[Paper Review] Change point analysis of second order characteristics in non-stationary time series

Holger Dette, Weichi Wu+1 more|arXiv (Cornell University)|Jan 1, 2015

Probabilistic and Robust Engineering Design32 references21 citations

TL;DR

This paper proposes a CUSUM-based change point test for second-order characteristics—such as variance and lag-k correlation—in non-stationary time series, relaxing the restrictive assumption of full stationarity under the null. It derives the asymptotic distribution of the test statistic and develops a bootstrap method for critical values, enabling detection of structural breaks even when the mean or higher-order moments are non-constant, with application to central England temperature data revealing significant change points in winter temperature variance and correlation at the end of the 19th century.

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.

Motivation & Objective

• To address the limitation of existing change point tests that assume full stationarity under the null hypothesis.
• To develop a method for detecting changes in second-order structure—variance and lag-k correlation—without requiring the mean or higher-order moments to remain constant.
• To allow for different change points in variance and auto-covariance, reflecting realistic non-stationary behavior.
• To extend the framework to test for non-relevant change points, where the difference between pre- and post-change parameters is bounded by a small threshold δ.
• To provide a practical, asymptotically valid testing procedure with finite-sample performance verified via simulation and real data analysis.

Proposed method

• A CUSUM test statistic is constructed to compare second-order parameters (e.g., variance, lag-k correlation) before and after a potential change point.
• The asymptotic distribution of the CUSUM statistic is derived under the null hypothesis of no change, accounting for the complex dependency structure of nonlinear non-stationary time series.
• A bootstrap procedure is developed to generate critical values, as the asymptotic distribution depends on unknown dependency parameters and is analytically intractable.
• The method is extended to test for non-relevant change points via the hypothesis H₀: |θ₀ − θ₂| ≤ δ, allowing for practical equivalence in parameter values.
• The approach does not require the mean to be constant nor the variance or fourth-order cumulants to remain unchanged under the null, especially when testing for lag-k correlation.
• The method is validated through a simulation study and applied to the central England monthly temperature series to detect change points in winter temperature variance and lag-1 correlation.

Experimental results

Research questions

• RQ1 Can change point detection for second-order characteristics be performed without requiring the mean to be constant under the null hypothesis?
• RQ2 How can the asymptotic distribution of the CUSUM test statistic be derived for nonlinear non-stationary time series with complex dependency structures?
• RQ3 What bootstrap method ensures accurate critical values when the asymptotic distribution is analytically intractable?
• RQ4 Can the framework detect change points in variance and auto-covariance even when they occur at different locations?
• RQ5 Is the proposed method robust and effective in finite samples, particularly for detecting non-relevant change points?

Key findings

• The CUSUM test statistic's asymptotic distribution depends on the dependency structure of the time series and cannot be expressed in a simple closed form.
• The bootstrap method successfully generates critical values for the test statistic, enabling valid inference under non-stationary conditions.
• The simulation study confirms that the proposed procedure has good finite-sample properties, maintaining correct size and reasonable power.
• The analysis of the central England monthly temperature series reveals a significant change point in winter temperature variance and lag-1 correlation around the late 19th century.
• The method successfully detects structural changes in second-order structure even when the mean is non-constant, demonstrating robustness to realistic non-stationarities.
• The extension to non-relevant change points allows for practical inference by testing whether parameter differences are within a negligible threshold δ.