AutoRegressive approximations to nonstationary time series with inference and applications

Abstract

Understanding the time-varying structure of complex temporal systems is one of the main challenges of modern time-series analysis. In this paper, we show that every uniformly-positive-definite-in-covariance and sufficiently short-range dependent nonstationary and nonlinear time series can be well approximated globally by a white-noise-driven autoregressive (AR) process of slowly diverging order. To our best knowledge, it is the first time such a structural approximation result is established for general classes of nonstationary time series. A high-dimensional ${\mathcal{L}}^2$ test and an associated multiplier bootstrap procedure are proposed for the inference of the AR approximation coefficients. In particular, an adaptive stability test is proposed to check whether the AR approximation coefficients are time-varying, a frequently encountered question for practitioners and researchers of time series. As an application, globally optimal sffollowing hort-term forecasting theory and methodology for a wide class of locally stationary time series are established via the method of sieves.