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When analysing time series an important issue is to decide whether the timeseries is stationary or a random walk. Relaxing these notions, we consider theproblem to decide in favor of the I(0)- or I(1)-property. Fixed-samplestatistical tests for that problem are well studied in the literature. In thispaper we provide first results for the problem to monitor sequentially a timeseries. Our stopping times are based on a sequential version of akernel-weighted variance-ratio statistic. The asymptotic distributions areestablished for I(1) processes, a rich class of stationary processes, possiblyaffected by local nonpara- metric alternatives, and the local-to-unity model.Further, we consider the two interesting change-point models where the timeseries changes its behaviour after a certain fraction of the observations andderive the associated limiting laws. Our Monte-Carlo studies show that theproposed detection procedures have high power when interpreted as a hypothesistest, and that the decision can often be made very early.

MONITORING PROCEDURES TO DETECT UNIT ROOTS AND STATIONARITY