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516 Chapter 12 Instrumental Variables Regression

We start by rewriting Equation (12.20). Because of the consistency of the sample average,

in large samples, Z is close to mZ, and X is close to mX. Thus the term in the denominator
n n
of Equation (12.20) is approximately (n1) g i= 1(Zi - mZ)(Xi - mX) = (n1) g i= 1 ri = r, where

ri = (Zi - mZ)(Xi - mX). Let s2r = var3(Zi - mZ)(Xi - mX)4, let s2r = sr2>n, and let

q, s2q, and sq2 be as defined in Appendix 12.3. Then Equation (12.20) implies that, in large

samples,

bnT1 SLS ≅ b1 + q = b1 + a sq b a q > sq b = b1 + a sq b a q > sq b . (12.21)
r sr r > sr sr r > sr

If the instrument is irrelevant, then E(ri) = cov(Zi, Xi) = 0. Thus r is the sample average
of the random variables ri, i = 1, c, n, which are i.i.d. (by the second least squares
assumption), have variance s2r = var3(Zi - mZ)(Xi - mX)4 (which is finite by the third IV
regression assumption), and have a mean of zero (because the instruments are irrelevant).
It follows that the central limit theorem applies to r, specifically, r>sr is approximately
distributed N(0, 1). Therefore, the final expression of Equation (12.21) implies that, in
large samples, the distribution of bn1TSLS - b1 is the distribution of aS, where a = sq>sr and
S is the ratio of two random variables, each of which has a standard normal distribution
(these two standard normal random variables are correlated).

In other words, when the instrument is irrelevant, the central limit theorem applies to
the denominator as well as the numerator of the TSLS estimator, so in large samples the
distribution of the TSLS estimator is the distribution of the ratio of two normal random
variables. Because Xi and ui are correlated, these normal random variables are correlated,
and the large-sample distribution of the TSLS estimator when the instrument is irrelevant
is complicated. In fact, the large-sample distribution of the TSLS estimator with irrelevant
instruments is centered on the probability limit of the OLS estimator. Thus, when the
instrument is irrelevant, TSLS does not eliminate the bias in OLS and, moreover, has a
nonnormal distribution, even in large samples.

A weak instrument represents an intermediate case between an irrelevant instrument
and the normal distribution derived in Appendix 12.3. When the instrument is weak but
not irrelevant, the distribution of the TSLS estimator continues to be nonnormal, so the
general lesson here about the extreme case of an irrelevant instrument carries over to weak
instruments.

Large-Sample Distribution of bn1TSLS When
the Instrument Is Endogenous

The numerator in the final expression in Equation (12.20) converges in probability to
cov(Zi, ui). If the instrument is exogenous, this is zero, and the TSLS estimator is consistent