A test for second order stationarity of a multivariate time series

It is well known that the discrete Fourier transforms (DFTs) of a second order stationary time series between two distinct Fourier frequencies are asymptotically uncorrelated. In contrast for a large class of second order nonstationary time series, including locally stationary time series, this property does not hold. In this paper these starkly differing properties are used to define a global test for stationarity based on the DFT of a vector time series. It is shown that the test statistic under the null of stationarity asymptotically has a chi-squared distribution, whereas under the alternative of local stationarity asymptotically it has a noncentral chi-squared distribution. Further, if the time series is Gaussian and stationary, the test statistic is pivotal. However, in many econometric applications, the assumption of Gaussianity can be too strong, but under weaker conditions the test statistic involves an unknown variance that is extremely difficult to directly estimate from the data. To overcome this issue, a scheme to estimate the unknown variance, based on the stationary bootstrap, is proposed. The properties of the stationary bootstrap under both stationarity and nonstationarity are derived. These results are used to show consistency of the bootstrap estimator under stationarity and to derive the power of the test under nonstationarity. The method is illustrated with some simulations. The test is also used to test for stationarity of FTSE 100 and DAX 30 stock indexes from January 2011-December 2012.