We consider a zero mean discrete time series, and define its discrete Fouriertransform at the canonical frequencies. It is well known that the discreteFourier transform is asymptotically uncorrelated at the canonical frequenciesif and if only the time series is second order stationary. Exploiting thisimportant property, we construct a Portmanteau type test statistic for testingstationarity of the time series. It is shown that under the null ofstationarity, the test statistic is approximately a chi square distribution. Toexamine the power of the test statistic, the asymptotic distribution under thelocally stationary alternative is established. It is shown to be a type ofnoncentral chi-square, where the noncentrality parameter measures the deviationfrom stationarity. The test is illustrated with simulations, where is it shownto have good power. Some real examples are also included to illustrate thetest.
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