A test for second order stationarity of a time series based on the Discrete Fourier Transform Yogesh Dwivedi and Suhasini Subba Rao Texas A&M University, College Station, Texas 77845, USA February 22, 2010 Abstract We consider a zero mean discrete time series, and define its discrete Fourier transform at the canonical frequencies. It can be shown 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 generalised 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. Keywords and phrases α-mixing, Discrete Fourier Transform, linear time series, local stationarity, Portmanteau test, test for second order stationarity. Primary 62M10, Secondary 62F10 1 Introduction An important assumption that is often made when analysing time series is that it is at least second order stationary. A large proportion of the time series literature is based on this assumption. If the assumption is not properly tested and the analysis is performed, the resulting model may be misspecified and the forecasts obtained may be inappropriate. Therefore, it is important to check whether the time series is second order stationary. Over the years, various statistical tests have been proposed. One of the first tests for stationarity is considered in Priestley and Subba Rao (1969), more recently tests have been proposed in Sachs and 1