COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1824R

Maximum Likelihood Estimation and Uniform Inference
with Sporadic Identification Failure

Donald W. K. Andrews and Xu Cheng

October 2011
Revised October 2012

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.

Keywords: Asymptotic size, Binary choice, Confidence set, Estimator, Identification, Likelihood, Nonlinear models, Test, Smooth transition threshold autoregression, Weak identification

JEL Classification: C12, C15