18.17 Consider the regression model in matrix form Y = XB + WG + U, where X and W are matrices of regressors and B and G are vectors of unknown regression coefficients. Let X

= MWX and Y

= MWY, where MW = I - W(WW)-1W.

a. Show that the OLS estimators of B and G can be written as

(Hint: Show that the product of the two matrices is equal to the identity matrix.)

c. Show that B n = (XMWX)-1XMWY.

d. The Frisch–Waugh theorem (Appendix 6.2) says that B n = (XX)-1XY. Use the result in

(c) to prove the Frisch–Waugh theorem.

b. Show that [] = X'X X'W X'Y W'X W'W W'Y X'X X'W w'X W'W (X'MwX)- = [-(ww) -1 (WW)W'X(X'MwX) (W'W) + (W'W) 'W'X(X'MwX) 'X' W(W'W)