Monitoring Procedures To Detect Unit Roots And Stationarity

When analyzing time series an important issue is to decide whether the time series is stationary or a random walk. Relaxing these notions, we consider the problem to decide in favor of the I(0) or I(1) property. Fixed-sample statistical tests for that problem are well studied in the literature. In this paper we provide first results for the problem of monitoring sequentially a time series. Our stopping times are based on a sequential version of a kernel-weighted variance-ratio statistic. The asymptotic distributions are established for I(1) processes, a rich class of stationary processes, possibly affected by local nonparametric alternatives, and the local-to-unity model. Further, we consider the two interesting change-point models where the time series changes its behavior after a certain fraction of the observations and derive the associated limiting laws. Our Monte Carlo studies show that the proposed detection procedures have high power when interpreted as a hypothesis test and that the decision can often be made very early.The financial support of the DFG (Deutsche Forschungsgemeinschaft, SFB 475, Reduction of Complexity in Multivariate Data Structures) is gratefully acknowledged. I thank two anonymous referees for constructive and helpful remarks that improved the paper and Dipl.-Math. Sabine Teller for proofreading a revised version.