Conditional Distribution Model Specification Testing Using Chi-Square Goodness-of-Fit Tests

TL;DR

This paper develops chi-square goodness-of-fit tests for conditional distribution models, utilizing the Rosenblatt transform and three test statistics, with proven asymptotic properties and validated through Monte Carlo simulations.

Contribution

It introduces a novel framework for testing conditional distribution models using chi-square tests based on the Rosenblatt transform and proposes three new test statistics with invariant asymptotic distributions.

Findings

Proposed tests perform well in Monte Carlo experiments.

Asymptotic distribution of tests is invariant to sample-dependent partitions.

New test statistics improve model specification assessment.

Abstract

This paper introduces chi-square goodness-of-fit tests to check for conditional distribution model specification. The data is cross-classified according to the Rosenblatt transform of the dependent variable and the explanatory variables, resulting in a contingency table with expected joint frequencies equal to the product of the row and column marginals, which are independent of the model parameters. The test statistics assess whether the difference between observed and expected frequencies is due to chance. We propose three types of test statistics: the classical trinity of tests based on the likelihood of grouped data, and two statistics based on the efficient raw data estimator -- namely, a Chernoff-Lehmann and a generalized Wald statistic. The asymptotic distribution of these statistics is invariant to sample-dependent partitions. Monte Carlo experiments demonstrate the good performance of the proposed tests.