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A test for second order stationarity of a time series based on the
Discrete Fourier Transform

We consider a zero mean discrete time series, and define its discrete Fourier
transform at the canonical Fourier frequencies. It is well known that the
discrete Fourier transform is asymptotically uncorrelated at the
canonical frequencies if and if only the time series is second order
stationary. Exploiting this important property, we construct a Portmanteau
type test statistic for testing stationarity of the time series.
It is shown that under the null of stationarity, the test statistic is
approximately a chi square distribution.
To examine the power of the test statistic, the asymptotic distribution
under the locally stationary alternative is established. It is shown to be
a type of noncentral chi-square, where the noncentrality parameter
measures the deviation from stationarity.
The test is illustrated with simulations, where is it shown to have
good power. Some real examples are also included to illustrate the test.

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