Page 520 -

TSLS with Control Variables 519

recommended; however, when instruments are strong (so TSLS is valid) and the coeffi-
cients are overidentified, the Anderson–Rubin test is inefficient in the sense that it is less
powerful than the TSLS t-test.

Estimation of b

If the instruments are irrelevant, it is not possible to obtain an unbiased estimator of b1, even
in large samples. Nevertheless, when instruments are weak, some IV estimators tend to be
more centered on the true value of b1 than is TSLS. One such estimator is the limited infor-
mation maximum likelihood (LIML) estimator. As its name implies, the LIML estimator is
the maximum likelihood estimator of b1 in the system of Equations (12.13) and (12.14). (For
a discussion of maximum likelihood estimation, see Appendix 11.2.) The LIML estimator
also is the value of b1,0 that minimizes the homoskedasticity-only Anderson–Rubin test
statistic. Thus, if the Anderson–Rubin confidence set is not empty, it will contain the LIML
estimator. In addition, the CLR confidence interval contains the LIML estimator.

If the instruments are weak, the LIML estimator is more nearly centered on the true value
of b1 than is TSLS. If instruments are strong, the LIML and TSLS estimators coincide in large
samples. A drawback of the LIML estimator is that it can produce extreme outliers. Confidence
intervals constructed around the LIML estimator using the LIML standard error are more
reliable than intervals constructed around the TSLS estimator using the TSLS standard error,
but are less reliable than Anderson–Rubin or CLR intervals when the instruments are weak.

The problems of estimation, testing, and confidence intervals in IV regression with
weak instruments constitute an area of ongoing research. To learn more about this topic,
visit the website for this book.

A p p e n di x

12.6 TSLS with Control Variables

In Key Concept 12.4, the W variables are assumed to be exogenous. This appendix considers
the case in which W is not exogenous, but instead is a control variable included to make Z
exogenous. The logic of control variables in TSLS parallels the logic in OLS: If a control
variable effectively controls for an omitted factor, then the instrument is uncorrelated with
the error term. Because the control variable is correlated with the error term, the coefficient
on a control variable does not have a causal interpretation. The mathematics of control
variables in TSLS also parallels the mathematics of control variables in OLS and entails
relaxing the assumption that the error has conditional mean zero, given Z and W, to be that
the conditional mean of the error does not depend on Z. This appendix draws on Appendix 7.2
(Conditional Mean Independence), which should be reviewed first.