Sieve bootstrap test for changes between unit root and long memory process with time trend

This paper studies the hypothesis testing problem where the null hypothesis is that a time series is an unit root process and the alternative hypothesis is that there is a change point so that the time series shifts from an unit root process to a long memory process or vice verse. We propose a residual based Dickey-Fuller type ratio statistic and derive its null distribution. We also establish the limiting distribution of the test statistic under the long memory process with no change point hypothesis. In particular, a sieve bootstrap method is designed to determine the critical value of the test statistic. Simulations show that the new test can control the empirical size well, even for the heavy-tailed long memory process. Our proposed test also gives competitive empirical power compared to the well known CUSUM of squares-based test. Finally, the validity of the new test is illustrated by a set of foreign exchange rate data between RMB and U.S. dollars.