Testing for a change of the innovation distribution in nonparametric autoregression - the sequential empirical process approach

TL;DR

This paper develops a nonparametric test for detecting changes in the distribution of innovations in autoregressive models with heteroscedasticity, using a sequential empirical process approach, and demonstrates its effectiveness through theoretical and simulation results.

Contribution

It introduces a new asymptotic distribution-free test for change points in innovation distributions in nonparametric autoregression models.

Findings

Test is asymptotically distribution-free under null hypothesis.

The test is consistent against fixed alternatives.

Simulation studies show good small sample performance.

Abstract

We consider a nonparametric autoregression model under conditional heteroscedasticity with the aim to test whether the innovation distribution changes in time. To this end we develop an asymptotic expansion for the sequential empirical process of nonparametrically estimated innovations (residuals). We suggest a Kolmogorov-Smirnov statistic based on the difference of the estimated innovation distributions built from the first ns and the last n-ns residuals, respectively. Weak convergence of the underlying stochastic process to a Gaussian process is proved under the null hypothesis of no change point. The result implies that the test is asymptotically distribution-free. Consistency against fixed alternatives is shown. The small sample performances of the proposed test is investigated in a simulation study and the test is applied to data examples.