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802 Chapter 18 The Theory of Multiple Regression

Z′Z > n ¡p QZZ and X′Z > n ¡p QXZ. Thus B′U = (Z′Z)-1>2′Z′U = (Z′Z/n)-1>2′
(Z′U> 2n) ¡d suz where z is distributed N(0m + r + 1, Im + r + 1). In addition, B′X > 2n =
(Z′Z>n)-1>2′(Z′X>n) ¡p QZ-1Z>2QZX, so MB′X ¡p I - QZ-1Z>2QZX(QXZ QZ-1Z>2 ′QZ-1Z>2QZX)-1
QXZQZ-1Z>2′= M .QZ-Z1>2QZX Thus

Un ′PZUn ¡d (z′MQXZ Q-Z1Z>2 z)s2u. (18.91)

Under the null hypothesis, the TSLS estimator is consistent and the coefficients in the
regression of Un on Z converge in probability to zero [an implication of Equation (18.91)],
so the denominator in the definition of the J-statistic is a consistent estimator of su2:

Un ′MZUn >(n - m - r - 1) ¡p su2. (18.92)

From the definition of the J-statistic and Equations (18.91) and (18.92), it follows that

J = Un ′PZUn - r - 1) ¡d z ′MQ-Z1Z>2 QXZ z. (18.93)
Un ′MZUn >(n - m

Because z is a standard normal random vector and MQ-Z1Z>2QZX is a symmetric idempotent
matrix, J is distributed as a chi-squared random variable with degrees of freedom that equal

the rank of MQZ-Z1>2QZX [Equation (18.78)]. Because QZ-1Z>2QZX is (m +r+ 1) * (k + r + 1)
and m > k, the rank of MQ-Z1Z>2QZX is m − k [Exercise 18.5]. Thus J ¡d x2m - k, which is the

result stated in Equation (18.64).

The Efficiency of the Efficient GMM Estimator

The infeasible efficient GMM estimator, BEff.GMM, is defined in Equation (18.66). The

proof that BEff.GMM is efficient entails showing that c′( IV - Eff.GMM)c Ú 0 for all vectors c.
A

The proof closely parallels the proof of the efficiency of the TSLS estimator in the first

section of this appendix, with the sole modification that H−1 replaces QZZsu2 in Equation

(18.85) and subsequently.

Distribution of the GMM J-Statistic

The GMM J-statistic is given in Equation (18.70). The proof that, under the null hypoth-
esis, JGMM ¡d x2m - k closely parallels the corresponding proof for the TSLS J-statistic
under homoskedasticity.