Functional Estimation and Change Detection for Nonstationary Time Series

Abstract

Tests for structural breaks in time series should ideally be sensitive to breaks in the parameter of interest, while being robust to nuisance changes. Statistical analysis thus needs to allow for some form of nonstationarity under the null hypothesis of no change. In this article, estimators for integrated parameters of locally stationary time series are constructed and a corresponding functional central limit theorem is established, enabling change-point inference for a broad class of parameters under mild assumptions. The proposed framework covers all parameters which may be expressed as nonlinear functions of moments, for example kurtosis, autocorrelation, and coefficients in a linear regression model. To perform feasible inference based on the derived limit distribution, a bootstrap variant is proposed and its consistency is established. The methodology is illustrated by means of a simulation study and by an application to high-frequency asset prices.

Keywords:

• Bootstrap inference
• Gradual change
• Locally stationary process
• p-Variation

Supplementary Material

The appendices A and B containing additional examples and simulation results of change point problems, and Appendix C containing all technical proofs, are available as a digital supplement. The R code used for the simulations, and the data used for the empirical example are also available as a supplement.

Acknowledgments

The author gratefully acknowledges the comments of the anonymous reviewers, which led to substantial improvements of this article.