Sequentially Updated Residuals and Detection of Stationary Errors in Polynomial Regression Models

TL;DR

This paper develops a sequential monitoring procedure using residuals in polynomial regression models to detect when error terms shift from a random walk to a stationary process, with theoretical and simulation validation.

Contribution

It introduces a new asymptotic distribution theory for a control chart-based test to detect changes in error process stationarity in polynomial regression models.

Findings

Asymptotic distribution theory for the monitoring procedure

Functional central limit theorem established

Finite sample properties validated through simulations

Abstract

The question whether a time series behaves as a random walk or as a station- ary process is an important and delicate problem, particularly arising in financial statistics, econometrics, and engineering. This paper studies the problem to detect sequentially that the error terms in a polynomial regression model no longer behave as a random walk but as a stationary process. We provide the asymptotic distribution theory for a monitoring procedure given by a control chart, i.e., a stopping time, which is related to a well known unit root test statistic calculated from sequentially updated residuals. We provide a functional central limit theorem for the corresponding stochastic process which implies a central limit theorem for the control chart. The finite sample properties are investigated by a simulation study.