\( \newcommand\R{\mathbb{R}} \newcommand\Z{\mathbb{Z}} \newcommand\LL{\mathbb{L}} \newcommand{\E}[1]{\mathbb{E}\left[#1\right]} \newcommand{\V}[1]{\mathbb{V}\left[#1\right]} \newcommand{\ps}[2]{\left\langle #1 \,,\, #2 \right\rangle} \newcommand\1{\mathbb{1}} \newcommand\N{\mathbb{N}} \newcommand\Norm{\mathcal{N}} \newcommand{\transp}[1]{{}^t\!#1} \newcommand\ud{\,\mathrm{d}} \DeclareMathOperator*{\argmax}{argmax} \DeclareMathOperator*{\argmin}{argmin} \DeclareMathOperator{\e}{e} \DeclareMathOperator{\Cov}{Cov} \DeclareMathOperator{\Determinant}{det} \newcommand{\determinant}[1]{\Determinant\left(#1\right)} \)

Abstract

Abstract

This paper describes and compares different approaches to build asymmetric filters: local polynomials filters, methods based on an optimization of filters’ properties (Fidelity-Smoothness-Timeliness, FST, approach and a data-dependent filter) and filters based on Reproducing Kernel Hilbert Space. It also describes how local polynomials filters can be extended to include a timeliness criterion to minimize phase shift. All these methods can be seen as a special case of a general unifying framework to derive linear filters.

This paper shows that, when the length of the filter is adapted to the variability of the series, constraining asymmetric filters to preserve constant trends (and not necessarily polynomial ones) reduce revision error and time lag. Therefore, future studies on the subject can focus on these filters. Moreover, with RKHS filters some optimisation issues can occurs and they might lead to erratic estimation. They might be able to produce satisfying results in terms of phase-shift and revisions, but they should be avoid for the last estimates of the trend-cycle component: other methods should then be prefered to reduce revisions with the final estimates.

All the methods are implemented in the package rjdfilters and the results can be easily reproduced. The programs used, and a web version of this report, are available at https://github.com/AQLT/articles.