Optimal Comparison of Misspecied Moment Restriction Models1

This paper considers optimal testing of model comparison hypotheses for misspeci…ed unconditional moment restriction models. We adopt the generalized Neyman-Pearson optimality criterion, which focuses on the convergence rates of the type I and II error probabilities under …xed global alternatives, and derive an optimal but practically infeasible test. We then propose feasible approximation test statistics to the optimal one. For linear instrumental variable regression models, the conventional empirical likelihood ratio test statistic emerges. For general nonlinear moment restrictions, we propose a new test statistic based on an iterative algorithm. We derive the asymptotic properties of these test statistics. JEL classi…cation: C12; C14; C52 Keywords: Moment restriction; Model comparison; Misspeci…cation; Generalized Neyman-Pearson optimality; Empirical likelihood; GMM 1 Introduction Econometric models are often de…ned in the form of moment restrictions and estimated by the generalized method of moments (GMM) (Hansen, 1982), empirical likelihood (EL) (Qin and Lawless, 1994), or their generalizations (see, e.g., Newey and Smith (2004)). In many applications, it is natural to assume that the model is misspeci…ed. While such a model will be rejected with probability approaching one by a consistent overidentifying restriction test, it nevertheless can be of interest as an approximation to the true data generating process. For example, Prescott (1991) argues that a model is only an approximation and should not be regarded as a null hypothesis to be statistically tested. Thus, choosing a model closest to the truth in some sense among several misspeci…ed models is of great importance for practitioners. Misspeci…ed models and inference procedures for such models have been discussed extensively in the econometric literature. White (1982) studies the properties of the maximum likelihood estimator under misspeci…cation. Vuong (1989) proposes a test of the null hypothesis that two misspeci…ed parametric models provide an equivalent approximation to the true distribution in terms of their Kullback-Leibler information criteria (KLIC). Rivers and Vuong (2002) extend such tests to a more general setting that allows to compare misspeci…ed moment restriction models. Kitamura (2000) develops an information theoretic test that compares misspeci…ed moment restriction models by closeness to the true distribution in terms of the KLIC. Kitamura (2003) extends the information theoretic approach to compare misspeci…ed conditional moment restriction models. Corradi and Swanson (2007) propose a Kolmogorov-type test to compare misspeci…ed dynamic stochastic general equilibrium models. Hall and Inoue (2003) discuss inference for misspeci…ed moment restriction models estimated by the GMM. This paper considers optimal testing of model comparison hypotheses for misspeci…ed unconditional moment restriction models. Our focus is not on the choice of the measure of …t to set up the model comparison hypotheses but on the choice of the test given the measure of …t. We adopt a GMM-type distance between a model and the true data generating process. To compare di¤erent tests, we employ the large deviation approach.1 In particular, we adopt the generalized Neyman-Pearson (GNP) optimality criterion, which focuses on the convergence rates of the type I and II error 1See, e.g., Dembo and Zeitouni (1998) for a review on large deviation theory.