Theory of Weak Identification in Semiparametric Models

Abstract

We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak identification occurs when a parameter is weakly regular, that is, 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.

Optimal Decision Rules for Weak GMM

“…11 Kaji (2021) studied weakly identified parameters in semiparametric models, and introduced a notion of weak efficiency for estimators. Weak efficiency is necessary, but not in general sufficient, for decision‐theoretic optimality (e.g., admissibility) in many contexts. …”

“…Prior work byKaji (2021) also analyzes weak identification using paths satisfying (2).6 The more general assumption that θ * is local to 0 yields a limit experiment similar to that derived below, at the cost of heavier notation. Hence, we focus on the case with θ * ∈ 0 .…”

Optimal Decision Rules for Weak GMM

“…3 Kaji (2020) studies weakly identified parameters in semiparametric models, and introduces a notion of weak efficiency for estimators. Weak efficiency is necessary, but not in general sufficient, for decisiontheoretic optimality (e.g.…”