Detecting heteroskedasticity in nonparametric regression using weighted empirical processes

TL;DR

This paper proposes a new, distribution-free test for heteroskedasticity in nonparametric regression models that uses residual-based empirical processes and is effective even with missing data.

Contribution

It introduces a novel heteroskedasticity test based on residuals and empirical processes that does not require specifying the variance function model.

Findings

Test is asymptotically distribution free.

Method is consistent at the root-n rate.

Extended to handle missing responses.

Abstract

Heteroskedastic errors can lead to inaccurate statistical conclusions if they are not properly handled. We introduce a test for heteroskedasticity for the nonparametric regression model with multiple covariates. It is based on a suitable residual-based empirical distribution function. The residuals are constructed using local polynomial smoothing. Our test statistic involves a detection function that can verify heteroskedasticity by exploiting just the independence-dependence structure between the detection function and model errors, i.e. we do not require a specific model of the variance function. The procedure is asymptotically distribution free: inferences made from it do not depend on unknown parameters. It is consistent at the parametric (root-n) rate of convergence. Our results are extended to the case of missing responses and illustrated with simulations.