Boosted Whittaker–Henderson Graduation

Abstract

The Whittaker–Henderson (WH) graduation is a smoothing method for equally spaced one-dimensional data such as time series. It includes the Bohlmann filter, the Hodrick–Prescott (HP) filter, and the Whittaker graduation as special cases. Among them, the HP filter is the most prominent trend-cycle decomposition method for macroeconomic time series such as real gross domestic product. Recently, a modification of the HP filter, the boosted HP (bHP) filter, has been developed, and several studies have been conducted. The basic idea of the modification is to achieve more desirable smoothing by extracting long-term fluctuations remaining in the smoothing residuals. Inspired by the modification, this paper develops the boosted version of the WH graduation, which includes the bHP filter as a special case. Then, we establish its properties that are fundamental for applied work. To investigate the properties, we use a spectral decomposition of the penalty matrix of the WH graduation