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Inference for Multiple Change-points in Piecewise Locally Stationary Time Series

Wai Leong Ng, Xinyi Tang, Mun Lau Cheung, Jiacheng Gao, Chun Yip Yau, Holger Dette

TL;DR

This paper proposes a novel likelihood-based procedure for the inference of multiple change-points in locally stationary time series and shows that the proposed method can consistently estimate the number, locations, and the types of change-points.

Abstract

Change-point detection and locally stationary time series modeling are two major approaches for the analysis of non-stationary data. The former aims to identify stationary phases by detecting abrupt changes in the dynamics of a time series model, while the latter employs (locally) time-varying models to describe smooth changes in dependence structure of a time series. However, in some applications, abrupt and smooth changes can co-exist, and neither of the two approaches alone can model the data adequately. In this paper, we propose a novel likelihood-based procedure for the inference of multiple change-points in locally stationary time series. In contrast to traditional change-point analysis where an abrupt change occurs in a real-valued parameter, a change in locally stationary time series occurs in a parameter curve, and can be classified as a jump or a kink depending on whether the curve is discontinuous or not. We show that the proposed method can consistently estimate the number, locations, and the types of change-points. Two different asymptotic distributions corresponding respectively to jump and kink estimators are also established.Extensive simulation studies and a real data application to financial time series are provided.

Inference for Multiple Change-points in Piecewise Locally Stationary Time Series

TL;DR

This paper proposes a novel likelihood-based procedure for the inference of multiple change-points in locally stationary time series and shows that the proposed method can consistently estimate the number, locations, and the types of change-points.

Abstract

Change-point detection and locally stationary time series modeling are two major approaches for the analysis of non-stationary data. The former aims to identify stationary phases by detecting abrupt changes in the dynamics of a time series model, while the latter employs (locally) time-varying models to describe smooth changes in dependence structure of a time series. However, in some applications, abrupt and smooth changes can co-exist, and neither of the two approaches alone can model the data adequately. In this paper, we propose a novel likelihood-based procedure for the inference of multiple change-points in locally stationary time series. In contrast to traditional change-point analysis where an abrupt change occurs in a real-valued parameter, a change in locally stationary time series occurs in a parameter curve, and can be classified as a jump or a kink depending on whether the curve is discontinuous or not. We show that the proposed method can consistently estimate the number, locations, and the types of change-points. Two different asymptotic distributions corresponding respectively to jump and kink estimators are also established.Extensive simulation studies and a real data application to financial time series are provided.
Paper Structure (17 sections, 5 theorems, 58 equations, 4 figures, 3 tables)

This paper contains 17 sections, 5 theorems, 58 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Backgrounds on Locally Stationary Time Series
  3. Locally Stationary Time Series
  4. Time-varying Autoregressive Models
  5. Multiple Change-point Estimation for Piecewise tvAR models
  6. Problem Settings
  7. First Step: Quick Scanning for Potential Change-points
  8. Second Step: Consistent Estimation by Model Selection Approach
  9. Third Step: Final Adjustment and Confidence Intervals Construction
  10. Final Adjustment and Confidence Interval Construction for a Jump
  11. Final Adjustment and Confidence Interval Construction for a Kink
  12. Asymptotic Properties
  13. Simulation Studies
  14. Choice of Window Radius
  15. Performance of Estimation and Inference
  16. ...and 2 more sections

Key Result

Theorem 4.1

Let $\hat{J} = \{\hat{\tau}_1^{(J)}, \hat{\tau}_2^{(J)} , \dots , \hat{\tau}_{\hat{m}^{(J)}}^{(J)}\}$ and $\hat{K} = \{\hat{\tau}_1^{(K)}, \hat{\tau}_2^{(K)} , \dots , \hat{\tau}_{\hat{m}^{(K)}}^{(K)}\}$ be the sets of potential change-points selected in the first step, where $\hat{m}^{(J)}=|\hat{J

Figures (4)

  • Figure 1: Realizations of time series and AR(1) coefficients for Models 1 and 2.
  • Figure 2: ACE for Model 1 and Model 2 under different radii $h$ and $\widetilde{h}$ and sample size $T$.
  • Figure 3: Realizations of time series for Model 3 - 9.
  • Figure 4: Hang Seng Index price and corresponding log-return time series with detected change-points from both the proposed method, the LS and the MuBreD. The Kink (dashed line) and Jump (dashed line) change-points, along with their corresponding confidence intervals (gray shaded areas), are estimated using our proposed method. The LS and MuBreD change-points are represented by dotted lines and solid lines respectively.

Theorems & Definitions (6)

  • Remark 1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 4.5