Test of change point versus long‐range dependence in functional time series

Abstract

In the context of functional time series, we propose a significance test to distinguish between short memory with a change point and long range dependence. The test is based on coefficients of projections onto an optimal direction that captures the dependence structure of the latent stationary functions that are not observable due to a potential change point. The optimal direction must be estimated as well. The test statistic is constructed using the local Whittle estimator applied to these coefficients. It has standard normal distribution under the null hypothesis (change point) and diverges to infinity under the alternative (long range dependence). The article includes asymptotic theory, a simulation study and an application to curve‐valued time series derived from intraday asset prices.