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

Abstract

A restrictive 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. For example, if change point analysis for a particular parameter of the process (such as the variance) is performed, it is not necessary clear why other parameters (such as the mean or higher order moments) have to stay constant under the hypothesis that there is no change point in the parameter of interest. 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 H₀ : θ₁ = θ₂ vs. H₁ : θ₀ ≠ θ₂, where θ₁ and θ₂ denote second order parameters (such as the variance or the lag k-correlation) 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 non relevant change point, i.e. H₀ : | θ₀ - θ₂ | ≤ δ , 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. The results are illustrated by means of a simulation study, which shows that the new procedures have nice finite sample properties. The central England monthly temperature series are analyzed and significant change points in the variance and lag 1-correlation are found in the winter monthly temperature at the late 19th century.

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