See discussions, stats, and author profiles for this publication at: https://www.researchgate.

net/publication/24130215

The Error Correction Model as a Test for Cointegration

Article · April 2003

Source: RePEc

CITATIONS READS
8 5,365

2 authors, including:

Paul M. Turner
Retired
88 PUBLICATIONS 896 CITATIONS

SEE PROFILE

All content following this page was uploaded by Paul M. Turner on 03 June 2014.

The user has requested enhancement of the downloaded file.

Sheffield Economic Research Paper Series

SERP Number: 2003001

Athina Kanioura and Paul Turner

The Error Correction Model as a Test for Cointegration

March 2003

Department of Economics
University of Sheffield
9 Mappin Street
Sheffield
S1 4DT
United Kingdom
www.shef.ac.uk/economics
Abstract

In this paper we generate critical values for a test for cointegration based on the joint
significance of the levels terms in an error correction equation. We show that the
appropriate critical values are higher than those derived from the standard F-distribution.
We compare the power properties of this test with those of the Engle-Granger test and
Kremers et al’s t-test based on the t-statistic from an error correction equation. The F-test
has higher power than the Engle-Granger test but lower power than the t-form of the error
correction test. However, the F-form of the test has the advantage that its distribution is
independent of the parameters of the problem being considered. Finally, we consider a
test for cointegration between UK and US interest rates. We show that the F-test rejects
the null of no cointegration between these variables although the Engle-Granger test fails
to do so.

Keywords: Cointegration, error correction.

JEL Numbers: C12, C15

Correspondence Address:

Dr P. M. Turner, Department of Economics, University of Sheffield, 9 Mappin Street,

Sheffield, S1 4DT, UNITED KINGDOM, Tel: 0114 2223404, e-mail:
p.turner@sheffield.ac.uk

1
I. INTRODUCTION

The use of the error correction model in applied econometrics goes back to Sargan
(1964). However, its integration into modern time series econometrics began with the
publication of two important papers in the mid 1970s. These were the analysis of the UK
consumption function by Davidson, Hendry, Srba and Yeo – DHSY - (1978) and that of
the UK demand for broad money by Hendry and Mizon (1978). These papers were
important because they emphasised the potential importance of levels terms within a time
series regression framework as a means of capturing the equilibrium interactions between
variables.

Consider the following model set out by Hendry and Mizon which relates a variable yt to

its own lagged value and the current and lagged values of another variable xt :

yt = β1 yt −1 + γ 0 xt + γ 1 xt −1 + vt (1)

where β1 < 1 and vt has zero mean, constant variance σ 2 and is serially independent. It

is straightforward to see that equation (1) can be reformulated to give:

∆yt = γ 0 ∆xt + ( β1 − 1) yt −1 + (γ 0 + γ 1 ) xt −1 + vt (2)

Given this formulation, a test of the joint hypothesis that the coefficients on yt −1 and xt −1

are zero is effectively a test of the hypothesis that the y and x processes have a common
root equal to one i.e. it is appropriate to estimate the equation in first differences. Hendry
and Mizon applied a model of this type to the UK demand for money function while

2
DHSY applied a similar model to the UK consumption function and in both cases
rejected the common unit root assumption.

At the time this approach to time series model building was criticised by Williams (1978)
on the grounds that “A recent paper by Hendry and Mizon (1978) has suggested that it is
possible to test whether a particular relationship should be estimated in levels or first
differences by re-arranging the levels formulation into a first-difference formulation and
a ‘remainder’ and then testing whether or not the coefficients in the ‘remainder’ are
statistically significant. The fallacy in this approach is that in order for the estimation
technique to be valid, it must be assumed that the error structure in the levels formulation
is stationary”. In retrospect this statement is a neat summary of the cointegration problem
which went on to become the major topic of research in time series econometrics for the
next two decades.

Since the publication of the above articles our understanding of the theoretical and
empirical properties of cointegrating relationships has increased enormously. Testing
procedures have been developed by Engle and Granger (1987) and Kremers et al (1992)
for single equation models and Johansen (1988) for multiple equation systems. The
Engle-Granger procedure is to apply the Augmented Dickey-Fuller test to the residuals
from a least squares regression between the levels of the variables. Appropriate critical
values for this test have been computed by MacKinnon (1991). Kremers et al estimate an
error correction model and use the t-ratio for the error correction term as their test
statistic. Unfortunately the distribution of the test statistic depends on unobservable
parameters of the specific problem and thus it may not be possible to implement the test
in practice. This test has been investigated further by Zivot (1996) and Ericsson and
MacKinnon (2002). Implementation of this test is considerably easier when the
cointegrating vector is know prior to estimation. However, in many circumstances this is
not a reasonable starting assumption. Finally, Banerjee et al.(1998) estimate an error
correction model test, based on the null hypothesis of non-cointegration, but with the t-

3
ratio version of the test suffering in finite samples when one tries to impose potentially
invalid common-factor restrictions.

The focus of this paper is on testing for a single cointegrating vector. In section II we
derive critical values for an alternative test based on the joint significance of the levels
terms in an error correction model. We demonstrate that these critical values differ from
those derived from the standard F-distribution in that they are consistently larger. In
section III we compare the performance of our test statistic with the Engle-Granger test
and the Kremers et al test. We show that our test has more power in rejecting a false null
hypothesis when compared with the Engle-Granger test. We also show that our testing
procedure has an advantage over the Kremers et al test in that it generates critical values
which are not sensitive to the parameters of the particular error correction model we
estimate. Section IV presents an example using monthly data for UK and US interest
rates and Section V gives our conclusions.

II. CRITICAL VALUES FOR AN F-TEST FOR COINTEGRATION

We begin by assuming a general bivariate data generation process for yt and xt ,

t = 1,…T . This is set out in equations (3)-(5).

∆yt = α1∆xt + α 2 yt −1 + α 3 xt −1 + ε1,t (3)

∆xt = ε 2,t (4)

 ε1,t    0   σ 12 0  
  ∼ IN    ,  2 

(5)
 ε 2,t  0  0 σ2 

4
Next, we consider the simplest possible case in which y and x are independent random
walks. This means we set α i = 0; i = 1, 2,3 . Based on these parameters we generate 10,000

replications using seeded pseudo random values for ε1 and ε 2 generated by the EViews

random number generator. We then estimate equation (3) by OLS and perform an F-test
for the null hypothesis H 0 :α 2 = α 3 = 0 against the alternative H1 : α 2 ≠ 0 or α 3 ≠ 0 for

various sample sizes. In each case we discard the first 50 observations to avoid the danger
of the start-up values biasing the results. The results are then tabulated and used to
calculate empirical 10%, 5% and 1% critical values for the test statistic. The results are
presented in Table One along with the critical values from the standard F-distribution.
The values calculated are all considerable higher than those given by the critical values
from the conventional F-distribution reflecting the fact that under the null hypothesis the
series are not stationary and therefore classical statistical distribution results do not apply.

[Insert Table One here]

The critical values given in Table One are calculated on the basis of particular parameter
values for the DGP. We also investigated the sensitivity of our results to changes in these
parameter values in two ways. First, we allowed the relative variances of the y and x
σ 12
processes to change. We varied within the range 10−8 and 108 and found that the
σ 22
Monte Carlo critical values were completely insensitive to this ratio. Second, we
experimented with values for α1 within the range −1 ≤ α1 ≤ 1 and again found that our
critical values were insensitive to the value chosen. All results in Table One are for a
‘large’ sample of 500 observations. Our conclusion is that the critical values we derived
are not sensitive to the particular experimental design we have adopted.

Kremers et al adopt an alternative approach to testing for cointegration within an error

correction framework. They begin by assuming that there is a ‘natural’ cointegrating
vector for which we may wish to test. This often arises when there is a unit elasticity

5
restriction which defines a constant ratio between the variables which make up the
cointegrating relationship. For example, the error correction term for the DHSY
consumption function consists of the lagged difference of the logarithms of consumption
and disposable income. Given a natural cointegrating vector of this type we can impose
the unit elasticity and then base a cointegration test on the t-ratio for the error correction
term. In terms of our DGP we impose the restriction α 2 = −α 3 estimate an equation of the

form given in equation (6) and test H 0 : α 2 = 0 against the alternative H1 : α 2 < 0 .

∆yt = α1∆xt + α 2 ( yt −1 − xt −1 ) + ut (6)

Kremers et al derive the distribution of the test statistic under the null hypothesis and note
σ2
that it depends on the ‘signal to noise’ ratio where this is defined as q = − (α1 − 1) .
σ1

The problem is that the critical values for their test (based on the t ratio for α 2 ) depend

on this ratio and therefore on the specific parameters of the problem in question. Thus the
Kremer et al test has the unattractive feature that critical values for the test depend on the
specific values of the parameters of the problem being examined. This marks a distinct
advantage of the F-test approach for which this problem does not apply.

Table Two gives Monte Carlo 5% critical values for the F-form and the t-form of the
σ2
ECM test for different values of α1 and the ratio . Whereas the F-form critical
σ1
values are independent of these parameters, there is considerable variation in the t-form
values. One interesting feature is that (provided α1 < 1 ) the critical value for the t-test

σ2
always converges on the standard normal value as → ∞ . This confirms the result
σ1
derived by Kremers et al who show that as the signal to noise ratio increases then the
distribution of the test statistic converges on the standard normal. Similarly the

6
distribution of the test statistic gets closer to the standard normal for small values of the
α1 parameter.

[Insert Table Two here]

III. THE POWER OF ALTERNATIVE COINTEGRATION TESTS

The low power of the Engle-Granger test has proved to be a major drawback in applied
work. This provides the motivating force behind Kremers et al’s development of the
ECM t-test. In this section we investigate the relative power properties of the three tests
using a DGP in which the y and x processes are cointegrated. The DGP we assume takes
the form:

yt = 0.9 yt −1 + 0.1 xt + ε1,t (7)

xt = xt −1 + ε 2,t (8)

As before we assume var ( ε1 ) = σ 12 , var ( ε 2 ) = σ 22 with cov ( ε1 , ε 2 ) = 0 . Based on this

experimental design we carry out 10,000 replications for a variety of different sample
sizes and compute the percentage of rejections of the false null hypothesis that y and x are
not cointegrated. The critical values used were the MacKinnon critical values for the
Engle-Granger test and the Monte Carlo critical values from Table One for the F-form of
the ECM test. For the t-form of the ECM test, we made the additional assumption that
σ 12 = σ 22 and used Monte Carlo methods to calculate a set of critical values for the

particular sample design given above. These are given for reference in Table Four.

The results of our experiments are reported in Table Three. These show a consistent
ranking in terms of the power of the test. The Engle-Granger test consistently has the

7
lowest power and performs very badly in small samples. The F-form of the ECM test has
higher power but the t-form of the ECM test has consistently higher power than the other
two tests. As the sample size increases then the power of all three tests increases and
eventually converges on 100%.

Our results therefore indicate that the error correction approach to testing for
cointegration is consistently more powerful than the Engle-Granger approach. Within the
class of error correction tests, we find that the t-form of the test is more powerful than the
F-form of the test when we know the correct set of critical values to apply. However, the
critical values for the t-test can vary considerably depending on the particular nature of
the problem being considered. Even if these parameters are known, then we need to
generate appropriate critical values using Monte Carlo simulations and in practice these
parameters may not be observable. Although the F-form of the test has lower power it
does have the advantage that the critical values do not vary with the sample design and
therefore this test is considerably easier to apply in practice.

[Insert Tables Three and Four here]

IV. EXAMPLE: UK AND US INTEREST RATES

In this section we present an example which illustrates the relative performance of the
tests we have discussed in the previous sections. Our aim is to test for the existence of a
cointegrating vector linking UK and US nominal interest rates. The rationale for the
existence of a cointegrating vector between these two variables derives from the
uncovered interest parity (UIP) condition. This states that the interest rate differential
between similar assets in different countries should equal the expected rate of change of
the exchange rate. Now it is a well established empirical fact that, in the vast majority of
cases, exchange rates are integrated of order one. It follows therefore that, to be

8
consistent with UIP, interest rates in the two countries should be cointegrated and that the
cointegrating slope coefficient should be unity.

[Insert Figure One here]

Our data is illustrated in Figure One which shows the Treasury Bill rate for the UK and
the US over the period February 1977 to December 2002. The data are taken from the
International Monetary Fund International Financial Statistics database. Although the
two series clearly exhibit some common features it is not clear whether they are
cointegrated. Preliminary investigation of the data indicates that both the interest rate
series and the sterling dollar exchange rate contain single unit roots. Therefore we
proceed to the next stage of the analysis and perform cointegration tests for the two
interest rates.

As a first step, we apply the two step Engle-Granger procedure. In the first stage we
obtain the results reported in equation (9) for a regression of the UK rate on a constant
and the US rate:

iUK ,t = 3.3162 + 0.8448 iUS ,t + uˆ1,t

( 0.623) ( 0.093) (9)
T = 311 σˆ = 2.20 DW = 0.10

uˆ1,t : t = 1,… 311 is the vector of OLS residuals. σˆ is the standard error of the regression

and DW is the Durbin-Watson test statistic for first order autocorrelation. The standard
errors reported in parentheses below coefficients are the Newey-West heteroscedasticity
and autocorrelation adjusted standard errors. Equation (9) can be interpreted sensibly in
terms of economic theory in that the slope coefficient is within two standard errors of
one. However, the intercept term is apparently significant which indicates the possibility
of a positive risk premium on UK assets. In the second stage of the analysis we apply an

9
Augmented Dickey Fuller (ADF) test to the residuals uˆ1 . Using the Schwarz criterion we
determined the optimal number of lagged differenced terms in the auxiliary regression to
be two and obtained a test statistic of −3.10 . From the MacKinnon response surfaces we
obtain critical values of –3.93, -3.36 and –3.06 at the 1%, 5% and 10% significance levels
respectively. Thus the Engle-Granger test indicates that we cannot reject the null
hypothesis that the two interest rates are not cointegrated at the 5% level, though it is
possible to reject at the 10% level.

Next we consider the F-form of the error correction test. Estimation of an error correction
model for the UK interest rate yields the results given in equation (10):

∆iUK ,t = 0.0309 + 0.0998 ∆iUS ,t − 0.0499 iUK ,t −1 + 0.0586 iUS ,t −1 + uˆ2,t

( 0.086 ) ( 0.049 ) ( 0.013) ( 0.015) (10)
T = 311 σˆ = 0.52 DW = 1.34

The F-test for the joint significance of the two interest rate levels in equation (10) yields a
value of 8.22. This compares with a 5% critical value of about 5.84 obtained from the
Monte Carlo simulations reported in Table One. Indeed, the critical values reported in
Table One indicate that in this case the F-test would also reject the null at the 1%
significance level. Therefore, in this example, our results indicate that the F-test is more
powerful in detecting a cointegrating vector than the Engle-Granger method.

Finally, we consider the t-form of the error correction test. First, we reparameterise
equation (10) and obtain the results reported in equation (11):

∆iUK ,t = 0.0309 + 0.0998 ∆iUS ,t − 0.0499 ( iUK ,t −1 − iUS ,t −1 ) + 0.0087 iUS ,t −1 + uˆ2,t
( 0.086 ) ( 0.049 ) ( 0.013) ( 0.010) (11)
T = 311 σˆ = 0.52 DW = 1.34

10
Since, the coefficient on the lagged US interest rate in (11) is statistically insignificant, it
appears that the model accepts the restriction of a cointegrating slope coefficient equal to
unity. Therefore we impose this restriction and obtain the results reported in equation
(12):

∆iUK ,t = 0.0933 + 0.0970 ∆iUS ,t − 0.0522 ( iUK ,t −1 − iUS ,t −1 ) + uˆ3,t

( 0.043) ( 0.049 ) ( 0.013) (12)
T = 311 σˆ = 0.52 DW = 1.34

The t-ratio for the error correction term in equation (12) is –3.97. We need to determine
what are the appropriate critical values since we have seen that these are affected by the
nuisance parameters which determine the ‘signal to noise’ ratio for this problem. The
Monte Carlo critical values reported in Table One are for a sample size of 500. However,
further simulations show that the 5% critical value of –2.89 for the most conservative
σ2
case ( = 10−4 , α = 0.9 ) does not change when we reduce the sample size to 311 as
σ1
in our empirical problem.. Therefore, even using the most conservative critical values, we
can safely state that in this case the t-form of the error correction test will also reject the
null of no cointegration.

11
V. CONCLUSIONS

In this paper we have used Monte Carlo methods to investigate the empirical distribution
of the levels terms in the error correction relationship between a set of I(1) variables. We
generate critical values for the conventional F-test for the joint significance of the levels
terms in such a regression and show that these are generally higher than the critical
values from the F-distribution. Investigation of the power properties of this test indicate
that it has higher power than the Engle-Granger test but lower power than a t-test based
on the error correction model. However, the F-form of the test has the advantage that its
distribution does not depend on the specific parameters of the problem being considered.
Finally, we illustrate the value of our approach by considering the relationship between
UK and US interest rates over the period 1977.02 to 2002.12. We show that it is not
possible to reject the null of no cointegration between these variables at the 5% level
using the Engle-Granger test. However, our alternative F-test rejects the null
convincingly as does the t-form of the error correction test

12
References:

Banerjee, A., Dolado, J.J., and Mestre, Ricardo (1998) ‘Error-Correction Mechanism
Tests for Cointegration in a Single-Equation Framework’, Journal of Time Series
Analysis, Vol. 19, No.3, pp. 267-283.
Davidson, J.E.H., Hendry, D.F., Srba, F. and Yeo, J.S. (1978) ‘Econometric Modelling of
the Aggregate Time-Series Relationship between Consumer’s Expenditure and Income in
the United Kingdom’, Economic Journal, Vol 88, No 352, pp. 661-692.
Ericsson, N. R. and MacKinnon, J. G. (2002) ‘Distributions of Error Correction Tests for
Cointegration’, Econometrics Journal, Vol 5, pp. 285-318.
Engle, R.F. and Granger, C.W.J. (1987) ‘Cointegration and Error Correction:
Representation, Estimation and Testing’, Econometrica, Vol 55, pp 251-276.
Hendry, D.F. and Mizon, G.E. (1978) ‘Serial Correlation as a Convenient Simplification,
not a Nuisance: A Comment on a Study of the Demand for Money by the Bank of
England’, Economic Journal, Vol 88, No 351, pp. 549-563.
Johansen, S. (1988) ‘Statistical Analysis of Cointegration Vectors’, Journal of Economic
Dynamics and Control, Vol 12, pp 231-254.
Kremers, J.J.M, Ericsson, N. and Dolado, J.J. (1992) ‘The Power of Cointegration Tests’,
Oxford Bulletin of Economics and Statistics, Vol 54, No 3, pp. 325-348.
MacKinnon, J.G. (1991) ‘Critical Values for Cointegration Tests’, in Engle, R.F. and
Granger, C.W.J. (eds) Long-Run Economic Relationships, Oxford: Oxford University
Press.
Sargan, J.D (1964) ‘Wages and Prices in the United Kingdom: A Study in Econometric
Methodology (with discussion)’ in P.E. Hart, G. Mills and J.K. Whitaker (eds),
Econometric Analysis for National Economic Planning, Colston Papers, vol. 16. pp. 25-
63. London: Butterworth and Co.
Williams, D. (1978) ‘Estimating in Levels or First Differences: A Defence of the Method
Used for Certain Demand for Money Equations’, Economic Journal, Vol 88, No 351, pp
564-568.
Zivot, E. (1996) ‘The Power of Single Equation Tests for Cointegrating when the
Cointegrating Vector is Prespecified’, (Mimeo)

13
Table One: Critical Values for F-Form of Cointegration Test

k=2

Monte Carlo Critical Standard F-

Values distribution Critical
Values
Sample Size 1% 5% 10% 1% 5% 10%
50 8.86 6.07 4.95 5.08 3.19 2.42
100 8.12 5.94 4.86 4.83 3.09 2.36
200 7.97 5.84 4.79 4.71 3.04 2.33
500 7.86 5.83 4.85 4.65 3.01 2.31

k=3

Monte Carlo Critical Standard F-

Values distribution Critical
Values
Sample Size 1% 5% 10% 1% 5% 10%
50 7.25 5.26 4.35 4.23 2.80 2.20
100 6.61 5.00 4.22 3.99 2.70 2.14
200 6.39 4.89 4.19 3.88 2.65 2.11
500 6.39 4.86 4.15 3.82 2.62 2.09

k=4

Monte Carlo Critical Standard F-

Values distribution Critical
Values
Sample Size 1% 5% 10% 1% 5% 10%
50 6.65 4.76 4.01 3.76 2.57 2.07
100 6.05 4.51 3.86 3.52 2.47 2.00
200 5.62 4.39 3.79 3.42 2.42 1.97
500 5.49 4.39 3.77 3.36 2.39 1.96

14
Table Two: 5% Critical Values for Alternative Error Correction Tests

F-Test T-Test
α1 α1
0 0.5 0.9 0 0.5 0.9
10−4 5.83 5.83 5.83 -2.89 -2.89 -2.89
σ2
σ1 1 5.83 5.83 5.83 -2.60 -2.79 -2.89
104 5.83 5.83 5.83 -1.61 -1.61 -1.61

15
Table Three: Comparison of Power of the ECM test with the Engle-Granger Test

Rejection Frequency 10% Level

Sample Size Engle-Granger Test ECM (T-test) ECM (F-Test)

50 11.8 51.2 28.4
100 25.5 89.4 65.2
200 71.5 99.9 98.3
500 100.0 100.0 100.0

Rejection Frequency 5% Level

Sample Size Engle-Granger Test ECM (T-test) ECM (F-Test)

50 6.1 35.2 16.3
100 13.4 77.7 47.6
200 55.6 98.2 95.1
500 99.9 100.0 100.0

Rejection Frequency 1% Level

Sample Size Engle-Granger Test ECM (T-test) ECM (F-Test)

50 1.3 11.2 4.1
100 3.2 41.3 20.4
200 18.7 95.8 79.7
500 99.2 100.0 100.0

The rejection frequencies reported above are calculated using the critical values from
MacKinnon for the Engle-Granger test and the authors’ Monte Carlo estimates for the
error correction tests.

16
Table Four: Critical Values for the t-form of the ECM test

Sample Size 10% 5% 1%

50 -2.32 -2.65 -3.37
100 -2.28 -2.64 -3.36
200 -2.26 -2.60 -3.26
500 -2.25 -2.60 -3.26

These critical values are based on the particular sample design given in equations (7) and
(8) and the assumption that σ 12 = σ 22 .

17
 Figure One: UK and US Treasury Bill Rates 1977.02-2002.12

20

16

12

0
1980 1985 1990 1995 2000

UK US

18

View publication stats