On discriminating between long-range dependence and changes in mean

Abstract

We develop a testing procedure for distinguishing between a long-range dependent time series and a weakly dependent time series with change-points in the mean. In the simplest case, under the null hypothesis the time series is weakly dependent with one change in mean at an unknown point, and under the alternative it is long-range dependent. We compute the CUSUM statistic T_n, which allows us to construct an estimator k̂ of a change-point. We then compute the statistic T_n,1 based on the observations up to time k̂ and the statistic T_n,2 based on the observations after time k̂. The statistic M_n=max [T_n,1,T_n,2] converges to a well-known distribution under the null, but diverges to infinity if the observations exhibit long-range dependence. The theory is illustrated by examples and an application to the returns of the Dow Jones index.