. Introduction . Estimator, Criterion Function, and Examples. GMM Estimators . Example 1: Nonlinear Regression with Endogeneity. Example 2: Probit Model with Endogeneity andPossibly Weak Instruments. Confidence Sets and Tests. Drifting Sequences of Distributions . Assumptions . Assumption GMM1. Assumption GMM2. Assumption GMM3. Assumption GMM4. Assumption GMM5. Minimum Distance Estimators. Parameter Space Assumptions . GMM Estimation Results . Wald Confidence Sets and Tests. Wald Statistics. Rotation. Function r(0=x"0112) of Interest. Variance Matrix Estimators. Asymptotic Null Distribution of the Wald Statistic. Asymptotic Distribution of the Wald StatisticUnder the Alternative. Asymptotic Size of Standard Wald Confidence Sets. Robust Wald Confidence Sets. Least Favorable Critical Value. Type 2 Robust Critical Value. Asymptotic Size of Robust Wald CS's . QLR Confidence Sets and Tests . Numerical Results: Nonlinear Regression Model with Endogeneity . Probit Model with Endogeneity: Verificationof Assumptions. Verification of Assumption GMM1. Verification of Assumption GMM2. Verification of Assumption GMM3. Verification of Assumption GMM4. Verification of Assumption GMM5. Verification of Assumptions V1 and V2 (Vector 0=x"010C) . Appendix A: Proofs of GMM Estimation Results. Lemmas. Minimum Distance Estimators. Proofs of Lemmas. Proofs of Section 3 Lemmas . Appendix B: Proofs for Wald Tests. Proofs of Asymptotic Distributions. Proofs of Asymptotic Size ResultsGMM ESTIMATION AND UNIFORM SUBVECTOR INFERENCE WITH POSSIBLE IDENTIFICATION FAILURE By Donald W. K. Andrews and Xu Cheng October 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1828 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/GMM Estimation andUniform Subvector Inferencewith Possible Identi…cation FailureDonald W. K. Andrews Cowles FoundationYale UniversityXu ChengDepartment of EconomicsUniversity of PennsylvaniaFirst Draft: August, 2007Revised: October 24, 2011 Andrews gratefully acknowledges the research support of the NationalScience Foundation via grant numbers SES-0751517 and SES-1058376.The authors thank Xiaohong Chen, Sukjin Han, Yuichi Kitamura, PeterPhillips, Eric Renault, Frank Schorfheide, and Ed Vytlacil for helpfulcomments.AbstractThis paper determines the properties of standard generalized method of moments(GMM) estimators, tests, and con…dence sets (CS’s) in moment condition models inwhich some parameters are unidenti…ed or weakly identi…ed in part of the parameterspace. The asymptotic distributions of GMM estimators are established under a fullrange of drifting sequences of true parameters and distributions. The asymptotic sizes(in a uniform sense) of standard GMM tests and CS’s are established.The paper also establishes the correct asymptotic sizes of “robust” GMM-basedWald, t; and quasi-likelihood ratio tests and CS’s whose critical values are designed toyield robustness to identi…cation problems.The results of the paper are applied to a nonlinear regression model with endogeneityand a probit model with endogeneity and possibly weak instrumental variables.Keywords: Asymptotic size, con…dence set, generalized method of moments, GMM es-timator, identi…cation, nonlinear models, test, Wald test, weak identi…cation.JEL Classi…cation Numbers: C12, C15.1. IntroductionThis paper gives a set of GMM regularity conditions that are akin to the classicconditions in Hansen (1982) and Pakes and Pollard (1989). But, they allow for singu-larity of the GMM estimator’s variance matrix due to the lack of identi…cation of someparameters in part of the parameter space. Under the conditions given, the asymptoticdistributions of GMM estimators and Wald and quasi-likelihood ratio (QLR) test sta-tistics are established. The asymptotic sizes of standard GMM tests and con…dence sets(CS’s) are established. In many cases, their asymptotic sizes are not correct. We showthat Wald and QLR statistics combined with “identi…cation robust”critical values havecorrect asymptotic sizes (in a uniform sense).In contrast to standard GMM results in the literature, the results given here covera full range of drifting sequences of true parameters and distributions. Such results areneeded to establish the (uniform) asymptotic size properties of tests and CS’s and togive good approximations to the …nite-sample properties of estimators, tests, and CS’sunder weak identi…cation. Non-smooth sample moment conditions are allowed, as inPakes and Pollard (1989) and Andrews (2002).This paper is a sequel to Andrews and Cheng (2007a) (AC1), which provides resultsfor extremum estimators, t tests, and QLR tests under high-level conditions. Here weprovide more primitive conditions for GMM statistics by verifying the high-level condi-tions of AC1. Some results also are given here for minimum distance (MD) estimators,tests, and CS’s. The paper provides results for Wald tests and CS’s that apply not onlyto GMM estimators, but also to other extremum estimators covered by AC1.We consider moment condition models where the parameter  is of the form  =(; ; ); where  is identi…ed if and only if  6= 0;  is not related to the identi…cationof ; and = (; ) is always identi…ed. The parameters ; ; and  may be scalarsor vectors. For example, this framework applies to the nonlinear regression model Yi=  h (X1;i; ) + X02;i + Uiwith endogenous variables X1;ior X2;iand instruments (IV’s)Zi: Here lack of identi…cation of  when  = 0 occurs because of nonlinearity. Thisframework also applies to the probit model with endogeneity: y i= Yi + X0i 1+ U i;where one observes yi= 1(y i> 0); the endogenous variable Yi; and the exogenousregressor vector Xiand the reduced form for Yiis Yi= Z0i + X0i2+ Vi: In this case,lack of identi…cation of  occurs when  = 0 because the IV’s are irrelevant.We determine the asymptotic properties of GMM estimators and tests under drifting1sequences of true parameters n= (n; n; n) for n  1; where n indexes the samplesize. The behavior of GMM estimators and tests depends on the magnitude of jjnjj:The asymptotic behavior of these statistics varies across three categories of sequencesfn: n  1g : Category I(a) n= 0 8n  1;  is unidenti…ed; Category I(b) n6= 0 andn1=2n! b 2 Rd;