  An important problem in time series analysis is the discrimination between
non-stationarity and longrange dependence. Most of the literature considers the
problem of testing specific parametric hypotheses of non-stationarity (such as
a change in the mean) against long-range dependent stationary alternatives. In
this paper we suggest a simple approach, which can be used to test the
null-hypothesis of a general non-stationary short-memory against the
alternative of a non-stationary long-memory process. The test procedure works
in the spectral domain and uses a sequence of approximating tvFARIMA models to
estimate the time varying long-range dependence parameter. We prove uniform
consistency of this estimate and asymptotic normality of an averaged version.
These results yield a simple test (based on the quantiles of the standard
normal distribution), and it is demonstrated in a simulation study that -
despite of its semi-parametric nature - the new test outperforms the currently
available methods, which are constructed to discriminate between specific
parametric hypotheses of non-stationarity short- and stationarity long-range
dependence.
