Provides an asymmetric filter based on the given reference filter (usually symmetric) minimizing the mean square revision error.
Usage
mmsre_filter(
ref_filter,
q,
U,
Z = NULL,
delta = NULL,
kernel = NULL,
tweight = 0,
passband = pi/12
)Arguments
- ref_filter
The reference filter (a
moving_average()object).- q
The horizon of the asymmetric filter.
- U
Matrix of the constraints.
- Z
Matrix of the bias (can be
NULL).- delta
Coefficients of the linear model.
- kernel
The kernel used for weighting factors, by default, no weight is used. See
lp_filter()for the available kernels.- tweight
timeliness weight.
- passband
passband threshold.
Details
The asymmetric filter \(\boldsymbol v=(v_{-h},\dots,v{q})'\) minimizes the mean square revision error (mmsre) relative to the reference filter \(\boldsymbol \theta=(\theta_{-h},\dots,\theta_{h'})'\). The series follows the model $$ \boldsymbol y=\boldsymbol U \boldsymbol \gamma + \boldsymbol Z \boldsymbol \delta + \boldsymbol \varepsilon, \quad \boldsymbol \varepsilon \sim \mathcal N(0,\sigma^2 \boldsymbol K^{-1}). $$
With \(K\) a set of weights (kernel), by default (kernel = NULL) no weight is used.
The matrix \(U\) represents the constraints of the symmetric filter (usually polynomials preservations), \(\boldsymbol \theta\),
imposed to the asymmetric filter, \(\boldsymbol v\).
Partitionning the matrix \(\boldsymbol U=\begin{pmatrix} \boldsymbol U_p' & \boldsymbol U_f'\end{pmatrix}'\)
with \(\boldsymbol U_p\) the first \(h+q+1\) rows and \(\boldsymbol U_f\) the remaining rows, the constraints are
\(\boldsymbol U_p'\boldsymbol v=\boldsymbol U'\boldsymbol \theta\).
The matrix \(\boldsymbol Z\) represents the bias of the asymmetric filter: usually constraints imposed to the symmetric filter but not to the asymmetric filter.
References
Proietti, Tommaso and Alessandra Luati (2008). “Real time estimation in local polynomial regression, with application to trend-cycle analysis”.
Examples
QL <- lp_filter(endpoints = "QL", ic = 3.5)
LC <- lp_filter(endpoints = "LC", ic = 3.5)
DAF <- lp_filter(endpoints = "DAF")
h6 <- QL[, "q=6"]
# To reproduce DAF filter
mmsre_filter(
ref_filter = h6, q = 0,
U = polynomial_matrix(l = - 6, d0 = 0, d1 = 3),
kernel = "Henderson"
)
#> [1] " - 0.0172 B^6 + 0.0219 B^5 + 0.0400 B^4 - 0.0341 B^3 - 0.0979 B^2 + 0.1322 B + 0.9552"
DAF[, "q=0"]
#> [1] " - 0.0172 B^6 + 0.0219 B^5 + 0.0400 B^4 - 0.0341 B^3 - 0.0979 B^2 + 0.1322 B + 0.9552"
# To reproduce QL filter
mmsre_filter(
ref_filter = h6, q = 1,
delta = 2 / (sqrt(pi) * 3.5),
U = polynomial_matrix(l = -6, d0 = 0, d1 = 1),
Z = polynomial_matrix(l = -6, d0 = 2, d1 = 2)
)
#> [1] " - 0.0083 B^6 - 0.0395 B^5 - 0.0216 B^4 + 0.0466 B^3 + 0.1440 B^2 + 0.2392 B + 0.3058 + 0.3337 F"
QL[, "q=1"]
#> [1] " - 0.0083 B^6 - 0.0395 B^5 - 0.0216 B^4 + 0.0466 B^3 + 0.1440 B^2 + 0.2392 B + 0.3058 + 0.3337 F"
# Or using the Uniform kernel
mmsre_filter(
ref_filter = h6, q = 2,
# we multiply by the square root of the inverse of weights (1/13)
# to get the same result as the QL filter
delta = 2 / (sqrt(pi) * 3.5) * (sqrt(13)),
U = polynomial_matrix(l = -6, d0 = 0, d1 = 0),
Z = polynomial_matrix(l = -6, d0 = 1, d1 = 1),
kernel = "Uniform"
)
#> [1] " - 0.0160 B^6 - 0.0249 B^5 + 0.0027 B^4 + 0.0678 B^3 + 0.1494 B^2 + 0.2160 B + 0.2414 + 0.2154 F + 0.1481 F^2"
LC[, "q=2"]
#> [1] " - 0.0160 B^6 - 0.0249 B^5 + 0.0027 B^4 + 0.0678 B^3 + 0.1494 B^2 + 0.2160 B + 0.2414 + 0.2154 F + 0.1481 F^2"