Partial Sum Processes of Residual-Based and Wald-type Break-Point Statistics in Time Series Regression Models

TL;DR

This paper analyzes the asymptotic behavior of residual-based and Wald-type break-point tests in linear regression models, revealing differences in their distributions under stationary and nonstationary conditions, with implications for testing accuracy.

Contribution

It establishes the Brownian bridge limit for these test statistics and investigates how nuisance parameters affect their asymptotic behavior in nonstationary models.

Findings

Brownian bridge distribution for stationary models

Size distortions in nonstationary models

Nuisance parameters influence asymptotic behavior

Abstract

We revisit classical asymptotics when testing for a structural break in linear regression models by obtaining the limit theory of residual-based and Wald-type processes. First, we establish the Brownian bridge limiting distribution of these test statistics. Second, we study the asymptotic behaviour of the partial-sum processes in nonstationary (linear) time series regression models. Although, the particular comparisons of these two different modelling environments is done from the perspective of the partial-sum processes, it emphasizes that the presence of nuisance parameters can change the asymptotic behaviour of the functionals under consideration. Simulation experiments verify size distortions when testing for a break in nonstationary time series regressions which indicates that the Brownian bridge limit cannot provide a suitable asymptotic approximation in this case. Further research is required to establish the cause of size distortions under the null hypothesis of parameter stability.