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 coe cients of the instruments in the rst 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 ) 100% con dence 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 justidenti ed, thus providing valid con dence intervals. Even when the model is over-identi ed, we show that the distributions of the likelihood type test statistics are bounded from above by a 2 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, con dence intervals. The con dence intervals obtained by using this bounding distribution are compared with those obtained by using the standard (1) asymptotic distribution and an alternative bounding distribution, a transformation of the distribution of the Wilks statistic, suggested by Dufour (1994). Con dence intervals based on our 2 bounding distribution are shown to be tighter than those based on the Wilks bounding distribution by Monte Carlo experiments.