Change point detection in autoregressive models with no moment assumptions

Abstract: In this paper we consider the problem of detecting a change in the parameters of an autoregressive process, where the moments of the innovation process do not necessarily exist. An empirical likelihood ratio test for the existence of a change point is proposed and its asymptotic properties are studied. In contrast to other work on change point tests using empirical likelihood, we do not assume knowledge of the location of the change point. In particular, we prove that the maximizer of the empirical likelihood is a consistent estimator for the parameters of the autoregressive model in the case of no change point and derive the limiting distribution of the corresponding test statistic under the null hypothesis. We also establish consistency of the new test. A nice feature of the method consists in the fact that the resulting test is asymptotically distribution free and does not require an estimate of the long run variance. The asymptotic properties of the test are investigated by means of a small simulation study, which demonstrates good finite sample properties of the proposed method.