Conditional Distribution Specification Testing Based on Data-Dependent Partitions

This article introduces a Pearson-type goodness-of-fit test for the parametric specification of conditional distribution models with continuous responses. Under correct specification, the Rosenblatt transform is uniformly distributed on $[0,1]$ conditionally on the explanatory variables. The test exploits this characterization by cross-classifying the transformed observations and the explanatory variables according to partitions of $[0,1]$ and their support, respectively. The resulting Pearson statistic has a chi-squared limiting distribution with known degrees of freedom and detects local alternatives converging to the null at the $n^{-1/2}$ rate. These results remain valid for the class of data-dependent partitions considered. Monte Carlo simulations indicate accurate size control and favorable power relative to existing bootstrap-based tests, particularly in higher-dimensional settings.