling case, the moment conditions
m()

"
#
n
H
1 X
1X
h
K(yt , xt )
k(e
yt , xt ) zt
n i=1
H
h=1
"
#
n
H
X
1X
1
k(xt , 0 ) + t
[k(xt , ) + ht ] zt
n i=1
H
h=1

(21.5)

(21.6)

converge almost surely to

Z
m
() =


k(x, 0 ) k(x, ) z(x)d(x).

(note: zt is assume to be made up of functions of xt ). The objective function converges to

s () = m
()0 1
()
m
which obviously has a minimum at 0 , henceforth consistency.
If you look at equation 21.6 a bit, you will see why the variance inflation factor is (1 +

21.4

1
H ).

Efficient method of moments (EMM)

The choice of which moments upon which to base a GMM estimator can have very pronounced
effects upon the efficiency of the estimator.
A poor choice of moment conditions may lead to very inefficient estimators, and can even
cause identification problems (as weve seen with the GMM problem set).
The drawback of the above approach MSM is that the moment conditions used in estimation
are selected arbitrarily. The asymptotic efficiency of the estimator may be low.
The asymptotically optimal choice of moments would be the score vector of the likelihood
function,
mt () = D ln pt ( | It )
As before, this choice is unavailable.
The efficient method of moments (EMM) (see Gallant and Tauchen (1996), Which Moments
to Match?, ECONOMETRIC THEORY, Vol. 12, 1996, pages 657-681) seeks to provide moment
conditions that closely mimic the score vector. If the approximation is very good, the resulting
estimator will be very nearly fully efficient.
The DGP is characterized by random sampling from the density
p(yt |xt , 0 ) pt (0 )
We can define an auxiliary model, called the score generator, which simply provides a (misspecified) parametric density
f (y|xt , ) ft ()
This density is known up to a parameter . We assume that this density function is calculable.
Therefore quasi-ML estimation is possible. Specifically,
n

X
= arg max sn () = 1
ln ft ().

n t=1
we can calculate the score functions D ln f (yt |xt , ).

After determining
The important point is that even if the density is misspecified, there is a pseudo-true 0
for which the true expectation, taken with respect to the true but unknown density of y,
p(y|xt , 0 ), and then marginalized over x is zero:


0 : EX EY |X D ln f (y|x, 0 ) =

Z Z
X

Y |X

D ln f (y|x, 0 )p(y|x, 0 )dyd(x) = 0

0 ; this suggests using the moment conditions

We have seen in the section on QML that
n

=
mn (, )

1X
n t=1

t ()dy
D ln ft ()p

(21.7)

These moment conditions are not calculable, since pt () is not available, but they are simulable using
n
H
X
1 X
= 1

m
fn (, )
D ln f (e
yth |xt , )
n t=1 H
h=1

where

yth

converges
is a draw from DGP (), holding xt fixed. By the LLN and the fact that

to ,
m
e (0 , 0 ) = 0.
This is not the case for other values of , assuming that 0 is identified.
The advantage of this procedure is that if f (yt |xt , ) closely approximates p(y|xt , ), then
will closely approximate the optimal moment conditions which characterize maxim
e n (, )
mum likelihood estimation, which is fully efficient.
If one has prior information that a certain density approximates the data well, it would be a
good choice for f ().
If one has no density in mind, there exist good ways of approximating unknown distributions
parametrically: Philips ERAs (Econometrica, 1983) and Gallant and Nychkas (Econometrica,
1987) SNP density estimator which we saw before. Since the SNP density is consistent, the
efficiency of the indirect estimator is the same as the infeasible ML estimator.

Optimal weighting matrix

I will present the theory for H finite, and possibly small. This is done because it is sometimes
impractical to estimate with H very large. Gallant and Tauchen give the theory for the case of
H so large that it may be treated as infinite (the difference being irrelevant given the numerical
precision of a computer). The theory for the case of H infinite follows directly from the results
presented here.
depends on the pseudo-ML estimate .
We can apply Theorem
The moment condition m(,
e )
31 to conclude that





d
0
N 0, J (0 )1 I(0 )J (0 )1
n

(21.8)

were in fact the true density p(y|xt , ), then

would be the maximum
If the density f (yt |xt , )
likelihood estimator, and J (0 )1 I(0 ) would be an identity matrix, due to the information matrix
is only an approximation to
equality. However, in the present case we assume that f (yt |xt , )
p(y|xt , ), so there is no cancellation.

Recall that J (0 ) p lim

2
0
0 sn ( )

. Comparing the definition of sn () with the definition

of the moment condition in Equation 21.7, we see that

J (0 ) = D0 m(0 , 0 ).
As in Theorem 31,

 

sn ()  sn () 
I( ) = lim E n
.
n
0 0 0
0

In this case, this is simply the asymptotic variance covariance matrix of the moment conditions, .

about 0 :
Now take a first order Taylors series approximation to nmn (0 , )


First consider

=
nm
n (0 , )

nm
n (0 , 0 ) +



0 + op (1)
nD0 m(
0 , 0 )

nm
n (0 , 0 ). It is straightforward but somewhat tedious to show that the

1
0
H I ( ).



a.s.
0 . Note that D0 m
Next consider the second term nD0 m(
0 , 0 )
n (0 , 0 ) J (0 ),

asymptotic variance of this term is

so we have





0 , a.s.
0 = nJ (0 )
nD0 m(
0 , 0 )

But noting equation 21.8





a
0
N 0, I(0 )
nJ (0 )

Now, combining the results for the first and second terms,

 


1
a
0

nm
n ( , ) N 0, 1 +
I( )
H
0

0 ) is a consistent estimator of the asymptotic variance-covariance matrix of the

\
Suppose that I(

moment conditions. This may be complicated if the score generator is a poor approximator, since
the individual score contributions may not have mean zero in this case (see the section on QML)
. Even if this is the case, the individuals means can be calculated by simulation, so it is always
possible to consistently estimate I(0 ) when the model is simulable. On the other hand, if the
score generator is taken to be correctly specified, the ordinary estimator of the information matrix
is consistent. Combining this with the result on the efficient GMM weighting matrix in Theorem
46, we see that defining as
0
= arg min mn (, )



1
1+
H

1


0)
\
I(

mn (, )

is the GMM estimator with the efficient choice of weighting matrix.

If one has used the Gallant-Nychka ML estimator as the auxiliary model, the appropriate
weighting matrix is simply the information matrix of the auxiliary model, since the scores
are uncorrelated. (e.g., it really is ML estimation asymptotically, since the score generator
can approximate the unknown density arbitrarily well).

Asymptotic distribution
Since we use the optimal weighting matrix, the asymptotic distribution is as in Equation 14.3, so
we have (using the result in Equation 21.8):

!1


1


1
d
0
,
n 0 N 0, D 1 +
I(0 )
D
H
where


D = lim E D m0n (0 , 0 ) .
n

This can be consistently estimated using

)

= D m0 (,
D
n

Diagnotic testing
The fact that

 


1
a

nmn (0 , )
N 0, 1 +
I(0 )
H

implies that
)
0
nmn (,



1
1+
H

1
a
)

mn (,
2 (q)
I()

where q is dim() dim(), since without dim() moment conditions the model is not identified,
so testing is impossible. One test of the model is simply based on this statistic: if it exceeds the
2 (q) critical point, something may be wrong (the small sample performance of this sort of test
would be a topic worth investigating).
Information about what is wrong can be gotten from the pseudo-t-statistics:


1/2 !1

diag 1 +
I()
nmn (,
H
can be used to test which moments are not well modeled. Since these moments are related
to parameters of the score generator, which are usually related to certain features of the
model, this information can be used to revise the model. These arent actually distributed as

and nmn (,
)
have different distributions (that of nmn (,
)

N (0, 1), since nmn (0 , )

is somewhat more complicated). It can be shown that the pseudo-t statistics are biased
toward nonrejection. See Gourieroux et. al. or Gallant and Long, 1995, for more details.

21.5

Examples

SML of a Poisson model with latent heterogeneity

We have seen (see equation 18.1) that a Poisson model with latent heterogeneity that follows an
exponential distribution leads to the negative binomial model. To illustrate SML, we can integrate
out the latent heterogeneity using Monte Carlo, rather than the analytical approach which leads to
the negative binomial model. In actual practice, one would not want to use SML in this case, but it
is a nice example since it allows us to compare SML to the actual ML estimator. The Octave function
defined by PoissonLatentHet.m calculates the simulated log likelihood for a Poisson model where
= exp x0t +), where N (0, 1). This model is similar to the negative binomial model, except
that the latent variable is normally distributed rather than gamma distributed. The Octave script
EstimatePoissonLatentHet.m estimates this model using the MEPS OBDV data that has already
been discussed. Note that simulated annealing is used to maximize the log likelihood function.
Attempting to use BFGS leads to trouble. I suspect that the log likelihood is approximately nondifferentiable in places, around which it is very flat, though I have not checked if this is true. If
you run this script, you will see that it takes a long time to get the estimation results, which are:

******************************************************
Poisson Latent Heterogeneity model, SML estimation, MEPS 1996 full data set
MLE Estimation Results
BFGS convergence: Max. iters. exceeded
Average Log-L: -2.171826
Observations: 4564

constant
pub. ins.
priv. ins.
sex
age
edu
inc
lnalpha

estimate
-1.592
1.189
0.655
0.615
0.018
0.024
-0.000
0.203

st. err
0.146
0.068
0.065
0.044
0.002
0.010
0.000
0.014

t-stat
-10.892
17.425
10.124
13.888
10.865
2.523
-0.531
14.036

p-value
0.000
0.000
0.000
0.000
0.000
0.012
0.596
0.000

Information Criteria
CAIC : 19899.8396
Avg. CAIC:
4.3602
BIC : 19891.8396
Avg. BIC:
4.3584
AIC : 19840.4320
Avg. AIC:
4.3472
******************************************************
octave:3>

If you compare these results to the results for the negative binomial model, given in subsection
(18.2), you can see that the present model fits better according to the CAIC criterion. The present
model is considerably less convenient to work with, however, due to the computational requirements. The chapter on parallel computing is relevant if you wish to use models of this sort.

SMM
To be added in future: do SMM using unconditional moments for SV model (compare to Andersen
et al and others)

SNM
To be added.

EMM estimation of a discrete choice model

In this section consider EMM estimation. There is a sophisticated package by Gallant and Tauchen
for this, but here well look at some simple, but hopefully didactic code. The file probitdgp.m
generates data that follows the probit model. The file emm_moments.m defines EMM moment
conditions, where the DGP and score generator can be passed as arguments. Thus, it is a general
purpose moment condition for EMM estimation. This file is interesting enough to warrant some
discussion. A listing appears in Listing 19.1. Line 3 defines the DGP, and the arguments needed
to evaluate it are defined in line 4. The score generator is defined in line 5, and its arguments
are defined in line 6. The QML estimate of the parameter of the score generator is read in line 7.
Note in line 10 how the random draws needed to simulate data are passed with the data, and are
thus fixed during estimation, to avoid chattering. The simulated data is generated in line 16, and
the derivative of the score generator using the simulated data is calculated in line 18. In line 20
we average the scores of the score generator, which are the moment conditions that the function
returns.
1
2

function scores = emm_moments(theta, data, momentargs)

k = momentargs{1};

3
4
5
6
7
8
9
10

11
12
13
14
15
16
17
18
19
20
21

dgp = momentargs{2}; # the data generating process (DGP)

dgpargs = momentargs{3}; # its arguments (cell array)
sg = momentargs{4}; # the score generator (SG)
sgargs = momentargs{5}; # SG arguments (cell array)
phi = momentargs{6}; # QML estimate of SG parameter
y = data(:,1);
x = data(:,2:k+1);
rand_draws = data(:,k+2:columns(data)); # passed with data to ensure fixed across
iterations
n = rows(y);
scores = zeros(n,rows(phi)); # container for moment contributions
reps = columns(rand_draws); # how many simulations?
for i = 1:reps
e = rand_draws(:,i);
y = feval(dgp, theta, x, e, dgpargs); # simulated data
sgdata = [y x]; # simulated data for SG
scores = scores + numgradient(sg, {phi, sgdata, sgargs}); # gradient of SG
endfor
scores = scores / reps; # average over number of simulations
endfunction

Listing 21.1: emm_moments.m

The file emm_example.m performs EMM estimation of the probit model, using a logit model as
the score generator. The results we obtain are

Score generator results:

=====================================================
BFGSMIN final results
Used analytic gradient
-----------------------------------------------------STRONG CONVERGENCE
Function conv 1 Param conv 1 Gradient conv 1
-----------------------------------------------------Objective function value 0.281571
Stepsize 0.0279
15 iterations
-----------------------------------------------------param
1.8979
1.6648
1.9125
1.8875
1.7433

gradient
0.0000
-0.0000
-0.0000
-0.0000
-0.0000

change
0.0000
0.0000
0.0000
0.0000
0.0000

======================================================
Model results:
******************************************************
EMM example
GMM Estimation Results
BFGS convergence: Normal convergence

Objective function value: 0.000000

Observations: 1000
Exactly identified, no spec. test
estimate
st. err
t-stat
p-value
p1
1.069
0.022
47.618
0.000
p2
0.935
0.022
42.240
0.000
p3
1.085
0.022
49.630
0.000
p4
1.080
0.022
49.047
0.000
p5
0.978
0.023
41.643
0.000
******************************************************
It might be interesting to compare the standard errors with those obtained from ML estimation,
to check efficiency of the EMM estimator. One could even do a Monte Carlo study.

21.6

Exercises

1. (basic) Examine the Octave script and function discussed in subsection 21.5 and describe
what they do.
2. (basic) Examine the Octave scripts and functions discussed in subsection 21.5 and describe
what they do.
3. (advanced, but even if you dont do this you should be able to describe what needs to be
done) Write Octave code to do SML estimation of the probit model. Do an estimation using data generated by a probit model ( probitdgp.m might be helpful). Compare the SML
estimates to ML estimates.
4. (more advanced) Do a little Monte Carlo study to compare ML, SML and EMM estimation of
the probit model. Investigate how the number of simulations affect the two simulation-based
estimators.

Chapter 22

Parallel programming for

econometrics
The following borrows heavily from Creel (2005).
Parallel computing can offer an important reduction in the time to complete computations. This
is well-known, but it bears emphasis since it is the main reason that parallel computing may be
attractive to users. To illustrate, the Intel Pentium IV (Willamette) processor, running at 1.5GHz,
was introduced in November of 2000. The Pentium IV (Northwood-HT) processor, running at
3.06GHz, was introduced in November of 2002. An approximate doubling of the performance
of a commodity CPU took place in two years. Extrapolating this admittedly rough snapshot of
the evolution of the performance of commodity processors, one would need to wait more than
6.6 years and then purchase a new computer to obtain a 10-fold improvement in computational
performance. The examples in this chapter show that a 10-fold improvement in performance can
be achieved immediately, using distributed parallel computing on available computers.
Recent (this is written in 2005) developments that may make parallel computing attractive
to a broader spectrum of researchers who do computations. The first is the fact that setting up
a cluster of computers for distributed parallel computing is not difficult. If you are using the
ParallelKnoppix bootable CD that accompanies these notes, you are less than 10 minutes away
from creating a cluster, supposing you have a second computer at hand and a crossover ethernet
cable. See the ParallelKnoppix tutorial. A second development is the existence of extensions to
some of the high-level matrix programming (HLMP) languages1 that allow the incorporation of
parallelism into programs written in these languages. A third is the spread of dual and quad-core
CPUs, so that an ordinary desktop or laptop computer can be made into a mini-cluster. Those cores
wont work together on a single problem unless they are told how to.
Following are examples of parallel implementations of several mainstream problems in econometrics. A focus of the examples is on the possibility of hiding parallelization from end users of
programs. If programs that run in parallel have an interface that is nearly identical to the interface
of equivalent serial versions, end users will find it easy to take advantage of parallel computings
performance. We continue to use Octave, taking advantage of the MPI Toolbox (MPITB) for Octave, by by Fernndez Baldomero et al. (2004). There are also parallel packages for Ox, R, and
Python which may be of interest to econometricians, but as of this writing, the following examples
are the most accessible introduction to parallel programming for econometricians.
1 By high-level matrix programming language I mean languages such as MATLAB (TM the Mathworks, Inc.), Ox (TM
OxMetrics Technologies, Ltd.), and GNU Octave (www.octave.org), for example.

313

22.1

Example problems

This section introduces example problems from econometrics, and shows how they can be parallelized in a natural way.

Monte Carlo
A Monte Carlo study involves repeating a random experiment many times under identical conditions. Several authors have noted that Monte Carlo studies are obvious candidates for parallelization (Doornik et al. 2002; Bruche, 2003) since blocks of replications can be done independently
on different computers. To illustrate the parallelization of a Monte Carlo study, we use same trace
test example as do Doornik, et. al. (2002). tracetest.m is a function that calculates the trace test
statistic for the lack of cointegration of integrated time series. This function is illustrative of the
format that we adopt for Monte Carlo simulation of a function: it receives a single argument of
cell type, and it returns a row vector that holds the results of one random simulation. The single
argument in this case is a cell array that holds the length of the series in its first position, and
the number of series in the second position. It generates a random result though a process that
is internal to the function, and it reports some output in a row vector (in this case the result is a
scalar).
mc_example1.m is an Octave script that executes a Monte Carlo study of the trace test by
repeatedly evaluating the tracetest.m function. The main thing to notice about this script is
that lines 7 and 10 call the function montecarlo.m. When called with 3 arguments, as in line
7, montecarlo.m executes serially on the computer it is called from. In line 10, there is a fourth
argument. When called with four arguments, the last argument is the number of slave hosts to
use. We see that running the Monte Carlo study on one or more processors is transparent to the
user - he or she must only indicate the number of slave computers to be used.

ML
For a sample {(yt , xt )}n of n observations of a set of dependent and explanatory variables, the
maximum likelihood estimator of the parameter can be defined as
= arg max sn ()
where

sn () =

1X
ln f (yt |xt , )
n t=1

Here, yt may be a vector of random variables, and the model may be dynamic since xt may contain
lags of yt . As Swann (2002) points out, this can be broken into sums over blocks of observations,
for example two blocks:
1
sn () =
n

n1
X

!
ln f (yt |xt , )

t=1

n
X

!)
ln f (yt |xt , )

t=n1 +1

Analogously, we can define up to n blocks. Again following Swann, parallelization can be done by
calculating each block on separate computers.
mle_example1.m is an Octave script that calculates the maximum likelihood estimator of the
parameter vector of a model that assumes that the dependent variable is distributed as a Poisson
random variable, conditional on some explanatory variables. In lines 1-3 the data is read, the name
of the density function is provided in the variable model, and the initial value of the parameter

vector is set. In line 5, the function mle_estimate performs ordinary serial calculation of the
ML estimator, while in line 7 the same function is called with 6 arguments. The fourth and fifth
arguments are empty placeholders where options to mle_estimate may be set, while the sixth
argument is the number of slave computers to use for parallel execution, 1 in this case. A person
who runs the program sees no parallel programming code - the parallelization is transparent to
the end user, beyond having to select the number of slave computers. When executed, this script
prints out the estimates theta_s and theta_p, which are identical.
It is worth noting that a different likelihood function may be used by making the model variable
point to a different function. The likelihood function itself is an ordinary Octave function that is
not parallelized. The mle_estimate function is a generic function that can call any likelihood
function that has the appropriate input/output syntax for evaluation either serially or in parallel.
Users need only learn how to write the likelihood function using the Octave language.

GMM
For a sample as above, the GMM estimator of the parameter can be defined as
arg min sn ()

where
sn () = mn ()0 Wn mn ()
and

mn () =

1X
mt (yt |xt , )
n t=1

Since mn () is an average, it can obviously be computed blockwise, using for example 2 blocks:
1
mn () =
n

n1
X

!
mt (yt |xt , )

t=1

n
X

!)
mt (yt |xt , )

(22.1)

t=n1 +1

Likewise, we may define up to n blocks, each of which could potentially be computed on a different
machine.
gmm_example1.m is a script that illustrates how GMM estimation may be done serially or in
parallel. When this is run, theta_s and theta_p are identical up to the tolerance for convergence
of the minimization routine. The point to notice here is that an end user can perform the estimation
in parallel in virtually the same way as it is done serially. Again, gmm_estimate, used in lines 8
and 10, is a generic function that will estimate any model specified by the moments variable - a
different model can be estimated by changing the value of the moments variable. The function
that moments points to is an ordinary Octave function that uses no parallel programming, so users
can write their models using the simple and intuitive HLMP syntax of Octave. Whether estimation
is done in parallel or serially depends only the seventh argument to gmm_estimate - when it is
missing or zero, estimation is by default done serially with one processor. When it is positive, it
specifies the number of slave nodes to use.

Figure 22.1: Speedups from parallelization

11
10

MONTECARLO
BOOTSTRAP
MLE
GMM
KERNEL

9
8
7
6
5
4
3
2
1
2

10

12

nodes

Kernel regression
The Nadaraya-Watson kernel regression estimator of a function g(x) at a point x is
Pn
yt K [(x xt ) /n ]
Pt=1
g(x) =
n
t=1 K [(x xt ) /n ]
n
X

wt yy
t=1

We see that the weight depends upon every data point in the sample. To calculate the fit at every
point in a sample of size n, on the order of n2 k calculations must be done, where k is the dimension
of the vector of explanatory variables, x. Racine (2002) demonstrates that MPI parallelization can
be used to speed up calculation of the kernel regression estimator by calculating the fits for portions
of the sample on different computers. We follow this implementation here. kernel_example1.m is
a script for serial and parallel kernel regression. Serial execution is obtained by setting the number
of slaves equal to zero, in line 15. In line 17, a single slave is specified, so execution is in parallel
on the master and slave nodes.
The example programs show that parallelization may be mostly hidden from end users. Users
can benefit from parallelization without having to write or understand parallel code. The speedups
one can obtain are highly dependent upon the specific problem at hand, as well as the size of the
cluster, the efficiency of the network, etc. Some examples of speedups are presented in Creel
(2005). Figure 22.1 reproduces speedups for some econometric problems on a cluster of 12 desktop computers. The speedup for k nodes is the time to finish the problem on a single node divided
by the time to finish the problem on k nodes. Note that you can get 10X speedups, as claimed in
the introduction. Its pretty obvious that much greater speedups could be obtained using a larger
cluster, for the embarrassingly parallel problems.

Bibliography
[1] Bruche, M. (2003) A note on embarassingly parallel computation using OpenMosix
and Ox, working paper, Financial Markets Group, London School of Economics.
[2] Creel, M. (2005) User-friendly parallel computations with econometric examples,
Computational Economics, V. 26, pp. 107-128.
[3] Doornik, J.A., D.F. Hendry and N. Shephard (2002) Computationally-intensive
econometrics using a distributed matrix-programming language, Philosophical
Transactions of the Royal Society of London, Series A, 360, 1245-1266.
[4] Fernndez Baldomero, J. (2004) LAM/MPI parallel computing under GNU Octave,

atc.ugr.es/javier-bin/mpitb.
[5] Racine, Jeff (2002) Parallel distributed kernel estimation, Computational Statistics
& Data Analysis, 40, 293-302.
[6] Swann, C.A. (2002) Maximum likelihood estimation using parallel computing: an
introduction to MPI, Computational Economics, 19, 145-178.

317

Chapter 23

Final project: econometric

estimation of a RBC model
THIS IS NOT FINISHED - IGNORE IT FOR NOW
In this last chapter well go through a worked example that combines a number of the topics
weve seen. Well do simulated method of moments estimation of a real business cycle model,
similar to what Valderrama (2002) does.

23.1

Data

Well develop a model for private consumption and real gross private investment. The data are
obtained from the US Bureau of Economic Analysis (BEA) National Income and Product Accounts
(NIPA), Table 11.1.5, Lines 2 and 6 (you can download quarterly data from 1947-I to the present).
The data we use are in the file rbc_data.m. This data is real (constant dollars).
The program plots.m will make a few plots, including Figures 23.1 though 23.3. First looking
at the plot for levels, we can see that real consumption and investment are clearly nonstationary
(surprise, surprise). There appears to be somewhat of a structural change in the mid-1970s.
Looking at growth rates, the series for consumption has an extended period of high growth in the
1970s, becoming more moderate in the 90s. The volatility of growth of consumption has declined
somewhat, over time. Looking at investment, there are some notable periods of high volatility in
the mid-1970s and early 1980s, for example. Since 1990 or so, volatility seems to have declined.
Economic models for growth often imply that there is no long term growth (!) - the data that
the models generate is stationary and ergodic. Or, the data that the models generate needs to be
passed through the inverse of a filter. Well follow this, and generate stationary business cycle data
by applying the bandpass filter of Christiano and Fitzgerald (1999). The filtered data is in Figure
23.3. Well try to specify an economic model that can generate similar data. To get data that look
like the levels for consumption and investment, wed need to apply the inverse of the bandpass
filter.
Figure 23.1: Consumption and Investment, Levels

318

Figure 23.2: Consumption and Investment, Growth Rates

Figure 23.3: Consumption and Investment, Bandpass Filtered

23.2

An RBC Model

Consider a very simple stochastic growth model (the same used by Maliar and Maliar (2003), with
minor notational difference):
E0
max{ct ,kt }
t=0

t=0

t U (ct )

ct + kt

(1 ) kt1 + t kt1

log t

log t1 + t

t

IIN (0, 2 )

Assume that the utility function is

U (ct ) =

c1
1
t
1

is the discount rate

is the depreciation rate of capital
is the elasticity of output with respect to capital
is a technology shock that is positive. t is observed in period t.
is the coefficient of relative risk aversion. When = 1, the utility function is logarithmic.
gross investment, it , is the change in the capital stock:
it = kt (1 ) kt1
we assume that the initial condition (k0 , 0 ) is given.
We would like to estimate the parameters = , , , , , 2

0

using the data that we have on

consumption and investment. This problem is very similar to the GMM estimation of the portfolio
model discussed in Sections 14.16 and 14.17. Once can derive the Euler condition in the same way
we did there, and use it to define a GMM estimator. That approach was not very successful, recall.
Now well try to use some more informative moment conditions to see if we get better results.

23.3

A reduced form model

Macroeconomic time series data are often modeled using vector autoregressions. A vector autogression is just the vector version of an autoregressive model. Let yt be a G-vector of jointly
dependent variables. A VAR(p) model is
yt = c + A1 yt1 + A2 yt2 + ... + Ap ytp + vt
where c is a G-vector of parameters, and Aj , j=1,2,...,p, are G G matrices of parameters. Let
0

vt = Rt t , where t IIN (0, I2 ), and Rt is upper triangular. So V (vt |yt1 , ...ytp ) = Rt Rt . You
can think of a VAR model as the reduced form of a dynamic linear simultaneous equations model

where all of the variables are treated as endogenous. Clearly, if all of the variables are endogenous,
one would need some form of additional information to identify a structural model. But we already
have a structural model, and were only going to use the VAR to help us estimate the parameters.
A well-fitting reduced form model will be adequate for the purpose.
Were seen that our data seems to have episodes where the variance of growth rates and filtered
data is non-constant. This brings us to the general area of stochastic volatility. Without going
into details, well just consider the exponential GARCH model of Nelson (1991) as presented in
Hamilton (1994, pg. 668-669).
Define ht = vec (Rt ), the vector of elements in the upper triangle of Rt (in our case this is a
3 1 vector). We assume that the elements follow
n
o
p
log hjt = j + P(j,.) |vt1 | 2/ + (j,.) vt1 + G(j,.) log ht1
The variance of the VAR error depends upon its own past, as well as upon the past realizations of
the shocks.
This is an EGARCH(1,1) specification. The obvious generalization is the EGARCH(r, m) specification, with longer lags (r for lags of v, m for lags of h).
The advantage of the EGARCH formulation is that the variance is assuredly positive without
parameter restrictions
The matrix P has dimension 3 2.
The matrix G has dimension 3 3.
The matrix (reminder to self: this is an aleph) has dimension 2 2.
The parameter matrix allows for leverage, so that positive and negative shocks can have
asymmetric effects upon volatility.
We will probably want to restrict these parameter matrices in some way. For instance, G
could plausibly be diagonal.
With the above specification, we have
t

IIN (0, I2 )

R1
t vt

and we know how to calculate Rt and vt , given the data and the parameters. Thus, it is straighforward to do estimation by maximum likelihood. This will be the score generator.

23.4

Results (I): The score generator

23.5

Solving the structural model

The first order condition for the structural model is

1
c
= Et c
t
t+1 1 + t+1 kt

or




 1
1

ct = Et c
t+1 1 + t+1 kt

The problem is that we cannot solve for ct since we do not know the solution for the expectation
in the previous equation.
The parameterized expectations algorithm (PEA: den Haan and Marcet, 1990), is a means of
solving the problem. The expectations term is replaced by a parametric function. As long as the
parametric function is a flexible enough function of variables that have been realized in period t,
there exist parameter values that make the approximation as close to the true expectation as is
desired. We will write the approximation



1
Et c
' exp (0 + 1 log t + 2 log kt1 )
t+1 1 + t+1 kt
For given values of the parameters of this approximating function, we can solve for ct , and then
for kt using the restriction that

ct + kt = (1 ) kt1 + t kt1

This allows us to generate a series {(ct , kt )}. Then the expectations approximation is updated by
fitting


1
c
= exp (0 + 1 log t + 2 log kt1 ) + t
t+1 1 + t+1 kt
by nonlinear least squares. The 2 step procedure of generating data and updating the parameters
of the approximation to expectations is iterated until the parameters no longer change. When this
is the case, the expectations function is the best fit to the generated data. As long it is a rich enough
parametric model to encompass the true expectations function, it can be made to be equal to the
true expectations function by using a long enough simulation.

0
Thus, given the parameters of the structural model, = , , , , , 2 , we can generate data
{(ct , kt )} using the PEA. From this we can get the series {(ct , it )} using it = kt (1 ) kt1 . This
can be used to do EMM estimation using the scores of the reduced form model to define moments,
using the simulated data from the structural model.

Bibliography
[1] Creel. M (2005) A Note on Parallelizing the Parameterized Expectations Algorithm.
[2] den Haan, W. and Marcet, A. (1990) Solving the stochastic growth model by parameterized expectations, Journal of Business and Economics Statistics, 8, 31-34.
[3] Hamilton, J. (1994) Time Series Analysis, Princeton Univ. Press
[4] Maliar, L. and Maliar, S. (2003) Matlab code for Solving a Neoclassical Growh
Model with a Parametrized Expectations Algorithm and Moving Bounds
[5] Nelson, D. (1991) Conditional heteroscedasticity is asset returns: a new approach,
Econometrica, 59, 347-70.
[6] Valderrama, D. (2002) Statistical nonlinearities in the business cycle: a challenge
for the canonical RBC model, Economic Research, Federal Reserve Bank of San
Francisco. http://ideas.repec.org/p/fip/fedfap/2002-13.html

322

Chapter 24

Introduction to Octave
Why is Octave being used here, since its not that well-known by econometricians? Well, because
it is a high quality environment that is easily extensible, uses well-tested and high performance
numerical libraries, it is licensed under the GNU GPL, so you can get it for free and modify it if you
like, and it runs on both GNU/Linux, Mac OSX and Windows systems. Its also quite easy to learn.

24.1

Getting started

Get the ParallelKnoppix CD, as was described in Section 1.5. Then burn the image, and boot your
computer with it. This will give you this same PDF file, but with all of the example programs ready
to run. The editor is configure with a macro to execute the programs using Octave, which is of
course installed. From this point, I assume you are running the CD (or sitting in the computer
room across the hall from my office), or that you have configured your computer to be able to run
the *.m files mentioned below.

24.2

A short introduction

The objective of this introduction is to learn just the basics of Octave. There are other ways to use
Octave, which I encourage you to explore. These are just some rudiments. After this, you can look
at the example programs scattered throughout the document (and edit them, and run them) to
learn more about how Octave can be used to do econometrics. Students of mine: your problem
sets will include exercises that can be done by modifying the example programs in relatively minor
ways. So study the examples!
Octave can be used interactively, or it can be used to run programs that are written using a
text editor. Well use this second method, preparing programs with NEdit, and calling Octave from
within the editor. The program first.m gets us started. To run this, open it up with NEdit (by
finding the correct file inside the /home/knoppix/Desktop/Econometrics folder and clicking on
the icon) and then type CTRL-ALT-o, or use the Octave item in the Shell menu (see Figure 24.1).
Note that the output is not formatted in a pleasing way. Thats because printf() doesnt automatically start a new line. Edit first.m so that the 8th line reads printf(hello world\n);
and re-run the program.
We need to know how to load and save data. The program second.m shows how. Once you
have run this, you will find the file x in the directory Econometrics/Examples/OctaveIntro/
You might have a look at it with NEdit to see Octaves default format for saving data. Basically,
if you have data in an ASCII text file, named for example myfile.data, formed of numbers

323

Figure 24.1: Running an Octave program

separated by spaces, just use the command load myfile.data. After having done so, the matrix
myfile (without extension) will contain the data.
Please have a look at CommonOperations.m for examples of how to do some basic things in
Octave. Now that were done with the basics, have a look at the Octave programs that are included
as examples. If you are looking at the browsable PDF version of this document, then you should be
able to click on links to open them. If not, the example programs are available here and the support
files needed to run these are available here. Those pages will allow you to examine individual files,
out of context. To actually use these files (edit and run them), you should go to the home page
of this document, since you will probably want to download the pdf version together with all the
support files and examples. Or get the bootable CD.
There are some other resources for doing econometrics with Octave. You might like to check
the article Econometrics with Octave and the Econometrics Toolbox , which is for Matlab, but
much of which could be easily used with Octave.

24.3

If youre running a Linux installation...

Then to get the same behavior as found on the CD, you need to:
Get the collection of support programs and the examples, from the document home page.
Put them somewhere, and tell Octave how to find them, e.g., by putting a link to the MyOctaveFiles directory in /usr/local/share/octave/site-m
Make sure nedit is installed and configured to run Octave and use syntax highlighting. Copy
the file /home/econometrics/.nedit from the CD to do this. Or, get the file NeditConfiguration and save it in your $HOME directory with the name .nedit. Not to put too fine a
point on it, please note that there is a period in that name.
Associate *.m files with NEdit so that they open up in the editor when you click on them.
That should do it.

Chapter 25

Notation and Review

All vectors will be column vectors, unless they have a transpose symbol (or I forget to apply
this rule - your help catching typos and er0rors is much appreciated). For example, if xt is a
p 1 vector, x0t is a 1 p vector. When I refer to a p-vector, I mean a column vector.

25.1

Notation for differentiation of vectors and matrices

[3, Chapter 1]
Let s() : <p < be a real valued function of the p-vector . Then

s()

is organized as a

p-vector,

s()
1
s()
2

s()

..
.

s()
p

Following this convention, s()

0 is a 1 p vector, and

2 s()
=
0

s()
0

2 s()
0


=

Exercise 68. For a and x both p-vectors, show that

a0 x
x

is a p p matrix. Also,


s()


.

= a.

Let f ():< < be a n-vector valued function of the p-vector . Let f ()0 be the 1 n valued

0

f ()0 =
transpose of f . Then
0 f ().
p

Product rule: Let f ():<p <n and h():<p <n be n-vector valued functions of the
p-vector . Then

h()0 f () = h0
0

f
0


+f

h
0

has dimension 1 p. Applying the transposition rule we get






0
0
0
h() f () =
f h+
h f

which has dimension p 1.

Exercise 69. For A a p p matrix and x a p 1 vector, show that

x0 Ax
x

= A + A0 .

Chain rule: Let f ():<p <n a n-vector valued function of a p-vector argument, and let
g():<r <p be a p-vector valued function of an r-vector valued argument . Then




f
[g
()]
=
f
()
g()

0
0
0

=g()
326

has dimension n r.
Exercise 70. For x and both p 1 vectors, show that

25.2

exp(x0 )

= exp(x0 )x.

Convergenge modes

Readings: [1, Chapter 4];[4, Chapter 4].

We will consider several modes of convergence. The first three modes discussed are simply for
background. The stochastic modes are those which will be used later in the course.
Definition 71. A sequence is a mapping from the natural numbers {1, 2, ...} = {n}
n=1 = {n} to
some other set, so that the set is ordered according to the natural numbers associated with its
elements.

Real-valued sequences:
Definition 72. [Convergence] A real-valued sequence of vectors {an } converges to the vector a if
for any > 0 there exists an integer N such that for all n > N , k an a k< . a is the limit of
an , written an a.

Deterministic real-valued functions

Consider a sequence of functions {fn ()} where
fn : T <.
may be an arbitrary set.
Definition 73. [Pointwise convergence] A sequence of functions {fn ()} converges pointwise on
to the function f () if for all > 0 and there exists an integer N such that
|fn () f ()| < , n > N .
Its important to note that N depends upon , so that converge may be much more rapid for
certain than for others. Uniform convergence requires a similar rate of convergence throughout
.
Definition 74. [Uniform convergence] A sequence of functions {fn ()} converges uniformly on
to the function f () if for any > 0 there exists an integer N such that
sup |fn () f ()| < , n > N.

(insert a diagram here showing the envelope around f () in which fn () must lie).

Stochastic sequences
In econometrics, we typically deal with stochastic sequences. Given a probability space (, F, P ) ,
recall that a random variable maps the sample space to the real line, i.e., X() : <. A sequence
of random variables {Xn ()} is a collection of such mappings, i.e., each Xn () is a random variable
with respect to the probability space (, F, P ) . For example, given the model Y = X 0 + , the
1
OLS estimator n = (X 0 X) X 0 Y, where n is the sample size, can be used to form a sequence of
random vectors {n }. A number of modes of convergence are in use when dealing with sequences
of random variables. Several such modes of convergence should already be familiar:

Definition 75. [Convergence in probability] Let Xn () be a sequence of random variables, and let
X() be a random variable. Let An = { : |Xn () X()| > }. Then {Xn ()} converges in
probability to X() if
lim P (An ) = 0, > 0.

Convergence in probability is written as Xn X, or plim Xn = X.

Definition 76. [Almost sure convergence] Let Xn () be a sequence of random variables, and let
X() be a random variable. Let A = { : limn Xn () = X()}. Then {Xn ()} converges
almost surely to X() if
P (A) = 1.
In other words, Xn () X() (ordinary convergence of the two functions) except on a set
a.s.

C = A such that P (C) = 0. Almost sure convergence is written as Xn X, or Xn X, a.s.

One can show that
p

a.s.

Xn X Xn X.
Definition 77. [Convergence in distribution] Let the r.v. Xn have distribution function Fn and the
r.v. Xn have distribution function F. If Fn F at every continuity point of F, then Xn converges
in distribution to X.
d

Convergence in distribution is written as Xn X. It can be shown that convergence in probability

implies convergence in distribution.

Stochastic functions
a.s.
Simple laws of large numbers (LLNs) allow us to directly conclude that n 0 in the OLS

example, since
n = 0 +
and

X0
n

X 0X
n

1 

X 0
n


,

a.s.

0 by a SLLN. Note that this term is not a function of the parameter . This easy proof

is a result of the linearity of the model, which allows us to express the estimator in a way that
separates parameters from random functions. In general, this is not possible. We often deal with
the more complicated situation where the stochastic sequence depends on parameters in a manner
that is not reducible to a simple sequence of random variables. In this case, we have a sequence
of random functions that depend on : {Xn (, )}, where each Xn (, ) is a random variable with
respect to a probability space (, F, P ) and the parameter belongs to a parameter space .
Definition 78. [Uniform almost sure convergence] {Xn (, )} converges uniformly almost surely in
to X(, ) if
lim sup |Xn (, ) X(, )| = 0, (a.s.)

Implicit is the assumption that all Xn (, ) and X(, ) are random variables w.r.t. (, F, P )
u.a.s.

for all . Well indicate uniform almost sure convergence by and uniform convergence in
u.p.

probability by .
An equivalent definition, based on the fact that almost sure means with probability one
is


Pr


lim sup |Xn (, ) X(, )| = 0 = 1

This has a form similar to that of the definition of a.s. convergence - the essential difference
is the addition of the sup.

25.3

Rates of convergence and asymptotic equality

Its often useful to have notation for the relative magnitudes of quantities. Quantities that are small
relative to others can often be ignored, which simplifies analysis.
Definition 79. [Little-o] Let f (n) and g(n) be two real-valued functions. The notation f (n) =
o(g(n)) means limn

f (n)
g(n)

= 0.

Definition 80. [Big-O] Let f (n) and g(n) be two real-valued

 functions. The notation f (n) =

 f (n) 
O(g(n)) means there exists some N such that for n > N,  g(n)  < K, where K is a finite constant.
This definition doesnt require that

f (n)
g(n)

have a limit (it may fluctuate boundedly).

If {fn } and {gn } are sequences of random variables analogous definitions are
Definition 81. The notation f (n) = op (g(n)) means

f (n) p
g(n)

0.

Example 82. The least squares estimator = (X 0 X)1 X 0 Y = (X 0 X)1 X 0 X0 + = 0 +
0
1
0
(X 0 X)1 X 0 . Since plim (X X) X = 0, we can write (X 0 X)1 X 0 = op (1) and = 0 + op (1).
1

Asymptotically, the term op (1) is negligible. This is just a way of indicating that the LS estimator is
consistent.
Definition 83. The notation f (n) = Op (g(n)) means there exists some N such that for > 0 and
all n > N ,
P




 f (n) 
 < K > 1 ,

 g(n) 

where K is a finite constant.

Example 84. If Xn N (0, 1) then Xn = Op (1), since, given , there is always some K such that
P (|Xn | < K ) > 1 .
Useful rules:
Op (np )Op (nq ) = Op (np+q )
op (np )op (nq ) = op (np+q )
Example 85. Consider a random sample of iid r.v.s with mean 0 and variance 2 . The estimator
Pn
A
of the mean = 1/n
xi is asymptotically normally distributed, e.g., n1/2 N (0, 2 ). So
i=1

n1/2 = Op (1), so = Op (n1/2 ). Before we had = op (1), now we have have the stronger result
that relates the rate of convergence to the sample size.

Example 86. Now consider a random sample of iid r.v.s with mean and variance 2 . 
The
Pn
A
1/2

estimator of the mean = 1/n i=1 xi is asymptotically normally distributed, e.g., n



N (0, 2 ). So n1/2 = Op (1), so = Op (n1/2 ), so = Op (1).
These two examples show that averages of centered (mean zero) quantities typically have plim
0, while averages of uncentered quantities have finite nonzero plims. Note that the definition of
Op does not mean that f (n) and g(n) are of the same order. Asymptotic equality ensures that this
is the case.
Definition 87. Two sequences of random variables {fn } and {gn } are asymptotically equal (writa

ten fn = gn ) if

plim

f (n)
g(n)


=1

Finally, analogous almost sure versions of op and Op are defined in the obvious way.

For a and x both p 1 vectors, show that Dx a0 x = a.

For A a p p matrix and x a p 1 vector, show that Dx2 x0 Ax = A + A0 .
For x and both p 1 vectors, show that D exp x0 = exp(x0 )x.
For x and both p 1 vectors, find the analytic expression for D2 exp x0 .
Write an Octave program that verifies each of the previous results by taking numeric derivatives.
For a hint, type help numgradient and help numhessian inside octave.

Chapter 26

Licenses
This document and the associated examples and materials are copyright Michael Creel, under the
terms of the GNU General Public License, ver. 2., or at your option, under the Creative Commons
Attribution-Share Alike License, Version 2.5. The licenses follow.

26.1

The GPL

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331

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Also add information on how to contact you by electronic and paper mail.
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Chapter 27

The attic
This holds material that is not really ready to be incorporated into the main body, but that I dont
want to lose. Basically, ignore it, unless youd like to help get it ready for inclusion.

Optimal instruments for GMM

PLEASE IGNORE THE REST OF THIS SECTION: there is a flaw in the argument that needs correction. In particular, it may be the case that E(Zt t ) 6= 0 if instruments are chosen in the way
suggested here.
An interesting question that arises is how one should choose the instrumental variables Z(wt )
to achieve maximum efficiency.
Note that with this choice of moment conditions, we have that Dn

0
m ()

(a K g matrix)

is
Dn ()

=
=

1 0
0
(Z hn ())
n n

1 0
h () Zn
n n

which we can define to be

Dn () =

1
Hn Zn .
n

where Hn is a K n matrix that has the derivatives of the individual moment conditions as its
columns. Likewise, define the var-cov. of the moment conditions


E nmn (0 )mn (0 )0


1 0
0
0 0
= E
Z hn ( )hn ( ) Zn
n n


1
= Zn0 E
hn (0 )hn (0 )0 Zn
n
n
Zn0
Zn
n


where we have defined n = V hn (0 ) . Note that the dimension of this matrix is growing with
n

the sample size, so it is not consistently estimable without additional assumptions.

The asymptotic normality theorem above says that the GMM estimator using the optimal
weighting matrix is distributed as


d
n 0 N (0, V )

343

where

V = lim

Hn Zn
n



Zn0 n Zn
n

1 

Zn0 Hn0
n

!1
(27.1)

Using an argument similar to that used to prove that 1

is the efficient weighting matrix, we can
show that putting
0
Zn = 1
n Hn

causes the above var-cov matrix to simplify to


V = lim

0
Hn 1
n Hn
n

1
(27.2)

and furthermore, this matrix is smaller that the limiting var-cov for any other choice of instrumental variables. (To prove this, examine the difference of the inverses of the var-cov matrices with the
optimal intruments and with non-optimal instruments. As above, you can show that the difference
is positive semi-definite).
Note that both Hn , which we should write more properly as Hn (0 ), since it depends on 0 ,
and must be consistently estimated to apply this.
Usually, estimation of Hn is straightforward - one just uses
 
b = h0 ,
H
n
where is some initial consistent estimator based on non-optimal instruments.
Estimation of n may not be possible. It is an n n matrix, so it has more unique elements
than n, the sample size, so without restrictions on the parameters it cant be estimated consistently. Basically, you need to provide a parametric specification of the covariances of the
ht () in order to be able to use optimal instruments. A solution is to approximate this matrix
parametrically to define the instruments. Note that the simplified var-cov matrix in equation
27.2 will not apply if approximately optimal instruments are used - it will be necessary to
use an estimator based upon equation 27.1, where the term n1 Zn0 n Zn must be estimated
consistently apart, for example by the Newey-West procedure.

27.1

Hurdle models

Returning to the Poisson model, lets look at actual and fitted count probabilities. Actual relative freP
Pn

quencies are f (y = j) =
1(yi = j)/n and fitted frequencies are f(y = j) =
fY (j|xi , )/n
i=1

We see that for the OBDV measure, there are many more actual zeros than predicted. For ERV,
Table 27.1: Actual and Poisson fitted frequencies
Count
Count
0
1
2
3
4
5

OBDV
Actual Fitted
0.32
0.06
0.18
0.15
0.11
0.19
0.10
0.18
0.052
0.15
0.032
0.10

ERV
Actual Fitted
0.86
0.83
0.10
0.14
0.02
0.02
0.004
0.002
0.002 0.0002
0
2.4e-5

there are somewhat more actual zeros than fitted, but the difference is not too important.

Why might OBDV not fit the zeros well? What if people made the decision to contact the doctor
for a first visit, they are sick, then the doctor decides on whether or not follow-up visits are needed.
This is a principal/agent type situation, where the total number of visits depends upon the decision
of both the patient and the doctor. Since different parameters may govern the two decision-makers
choices, we might expect that different parameters govern the probability of zeros versus the other
counts. Let p be the parameters of the patients demand for visits, and let d be the paramter
of the doctors demand for visits. The patient will initiate visits according to a discrete choice
model, for example, a logit model:

Pr(Y = 0)

= fY (0, p ) =

Pr(Y > 0)

1 1/ [1 + exp(p )]
1/ [1 + exp(p )] ,

The above probabilities are used to estimate the binary 0/1 hurdle process. Then, for the observations where visits are positive, a truncated Poisson density is estimated. This density is
fY (y, d |y > 0)

=
=

fY (y, d )
Pr(y > 0)
fY (y, d )
1 exp(d )

since according to the Poisson model with the doctors paramaters,

Pr(y = 0) =

exp(d )0d
.
0!

Since the hurdle and truncated components of the overall density for Y share no parameters, they
may be estimated separately, which is computationally more efficient than estimating the overall
model. (Recall that the BFGS algorithm, for example, will have to invert the approximated Hessian.
The computational overhead is of order K 2 where K is the number of parameters to be estimated)
. The expectation of Y is
E(Y |x)

=
=

Pr(Y > 0|x)E(Y |Y > 0, x)




1
d
1 + exp(p )
1 exp(d )

Here are hurdle Poisson estimation results for OBDV, obtained from this estimation program

**************************************************************************
MEPS data, OBDV
logit results
Strong convergence
Observations = 500
Function value
-0.58939
t-Stats
params
t(OPG)
t(Sand.)
t(Hess)
constant
-1.5502
-2.5709
-2.5269
-2.5560
pub_ins
1.0519
3.0520
3.0027
3.0384
priv_ins
0.45867
1.7289
1.6924
1.7166
sex
0.63570
3.0873
3.1677
3.1366
age
0.018614
2.1547
2.1969
2.1807
educ
0.039606
1.0467
0.98710
1.0222
inc
0.077446
1.7655
2.1672
1.9601
Information Criteria
Consistent Akaike
639.89
Schwartz
632.89
Hannan-Quinn
614.96
Akaike
603.39
**************************************************************************

The results for the truncated part:

**************************************************************************
MEPS data, OBDV
tpoisson results
Strong convergence
Observations = 500
Function value
-2.7042
t-Stats
params
t(OPG)
t(Sand.)
t(Hess)
constant
0.54254
7.4291
1.1747
3.2323
pub_ins
0.31001
6.5708
1.7573
3.7183
priv_ins
0.014382
0.29433
0.10438
0.18112
sex
0.19075
10.293
1.1890
3.6942
age
0.016683
16.148
3.5262
7.9814
educ
0.016286
4.2144
0.56547
1.6353
inc
-0.0079016
-2.3186
-0.35309
-0.96078
Information Criteria
Consistent Akaike
2754.7
Schwartz
2747.7
Hannan-Quinn
2729.8
Akaike
2718.2
**************************************************************************

Fitted and actual probabilites (NB-II fits are provided as well) are:
Table 27.2: Actual and Hurdle Poisson fitted frequencies
Count
Count
0
1
2
3
4
5

Actual
0.32
0.18
0.11
0.10
0.052
0.032

OBDV
Fitted HP Fitted NB-II
0.32
0.34
0.035
0.16
0.071
0.11
0.10
0.08
0.11
0.06
0.10
0.05

Actual
0.86
0.10
0.02
0.004
0.002
0

ERV
Fitted HP
0.86
0.10
0.02
0.006
0.002
0.0005

Fitted NB-II
0.86
0.10
0.02
0.006
0.002
0.001

For the Hurdle Poisson models, the ERV fit is very accurate. The OBDV fit is not so good. Zeros
are exact, but 1s and 2s are underestimated, and higher counts are overestimated. For the NB-II
fits, performance is at least as good as the hurdle Poisson model, and one should recall that many
fewer parameters are used. Hurdle version of the negative binomial model are also widely used.

Finite mixture models

The following are results for a mixture of 2 negative binomial (NB-I) models, for the OBDV data,
which you can replicate using this estimation program

**************************************************************************
MEPS data, OBDV
mixnegbin results
Strong convergence
Observations = 500
Function value
-2.2312
t-Stats
params
t(OPG)
t(Sand.)
t(Hess)
constant
0.64852
1.3851
1.3226
1.4358
pub_ins
-0.062139
-0.23188
-0.13802
-0.18729
priv_ins
0.093396
0.46948
0.33046
0.40854
sex
0.39785
2.6121
2.2148
2.4882
age
0.015969
2.5173
2.5475
2.7151
educ
-0.049175
-1.8013
-1.7061
-1.8036
inc
0.015880
0.58386
0.76782
0.73281
ln_alpha
0.69961
2.3456
2.0396
2.4029
constant
-3.6130
-1.6126
-1.7365
-1.8411
pub_ins
2.3456
1.7527
3.7677
2.6519
priv_ins
0.77431
0.73854
1.1366
0.97338
sex
0.34886
0.80035
0.74016
0.81892
age
0.021425
1.1354
1.3032
1.3387
educ
0.22461
2.0922
1.7826
2.1470
inc
0.019227
0.20453
0.40854
0.36313
ln_alpha
2.8419
6.2497
6.8702
7.6182
logit_inv_mix
0.85186
1.7096
1.4827
1.7883
Information Criteria
Consistent Akaike
2353.8
Schwartz
2336.8
Hannan-Quinn
2293.3
Akaike
2265.2
**************************************************************************
Delta method for mix parameter st. err.
mix
se_mix
0.70096
0.12043
The 95% confidence interval for the mix parameter is perilously close to 1, which suggests
that there may really be only one component density, rather than a mixture. Again, this is
not the way to test this - it is merely suggestive.
Education is interesting. For the subpopulation that is healthy, i.e., that makes relatively
few visits, education seems to have a positive effect on visits. For the unhealthy group,
education has a negative effect on visits. The other results are more mixed. A larger sample
could help clarify things.
The following are results for a 2 component constrained mixture negative binomial model where
all the slope parameters in j = exj are the same across the two components. The constants and
the overdispersion parameters j are allowed to differ for the two components.

**************************************************************************
MEPS data, OBDV
cmixnegbin results
Strong convergence
Observations = 500
Function value
-2.2441
t-Stats
params
t(OPG)
t(Sand.)
t(Hess)
constant
-0.34153
-0.94203
-0.91456
-0.97943
pub_ins
0.45320
2.6206
2.5088
2.7067
priv_ins
0.20663
1.4258
1.3105
1.3895
sex
0.37714
3.1948
3.4929
3.5319
age
0.015822
3.1212
3.7806
3.7042
educ
0.011784
0.65887
0.50362
0.58331
inc
0.014088
0.69088
0.96831
0.83408
ln_alpha
1.1798
4.6140
7.2462
6.4293
const_2
1.2621
0.47525
2.5219
1.5060
lnalpha_2
2.7769
1.5539
6.4918
4.2243
logit_inv_mix
2.4888
0.60073
3.7224
1.9693
Information Criteria
Consistent Akaike
2323.5
Schwartz
2312.5
Hannan-Quinn
2284.3
Akaike
2266.1
**************************************************************************
Delta method for mix parameter st. err.
mix
se_mix
0.92335
0.047318
Now the mixture parameter is even closer to 1.
The slope parameter estimates are pretty close to what we got with the NB-I model.

Bibliography
[1] Davidson, R. and J.G. MacKinnon (1993) Estimation and Inference in Econometrics,
Oxford Univ. Press.
[2] Davidson, R. and J.G. MacKinnon (2004) Econometric Theory and Methods, Oxford
Univ. Press.
[3] Gallant, A.R. (1985) Nonlinear Statistical Models, Wiley.
[4] Gallant, A.R. (1997) An Introduction to Econometric Theory, Princeton Univ. Press.
[5] Hamilton, J. (1994) Time Series Analysis, Princeton Univ. Press
[6] Hayashi, F. (2000) Econometrics, Princeton Univ. Press.
[7] Wooldridge (2003), Introductory Econometrics, Thomson. (undergraduate level, for
supplementary use only).

351

Index
A

Product rule, 326

ARCH, 294
asymptotic equality, 329

R
R- squared, uncentered, 23

residuals, 18

Chain rule, 326

R-squared, centered, 23

Cobb-Douglas model, 16
conditional heteroscedasticity, 294
convergence, almost sure, 328
convergence, in distribution, 328
convergence, in probability, 328
Convergence, ordinary, 327
convergence, pointwise, 327
convergence, uniform, 327
convergence, uniform almost sure, 328
E
estimator, linear, 21, 27
estimator, OLS, 18
extremum estimator, 164
F
fitted values, 18
G
GARCH, 294
L
leptokurtosis, 294
leverage, 21
likelihood function, 177
M
matrix, idempotent, 20
matrix, projection, 19
matrix, symmetric, 20
O
observations, influential, 21
outliers, 21
own influence, 21
P
parameter space, 177
352