A Unified View of Trend-Cycle Predictors in Reproducing Kernel Hilbert Spaces (RKHS)

A common approach for trend-cycle estimation is to use nonparametric smoothing estimators which can be finite or infinite in length. The main methods presented in this chapter are based on different criteria of fitting and smoothing, namely: (1) density functions, (2) local polynomial fitting, (3) graduation theory, and (4) smoothing spline regression. A unified approach for all of these different trend-cycle nonparametric estimators is provided by means of the Reproducing Kernel Hilbert Space (RKHS) methodology. It is shown how nonparametric estimators can be transformed into kernel functions of order two, that are probability densities, and from which corresponding higher order kernels are derived. This kernel representation enables the comparison of estimators based on different smoothing criteria, and has important consequences in the derivation of the asymmetric filters which are applied to the most recent seasonally adjusted data for real time trend-cycle estimation.