Model Checking for Regressions Based on Weighted Residual Processes with Diverging Number of Predictors

TL;DR

This paper introduces a new high-dimensional regression model specification test based on weighted residual processes, addressing limitations of classical tests when the number of predictors diverges.

Contribution

It develops a novel test with proven asymptotic properties and a bootstrap method suitable for high-dimensional settings, improving size control and power.

Findings

The test maintains the nominal significance level in high-dimensional regimes.

It can detect local alternatives at the parametric rate $1/\sqrt{n}$.

Simulation and real-data studies demonstrate its effectiveness.

Abstract

The integrated conditional moment (ICM) test is a classical and widely used method for assessing the adequacy of regression models. Although it performs well in fixed-dimension settings, its behavior changes dramatically when the predictor dimension diverges: in such regimes, the limiting null and alternative distributions of the ICM statistic degenerate to fixed constants. Moreover, when the number of predictors diverges, the commonly used wild bootstrap no longer approximates the null distribution of the ICM statistic well, leading to size distortion and substantial power loss. To address these challenges, we propose a new specification test based on weighted residual processes for evaluating the parametric form of the regression mean function in high-dimensional settings where the number of predictors increases with the sample size. We establish the asymptotic properties of the test statistic under the null hypothesis and under global and local alternatives. The proposed test maintains the nominal significance level and can detect local alternatives that deviate from the null hypothesis at the parametric rate $1/\sqrt{n}$. Furthermore, we propose a smooth residual bootstrap to approximate the limiting null distribution and establish its validity in high-dimensional settings. Two simulation studies and a real-data example are conducted to evaluate the finite-sample performance of the proposed test.