Modelling Time-Varying First and Second-Order Structure of Time Series via Wavelets and Differencing

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 se-
ries). 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 de-
velop 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.