Maximum likelihood estimation and uniform inference with sporadic identification failure

This paper analyzes the properties of a class of estimators, tests, and confidence sets (CSs) when the parameters are not identified in parts of the parameter space. Specifically, we consider estimator criterion functions that are sample averages and are smooth functions of a parameter θ . This includes log likelihood, quasi-log likelihood, and least squares criterion functions. ermine the asymptotic distributions of estimators under lack of identification and under weak, semi-strong, and strong identification. We determine the asymptotic size (in a uniform sense) of standard t and quasi-likelihood ratio (QLR) tests and CSs. We provide methods of constructing QLR tests and CSs that are robust to the strength of identification. sults are applied to two examples: a nonlinear binary choice model and the smooth transition threshold autoregressive (STAR) model.