Change point analysis of second order characteristics in non-stationary time series

Abstract

An important assumption in the work on testing for structural breaks in time series consists in the fact that the model is formulated such that the stochastic process under the null hypothesis of "no change-point" is stationary. This assumption is crucial to derive (asymptotic) critical values for the corresponding testing procedures using an elegant and powerful mathematical theory, but it might be not very realistic from a practical point of view. This paper develops change point analysis under less restrictive assumptions and deals with the problem of detecting change points in the marginal variance and correlation structures of a non-stationary time series. A CUSUM approach is proposed, which is used to test the "classical" hypothesis of the form $H_0: \theta_1=\theta_2$ vs. $H_1: \theta_1 \not =\theta_2$, where $\theta_1$ and $\theta_2$ denote second order parameters of the process before and after a change point. The asymptotic distribution of the CUSUM test statistic is derived under the null hypothesis. This distribution depends in a complicated way on the dependency structure of the nonlinear non-stationary time series and a bootstrap approach is developed to generate critical values. The results are then extended to test the hypothesis of a {\it non relevant change point}, i.e. $H_0: | \theta_1-\theta_2 | \leq \delta$, which reflects the fact that inference should not be changed, if the difference between the parameters before and after the change-point is small. In contrast to previous work, our approach does neither require the mean to be constant nor - in the case of testing for lag $k$-correlation - that the mean, variance and fourth order joint cumulants are constant under the null hypothesis. In particular, we allow that the variance has a change point at a different location than the auto-covariance.