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Proof of Selected Results for IV and GMM Estimation 801

where the second expression in brackets uses F′F -1′ = I. Thus

c′( IV - TSLS)c = d′[I - D(D′D) - 1D′]dsu2, (18.87)
A

where d = FAQZX(QXZAQZX)-1c and D = F -1′QZX. Now I - D(D′D)-1D′ is a symmetric

idempotent matrix (Exercise 18.5). As a result, I - D(D′D)-1D′ has eigenvalues that are

either 0 or 1 and d′[I - D(D′D)-1D′]d Ú 0 (Exercise 18.10). Thus c′( IV - TSLS)c Ú 0,
A

proving that TSLS is efficient under homoskedasticity.

Asymptotic Distribution of the J-Statistic Under
Homoskedasticity

The J-statistic is defined in Equation (18.63). First note that

Un = Y - XBnTSLS
= Y - X(X′PZX )-1X′PZY
= (XB + U ) - X(X′PZX )-1X′PZ(XB + U )
= U - X(X′PZX )-1 X′PZU
= [I - X(X′PZX )-1X′PZ]U. (18.88)

Thus

Un PZUn = U′[I - PZX(X′PZX)-1X′]PZ[I - X(X′PZX)-1X′PZ]U
= U′[PZ - PZX(X′PZX )-1X′PZ]U, (18.89)

where the second equality follows by simplifying the preceding expression. Because Z′Z is

symmetric and positive definite, it can be written in terms of its matrix square root,
Z′Z = (Z′Z)1>2′(Z′Z)1>2, and this matrix square root is invertible, so (Z′Z)-1 =
(Z′Z)-1>2(Z′Z)-1>2′, where (Z′Z)-1>2 = [(Z′Z)1>2]-1. Thus PZ can be written as PZ =
Z(Z′Z)-1Z′ = BB′, where B = Z(Z′Z)-1>2. Substituting this expression for PZ into the
final expression in Equation (18.89) yields

Un ′PZUn = U′[BB′ - BB′X(X′BB′X)-1 X′BB′]U

= U′B [I - B′X(X′BB′X)-1X′B]B′U

= U′BMB′XB′U, (18.90)

where MB′X = I - B′X(X′BB′X)-1X′B is a symmetric idempotent matrix.
The asymptotic null distribution of Un ′PZUn is found by computing the limits in probability

and in distribution of the various terms in the final expression in Equation (18.90) under the

null hypothesis. Under the null hypothesis that E(Ziui) = 0, Z′U > 2n has mean zero and
the central limit theorem applies, so Z′U > 2n ¡d N(0, QZZs2u). In addition,