Introductory Econometrics 4th Edition by Jeffrey Wooldridge

Exercise 3

Let be the OLS estimate from the regression of y on X. Let A be a (k + 1) × (k + 1) nonsingular matrix and define zt = xtA, t = 1,….,n. Therefore, zt is 1 × (k + 1) and is a nonsingular linear combination of xt. Let Z be the n × (k + 1) matrix with rows zt. Let denote the OLS estimate from a regression of y on Z.
(i) Show that =A-1 .
(ii) L et be the fitted values from the original regression and let. y t be the fitted values from regressing y on Z. Show that = , for all t =- 1, 2,…., n. How do the residuals from the two regressions compare
(iii) Show that the estimated variance matrix for is A1(X X)1A1 , where is the usual variance estimate from regressing y on X.
(iv) L et the be the OLS estimates from regressing yt on 1, xt1,...,xtk, and let the be the OLS estimates from the regression of yt on 1, a1xt1,…,ak xtk, where aj 0, j =1,…,k. Use the results from part (i) to find the relationship between the and the
(v) Assuming the setup of part (iv), use part (iii) to show that se( ) = se( )/aj.
(vi) Assuming the setup of part (iv), show that the absolute values of the t statistics for and are identical.