  Statistical inference for stochastic processes with time-varying spectral
characteristics has received considerable attention in recent decades. We
develop a nonparametric test for stationarity against the alternative of a
smoothly time-varying spectral structure. The test is based on a comparison
between the sample spectral density calculated locally on a moving window of
data and a global spectral density estimator based on the whole stretch of
observations. Asymptotic properties of the nonparametric estimators involved
and of the test statistic under the null hypothesis of stationarity are
derived. Power properties under the alternative of a time-varying spectral
structure are discussed and the behavior of the test for fixed alternatives
belonging to the locally stationary processes class is investigated.
