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

We propose a cross-classification rule for the dependent and explanatory variables resulting in a contingency table such that the classical trinity of chi-square statistics can be used to check for conditional distribution specification. The resulting Pearson statistic is equal to the Lagrange multiplier statistic. We also provide a Chernoff-Lehmann result for the Pearson statistic using the raw data maximum likelihood estimator, which is applied to show that the corresponding limiting distribution of the Wald statistic does not depend on the number of parameters. The asymptotic distribution of the proposed statistics does not change when the grouping is data dependent. An algorithm allowing to control the number of observations per cell is developed. Monte Carlo experiments provide evidence of the excellent size accuracy of the proposed tests and their good power performance, compared to omnibus tests, in high dimensions.