18.9 This exercise shows that the OLS estimator of a subset of the regression coefficients is consistent under the conditional mean independence assumption stated in Appendix 7.2. Consider the multiple regression model in matrix form Y = XB + WG + U, where X and W are, respectively, n * k1 and n * k2 matrices of regressors. Let Xi and Wi denote the ith rows of X and W [as in Equation (18.3)]. Assume that (i) E(ui|Xi, Wi) =

WiD, where D is a k2 * 1 vector of unknown parameters; (ii) (Xi, Wi, Yi)

are i.i.d.; (iii) (Xi, Wi, ui) have four finite, nonzero moments;

and (iv) there is no perfect multicollinearity. These are Assumptions

#1 through #4 of Key Concept 18.1, with the conditional mean independence assumption (i) replacing the usual conditional mean zero assumption.

a. Use the expression for B n given in Exercise 18.6 to write B n - B =

(n-1XMWX )-1(n-1XMWU ).

b. Show that n-1XMWX ¡p XX - XW-1 WWWX, where XX =

E(XiXi ), XW = E(XiWi ), and so forth. [The matrix An ¡p A if An,ij ¡p Aij for all i, j, where An,ij and Aij are the (i, j) elements of An and A.]

c. Show that assumptions (i) and (ii) imply that E(U|X, W) = WD.

d. Use

(c) and the law of iterated expectations to show that n-1XMWU ¡p 0k1 * 1.

e. Use

(a) through

(d) to conclude that, under conditions (i) through (iv), B n ¡p B.