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Suppose that the econometrician is interested in comparing two misspeciﬁed moment restriction models, where the comparison is performed in terms of some chosen measure of ﬁt. This paper is concerned with describing an optimal test of the Vuong (1989) and Rivers and Vuong (2002) type null hypothesis that the two models are equivalent under the given measure of ﬁt (the ranking may vary for diﬀerent measures). We adopt the generalized Neyman-Pearson optimality criterion, which focuses on the decay rates of the type I and II error probabilities under ﬁxed non-local alternatives, and derive an optimal but practically infeasible test. Then, as an illustration, by considering the model comparison hypothesis deﬁned by the weighted Euclidean norm of moment restrictions, we propose a feasible approximate test statistic to the optimal one and study its asymptotic properties. Local power properties, one-sided test, and comparison under the generalized empirical likelihood-based measure of ﬁt are also investigated. A simulation study illustrates that our approximate test is more powerful than the Rivers-Vuong test.
Marmer, Vadim and Otsu, Taisuke, "Optimal Comparison of Misspecified Moment Restriction Models under a Chosen Measure of Fit" (2009). Cowles Foundation Discussion Papers. 2043.