This paper analyzes the properties of standard estimators, tests, and conﬁdence sets (CS’s) in a class of models in which the parameters are unidentiﬁed or weakly identiﬁed in some parts of the parameter space. The paper also introduces methods to make the tests and CS’s robust to such identiﬁcation problems. The results apply to a class of extremum estimators and corresponding tests and CS’s, including maximum likelihood (ML), least squares (LS), quantile, generalized method of moments (GMM), generalized empirical likelihood (GEL), minimum distance (MD), and semi-parametric estimators. The consistency/lack-of-consistency and asymptotic distributions of the estimators are established under a full range of drifting sequences of true distributions. The asymptotic size (in a uniform sense) of standard tests and CS’s is established. The results are applied to the ML estimator of an ARMA(1, 1) model and to the LS estimator of a nonlinear regression model.
Andrews, Donald W.K. and Cheng, Xu, "Estimation and Inference with Weak, Semi-strong, and Strong Identification" (2010). Cowles Foundation Discussion Papers. 2113.