Minimum chi-square estimation and tests for model selection

This paper studies the problem of model selection based on Pearson chi-square type statistics. Such goodness-of-fit statistics have been considered by Moore (1978) and diagnostic tests based on them have recently been extended to general econometric models by Andrews (1988a, b). Here we consider another important use of these statistics. Specifically, we propose some convenient asymptotically standard normal tests for model selection based on chi-square type statistics. Following Vuong (1989), the null hypothesis is that the competing models are equally close to the data-generating process (DGP) vs. the alternative hypotheses that one model is closer to the DGP where closeness of a model is measured according to the discrepancy implicit in the Pearson type statistic used. Our model selection tests have the desirable feature that neither model needs to be correctly specified nor nested in each other. As a prerequisite, we study the corresponding class of minimum chi-square estimators.