A test for second order stationarity of a multivariate time series

It is well known that the discrete Fourier transform (DFT) of a second order stationary time series between two distinct Fourier frequencies are asymptotically uncorrelated. In contrast we show that for a large class of second order nonstationary time series, including locally stationary time series, the covariance between the Fourier frequencies is non-zero. Indeed the correlations between the DFTs contain important information about the underlying nonstationary time series. In this paper we use these starkly differing properties to define a global test for stationarity based on the DFT of a vector time series. We show under the null of stationarity the test statistic has a chi-squared distribution and under the alternative of local stationary a noncentral chi-squared distribution. In addition, under the assumption of Gaussianity of the time series, the test statistic is pivotal. However, in many econometric applications, such assumptions can be too strong, therefore the proposed test does not make any model assumptions (such as linearity) on the underlying time series. Under these general conditions the test statistic involves an unknown variance that is extremely difficult to directly estimate from the data. To overcome this issue, we propose a scheme based on the stationary bootstrap to estimate the unknown variance and show that the resulting estimator is a consistent estimator of the variance.