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Keywords:

  • conditional heteroscedasticity;
  • empirical distribution function;
  • hypothesis testing;
  • kernel estimation;
  • nonparametric AR-ARCH model;
  • nonparametric CHARN model;
  • partial sum process;
  • time series

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 (0 ≤ s ≤ 1). 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 performance of the proposed test is investigated in a simulation study and the test is applied to a data example.