  We develop an estimator for the high-dimensional covariance matrix of a
locally stationary process with a smoothly varying trend and use this statistic
to derive consistent predictors in non-stationary time series. In contrast to
the currently available methods for this problem the predictor developed here
does not rely on fitting an autoregressive model and does not require a
vanishing trend. The finite sample properties of the new methodology are
illustrated by means of a simulation study and a financial indices study.
