A wavelet-based test for stationarity

We develop a test for stationarity of a time series against the alternative of a time-varying covariance structure. Using localized versions of the periodogram, we obtain empirical versions of a reasonable notion of a time-varying spectral density. Coefficients with respect to a Haar wavelet series expansion of such a time-varying periodogram are an indicator of whether there is some deviation from covariance stationarity. We propose a test based on the limit distribution of these empirical coefficients.