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 = Xβ + Wγ + u, where X and W are, respectively, n × k1 and n × k2 matrices of regressors. Let Xi€² and Wi€² denote the ith rows of X and W[as in Equation (18.3)]. Assume that (i) E(ui|Xi, Wi) = Wi€²Î´, where δ 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 #l #4of Key Concept 18.1, with the conditional mean independence assumption (i) replacing the usual conditional mean zero assumption.
a. Use the expression for β given in Exercise 18.6 to write β - β =

b. Show that

Where ˆ‘XX =

and so forth. [The matrix

if

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) = Wδ.
d. Use (c) and the law of iterated expectations to show that

e. Use (a) through (d) to conclude that, under conditions (i) through (iv),

4,, A 4,44,