# convergence in distribut

various statistics. We Örst show that the statistic (12.66) is not altered if based on the full contrast b e. Indeed, b 1 e 1 is a linear function of b 2 e 2 , so there is no extra information in the full contrast. To see this, observe that given b 2 , we can solve by least-squares to Önd b 1 = X0 1X1 1 CHAPTER 12. INSTRUMENTAL VARIABLES 447 and similarly e 1 = X0 1X1 1 X0 1 y P ZX2e = X0 1X1 1 X0 1 y X2e the second equality since P ZX1 = X1. Thus b 1 e 1 = X0 1X1 1 X0 1 y X2b 2 X0 1X1 1 X0 1 y P ZX2e = X0 1X1 1 X0 1X2 e 2 b 2 as claimed. We next show that T in (12.66) equals the homoskedastic Wald statistic W0 for b from the regression (12.65). Consider the latter regression. Since X2 is contained in X, the coe¢ cient estimate b is invariant to replacing Ub 2 = X2Xc2 with Xc2 = P ZX2. By the FWL representation, setting MX = In X (X0X) 1 X0 b = Xc0 2MXXc2 1 Xc0 2MXy = X0 2P ZMXP ZX2 1 X0 2P ZMXy: It follows that W0 = y 0MXP ZX2 (X0 2P ZMXP ZX2) 1 X0 2P ZMXy b 2 : Our goal is to show that T = W0 . DeÖne Xf2 = (In P 1) X2 so b 2 = Xf0 2Xf2 1 Xf0 2y. Then deÖning using (P Z P 1) (In P 1) = (P Z P 1) and deÖning Q = Xf2 Xf0 2Xf2 1 Xf0 2 def = X0 2 (P Z P 1) X2 e 2 b 2 = X0 2 (P Z P 1) y X0 2 (P Z P 1) X2 Xf0 2Xf2 1 Xf0 2y = X0 2 (P Z P 1) (In Q) y = X0 2 (P Z P 1 P ZQ) y = X0 2P Z (In P 1 Q) y = X0 2P ZMXy: The third-to-last equality is P 1Q = 0 and the Önal uses MX = In P 1 Q. We also calculate that Q def = X0 2 (P Z P 1) X2 X0 2 (P Z P 1) X2 1 X0 2M1X2 1 X0 2 (P Z P 1) X2 = X0 2 (P Z P 1 (P Z P 1) Q (P Z P 1)) X2 = X0 2 (P Z P 1 P ZQP Z) X2 = X0 2P ZMXP ZX2: Thus T = 0Q1 b 2 = y CHAPTER 12. INSTRUMENTAL VARI