Additive Decomposition and Boundary Conditions in Wavelet-Based Forecasting Approaches

An interesting approach to economic and financial time series forecasting consists of decomposing an input time series additively into several components, each component capturing the dynamics of a different frequency range. Consequently, each component is modelled and forecasted separately, the predictions being summed up to form an overall forecast of the input time series. The present paper considers one very important aspect of the forecasting procedure. More specifically, it provides a better understanding of how an additive decomposition of the input time series into several components can be obtained using the wavelet transform and how boundary conditions in the individual components should be properly treated. Even though these aspects are presented as a part of the wavelet theory in several books on wavelets, their implementation is prone to misinterpretations in the literature on applied time series forecasting, possibly due to the complexity of the wavelet transform. Since our exposition is focused predominantly on these aspects, it provides a concise explanation which may be helpful to practitioners. The maximal overlap discrete wavelet transform is employed, other types of wavelet transforms also being briefly discussed.