Testing for a break in persistence under long-range dependencies

We show that tests for a break in the persistence of a time series in the classical I(0) - I(1) framework have serious size distortions when the actual data generating process exhibits long-range dependencies. We prove that the limiting distribution of a CUSUM of squares based test depends on the true memory parameter if the DGP exhibits long memory. We propose adjusted critical values for the test and give finite sample response curves which allow the practitioner to easily implement the test and to compute the relevant critical values. We furthermore prove consistency of the test and prove consistency for a simple break point estimator also under long memory. We show that the test has satisfying power properties when the correct critical values are used.