Exam 17: The Theory of Linear Regression With One Regressor

Consider the model Yi = β1Xi + ui,where ui = c ei and all of the X's and e's are i.i.d.and distributed N(0,1). (a)Which of the Extended Least Squares Assumptions are satisfied here? Prove your assertions. (b)Would an OLS estimator of β1 be efficient here? (c)How would you estimate β1 by WLS?

(Requires Appendix material)State and prove the Cauchy-Schwarz Inequality.

"One should never bother with WLS.Using OLS with robust standard errors gives correct inference,at least asymptotically." True,false,or a bit of both? Explain carefully what the quote means and evaluate it critically.

(Requires Appendix material)If the Gauss-Markov conditions hold,then OLS is BLUE.In addition,assume here that X is nonrandom.Your textbook proves the Gauss-Markov theorem by using the simple regression model Yi = β0 + β1Xi + ui and assuming a linear estimator .Substitution of the simple regression model into this expression then results in two conditions for the unbiasedness of the estimator: = 0 and = 1. The variance of the estimator is var( X1,…,Xn)= . Different from your textbook,use the Lagrangian method to minimize the variance subject to the two constraints.Show that the resulting weights correspond to the OLS weights.

What does the Gauss-Markov theorem prove? Without giving mathematical details,explain how the proof proceeds.What is its importance?

If,in addition to the least squares assumptions made in the previous chapter on the simple regression model,the errors are homoskedastic,then the OLS estimator is

One of the earlier textbooks in econometrics,first published in 1971,compared "estimation of a parameter to shooting at a target with a rifle.The bull's-eye can be taken to represent the true value of the parameter,the rifle the estimator,and each shot a particular estimate." Use this analogy to discuss small and large sample properties of estimators.How do you think the author approached the n → ∞ condition? (Dependent on your view of the world,feel free to substitute guns with bow and arrow,or missile. )

In order to use the t-statistic for hypothesis testing and constructing a 95% confidence interval as 1.96 standard errors,the following three assumptions have to hold:

Suppose that the conditional variance is var(ui|Xi)= λh(Xi)where λ is a constant and h is a known function.The WLS estimator is

(Requires Appendix material)If X and Y are jointly normally distributed and are uncorrelated,

Consider the model Yi - β1Xi + ui,where the Xi and ui the are mutually independent i.i.d.random variables with finite fourth moment and E(ui)= 0. (a)Let 1 denote the OLS estimator of β1.Show that ( 1- β1)= . (b)What is the mean and the variance of ? Assuming that the Central Limit Theorem holds,what is its limiting distribution? (c)Deduce the limiting distribution of ( 1 - β1)? State what theorems are necessary for your deduction.

The class of linear conditionally unbiased estimators consists of

You need to adjust by the degrees of freedom to ensure that is

The following is not part of the extended least squares assumptions for regression with a single regressor:

Homoskedasticity means that

Your textbook states that an implication of the Gauss-Markov theorem is that the sample average, ,is the most efficient linear estimator of E(Yi)when Y1,... ,Yn are i.i.d.with E(Yi)= μY and var(Yi)= .This follows from the regression model with no slope and the fact that the OLS estimator is BLUE. Provide a proof by assuming a linear estimator in the Y's, (a)State the condition under which this estimator is unbiased. (b)Derive the variance of this estimator. (c)Minimize this variance subject to the constraint (condition)derived in (a)and show that the sample mean is BLUE.

The extended least squares assumptions are of interest,because

The WLS estimator is called infeasible WLS estimator when

Consider the simple regression model Yi = β0 + β1Xi + ui where Xi > 0 for all i,and the conditional variance is var(ui Xi)= θX where θ is a known constant with θ > 0. (a)Write the weighted regression as i = β0 0i + β1 1i + i.How would you construct i, 0i and 1i? (b)Prove that the variance of is i homoskedastic. (c)Which coefficient is the intercept in the modified regression model? Which is the slope? (d)When interpreting the regression results,which of the two equations should you use,the original or the modified model?

The following is not one of the Gauss-Markov conditions: