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 explanatory endogenous 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 infomation framework. We show that Wald-type test statistics are not asymptotically pivotal, thus (1 - a)*100% confidence intervals implied by those test statistics can have zero coverage probability if standard asymptotic distribution theory is used. In contrast, likelihood-type test statistics are asymptotically pivotal when the model is just identified thus providing valid confidence intervals. Even when the model is overidentified, we show that the distributions of the likelihood-type test statistics are asymptotically bounded from above by a chi- square distribution with degrees of freedom given by the number of instruments. Hence, we can always invert the likelihood-type 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). Using Monte Carlo methods, confidence intervals based on our chi-square bounding distribution are shown to be tighter in finite samples than those based on the Wilks bounding distribution