m-Testing of Model Specification in Many Groups

The m-testing of model specification pioneered by Newey (1985) and Tauchen (1985) and extended by White (1987) offers a general framework that unifies many specification tests in the literature. Within the m-testing framework, we consider how to test the hypothesis that a model is correctly specified in each of many groups in the population under study, where each of the groups offers only a few observations. In our problem setup, each group may have a different population distribution. This means that even if the moment condition is grossly violated under some group populations, the moment condition may nearly hold on average taken over the groups. Thus, incorrect specification can be masked in the usual form of m-tests based on the moment functions averaged over all groups. To avoid this masking problem, one might set up moment conditions for each group separately. Nevertheless, such approach is impossible or unreliable, due to small group sample sizes. We here propose a novel implementation of m-testing that works in the above-mentioned setup. In the proposed approach, we unbiasedly estimate a quadratic form of the moments in question for each group, where the quadratic form has a positive definite weighting matrix, and then take the average of the estimated quadratic form over the groups to obtain a statistic. We allow the moments to depend on nuisance parameters. We consider both the situation in which the nuisance parameters are estimated