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 such series, order estimate second-order structure (of differenced series). However, often we require inference on behaviour of original for example, when performing estimation. In this article, propose method, using differencing, jointly nonstationary within locally stationary wavelet modelling framework. We develop wavelet-based estimator based estimate, show how can be incorporated into estimation series. perform simulation study investigate performance methodology, demonstrate utility method by analysing data examples from environmental biomedical science.