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Validating Stationarity Assumptions in Time Series Analysis by Rolling Local Periodograms

We propose a simple and powerful procedure to validate the assumption of weak stationarity in time series analysis. Our focus is on processes with a slowly varying autocovariance structure. The procedure evaluates the supremum over time of the L2-distance between the local sample spectral density (local periodogram) calculated using a segment of observations falling within a rolling window and an estimator of the spectral density obtained using the entire time series at hand. Large sample properties of a basic deviation process are investigated and critical values of a supremum type test are obtained using an appropriate bootstrap procedure. The finite sample size and power properties of the procedure are investigated by means of simulations. Real data examples demonstrate the ability of the procedure to detect (possible) changes in the autocovariance structure of a time series and to understand their nature.