Estimation of a filter using the Fidelity-Smoothness-Timeliness criteria

Usage

fst_filter(
  lags = 6,
  leads = 0,
  pdegree = 2,
  smoothness.weight = 1,
  smoothness.degree = 3,
  timeliness.weight = 0,
  timeliness.passband = pi/6,
  timeliness.antiphase = TRUE
)

Arguments

lags: Lags of the filter (should be positive).
leads: Leads of the filter (should be positive or 0).
pdegree: Local polynomials preservation: max degree.
smoothness.weight: Weight for the smoothness criterion (in $[0, 1]$).
smoothness.degree: Degree of the smoothness criterion (3 for Henderson).
timeliness.weight: Weight for the Timeliness criterion (in $[0, 1[$). sweight+tweight should be in $[0,1]$.
timeliness.passband: Passband for the timeliness criterion (in radians). The phase effect is computed in $[0, passband]$.
timeliness.antiphase: boolean indicating if the timeliness should be computed analytically (TRUE) or numerically (FALSE).

Details

Moving average computed by a minimisation of a weighted mean of three criteria under polynomials constraints. Let $\boldsymbol \theta=(\theta_{-p},\dots,\theta_{f})'$ be a moving average where $p$ and $f$ are two integers defined by the parameter lags and leads. The three criteria are:

Fidelity, $F_g$: it's the variance reduction ratio. $$ F_g(\boldsymbol \theta) = \sum_{k=-p}^{+f}\theta_{k}^{2} $$
Smoothness, $S_g$: it measures the flexibility of the coefficient curve of a filter and the smoothness of the trend. $$ S_g(\boldsymbol \theta) = \sum_{j}(\nabla^{q}\theta_{j})^{2} $$ The integer $q$ is defined by parameter smoothness.degree. By default, the Henderson criteria is used (smoothness.degree = 3).
Timeliness, $T_g$ : $$ T_g(\boldsymbol\theta)=\int_{0}^{\omega_{2}}f(\rho_{\boldsymbol\theta}(\omega),\varphi_{\boldsymbol\theta}(\omega))d\omega $$ with $\rho_{\boldsymbol\theta}$ and $\varphi_{\boldsymbol\theta}$ the gain and phase shift functions of $\boldsymbol \theta$, and $f$ a penalty function defined as $f\colon(\rho,\varphi)\mapsto\rho^2\sin(\varphi)^2$ to have an analytically solvable criterium. $\omega_{2}$ is defined by the parameter timeliness.passband and is it by default equal to $2\pi/12$: for monthly time series, we focus on the timeliness associated to cycles of 12 months or more.

The moving average is then computed solving the problem: $$ \begin{cases} \underset{\theta}{\min} & J(\theta)= (1-\beta-\gamma) F_g(\theta)+\beta S_g(\theta)+\gamma T_g(\theta)\\ s.t. & C\theta=a \end{cases} $$ Where $C\theta=a$ represents linear constraints to have a moving average that preserve polynomials of degree $q$ (pdegree): $$ C=\begin{pmatrix} 1 & \cdots&1\\ -h & \cdots&h \\ \vdots & \cdots & \vdots \\ (-h)^d & \cdots&h^d \end{pmatrix},\quad a=\begin{pmatrix} 1 \\0 \\ \vdots\\0 \end{pmatrix} $$

References

Grun-Rehomme, Michel, Fabien Guggemos, and Dominique Ladiray (2018). “Asymmetric Moving Averages Minimizing Phase Shift”. In: Handbook on Seasonal Adjustment, https://ec.europa.eu/eurostat/web/products-manuals-and-guidelines/-/ks-gq-18-001.

Examples

filter <- fst_filter(lags = 6, leads = 0)
filter
#> [1] "0.1678 B^6 - 0.3147 B^4 - 0.2797 B^3 + 0.2098 B^2 + 0.6713 B + 0.5455"