Chi-Square Goodness-of-Fit Tests for Conditional Distributions

This paper proposes $\chi^2$ goodness-of-fit tests for checking conditional distribution model's specifications. The method involves partitioning the sample into classes based on a cross-classification of the dependent and explanatory variables, resulting in a contingency table with expected frequencies that are independent of the parameters in the model and equal to the product of the marginals. Test statistics are computed using the trinity of tests, based on the likelihood of grouped data, to test whether the expected frequencies satisfy the model's restrictions. We also present a Chernoff-Lehman result that enables us to derive the asymptotic distribution of a Wald statistic using the efficient raw data estimator. The asymptotic distribution of the test statistics remains the same even when partitions are sample-dependent. An algorithm is developed to control the number of observations per cell. Monte Carlo experiments demonstrate the proposed tests' excellent size accuracy and good power properties.