GMM with Weak Identification

This paper develops asymptotic distribution theory for GMM estimators and test
statistics when some or all of the parameters are weakly identified. General results are
obtained and are specialized to two important cases: linear instrumental variables regression
and Euler equations estimation of the CCAPM. Numerical results for the CCAPM
demonstrate that weak-identification asymptotics explains the breakdown of conventional
GMM procedures documented in previous Monte Carlo studies. Confidence sets immune
to weak identification are proposed. We use these results to inform an empirical
investigation of various CCAPM specifications; the substantive conclusions reached differ
from those obtained using conventional methods.
KEYWORDS: Instrumental variables, empirical processes, EuIer equation estimation,
asset pricing.

1. INTRODUCTION
THEREIS CONSIDERABLE EVIDENCE that asymptotic normality often provides a
poor approximation to the sampling distributions of generalized method of
moments (GMM) estimators and test statistics in designs and sample sizes of
empirical relevance in economics. Examples of this discrepancy in estimation of
stochastic Euler equations are investigated by Tauchen (1986), Kocherlakota
(19901, Neeley (19941, West and Wilcox (1994), Fuhrer, Moore, and Schuh
(19951, and Hansen, Heaton, and Yaron (1996); also see the articles in the 1996
special issue of the Journal of Business and Economic Statistics on GMM
estimation. Depending on the design, the sampling distributions of GMM
estimators can be skewed and can have heavy tails, and likelihood ratio tests of
the parameter values and tests of overidentifying restrictions can exhibit substantial
size distortions. Although these problems are well documented, their
source is not well understood.