A Theoretical Comparison Between Classical and Reproducing Kernel Hilbert Space Henderson Predictors

The Henderson smoother (1916) has been traditionally applied for trend-cycle estimation in the context of nonparametric seasonal ad- justment softwares such as Census X11, X11/X12 ARIMA, officially adopted by statistical agencies. Particularly, the 13-term filter has shown to possess good properties to detect an upcoming turning point but it has the shortcoming of introducing large revisions for the most recent estimates when new observations are added to the series. This limitation is of serious consequences for short-term trend analysis. In this study we introduce a Henderson third order kernel representation by means of the reproducing kernel Hilbert space (RKHS) method- ology. Two density functions and corresponding orthonormal poly- nomials up to the second degree have been calculated. One is based on the Henderson weighting function applied in the least square fit- ting minimization procedure. The other is the biweight density with the associated Jacobi polynomials. Both are shown to give excellent representations for short and medium length filters. The asymmet- ric weights are derived by adapting the third order kernel functions to the length of the various filters. A comparison of the Henderson third order kernel asymmetric filters is made with the classical ones developed by Musgrave (1964a and 1964b). The former are shown to be superior in terms of signal passing, noise suppression and speed of convergence to the symmetric filter.