  In this paper, we develop new multiscale methods to test qualitative
hypotheses about the regression function m in a nonparametric regression model
with fixed design points and time series errors. In time series applications, m
represents a nonparametric time trend. Practitioners are often interested in
whether the trend m has certain shape properties. For example, they would like
to know whether m is constant or whether it is increasing/decreasing in certain
time regions. Our multiscale methods allow to test for such shape properties of
the trend m. In order to perform the methods, we require an estimator of the
long-run variance of the error process. We propose a new difference-based
estimator of the long-run error variance for the case that the error terms form
an AR(p) process. In the technical part of the paper, we derive asymptotic
theory for the proposed multiscale test and the estimator of the long-run error
variance. The theory is complemented by a simulation study and an empirical
application to climate data.
