Change-Point Analysis of Time Series with Evolutionary Spectra

Abstract

This paper develops change-point methods for the spectrum of a locally stationary time series. We focus on series with a bounded spectral density that change smoothly under the null hypothesis but exhibits change-points or becomes less smooth under the alternative. We address two local problems. The first is the detection of discontinuities (or breaks) in the spectrum at unknown dates and frequencies. The second involves abrupt yet continuous changes in the spectrum over a short time period at an unknown frequency without signifying a break. Both problems can be cast into changes in the degree of smoothness of the spectral density over time. We consider estimation and minimax-optimal testing. We determine the optimal rate for the minimax distinguishable boundary, i.e., the minimum break magnitude such that we are able to uniformly control type I and type II errors. We propose a novel procedure for the estimation of the change-points based on a wild sequential top-down algorithm and show its consistency under shrinking shifts and possibly growing number of change-points. Our method can be used across many fields and a companion program is made available in popular software packages.

Segmenting Time Series via Self-Normalisation

“…In the literature, there exists pure local-window-based segmentation algorithms, for example, SaRa in Niu and Zhang (2012) for change in mean, LRSM in Yau and Zhao (2016) for change in AR models. The pure local-window approach only considers the smallest local-window (k − h + 1, k + h) when constructing change-point tests for k given a window size h. Such an approach is also employed in the literature of 'piecewise smooth' change, see Wu and Zhao (2007), Bibinger et al (2017) and Casini and Perron (2021a).…”

“…For future research, it may be desirable to relax the piecewise constant assumption and allow the parameter to vary smoothly within each segment; see Wu and Zhou (2019) for such a formulation in non-parametric trend models and Casini and Perron (2021a) in locally stationary time series.…”

“…Existing non-parametric approaches for change-point estimation in temporally dependent observations primarily focus on first or second-order moments, see Bai and Perron (1998), Eichinger and Kirch (2018) for change-point estimation in mean, Aue et al (2009), Preuss et al (2015) in (auto)-covariance, and Cho and Fryzlewicz (2012), Casini and Perron (2021a) in spectral density function (thus second-order properties). However, for many applications, the key interest can go beyond mean or covariance.…”