Data-driven optimal decomposition of time series

A data-driven optimal decomposition of time series with trend-cyclical and seasonal components as well as the estimation of derivatives of the trend-cyclical is considered. The time series is smoothed by locally weighted regression with polynomials and trigonometric functions as local regressors. Two variates for the selection of the optimal bandwidths and the order of the polynomials are proposed with a particular approach for the estimation in the boundary areas of the time series. The second of these procedures can also be used for the selection of optimal bandwidths if only one component is considered. The smoothing of a time series without seasonal variations is just a special case for these procedures. The rate of convergence in the second procedure for this special case is discussed. A by-product of this work is the development of a seasonal-difference-based method to estimate the variance in a seasonal time series.