Auto-Regressive Approximations to Non-stationary Time Series, with Inference and Applications

TL;DR

This paper demonstrates that complex non-stationary time series can be globally approximated by a slowly diverging order auto-regressive process, enabling new inference methods and applications in forecasting.

Contribution

It establishes the first structural approximation result for general non-stationary time series using AR models and introduces novel inference and stability testing procedures.

Findings

Successful approximation of non-stationary series by AR models

Development of a high-dimensional $ ext{L}^2$ test and bootstrap for AR coefficients

Provision of optimal short-term forecasting methods for locally stationary series

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 non-stationary and nonlinear time series can be well approximated globally by a white-noise-driven auto-regressive (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 non-stationary 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 short-term forecasting theory and methodology for a wide class of locally stationary time series are established via the method of sieves.