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Large-Sample Distribution of the TSLS Estimator When the Instrument Is Not Valid 515

Large-Sample Distribution of bnT1SLS When the IV
Regression Assumptions in Key Concept 12.4 Hold

Equation (12.20) for the TSLS estimator is similar to Equation (4.30) in Appendix 4.3 for

the OLS estimator, with the exceptions that Z rather than X appears in the numerator and

the denominator is the covariance between Z and X rather than the variance of X. Because

of these similarities, and because Z is exogenous, the argument in Appendix 4.3 that the OLS

estimator is normally distributed in large samples extends to bn1TSLS.

Specifically, when the sample is large, Z ≅ mZ, so the numerator is approximately
n
q = (n1) g i= 1qi, where qi = (Zi - mZ)ui. Because the instrument is exogenous, E(qi) = 0.

By the IV regression assumptions in Key Concept 12.4, qi is i.i.d. with variance sq2 =

var3(Zi - mZ)ui4. It follows that var (q) = sq2 = sq2 >n, and, by the central limit theorem,

q>sq is, in large samples, distributed N(0, 1).
Because the sample covariance is consistent for the population covariance, sZX ¡p

cov(Zi, Xi), which, because the instrument is relevant, is nonzero. Thus, by Equation

(12.20) bnT1 SLS ≅ b1 + q>cov(Zi, Xi), so in large samples bn1TSLS is approximately distributed

N(b1,sb2n1TSLS) where s2 = s2q> 3cov(Zi, Xi)42 = (1 > n)var 3 (Zi - mZ)ui4 > 3cov(Zi, Xi)42,

bn 1TSLS
which is the expression given in Equation (12.8).

A p p e n di x

12.4 Large-Sample Distribution of the TSLS

Estimator When the Instrument Is Not Valid

This appendix considers the large-sample distribution of the TSLS estimator in the setup of
Section 12.1 (one X, one Z) when one or the other of the conditions for instrument validity
fails. If the instrument relevance condition fails, the large-sample distribution of the TSLS
estimator is not normal; in fact, its distribution is that of a ratio of two normal random vari-
ables. If the instrument exogeneity condition fails, the TSLS estimator is inconsistent.

Large-Sample Distribution of bn1TSLS When
the Instrument Is Weak

First consider the case that the instrument is irrelevant so that cov(Zi, Xi) = 0. Then the argu-
ment in Appendix 12.3 entails division by zero. To avoid this problem, we need to take a
closer look at the behavior of the term in the denominator of Equation (12.20) when the
population covariance is zero.