Theory of Weak Identification in Semiparametric Models

TL;DR

This paper develops a theoretical framework for understanding weak identification in semiparametric models, introduces concepts of efficiency, and proposes improved estimators, demonstrated through simulations.

Contribution

It provides a general formulation of weak identification, introduces the concept of weak efficiency, and develops new estimators for linear IV and nonlinear regression models.

Findings

Weak identification occurs when parameters are locally homogeneous of degree zero.

Existence of an underlying regular parameter characterizes weakly regular parameters.

New estimators improve upon 2SLS and optimal IV in simulations.

Abstract

We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak identification occurs when a parameter is weakly regular, i.e., when it is locally homogeneous of degree zero. When this happens, consistent or equivariant estimation is shown to be impossible. We then show that there exists an underlying regular parameter that fully characterizes the weakly regular parameter. While this parameter is not unique, concepts of sufficiency and minimality help pin down a desirable one. If estimation of minimal sufficient underlying parameters is inefficient, it introduces noise in the corresponding estimation of weakly regular parameters, whence we can improve the estimators by local asymptotic Rao-Blackwellization. We call an estimator weakly efficient if it does not admit such improvement. New weakly efficient estimators are presented in linear IV and nonlinear regression models. Simulation of a linear IV model demonstrates how 2SLS and optimal IV estimators are improved.