ECO311 Practice Questions 1
ECO311 Practice Questions 1
ECO311 Practice Questions 1
Question 2
Comment on the significance of the following Classical Linear Regression Model
(CLRM) assumptions:
i. Zero mean of the errors.
ii. Homoscedasticity.
iii. Zero correlation between the error term and the regressors.
Question 3
You are given the following regression with all the assumptions of the Classical
Normal Linear Regression Model (CNLRM) holding.
Yi 0 1 X i i
(a) Show that the maximum likelihood estimators of 0 and 1 are identical to the OLS
estimators
(b) Show that the maximum likelihood estimator of is biased
Question 4
Question 5
You are given the following regression with all the assumptions of the Classical
Normal Linear Regression Model (CNLRM) holding. The model is estimated using
OLS
Yi 0 1 X i i
1
(b) Suppose that 1 0 ; find the OLS estimator of 0 :
ˆ 2
(c) Show that the mean square error of the OLS estimator in (b) is MSEˆ 0
n
Question 6
You are given the following linear regression
e
Yi 0 1 X i u i , u 2
X
where e is a random variable with E e 0 and var( e) e2 . Assume that e is independent
of income.
(a) Show that E u X 0 , so that the key zero conditional mean assumption is
satisfied. [Hint: If e is independent of X, then E e X E e ]
(b) Show that so that the homoskedasticity Assumption is violated. [Hint:
var e X var e , if e and X are independent.]
1 cov X , Y covu , Y
(d) Show the following 1
var Y
Question 7
Suppose you are given the following regression which satisfies all the Gaussian assumptions.
Yi 0 1 X i i
You estimate the parameters 0 and 1 using two linear estimators. Using least squares, you
~ ~
obtain ˆ 0 and ˆ while the other technique you obtain 0 and .
1 1
(a) Assuming that the least squares weight is w and that of the other estimator is v , show
~ ~
that v must satisfy for 0 and 1 to be unbiased.
1
~
(b) Show that var ˆ var and var ˆ var
1 0
~
0
Question 8
Suppose you are given the following regression which satisfies all the Gaussian assumptions.
Yi 0 ui
yi2 Yi Y
n n
2
(c) Show that the residual sum of squares is
i 1 i 1
Question 9
Suppose you are given the following regression which satisfies all the Gaussian assumptions.
Yi X i i
2
(a) Show that ˆ ˆ X ; and deduce that E ˆ
1 X2 2X 2
(b) Show that var ˆ 2
n n xi2 nn xi2
i 1 i 1
PART 2: APPLICATIONS
Question 10
A researcher is interested in examining the relationship between corruption and income
inequality as measured by the gini coefficient. He collects data from 6 countries.
His empirical model is given as follows: Yi 0 1 X i i
where Y =is an index of corruption, higher values imply higher levels of corruption,
X = is a gini coefficient, and i is an error term. The sample data is summarised
as follows:
6 6 6
Yi 11.5
i 1
X i 3.13
i 1
Y X
i 1
i i 7.38
6 6
Yi 2 25.75
i 1
X
i 1
i
2
2.27
Question 11
The following results have been obtained from a sample of 20 observations on the value of
output (Y ) produced by a firm and the corresponding labour input (X).
20 20 20
i 1
X i 2786 Y 546
i 1 i
Y X i 68719
i 1 i
i=1
Y S 3218473
20
i 1 i
2
x X S 132585.7
20
i 1 i
2
y
3
(d) Compute the average elasticity of output with respect to labour and interpret your result.
(e) Find the standard errors of the coefficients of the production function.
(f) Assuming that inputs are paid their marginal products, how much does labour earn on
average?
(g) What is the estimated value of the conditional variance of output (Y )?
(h) What is the part of the variation in output produced which is not explained by the
regression?
(i) Assuming the firm adopts some labour augmenting technology, what happens to the slope
parameter?
Question 12
Below are regression results of a simple inflation model for Malawi where the exchange rate
(exrate) is the only independent variable. [25 Marks]
------------------------------------------------------------------------------
Inflation | Coef. Std. Err. t [95% Conf. Interval]
-------------+----------------------------------------------------------------
exrate| ( ) .3759982 -1.49 ( ) ( )
Constant| 221.3612 8.838372 25.05 203.9573 238.7651
------------------------------------------------------------------------------
(a) Fill in the missing numbers denoted as ( ).
(b) Interpret the coe¢cients.
(c) Evaluate the results on the basis of a priori economic theory and goodness of fit.
(d) The means of inflation and exrate are 488 and 13 respectively. What is the elasticity of
inflation with respect to exchange rate? Interpret your result.
(e) Is the relationship between exchange rate movements and in.ation in Malawi
economically and statistically significant?
(f) Can we conclude that there is a one-to-one relationship between inflation and
the exchange rate?
13. Below are regression results of an inflation model with the price of petrol as the
only regressor.
------------------------------------------------------------------------------
Inflation | Coef. Std. Err. t P>|t|
-------------+----------------------------------------------------------------
petrol| 0.001436 0.000880 ( ) 0.1040
Constant| -0.010810 ( ) 0.744217 0.4574
------------------------------------------------------------------------------
4
ii. Evaluate the results on the basis of a priori economic theory and goodness of fit.
iii. Interpret the coe¢cients.
iv. Comment on the statistical and economic significance of the results.
v. By using the test of significance and confidence interval approaches, test the hypothesis
that there is a one-to-one relationship between inflation and the price of petrol (Use 0.05
).