2. Consider the linear regression model Y = Bot Bi Xe tr t=1,2,...,n, where X₁0 is a non-stochastic variable and u, is a random error term such that E(u) =0 for all t. (a) Suppose that u = 0₁44-1 +0₂04-2+1, where ₁ and ₂ are known parameters and e, are random variables such that E() = 0 for all t, var(₁)=²> 0 for all t, and cov(t, ₂) = 0 for all t = 8. (i) What are the statistical properties of the ordinary least squares (OLS) estimators of 30 and 3₁? Explain. (ii) Is it possible to find estimators of 5 and 3, which are more efficient than the OLS estimators? If so, explain how to estimate 3 and 3₁ effi- ciently. Include in your answer a careful explanation of why your proposed estimation method would produce efficient estimates. (iii) Is a conventional OLS-based t-test for the hypothesis that 3₁ = 0 valid? If not, explain how to carry out a valid test of the hypothesis that 3₁ = 0. (iv) What do the hypotheses Ho: 01 02 = 0 and H₁: 010 and/or 02/0 imply about the regression errors u? Explain in detail how to test Ho against H, using the Breusch-Godfrey procedure. (b) Indicate whether you agree or disagree with each of the following statements and provide a brief justification for your answer. (i) "If the hypothesis that ₁ = 2 = 0 is rejected, it is advisable to use OLS with heteroskedasticity-consistent standard errors". (ii) "If the hypothesis that o₁ = 2 = 0 is rejected, a natural response is not to use an estimator other than OLS but to change the specification of the model". (iii) "If ₁ = 2 = 0 and var(t)=o² exp(X₁), o and 3₁ should be estimated by OLS in the model (Y₁/X₂) = (1/X₂) + B₁ + ₂". (iv) "OLS estimates of the coefficients of the model Y₁ = 30 +³₁X₁ + ₂ are accurate even if the true model is Y₁ = Bo + B₁X₁ + B₂X² + €₁".