Estimation and Inference with Weak Identication

This paper analyses the properties of standard estimators, tests, and con…dence sets (CS’s) in a class of models in which the parameters are unidenti…ed or weakly identi…ed in some parts of the parameter space. The paper also introduces a method to make the tests and CS’s robust to such identi…cation 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), and minimum distance (MD) 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 LS estimator of a nonlinear regression model, a LS estimator of a smooth transition threshold autoregressive model, the instrumental variables estimator of a nonlinear regression model with endogeneity, and the ML estimator of an ARMA (1, 1) model. Keywords: Asymptotic size, con…dence set, estimator, identi…cation, nonlinear models, test, weak identi…cation. JEL Classi…cation Numbers: C12, C15.