This paper analyzes the properties of a class of estimators, tests, and conﬁdence sets (CS’s) 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 theta. This includes log likelihood, quasi-log likelihood, and least squares criterion functions. We determine 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 CS’s. We provide methods of constructing QLR tests and CS’s that are robust to the strength of identiﬁcation. The results are applied to two examples: a nonlinear binary choice model and the smooth transition threshold autoregressive (STAR) model.
Andrews, Donald W.K. and Cheng, Xu, "Maximum Likelihood Estimation and Uniform Inference with Sporadic Identification Failure" (2011). Cowles Foundation Discussion Papers. 2176.