SECTION A
Answer all questions from this section
1. Consider the following regression model
Yi = Xi + ui , i = 1; :::; n:
The error term has a zero mean, variance equal to 2=X2 i ; and E (uiuj ) = 0 for i 6= j: You are given a sample of observations f(Yi ; Xi)g n i=1. You may treat Xi as being non-stochastic. Clearly annotating your answers:
(a) (5 marks) Derive the OLS estimator of : In the presence of heteroskedasticity, the OLS estimator remains unbiased (you are not asked to show this). Derive the variance of the OLS estimator of . (b)
(3 marks) Discuss how you can obtain the Best Linear Unbiased Estimator (BLUE) of given the heteroskedasticity.
2. Consider the simple linear regression model
Yi = + Xi + ui , i = 1; :::; n
in the presence of correlation between the error and regressor. The regressor exhibits variability in the sample, i.e., Pn i=1(Xi X) 2 6= 0: Under assumptions of homoskedasticity and the absence of autocorrelation, the IV estimator of that uses the instrument Z has the following (asymptotic) variance (no need to prove this statement)
V ar ^ IV = 2 P u n i=1(Xi X) 2 1 r 2 XZ ;
where rXZ 6= 0 is the sample correlation between X and Z and 2 u is the variance of the disturbance term u.
(a) (1 marks) Give the formula for ^ IV (you are not asked to derive it).
(b) (4 marks) Provide at least three factors that will help obtain more precise IV parameter estimates for : In your answer explain why the precision of parameter estimates is important.
(c) (3 marks) Discuss the following statement: “If X is not correlated with u; the best choice of instrument is using the regressor itself