[Paper Review] Structural Break Detection in High-Dimensional Non-Stationary VAR models

Abolfazl Safikhani, Ali Shojaie|arXiv (Cornell University)|Aug 9, 2017

Fault Detection and Control Systems28 references3 citations

TL;DR

This paper proposes a two-stage penalized least squares method with total variation LASSO for detecting structural breaks in high-dimensional non-stationary vector autoregressive (VAR) models under piecewise stationarity. The method consistently estimates the number and locations of change points, even when the number of break points grows with sample size, by combining variable selection with backward selection to correct overestimation from the LASSO penalty.

ABSTRACT

Assuming stationarity is unrealistic in many time series applications. A more realistic alternative is to allow for piecewise stationarity, where the model is allowed to change at given time points. In this article, the problem of detecting the change points in a high-dimensional piecewise vector autoregressive model (VAR) is considered. Reformulated the problem as a high-dimensional variable selection, a penalized least square estimation using total variation LASSO penalty is proposed for estimation of model parameters. It is shown that the developed method over-estimates the number of change points. A backward selection criterion is thus proposed in conjunction with the penalized least square estimator to tackle this issue. We prove that the proposed two-stage procedure consistently detects the number of change points and their locations. A block coordinate descent algorithm is developed for efficient computation of model parameters. The performance of the method is illustrated using several simulation scenarios.

Motivation & Objective

• To address the limitation of existing high-dimensional time series methods that assume stationarity, which is often unrealistic in real-world applications such as EEG data or financial time series.
• To develop a method capable of detecting multiple structural breaks in high-dimensional piecewise stationary VAR models where the dependence structure changes at unknown time points.
• To ensure consistent estimation of both the number and locations of change points, even when the number of breaks increases with sample size.
• To overcome the tendency of LASSO to overestimate the number of change points by introducing a backward selection criterion.

Proposed method

• Reformulates the structural break detection problem as a high-dimensional variable selection task by modeling the transition matrices as piecewise constant over time.
• Applies a penalized least squares estimator with a total variation LASSO penalty to estimate model parameters and detect change points.
• Uses a backward selection criterion to correct for the overestimation of change points inherent in the LASSO penalty, improving selection consistency.
• Develops a block coordinate descent algorithm for efficient computation of the penalized estimator in high-dimensional settings.
• Imposes regularity conditions on the design matrix and error structure to ensure theoretical consistency of the estimator.
• Employs an information criterion (IC) to select the optimal number of change points, balancing model fit and complexity.

Experimental results

Research questions

• RQ1 Can a high-dimensional VAR model with multiple structural breaks be consistently estimated under piecewise stationarity?
• RQ2 Does the use of total variation LASSO penalty lead to consistent detection of the number and locations of change points in high-dimensional non-stationary time series?
• RQ3 How can overestimation of change points by LASSO be corrected in a high-dimensional setting with multiple breaks?
• RQ4 Is the proposed two-stage procedure (LASSO + backward selection) consistent in estimating both the number and positions of structural breaks?
• RQ5 What are the theoretical conditions under which the method achieves consistent change point detection as the sample size increases?

Key findings

• The proposed two-stage method consistently estimates the number of change points, with probability tending to one as sample size increases.
• The method achieves consistent estimation of change point locations, with the maximum deviation between estimated and true break points bounded by $ n\gamma_n $.
• The backward selection criterion successfully corrects the overestimation of change points by the LASSO penalty, ensuring selection consistency.
• Theoretical analysis shows that the information criterion used in the backward selection step correctly identifies the true number of change points under regularity conditions.
• Simulation results confirm that the method accurately detects multiple change points in high-dimensional VAR models, even with increasing dimension and number of breaks.
• The block coordinate descent algorithm enables efficient computation of the penalized estimator, making the method scalable to high-dimensional time series.