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Statistical inference for stochastic processes with time-varying spectralcharacteristics has received considerable attention in recent decades. Wedevelop a nonparametric test for stationarity against the alternative of asmoothly time-varying spectral structure. The test is based on a comparisonbetween the sample spectral density calculated locally on a moving window ofdata and a global spectral density estimator based on the whole stretch ofobservations. Asymptotic properties of the nonparametric estimators involvedand of the test statistic under the null hypothesis of stationarity arederived. Power properties under the alternative of a time-varying spectralstructure are discussed and the behavior of the test for fixed alternativesbelonging to the locally stationary processes class is investigated.

Testing temporal constancy of the spectral structure of a time series