In the linear model estimated by GMMwith general weight matrixW the asymptotic variance of bgmm is V Q0WQ 1Q0WWQ Q0WQ 1 .

In the linear model estimated by GMMwith general weight matrixW the asymptotic variance of b¯gmm is V Æ

¡

Q0WQ

¢¡1Q0W­WQ

¡

Q0WQ

¢¡1 .

(a) Let V 0 be this matrix whenW Æ­¡1. Show that V 0 Æ

¡

Q0­¡1Q

¢¡1 .

(b) Wewant to showthat for anyW, V ¡V 0 is positive semi-definite (for then V 0 is the smaller possible covariance matrix andW Æ­¡1 is the efficient weight matrix). To do this start by finding matrices A and B such that V Æ A0­A and V 0 Æ B0­B.

(c) Show that B0­A Æ B0­B and therefore that B0­(A ¡B) Æ 0.

(d) Use the expressions V Æ A0­A, A Æ B Å(A ¡B) , and B0­(A ¡B) Æ 0 to show that V ¸V 0.