Testing the parametric form of the conditional variance in regressions based on distance covariance

TL;DR

This paper introduces a new, easy-to-implement test based on distance covariance for verifying the parametric form of conditional variance in nonlinear and nonparametric regressions, with proven consistency and good finite-sample performance.

Contribution

It develops a novel test leveraging distance covariance that is robust to high dimensionality and applicable in complex regression models, with a bootstrap method for distribution approximation.

Findings

Test is consistent against all alternatives.

Detects local alternatives at the parametric rate 1/√n.

Shows good finite-sample performance in simulations.

Abstract

In this paper, we propose a new test for checking the parametric form of the conditional variance based on distance covariance in nonlinear and nonparametric regression models. Inherit from the nice properties of distance covariance, our test is very easy to implement in practice and less effected by the dimensionality of covariates. The asymptotic properties of the test statistic are investigated under the null and alternative hypotheses. We show that the proposed test is consistent against any alternative and can detect local alternatives converging to the null hypothesis at the parametric rate 1/root(n) in both the nonlinear and nonparametric settings. As the limiting null distribution of the test statistic is intractable, we propose a residual bootstrap to approximate the limiting null distribution. Simulation studies are presented to assess the finite sample performance of the proposed test. We also apply the proposed test to a real data set for illustration.