Estimating change points in nonparametric time series regression models

TL;DR

This paper introduces a new nonparametric method for detecting change points in time series regression models with heteroscedasticity, demonstrating optimal convergence rates and practical effectiveness through simulations and real data.

Contribution

It develops a novel Kolmogorov-Smirnov based estimator for change points in nonparametric, heteroscedastic, and autoregressive time series models, extending existing methods.

Findings

Estimator is consistent with a convergence rate of O_P(n^{-1})

Method performs well in simulations for regression and autoregression

Real data application confirms practical utility

Abstract

In this paper we consider a regression model that allows for time series covariates as well as heteroscedasticity with a regression function that is modelled nonparametrically. We assume that the regression function changes at some unknown time $\lfloor ns_0\rfloor$, $s_0\in(0,1)$, and our aim is to estimate the (rescaled) change point $s_0$. The considered estimator is based on a Kolmogorov-Smirnov functional of the marked empirical process of residuals. We show consistency of the estimator and prove a rate of convergence of $O_P(n^{-1})$ which in this case is clearly optimal as there are only $n$ points in the sequence. Additionally we investigate the case of lagged dependent covariates, that is, autoregression models with a change in the nonparametric (auto-) regression function and give a consistency result. The method of proof also allows for different kinds of functionals such that Cram\'er-von Mises type estimators can be considered similarly. The approach extends existing literature by allowing nonparametric models, time series data as well as heteroscedasticity. Finite sample simulations indicate the good performance of our estimator in regression as well as autoregression models and a real data example shows its applicability in practise.