Page 801 -

800 Chapter 18 The Theory of Multiple Regression

Now let A = An + D so that D is the difference between the matrices A and An .
N o t e that An′A = (X′X)-1X′A = (X′X)-1 [by Equation (18.82)] and An′An =
(X′X)-1X′X(X′X)-1 = (X′X)-1, so An ′D = An ′(A - An ) = An ′A - An ′An = 0(k + 1) * (k + 1).
Substituting A = An + D into the formula for the conditional variance in Equation

(18.82) yields

var(B 0 X) = su2(An + D)′(An + D)

= s2u[An ′An + An ′D + D′An + D′D]
= su2(X′X)- 1 + s2uD′D, (18.83)

where the final equality uses the facts An ′An = (X′X)-1 and An′D = 0(k + 1) * (k + 1).
Because var(Bn ͉ X ) = su2(X′ X )-1, Equations (18.82) and (18.83) imply that

var(B 0 X) - var(Bn 0 X) = su2D′D. The difference between the variances of the two estima-

tors of the linear combination c′B thus is

var (c′B 0 X ) - var(c′Bn 0 X) = s2u c′D′Dc Ú 0. (18.84)

The inequality in Equation (18.84) holds for all linear combinations c′B, and the inequality

holds with equality for all nonzero c only if D = 0n * (k+1)—that is, if A = An or, equiva-
lently, B = Bn. Thus c′Bn has the smallest variance of all linear conditionally unbiased esti-

mators of c′B; that is, the OLS estimator is BLUE.

Appendix

18.6 Proof of Selected Results for IV

and GMM Estimation

The Efficiency of TSLS Under Homoskedasticity
[Proof of Equation (18.62)]

When the errors ui are homoskedastic, the difference between IV [Equation (18.61)] and
A

TSLS [Equation (18.55)] is given by

IV - TSLS = (QXZAQZX)-1QXZAQZZAQZX(QXZAQZX)-1su2 - (QXZQZ-1ZQZX)-1su2
A

= (QXZAQZX)-1QXZA[QZZ - QZX(QXZQZ-1Z QZX)-1QXZ]AQZX(QXZAQZX)-1su2,(18.85)

where the second term in brackets in the second equality follows from

(QXZAQZX)-1QXZAQZX = I(k + r + 1). Let F be the matrix square root of QZZ, so QZZ = F′F
and QZ-1Z = F -1F -1′. [The latter equality follows from noting that (F′F)-1 = F -1F′-1 and
F′-1 = F -1′.] Then the final expression in Equation (18.85) can be rewritten to yield

AIV - TSLS = (QXZAQZX)-1QXZAF′[I - F -1′QZX(QXZF -1F -1′QZX)-1QXZF -1]
* FAQZX(QXZAQZX)-1s2u, (18.86)