SOLVED Statistics 215B Assignment 4

 For this assignment you will use simulation to study the performance of OLS and IVLS coefficient estimators, as well as two different estimators of the error variance. Consider the model Yi D Xiˇ C i ; (1) Xi D Ui C 2Vi C ıi : Everything in sight is scalar. The vectors .Ui ; Vi ; i ; ıi/; i D 1; : : : ; n; are iid across i. Each vector is normal with mean zero. For each i we take the three random objects (1) Ui , (2) Vi , and (3) .i ; ıi/ to be mutually independent; furthermore, Var.Ui/ D Var.Vi/ D 1; Var.i/ D Var.ıi/ D  2 ; and Cov.i ; ıi/ D : In (1), the variable Xi is endogenous (explain why), and .Ui ; Vi/ are instruments (explain why). Suppose ˇ D 3,  2 D 1, and  D 3=4. For each of 1,000 simulation runs, generate n D 100 independent realizations of the vector .Ui ; Vi ; i ; ıi ; Xi ; Yi/ according to (1) and (2). Use OLS to obtain the estimate ˇO OLS, and IVLS to obtain ˇO IVLS. Plot the histogram for each estimator; report the mean, SD, and RMSE in each case. What are the relative merits of OLS versus IVLS? For each simulation, estimate the error variance  2 in two ways: first using the residuals obtained from plugging ˇO IVLS into (1), then using the residuals from the transformed equation .Z0Z/1=2Z 0Y D .Z0Z/1=2Z 0Xˇ C  : Here Z is the n 2 matrix of instruments, X is the n 1 design matrix, and Y is the n 1 vector of responses. What is the appropriate denominator in each case? Plot the histograms for the two estimators and report sample means and SDs. Comment briefl