19.17 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 cB n

Gn d = c XX XW WX WW d

-1 c

XY WY d

b. Show that J

XX XW WX WW R

-1

= c

(XMWX)-1 - (XMWX)-1XW(WW)-1

-(WW)-1WX(XMWX)-1 (WW)-1 + (WW)-1WX(XMWX)-1XW(WW)-1 d .

(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.

19.18 Consider the homoskedastic linear regression model with two regressors, and let rX1, X2 = corr(X1, X2). Show that corr(b n

1, b n

2) S -rX1,X2 [Equation (6.21)]

as n increases.