Joint Structural Break Detection and Parameter Estimation in High-Dimensional Nonstationary VAR Models

Abstract

Assuming stationarity is unrealistic in many time series applications. A more realistic alternative is to assume piecewise stationarity, where the model can change at potentially many change points. We propose a three-stage procedure for simultaneous estimation of change points and parameters of high-dimensional piecewise vector autoregressive (VAR) models. In the first step, we reformulate the change point detection problem as a high-dimensional variable selection one, and solve it using a penalized least square estimator with a total variation penalty. We show that the penalized estimation method over-estimates the number of change points, and propose a selection criterion to identify the change points. In the last step of our procedure, we estimate the VAR parameters in each of the segments. We prove that the proposed procedure consistently detects the number and location of change points, and provides consistent estimates of VAR parameters. The performance of the method is illustrated through several simulated and real data examples. Supplementary materials for this article are available online.

Keywords:

• High-dimensional time series
• Piecewise stationarity
• Structural breaks
• Total variation penalty

Supplementary Materials

The supplementary materials consist of technical lemmas needed to prove the main results and proofs of the main results (Appendices A and B), details of the algorithm for solving the optimization problem (5) and additional simulation results (Appendices C and D), a secondary analysis which sharpens the consistency rate together with an alternative procedure to our proposed third step on consistent parameter estimation (Appendices E and F), and finally, a data-driven method to select the tuning parameters (Appendix G).

Acknowledgments

The authors would like to thank the associate editor and two anonymous referees for their constructive feedback which led to improvements in the article.

Notes

¹ See also https://github.com/abolfazlsafikhani/SBDetection.

² http://www.nyc.gov/html/tlc/html/about/trip-record-data.shtml.

Additional information

Funding

This research was partially funded by grants from the National Science Foundation (DMS-1561814 and DMS-1722246) and the National Institute of Health (R01GM114029).