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 localto- 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+