A martingale-transform goodness-of-fit test for the form of the conditional variance

TL;DR

This paper introduces a new martingale-transform goodness-of-fit test for the parametric form of the variance function in nonparametric regression, providing asymptotically distribution-free tests with favorable finite sample properties.

Contribution

It develops a standardized process and applies a martingale transform to create distribution-free tests for the variance function in nonparametric regression.

Findings

The proposed tests are asymptotically distribution-free.

Simulation studies show good finite sample performance.

The method applies to Kolmogorov-Smirnov and Cramér-von-Mises functionals.

Abstract

In the common nonparametric regression model the problem of testing for a specific parametric form of the variance function is considered. Recently Dette and Hetzler (2008) proposed a test statistic, which is based on an empirical process of pseudo residuals. The process converges weakly to a Gaussian process with a complicated covariance kernel depending on the data generating process. In the present paper we consider a standardized version of this process and propose a martingale transform to obtain asymptotically distribution free tests for the corresponding Kolmogorov-Smirnov and Cram\'{e}r-von-Mises functionals. The finite sample properties of the proposed tests are investigated by means of a simulation study.