Weak identification in probit models with endogenous covariates

Weak identification is a well-known issue in the context of linear structural models. However, for probit models with endogenous explanatory variables, this problem has been little explored. In this paper, we study by simulating the behavior of the usual z-test and the LR test in the presence of weak identification. We find that the usual asymptotic z-test exhibits large level distortions (over-rejections under the null hypothesis). The magnitude of the level distortions depends heavily on the parameter value tested. In contrast, asymptotic LR tests do not over-reject and appear to be robust to weak identification.