  We consider a zero mean discrete time series, and define its discrete Fourier
transform at the canonical 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.
