  Most time series observed in practice exhibit time-varying trend
(first-order) and autocovariance (second-order) behaviour. Differencing is a
commonly-used technique to remove the trend in such series, in order to
estimate the time-varying second-order structure (of the differenced series).
However, often we require inference on the second-order behaviour of the
original series, for example, when performing trend estimation. In this
article, we propose a method, using differencing, to jointly estimate the
time-varying trend and second-order structure of a nonstationary time series,
within the locally stationary wavelet modelling framework. We develop a
wavelet-based estimator of the second-order structure of the original time
series based on the differenced estimate, and show how this can be incorporated
into the estimation of the trend of the time series. We perform a simulation
study to investigate the performance of the methodology, and demonstrate the
utility of the method by analysing data examples from environmental and
biomedical science.
