Estimation and Inference with Weak, Semi-strong, and Strong Identification
CFDP Revision Date
This paper analyzes the properties of standard estimators, tests, and conﬁdence sets (CS’s) for parameters that 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 that are based on criterion functions that satisfy certain asymptotic stochastic quadratic expansions and that depend on the parameter that determines the strength of identiﬁcation. This covers a class of models estimated using 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 sizes (in a uniform sense) of standard and identiﬁcation-robust tests and CS’s are established. The results are applied to the ARMA(1, 1) time series model estimated by ML and to the nonlinear regression model estimated by LS. In companion papers the results are applied to a number of other models.
Andrews, Donald W.K. and Cheng, Xu, "Estimation and Inference with Weak, Semi-strong, and Strong Identification" (2010). Cowles Foundation Discussion Papers. 2114.