Performance of Conditional Wald Tests in IV Regression with Weak Instruments

We compare the powers of five tests of the coefficient on a single endogenous
regressor in instrumental variables regression. Following Moreira (2003), all tests are
implemented using critical values that depend on a statistic which is sufficient under the
null hypothesis for the (unknown) concentration parameter, so these conditional tests are
asymptotically valid under weak instrument asymptotics. Four of the tests are based on
k-class Wald statistics (two-stage least squares, LIML, Fuller’s (1977), and bias-adjusted
TSLS); the fifth is Moreira’s (2003) conditional likelihood ratio (CLR) test. The
heretofore unstudied conditional Wald tests are found to perform poorly, compared to the
CLR test: in many cases, the conditional Wald tests have almost no power against a wide
range of alternatives. Our analysis is facilitated by a new algorithm, presented here, for
the computation of the asymptotic conditional p-value of the CLR test.