Non-Nested Testing in Models Estimated via Generalized Method of Moments

Rivers and Vuong (2002) develop a very general framework for choosing between two competing dynamic models. Within their framework, inference is based on a statistic that compares measures of goodness of fit between the two models. The null hypothesis is that the models have equal measures of goodness of fit; one model is preferred if its goodness of fit is statistically significantly smaller than its competitor. Under the null hypothesis, Rivers and Vuong (2002) show that their test statistic has a standard normal distribution under generic conditions that are argued to allow for a variety of estimation methods including Generalized Method of Moments (GMM). In this paper, we analyze the limiting distribution of Rivers and Vuong's (2002) statistic under the null hypothesis when inference is based on a comparison of GMM minimands evaluated at GMM estimators. It is shown that the limiting behaviour of this statistic depends on whether the models in question are correctly specified, locally misspecified or misspecified. Specifically, it is shown that: (i) if both models are correctly specified or locally misspecified then Rivers and Vuong's (2002) generic conditions are not satisfied, and the limiting distribution of the test statistic is non-standard under the null; (ii) if both models are misspecified then the generic conditions are satisfied, and so the statistic has a standard normal distribution under the null. In the latter case it is shown that the choice of weighting matrices affects the outcome of the test and thus the ranking of the models.