Weighted Dickey-Fuller Processes for Detecting Stationarity

TL;DR

This paper introduces weighted Dickey-Fuller control charts for real-time detection of stationarity in time series, providing theoretical foundations and practical methods for improved monitoring under various conditions.

Contribution

It develops new kernel-weighted sequential Dickey-Fuller processes and establishes their asymptotic properties, including methods to handle nuisance parameters for practical implementation.

Findings

Asymptotic distributions derived under null and alternative hypotheses.

Control limits can be estimated or transformed for invariance.

Simulation studies compare finite-sample performance of methods.

Abstract

Aiming at monitoring a time series to detect stationarity as soon as possible, we introduce monitoring procedures based on kernel-weighted sequential Dickey-Fuller (DF) processes, and related stopping times, which may be called weighted Dickey-Fuller control charts. Under rather weak assumptions, (functional) central limit theorems are established under the unit root null hypothesis and local-to-unity alternatives. For gen- eral dependent and heterogeneous innovation sequences the limit processes depend on a nuisance parameter. In this case of practical interest, one can use estimated control limits obtained from the estimated asymptotic law. Another easy-to-use approach is to transform the DF processes to obtain limit laws which are invariant with respect to the nuisance pa- rameter. We provide asymptotic theory for both approaches and compare their statistical behavior in finite samples by simulation.