6. (Instrumental variable estimation). Consider the model Yi = X:B + ui, where Xi and B are K x 1. Assuming that E(u;Xi) * 0 so that X; is endogenous. Further assume that there exists an instrumental variable (IV) Zi, of dimension K x 1, such that E(uiZi) = 0 and E(X;Z':) is non-singular. That is, the instrumental variable is uncorrelated with disturbance us, and correlated with the regressor Ci. (a) Show that = [E(Z;X:)]-'E(ZY;) (show your derivations). (b) From the result from (a), we can estimate S by Biv = [n-'Et, Z;X:]-in 'Eh ZiYi = (Z'X) -1Z'Y. Assuming iid data, show that BIV " B. (c) Derive the asymptotic distribution of Vn(Biv - B). (Hint: using LLN and CLT). (d) Provide a consistent estimator for the asymptotic variance of Vn(BIV - B).10. (General instrumental variance estimation. The same model as considered in problem 6 above except that we have more instruments than endogenous regressors. Z, is of dimension q x 1 with q > k. In this case E(X, Z/) is not a square matrix, hence, it is not invertible. We can estimate 3 by Biv = (XP,X) 'X'PY, where P. = Z(Z'Z)-12', the dimension of X, Z and Y are n x k, n x q and n x 1, respectively. (i) Show that Byv 4 3 (Hint: using LLN, but be careful, LLN applies to sum of iid random variables). (ii) Derive the asymptotic distribution of vn(B/v - B). (Hint: using LLN and CLT). (iii) Provide a consistent estimator for the asymptotic variance of vn(B,v - B). Solution to problem 10 For (i) we have Biv = (X'PX)-'X'PY = B+ (X'PX)-1X'Pu = 3+[X'z(Z'2)-'Z'x]-'x'z(Z'z)-'z'u = B+[(X'Z)(Z'Z) 1(Z'X)]](X'Z)(Z'Z) 'Z'u = 8+[(n-] Ex:Z))(n-] > zz!)-1 x( n 1 \\z x, ) ](n'Ex:Z)(n'>zz)-'n'yzqui 4 8 + E ( X Z)) ( E(ZZ)))'E(ZX))] E(X:Z)) (E(Z Z;))-'E(Zu;) = 8+0=3 -J because E(Z,u;) = E[Z;E(u.|Z.)] = 0. (ii) Using derivation similar to (i) above we have Vn(Biv - B) = [(X'Z) (Z'Z) 1(Z'X)]](X'z)(Z'Z)"'Z'u/vn [QQQ;]Q.=QIN(0, V) = A-'N(0, V) = N(0, A-VA-1 ), where for the last term, we used Z'w/vn = ,/ EL_, Zmu; 5 N(0, V) by the CLT, where V = E(u;Z;Z!), A = [Q.=Q_!Q:]-1Qx=Q:1, Qxz = E(X;Z!) and Q:2 = E(Z;Z!). (iii) We can consistently estimate A and V by A = [(X'Z)(Z'Z)-1(Z'X)]-1(X'Z)(Z'Z)-1 and V= n-12tuZ;Z, where u = Wi - XiBrv