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Question 1

Consider the regression model:

where ”  N (0; 2In) :
(a) Show that the ordinary least squares (OLS) estimator of

1 is

Y=X1 1+”; (n1) (n1)(11) (n1)

(1)

(2) [25 marks]

^1 = (X10 X1)1X10 Y:
(b) Derive the regression residual ^” and show that ^1 and ^” are independent.

[25 marks] (c) Discuss the conditions under which ^1 is an unbiased and e¢ cient estimator of 1:

(d) Suppose that the true model is

Y = X1 1 + X2 2 + v ; (n1) (n1)(11) (n1)(11) (n1)

so that the variable X2 is omitted in model (1). Discuss how the statistical properties of ^1 deÖned by (2) are a§ected in this case.

Question 2

Consider the regression model:

Exam code ECON41515WE01

(3)

Y=X +”; (n1) (nk)(k1) (n1)

where X and ” are correlated.
(a) Why is it problematic to use the OLS estimator, ^OLS = (X0X)1 X0Y , in this case?

[25 marks]

(b) Suppose there is a set of r instrumental variables:

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CONTINUED

26z1i 37 z =6z2i7;

i 4:::5

(r1)
zri

so that with sample size being n; the observations on these instrumental variables are 26 z 10 37

Z = 6 z 20 7 : 4:::5

zn0
What properties should zi have in order to be a valid instrument?

[25 marks] (c) Use the two stage least squares (TSLS) method to derive the instrumental variable

(IV) estimator ^IV .

[25 marks]

(d) Show that when r = k; ^IV derived in part (c) can be simpliÖed as ~IV = (Z0X)1 Z0Y . Assuming that n1 Z0X !p Qzx; where Qzx is a positive deÖnite matrix, show that ~IV is a consistent estimator of :

Question 3

(a) Consider the following linear model with k regressors: yi=x0i +”i

(4)

where xi = (x1i; x2i; :::; xki)0, = ( 1; 2; :::; k)0 ; and the error term “i has zero mean and unknown variance 2: Suppose all the k regressors are exogenous. Use the method of moments to estimate and 2 with a sample of size n:

[25 marks] (b) Now suppose xi is endogenous. Let zi be an r  1 (r  k) vector of instrumental

variables such that
Derive the generalized method of moments (GMM) estimator ^GMM of with a sample

E[zi(yix0i )]=0: (5) of size n: How the GMM estimator is related to the instrumental variable (IV) estimator?

(c) Discuss the advantages of GMM over OLS with some applications.

Question 4

Let Y1;:::;Yn be observations on the wages of n individuals. Assume that the Yi are independent, normally distributed (i.i.d.) with variance equal to 1:

Yi N( xi;1);
where xi is the number of years of schooling, and is an unknown parameter.

(a) Write down the log-likelihood of the data and derive the maximum likelihood estima- tor (MLE), denoted by ^; for .

[25 marks]

(b) Show that ^ is unbiased and Önd V ar(^):

[25 marks] (c) Discuss the advantages of maximum likelihood method over OLS with some applica-

tions. Explain how the MLE behaves when the error term “i is not normally distributed. [50 marks]

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