Optimal break tests for large linear time series models

TL;DR

This paper develops optimal tests for structural breaks occurring at unknown dates in large linear time series models, including infinite- and growing-dimensional settings like $AR(\infty)$ and nonparametric regression, using a growing-dimension sieve to approximate the problem. It derives a class of average-power optimal tests via a weighted exponential transform, and establishes a functional central limit theorem that accommodates nonlinear high-order serial dependence when both the sample size $T$ and the number of restrictions $p$ grow. To address the resulting nonstandard variance, the authors introduce a random-scaling (HLV-robust) correction and a bootstrap bias correction, yielding a pivotal test with improved size control in finite samples. Monte Carlo simulations and an empirical oil-output application illustrate substantial size improvements over conventional supremum and exponential tests in high-dimensional contexts, underscoring the method’s practical relevance for robust inference in large time-series systems.

Abstract

We develop a class of optimal tests for a structural break occurring at an unknown date in infinite and growing-order time series regression models, such as AR($\infty$), linear regression with increasingly many covariates, and nonparametric regression. Under an auxiliary i.i.d. Gaussian error assumption, we derive an average power optimal test, establishing a growing-dimensional analog of the exponential tests of Andrews and Ploberger (1994) to handle identification failure under the null hypothesis of no break. Relaxing the i.i.d. Gaussian assumption to a more general dependence structure, we establish a functional central limit theorem for the underlying stochastic processes, which features an extra high-order serial dependence term due to the growing dimension. We robustify our test both against this term and finite sample bias and illustrate its excellent performance and practical relevance in a Monte Carlo study and a real data empirical example.

Theorems & Definitions (23)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• Theorem 3.2
• Remark 1
• Proposition 4.1
• proof : Proof of Theorem \ref{['theorem:optimality']}
• proof : Proof of Theorem \ref{['thm:null_Chow']}:
• proof : Proof of Theorem \ref{['thm:local_power']}: