Imbens/Wooldridge, Lecture Notes 13, Summer ’07
1
What’s New in Econometrics
NBER, Summer 2007
Lecture 13, Wednesday, Aug 1st, 2.003.00pm
Weak Instruments and Many Instruments
1. Introduction
In recent years a literature has emerged that has raised concerns with the quality of
inferences based on conventional methods such as Two Stage Least Squares (TSLS) and
Limited Information Maximum Likelihood (LIML) in instrumental variables settings when
the instrument(s) is/are only weakly correlated with the endogenous regressor(s). Although
earlier work had already established the poor quality of conventional normal approximations
with weak or irrelevant instruments, the recent literature has been motivated by empirical
work where
ex post
conventional large sample approximations were found to be misleading.
The recent literature has aimed at developing better estimators and more reliable methods
for inference.
There are two aspects of the problem. In the justidentified case (with the number of
instruments equal to the number of endogenous regressors), or with low degrees of over
identification, the focus has largely been on the construction of confidence intervals that
have good coverage properties even if the instruments are weak. Even with very weak, or
completely irrelevant, instruments, conventional methods are rarely substantively mislead
ing, unless the degree of endogeneity is higher than one typically encounters in studies using
crosssection data. Conventional TSLS or LIML confidence intervals tend to be wide when
the instrument is very weak, even if those intervals do not have the correct nominal cov
erage for all parts of the parameter space. In this case better estimators are generally not
available. Improved methods for confidence intervals based on inverting test statistics have
been developed although these do not have the simple form of an estimate plus or minus a
constant times a standard error.
The second case of interest is that with a high degree of overidentification. These settings
often arise by interacting a set of basic instruments with exogenous covariates in order to
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Imbens/Wooldridge, Lecture Notes 13, Summer ’07
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improve precision. If there are many (weak) instruments, standard estimators can be severely
biased, and conventional methods for inference can be misleading. In particular TSLS has
been found to have very poor properties in these settings. Bootstrapping does not solve these
problems. LIML is generally much better, although conventional LIML standard errors are
too small. A simple to implement proportional adjustment to the LIML standard errors based
on the Bekker manyinstrument asymptotics or the ChamberlainImbens random coefficients
argument appears to lead to substantial improvements in coverage rates.
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 Econometrics, Normal Distribution, zi, TSLS, Yi /n, Summer ’07

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