Detecting at‐Most‐m Changes in Linear Regression Models

Abstract

In this article, we provide a new procedure to test for at‐most‐ frakturm changes in the time‐dependent regression model yt=boldxt⊤bold-italicβt+et,1⩽t⩽T, that is, β1 = β2 = ⋯ = βT under the no‐change null hypothesis against the alternative yt=bold-italicxt⊤bold-italicβ(i)+et, if ki−1∗<t⩽ki∗,1⩽i⩽frakturm+1 and β(j) ≠ β(ℓ) for some 1⩽j,ℓ⩽frakturm+1 with k0∗=0,1<k1∗<k2∗<⋯<kfrakturm∗<T,kfrakturm+1∗=T. Our procedure is based on weighted sums of the residuals, incorporating the possibility of frakturm changes. The weak limit of the proposed test statistic is the sum of two double‐exponential random variables. A small Monte Carlo simulation illustrates the applicability of the limit results in case of small and moderate sample sizes. We compare the new method to the cumulative sum control chart (CUSUM) and standardized (weighted) CUSUM procedures and obtain the power curves of the test statistics under the alternative. We apply our method to find changes in the unconditional four‐factor capital asset pricing model.

A data‐driven approach to detecting change points in linear regression models

“…Our testing procedure is based on the flexible framework for detecting at‐most‐ m changes, by Horváth et al (): where β ( i ) ( i =1,…, m +1) are regression coefficients in the i th subperiod; k 1 ,…, k m are the locations of m change points ( ); ( k 0 =0 and k m +1 = T ). The H 0 of no changes in the coefficients is rejected in all cases from at least one change point to at most m change points being present, where the m change point locations are specified before applying the test.…”

“…The assumptions for the test include stationarity of x t and ε t and that the sequence is a Bernoulli shift, which can be approximated with finitely dependent time series (see Horváth et al, , for more details).…”

“…Early approaches to the problem of changing parameters in time‐dependent linear regression have used a likelihood ratio test (Quandt, , ), theory of Markov processes (e.g., Vaman, ), and Bayes‐type statistics (Jandhyala & MacNeill, ; ). Most of the recent studies, however, are based on cumulative sum (CUSUM) statistics calculated on regression residuals (e.g., see Aue, Horváth, Hušková, & Kokoszka, ; Gombay, ; Horváth, Hušková, Kokoszka, & Steinebach, ; Horváth, Pouliot, & Wang, ); see more references in the reviews by Jandhyala, Zacks, and El‐Shaarawi (); Reeves, Chen, Wang, Lund, and Lu (); and Horváth and Rice (). Motivated by the applied questions from our Chesapeake Bay study, we favored the flexible framework of Horváth et al () for detecting at‐most‐ m change points in a linear regression model with potentially autocorrelated errors.…”

“…Most of the recent studies, however, are based on cumulative sum (CUSUM) statistics calculated on regression residuals (e.g., see Aue, Horváth, Hušková, & Kokoszka, ; Gombay, ; Horváth, Hušková, Kokoszka, & Steinebach, ; Horváth, Pouliot, & Wang, ); see more references in the reviews by Jandhyala, Zacks, and El‐Shaarawi (); Reeves, Chen, Wang, Lund, and Lu (); and Horváth and Rice (). Motivated by the applied questions from our Chesapeake Bay study, we favored the flexible framework of Horváth et al () for detecting at‐most‐ m change points in a linear regression model with potentially autocorrelated errors. The results by Horváth et al (), however, involve kernel‐based estimation of the long‐run variance function—an approach that can be sensitive to dominantly positive or negative autocorrelations.…”

“…Motivated by the applied questions from our Chesapeake Bay study, we favored the flexible framework of Horváth et al () for detecting at‐most‐ m change points in a linear regression model with potentially autocorrelated errors. The results by Horváth et al (), however, involve kernel‐based estimation of the long‐run variance function—an approach that can be sensitive to dominantly positive or negative autocorrelations. In practice, the estimation also requires selecting a kernel function and optimal bandwidth.…”