Inference on a Structural Parameter in Instrumental Variables Regression with Weak Instruments

In this paper we consider the problem of making inference on a structural parameter in instrumental variables regression when the instruments are only weakly correlated with the endogenous explanatory variables. Adopting a local-to-zero assumption as in Staiger and Stock (1994) on the coefficients of the instruments in the first stage equation, the asymptotic distributions of various test statistics are derived under a limited information framework. We show that Wald-type test statistics are not pivotal, thus (1-a)*100% confidence intervals implied by those test statistics can have zero coverage probability if the standard asymptotic distribution theory is used. In contrast, the likelihood type test statistics are pivotal when the model is just identified, thus providing valid confidence intervals. Even the model is overidentified, we show that the distributions of the likelihood type test statistics are bounded above by a Chi-Square distribution with degrees of freedom given by the number of instruments. Hence, we can always invert the likelihood type test statistics to obtain valid, although conservative, confidence intervals. The confidence intervals obtained by using this bounding distribution are compared with those obtained by using the standard Chi-Square 1 asymptotic distribution and an alternative bounding distribution, a transformation of the distribution of the Wilks statistic, suggested by Dufour (1994) . Confidence intervals based on our Chi-Square bounding distribution are shown to be tighter than those based on the Wilks bounding distribution by Monte Carlo experiments.