A model specification test for the variance function in nonparametric regression

TL;DR

This paper introduces a new nonparametric test for the conditional variance function in regression models, based on characteristic functions, with theoretical and empirical validation.

Contribution

It proposes a novel test statistic using weighted L2-distance of empirical characteristic functions, with asymptotic distribution and finite sample properties analyzed.

Findings

The test accurately detects deviations from the null hypothesis.

The asymptotic distribution approximates critical values well.

Finite sample simulations show good performance.

Abstract

The problem of testing for the parametric form of the conditional variance is considered in a fully nonparametric regression model. A test statistic based on a weighted $L_2$-distance between the empirical characteristic functions of residuals constructed under the null hypothesis and under the alternative is proposed and studied theoretically. The null asymptotic distribution of the test statistic is obtained and employed to approximate the critical values. Finite sample properties of the proposed test are numerically investigated in several Monte Carlo experiments. The developed results assume independent data. Their extension to dependent observations is also discussed.