README.md
In palatej/rjdfilters: Trend-Cycle Extraction with Linear Filters
rjd3filters
rjd3filters is R package on linear filters for real-time trend-cycle
estimates. It allows to create symmetric and asymmetric moving averages
with:
-
local polynomial filters, as defined by Proietti and Luati (2008);
-
the FST approach of Grun-Rehomme, Guggemos, and Ladiray (2018), based
on the optimization of the three criteria Fidelity, Smoothness and
Timeliness;
-
the Reproducing Kernel Hilbert Space (RKHS) of Dagum and Bianconcini
(2008).
Some quality criteria defined by Wildi and McElroy (2019) can also be
computed.
Installation
rjd3filters relies on the
rJava package and Java SE 17
or later version is required.
# Install development version from GitHub
# install.packages("devtools")
devtools::install_github("palatej/rjd3toolkit")
devtools::install_github("palatej/rj3dfilters")
Basic example
In this example we use the same symmetric moving average (Henderson),
but we use three different methods to compute asymmetric filters. As a
consequence, the filtered time series is the same, except at the
boundaries.
library(rjd3filters)
y <- window(retailsa$AllOtherGenMerchandiseStores,start = 2000)
musgrave <- lp_filter(horizon = 6, kernel = "Henderson",endpoints = "LC")
# we put a large weight on the timeliness criteria
fst_notimeliness_filter <- lapply(0:6, fst_filter,
lags = 6, smoothness.weight = 1/1000,
timeliness.weight = 1-1/1000, pdegree =2)
fst_notimeliness <- finite_filters(sfilter = fst_notimeliness_filter[[7]],
rfilters = fst_notimeliness_filter[-7],
first_to_last = TRUE)
# RKHS filters minimizing timeliness
rkhs_timeliness <- rkhs_filter(horizon = 6, asymmetricCriterion = "Timeliness")
trend_musgrave <- filter(y, musgrave)
trend_fst <- filter(y, fst_notimeliness)
trend_rkhs <- filter(y, rkhs_timeliness)
plot(ts.union(y, trend_musgrave,trend_fst,trend_rkhs),plot.type = "single",
col = c("black","orange", "lightblue", "red"),
main = "Filtered time series", ylab=NULL)
legend("topleft", legend = c("y","Musgrave", "FST", "RKHS"),
col= c("black","orange", "lightblue", "red"), lty = 1)
The last estimates can also be analysed with the implicit_forecast
function that retreive the implicit forecasts corresponding to the
asymmetric filters (i.e., the forecasts needed to have the same
end-points estimates but using the symmetric filter).
f_musgrave <- implicit_forecast(y, musgrave)
f_fst <- implicit_forecast(y, fst_notimeliness)
f_rkhs <- implicit_forecast(y, rkhs_timeliness)
plot(window(y, start = 2007),
xlim = c(2007,2012), ylim = c(3600, 4600),
main = "Last estimates and implicit forecast", ylab=NULL)
lines(trend_musgrave,
col = "orange")
lines(trend_fst,
col = "lightblue")
lines(trend_rkhs,
col = "red")
lines(ts(c(tail(y,1), f_musgrave), frequency = frequency(y), start = end(y)),
col = "orange", lty = 2)
lines(ts(c(tail(y,1), f_fst), frequency = frequency(y), start = end(y)),
col = "lightblue", lty = 2)
lines(ts(c(tail(y,1), f_rkhs), frequency = frequency(y), start = end(y)),
col = "red", lty = 2)
legend("topleft", legend = c("y","Musgrave", "FST", "RKHS", "Forecasts"),
col= c("black","orange", "lightblue", "red", "black"),
lty = c(1, 1, 1, 1, 2))
The real-time estimates (when no future points are available) can also
be compared:
trend_henderson<- filter(y, musgrave[,"q=6"])
trend_musgrave_q0 <- filter(y, musgrave[,"q=0"])
trend_fst_q0 <- filter(y, fst_notimeliness[,"q=0"])
trend_rkhs_q0 <- filter(y, rkhs_timeliness[,"q=0"])
plot(window(ts.union(y, trend_musgrave_q0,trend_fst_q0,trend_rkhs_q0),
start = 2007),
plot.type = "single",
col = c("black","orange", "lightblue", "red"),
main = "Real time estimates of the trend", ylab=NULL)
legend("topleft", legend = c("y","Musgrave", "FST", "RKHS"),
col= c("black","orange", "lightblue", "red"), lty = 1)
Comparison of the filters
Different quality criteria from Grun-Rehomme et al (2018) and Wildi
and McElroy(2019) can also be computed with the function
diagnostic_matrix():
q_0_coefs <- list(Musgrave = musgrave[,"q=0"],
fst_notimeliness = fst_notimeliness[,"q=0"],
rkhs_timeliness = rkhs_timeliness[,"q=0"])
sapply(q_0_coefs, diagnostic_matrix,
lags = 6, sweight = musgrave[,"q=6"])
#> Musgrave fst_notimeliness rkhs_timeliness
#> b_c 0.000000000 2.220446e-16 0.000000000
#> b_l -0.575984377 -1.554312e-15 -0.611459167
#> b_q -1.144593858 1.554312e-15 0.027626749
#> F_g 0.357509832 9.587810e-01 0.381135700
#> S_g 1.137610871 2.402400e+00 1.207752284
#> T_g 0.034088260 4.676398e-04 0.023197411
#> A_w 0.008306348 1.823745e-02 0.003677964
#> S_w 0.449956378 3.575634e+00 0.628156109
#> T_w 0.061789932 7.940547e-04 0.043540181
#> R_w 0.299548665 1.721377e-01 0.219948644
The filters can also be compared by plotting there coefficients
(plot_coef), gain function (plot_gain) and phase function
(plot_phase):
def.par <- par(no.readonly = TRUE)
par(mai = c(0.3, 0.3, 0.2, 0))
layout(matrix(c(1,1,2,3), 2, 2, byrow = TRUE))
plot_coef(fst_notimeliness, q = 0, col = "lightblue")
plot_coef(musgrave, q = 0, add = TRUE, col = "orange")
plot_coef(rkhs_timeliness, q = 0, add = TRUE, col = "red")
legend("topleft", legend = c("Musgrave", "FST", "RKHS"),
col= c("orange", "lightblue", "red"), lty = 1)
plot_gain(fst_notimeliness, q = 0, col = "lightblue")
plot_gain(musgrave, q = 0, col = "orange",add = TRUE)
plot_gain(rkhs_timeliness, q = 0,add = TRUE, col = "red")
legend("topright", legend = c("Musgrave", "FST", "RKHS"),
col= c("orange", "lightblue", "red"), lty = 1)
plot_phase(fst_notimeliness, q = 0, col = "lightblue")
plot_phase(musgrave, q = 0, col = "orange",add = TRUE)
plot_phase(rkhs_timeliness, q = 0,add = TRUE, col = "red")
legend("topright", legend = c("Musgrave", "FST", "RKHS"),
col= c("orange", "lightblue", "red"), lty = 1)
par(def.par)
Manipulate moving averages
You can also create and manipulate moving averages with the class
moving_average. In the next examples we show how to create the M2X12
moving average, the first moving average used to extract the trend-cycle
in X-11, and the M3X3 moving average, applied to each months to extract
seasonal component.
e1 <- moving_average(rep(1,12), lags = -6)
e1 <- e1/sum(e1)
e2 <- moving_average(rep(1/12, 12), lags = -5)
M2X12 <- (e1 + e2)/2
coef(M2X12)
#> t-6 t-5 t-4 t-3 t-2 t-1 t
#> 0.04166667 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333
#> t+1 t+2 t+3 t+4 t+5 t+6
#> 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333 0.04166667
M3 <- moving_average(rep(1/3, 3), lags = -1)
M3X3 <- M3 * M3
# M3X3 moving average applied to each month
M3X3
#> [1] "0,1111 B^2 + 0,2222 B + 0,3333 + 0,2222 F + 0,1111 F^2"
M3X3_seasonal <- to_seasonal(M3X3, 12)
# M3X3_seasonal moving average applied to the global series
M3X3_seasonal
#> [1] "0,1111 B^24 + 0,2222 B^12 + 0,3333 + 0,2222 F^12 + 0,1111 F^24"
def.par <- par(no.readonly = TRUE)
par(mai = c(0.5, 0.8, 0.3, 0))
layout(matrix(c(1,2), nrow = 1))
plot_gain(M3X3, main = "M3X3 applied to each month")
plot_gain(M3X3_seasonal, main = "M3X3 applied to the global series")
par(def.par)
# To apply the moving average
t <- y * M2X12
si <- y - t
s <- si * M3X3_seasonal
# or equivalently:
s_mm <- M3X3_seasonal * (1 - M2X12)
s <- y * s_mm
Manipulate finite filters
finite_filters object are a combination of a central filter (used for
the final estimates) and different asymmetric filters used for
intermediate estimates at the beginning/end of the series when the
central filter cannot be applied.
musgrave
#> q=6 q=5 q=4 q=3 q=2
#> t-6 -0.01934985 -0.016609040 -0.011623676 -0.009152423 -0.016139228
#> t-5 -0.02786378 -0.025914479 -0.022541271 -0.020981640 -0.024948087
#> t-4 0.00000000 0.001157790 0.002918842 0.003566851 0.002620762
#> t-3 0.06549178 0.065858066 0.066006963 0.065743350 0.067817618
#> t-2 0.14735651 0.146931288 0.145468029 0.144292794 0.149387420
#> t-1 0.21433675 0.213120014 0.210044599 0.207957742 0.216072726
#> t 0.24005716 0.238048915 0.233361346 0.230362866 0.241498208
#> t+1 0.21433675 0.211536998 0.205237273 0.201327171 0.215482871
#> t+2 0.14735651 0.143765257 0.135853376 0.131031652 0.148207710
#> t+3 0.06549178 0.061109020 0.051584983 0.045851637 0.000000000
#> t+4 0.00000000 -0.005174272 -0.016310464 0.000000000 0.000000000
#> t+5 -0.02786378 -0.033829557 0.000000000 0.000000000 0.000000000
#> t+6 -0.01934985 0.000000000 0.000000000 0.000000000 0.000000000
#> q=1 q=0
#> t-6 -0.037925830 -0.07371504
#> t-5 -0.035216813 -0.04601336
#> t-4 0.003869912 0.01806602
#> t-3 0.080584644 0.11977342
#> t-2 0.173672322 0.23785375
#> t-1 0.251875504 0.34104960
#> t 0.288818862 0.40298562
#> t+1 0.274321400 0.00000000
#> t+2 0.000000000 0.00000000
#> t+3 0.000000000 0.00000000
#> t+4 0.000000000 0.00000000
#> t+5 0.000000000 0.00000000
#> t+6 0.000000000 0.00000000
M3X3 <- macurves("S3X3")
musgrave * M3X3
#> q=8 q=7 q=6 q=5 q=4
#> t-8 -0.002149983 -0.002149983 -0.002149983 -0.001845449 -0.001291520
#> t-7 -0.007395941 -0.007395941 -0.007395941 -0.006570284 -0.005087625
#> t-6 -0.012641899 -0.012641899 -0.012641899 -0.011166476 -0.008559414
#> t-5 -0.006311026 -0.006311026 -0.006311026 -0.004754208 -0.002114058
#> t-4 0.022584742 0.022584742 0.022584742 0.023742532 0.025503585
#> t-3 0.075295705 0.075295705 0.075295705 0.075661988 0.075810884
#> t-2 0.137975973 0.137975973 0.137975973 0.137550748 0.136087489
#> t-1 0.188629568 0.188629568 0.188629568 0.187412835 0.184337420
#> t 0.208025720 0.208025720 0.208025720 0.206017479 0.201329910
#> t+1 0.188629568 0.188629568 0.188629568 0.185829819 0.179530094
#> t+2 0.137975973 0.137975973 0.137975973 0.134384717 0.127175208
#> t+3 0.075295705 0.075295705 0.075295705 0.068215408 0.065454326
#> t+4 0.022584742 0.022584742 0.020119429 0.013854891 0.021823700
#> t+5 -0.006311026 -0.007027687 -0.010926323 -0.008333999 0.000000000
#> t+6 -0.012641899 -0.013358560 -0.015107212 0.000000000 0.000000000
#> t+7 -0.007395941 -0.008112602 0.000000000 0.000000000 0.000000000
#> t+8 -0.002149983 0.000000000 0.000000000 0.000000000 0.000000000
#> q=3 q=2 q=1 q=0
#> t-8 -0.0010169359 -0.001793248 -0.004213981 -0.00819056
#> t-7 -0.0043651651 -0.006358505 -0.012340941 -0.02149372
#> t-6 -0.0073170774 -0.010632566 -0.020037912 -0.03278954
#> t-5 -0.0009303015 -0.003784842 -0.010353070 -0.01439608
#> t-4 0.0261515939 0.025205505 0.026454654 0.04065076
#> t-3 0.0755472713 0.077621540 0.090388566 0.12957734
#> t-2 0.1349122536 0.140006880 0.164291782 0.27095547
#> t-1 0.1822505629 0.190365547 0.257185423 0.35665854
#> t 0.1983314300 0.228425968 0.293999778 0.27902779
#> t+1 0.1838694336 0.217863738 0.214625702 0.00000000
#> t+2 0.1375457833 0.143079982 0.000000000 0.00000000
#> t+3 0.0750211512 0.000000000 0.000000000 0.00000000
#> t+4 0.0000000000 0.000000000 0.000000000 0.00000000
#> t+5 0.0000000000 0.000000000 0.000000000 0.00000000
#> t+6 0.0000000000 0.000000000 0.000000000 0.00000000
#> t+7 0.0000000000 0.000000000 0.000000000 0.00000000
#> t+8 0.0000000000 0.000000000 0.000000000 0.00000000
Bibliography
Dagum, Estela Bee and Silvia Bianconcini (2008). “The Henderson Smoother
in Reproducing Kernel Hilbert Space”. In: Journal of Business &
Economic Statistics 26, pp. 536–545. URL:
https://ideas.repec.org/a/bes/jnlbes/v26y2008p536-545.html.
Grun-Rehomme, Michel, Fabien Guggemos, and Dominique Ladiray (2018).
“Asymmetric Moving Averages Minimizing Phase Shift”. In: Handbook on
Seasonal Adjustment. URL:
https://ec.europa.eu/eurostat/web/products-manuals-and-guidelines/-/KS-GQ-18-001.
Proietti, Tommaso and Alessandra Luati (Dec. 2008). “Real time
estimation in local polynomial regression, with application to
trend-cycle analysis”. In: Ann. Appl. Stat. 2.4, pp. 1523–1553. URL:
https://doi.org/10.1214/08-AOAS195.
Wildi, Marc and Tucker McElroy (2019). “The trilemma between accuracy,
timeliness and smoothness in real-time signal extraction”. In:
International Journal of Forecasting 35.3, pp. 1072–1084. URL:
https://EconPapers.repec.org/RePEc:eee:intfor:v:35:y:2019:i:3:p:1072-1084.
palatej/rjdfilters documentation built on May 8, 2023, 6:28 a.m.
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rjd3filters
rjd3filters is R package on linear filters for real-time trend-cycle estimates. It allows to create symmetric and asymmetric moving averages with:
-
local polynomial filters, as defined by Proietti and Luati (2008);
-
the FST approach of Grun-Rehomme, Guggemos, and Ladiray (2018), based on the optimization of the three criteria Fidelity, Smoothness and Timeliness;
-
the Reproducing Kernel Hilbert Space (RKHS) of Dagum and Bianconcini (2008).
Some quality criteria defined by Wildi and McElroy (2019) can also be computed.
Installation
rjd3filters relies on the rJava package and Java SE 17 or later version is required.
# Install development version from GitHub
# install.packages("devtools")
devtools::install_github("palatej/rjd3toolkit")
devtools::install_github("palatej/rj3dfilters")
Basic example
In this example we use the same symmetric moving average (Henderson), but we use three different methods to compute asymmetric filters. As a consequence, the filtered time series is the same, except at the boundaries.
library(rjd3filters)
y <- window(retailsa$AllOtherGenMerchandiseStores,start = 2000)
musgrave <- lp_filter(horizon = 6, kernel = "Henderson",endpoints = "LC")
# we put a large weight on the timeliness criteria
fst_notimeliness_filter <- lapply(0:6, fst_filter,
lags = 6, smoothness.weight = 1/1000,
timeliness.weight = 1-1/1000, pdegree =2)
fst_notimeliness <- finite_filters(sfilter = fst_notimeliness_filter[[7]],
rfilters = fst_notimeliness_filter[-7],
first_to_last = TRUE)
# RKHS filters minimizing timeliness
rkhs_timeliness <- rkhs_filter(horizon = 6, asymmetricCriterion = "Timeliness")
trend_musgrave <- filter(y, musgrave)
trend_fst <- filter(y, fst_notimeliness)
trend_rkhs <- filter(y, rkhs_timeliness)
plot(ts.union(y, trend_musgrave,trend_fst,trend_rkhs),plot.type = "single",
col = c("black","orange", "lightblue", "red"),
main = "Filtered time series", ylab=NULL)
legend("topleft", legend = c("y","Musgrave", "FST", "RKHS"),
col= c("black","orange", "lightblue", "red"), lty = 1)
The last estimates can also be analysed with the implicit_forecast
function that retreive the implicit forecasts corresponding to the
asymmetric filters (i.e., the forecasts needed to have the same
end-points estimates but using the symmetric filter).
f_musgrave <- implicit_forecast(y, musgrave)
f_fst <- implicit_forecast(y, fst_notimeliness)
f_rkhs <- implicit_forecast(y, rkhs_timeliness)
plot(window(y, start = 2007),
xlim = c(2007,2012), ylim = c(3600, 4600),
main = "Last estimates and implicit forecast", ylab=NULL)
lines(trend_musgrave,
col = "orange")
lines(trend_fst,
col = "lightblue")
lines(trend_rkhs,
col = "red")
lines(ts(c(tail(y,1), f_musgrave), frequency = frequency(y), start = end(y)),
col = "orange", lty = 2)
lines(ts(c(tail(y,1), f_fst), frequency = frequency(y), start = end(y)),
col = "lightblue", lty = 2)
lines(ts(c(tail(y,1), f_rkhs), frequency = frequency(y), start = end(y)),
col = "red", lty = 2)
legend("topleft", legend = c("y","Musgrave", "FST", "RKHS", "Forecasts"),
col= c("black","orange", "lightblue", "red", "black"),
lty = c(1, 1, 1, 1, 2))
The real-time estimates (when no future points are available) can also be compared:
trend_henderson<- filter(y, musgrave[,"q=6"])
trend_musgrave_q0 <- filter(y, musgrave[,"q=0"])
trend_fst_q0 <- filter(y, fst_notimeliness[,"q=0"])
trend_rkhs_q0 <- filter(y, rkhs_timeliness[,"q=0"])
plot(window(ts.union(y, trend_musgrave_q0,trend_fst_q0,trend_rkhs_q0),
start = 2007),
plot.type = "single",
col = c("black","orange", "lightblue", "red"),
main = "Real time estimates of the trend", ylab=NULL)
legend("topleft", legend = c("y","Musgrave", "FST", "RKHS"),
col= c("black","orange", "lightblue", "red"), lty = 1)
Comparison of the filters
Different quality criteria from Grun-Rehomme et al (2018) and Wildi
and McElroy(2019) can also be computed with the function
diagnostic_matrix():
q_0_coefs <- list(Musgrave = musgrave[,"q=0"],
fst_notimeliness = fst_notimeliness[,"q=0"],
rkhs_timeliness = rkhs_timeliness[,"q=0"])
sapply(q_0_coefs, diagnostic_matrix,
lags = 6, sweight = musgrave[,"q=6"])
#> Musgrave fst_notimeliness rkhs_timeliness
#> b_c 0.000000000 2.220446e-16 0.000000000
#> b_l -0.575984377 -1.554312e-15 -0.611459167
#> b_q -1.144593858 1.554312e-15 0.027626749
#> F_g 0.357509832 9.587810e-01 0.381135700
#> S_g 1.137610871 2.402400e+00 1.207752284
#> T_g 0.034088260 4.676398e-04 0.023197411
#> A_w 0.008306348 1.823745e-02 0.003677964
#> S_w 0.449956378 3.575634e+00 0.628156109
#> T_w 0.061789932 7.940547e-04 0.043540181
#> R_w 0.299548665 1.721377e-01 0.219948644
The filters can also be compared by plotting there coefficients
(plot_coef), gain function (plot_gain) and phase function
(plot_phase):
def.par <- par(no.readonly = TRUE)
par(mai = c(0.3, 0.3, 0.2, 0))
layout(matrix(c(1,1,2,3), 2, 2, byrow = TRUE))
plot_coef(fst_notimeliness, q = 0, col = "lightblue")
plot_coef(musgrave, q = 0, add = TRUE, col = "orange")
plot_coef(rkhs_timeliness, q = 0, add = TRUE, col = "red")
legend("topleft", legend = c("Musgrave", "FST", "RKHS"),
col= c("orange", "lightblue", "red"), lty = 1)
plot_gain(fst_notimeliness, q = 0, col = "lightblue")
plot_gain(musgrave, q = 0, col = "orange",add = TRUE)
plot_gain(rkhs_timeliness, q = 0,add = TRUE, col = "red")
legend("topright", legend = c("Musgrave", "FST", "RKHS"),
col= c("orange", "lightblue", "red"), lty = 1)
plot_phase(fst_notimeliness, q = 0, col = "lightblue")
plot_phase(musgrave, q = 0, col = "orange",add = TRUE)
plot_phase(rkhs_timeliness, q = 0,add = TRUE, col = "red")
legend("topright", legend = c("Musgrave", "FST", "RKHS"),
col= c("orange", "lightblue", "red"), lty = 1)
par(def.par)
Manipulate moving averages
You can also create and manipulate moving averages with the class
moving_average. In the next examples we show how to create the M2X12
moving average, the first moving average used to extract the trend-cycle
in X-11, and the M3X3 moving average, applied to each months to extract
seasonal component.
e1 <- moving_average(rep(1,12), lags = -6)
e1 <- e1/sum(e1)
e2 <- moving_average(rep(1/12, 12), lags = -5)
M2X12 <- (e1 + e2)/2
coef(M2X12)
#> t-6 t-5 t-4 t-3 t-2 t-1 t
#> 0.04166667 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333
#> t+1 t+2 t+3 t+4 t+5 t+6
#> 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333 0.04166667
M3 <- moving_average(rep(1/3, 3), lags = -1)
M3X3 <- M3 * M3
# M3X3 moving average applied to each month
M3X3
#> [1] "0,1111 B^2 + 0,2222 B + 0,3333 + 0,2222 F + 0,1111 F^2"
M3X3_seasonal <- to_seasonal(M3X3, 12)
# M3X3_seasonal moving average applied to the global series
M3X3_seasonal
#> [1] "0,1111 B^24 + 0,2222 B^12 + 0,3333 + 0,2222 F^12 + 0,1111 F^24"
def.par <- par(no.readonly = TRUE)
par(mai = c(0.5, 0.8, 0.3, 0))
layout(matrix(c(1,2), nrow = 1))
plot_gain(M3X3, main = "M3X3 applied to each month")
plot_gain(M3X3_seasonal, main = "M3X3 applied to the global series")
par(def.par)
# To apply the moving average
t <- y * M2X12
si <- y - t
s <- si * M3X3_seasonal
# or equivalently:
s_mm <- M3X3_seasonal * (1 - M2X12)
s <- y * s_mm
Manipulate finite filters
finite_filters object are a combination of a central filter (used for
the final estimates) and different asymmetric filters used for
intermediate estimates at the beginning/end of the series when the
central filter cannot be applied.
musgrave
#> q=6 q=5 q=4 q=3 q=2
#> t-6 -0.01934985 -0.016609040 -0.011623676 -0.009152423 -0.016139228
#> t-5 -0.02786378 -0.025914479 -0.022541271 -0.020981640 -0.024948087
#> t-4 0.00000000 0.001157790 0.002918842 0.003566851 0.002620762
#> t-3 0.06549178 0.065858066 0.066006963 0.065743350 0.067817618
#> t-2 0.14735651 0.146931288 0.145468029 0.144292794 0.149387420
#> t-1 0.21433675 0.213120014 0.210044599 0.207957742 0.216072726
#> t 0.24005716 0.238048915 0.233361346 0.230362866 0.241498208
#> t+1 0.21433675 0.211536998 0.205237273 0.201327171 0.215482871
#> t+2 0.14735651 0.143765257 0.135853376 0.131031652 0.148207710
#> t+3 0.06549178 0.061109020 0.051584983 0.045851637 0.000000000
#> t+4 0.00000000 -0.005174272 -0.016310464 0.000000000 0.000000000
#> t+5 -0.02786378 -0.033829557 0.000000000 0.000000000 0.000000000
#> t+6 -0.01934985 0.000000000 0.000000000 0.000000000 0.000000000
#> q=1 q=0
#> t-6 -0.037925830 -0.07371504
#> t-5 -0.035216813 -0.04601336
#> t-4 0.003869912 0.01806602
#> t-3 0.080584644 0.11977342
#> t-2 0.173672322 0.23785375
#> t-1 0.251875504 0.34104960
#> t 0.288818862 0.40298562
#> t+1 0.274321400 0.00000000
#> t+2 0.000000000 0.00000000
#> t+3 0.000000000 0.00000000
#> t+4 0.000000000 0.00000000
#> t+5 0.000000000 0.00000000
#> t+6 0.000000000 0.00000000
M3X3 <- macurves("S3X3")
musgrave * M3X3
#> q=8 q=7 q=6 q=5 q=4
#> t-8 -0.002149983 -0.002149983 -0.002149983 -0.001845449 -0.001291520
#> t-7 -0.007395941 -0.007395941 -0.007395941 -0.006570284 -0.005087625
#> t-6 -0.012641899 -0.012641899 -0.012641899 -0.011166476 -0.008559414
#> t-5 -0.006311026 -0.006311026 -0.006311026 -0.004754208 -0.002114058
#> t-4 0.022584742 0.022584742 0.022584742 0.023742532 0.025503585
#> t-3 0.075295705 0.075295705 0.075295705 0.075661988 0.075810884
#> t-2 0.137975973 0.137975973 0.137975973 0.137550748 0.136087489
#> t-1 0.188629568 0.188629568 0.188629568 0.187412835 0.184337420
#> t 0.208025720 0.208025720 0.208025720 0.206017479 0.201329910
#> t+1 0.188629568 0.188629568 0.188629568 0.185829819 0.179530094
#> t+2 0.137975973 0.137975973 0.137975973 0.134384717 0.127175208
#> t+3 0.075295705 0.075295705 0.075295705 0.068215408 0.065454326
#> t+4 0.022584742 0.022584742 0.020119429 0.013854891 0.021823700
#> t+5 -0.006311026 -0.007027687 -0.010926323 -0.008333999 0.000000000
#> t+6 -0.012641899 -0.013358560 -0.015107212 0.000000000 0.000000000
#> t+7 -0.007395941 -0.008112602 0.000000000 0.000000000 0.000000000
#> t+8 -0.002149983 0.000000000 0.000000000 0.000000000 0.000000000
#> q=3 q=2 q=1 q=0
#> t-8 -0.0010169359 -0.001793248 -0.004213981 -0.00819056
#> t-7 -0.0043651651 -0.006358505 -0.012340941 -0.02149372
#> t-6 -0.0073170774 -0.010632566 -0.020037912 -0.03278954
#> t-5 -0.0009303015 -0.003784842 -0.010353070 -0.01439608
#> t-4 0.0261515939 0.025205505 0.026454654 0.04065076
#> t-3 0.0755472713 0.077621540 0.090388566 0.12957734
#> t-2 0.1349122536 0.140006880 0.164291782 0.27095547
#> t-1 0.1822505629 0.190365547 0.257185423 0.35665854
#> t 0.1983314300 0.228425968 0.293999778 0.27902779
#> t+1 0.1838694336 0.217863738 0.214625702 0.00000000
#> t+2 0.1375457833 0.143079982 0.000000000 0.00000000
#> t+3 0.0750211512 0.000000000 0.000000000 0.00000000
#> t+4 0.0000000000 0.000000000 0.000000000 0.00000000
#> t+5 0.0000000000 0.000000000 0.000000000 0.00000000
#> t+6 0.0000000000 0.000000000 0.000000000 0.00000000
#> t+7 0.0000000000 0.000000000 0.000000000 0.00000000
#> t+8 0.0000000000 0.000000000 0.000000000 0.00000000
Bibliography
Dagum, Estela Bee and Silvia Bianconcini (2008). “The Henderson Smoother in Reproducing Kernel Hilbert Space”. In: Journal of Business & Economic Statistics 26, pp. 536–545. URL: https://ideas.repec.org/a/bes/jnlbes/v26y2008p536-545.html.
Grun-Rehomme, Michel, Fabien Guggemos, and Dominique Ladiray (2018). “Asymmetric Moving Averages Minimizing Phase Shift”. In: Handbook on Seasonal Adjustment. URL: https://ec.europa.eu/eurostat/web/products-manuals-and-guidelines/-/KS-GQ-18-001.
Proietti, Tommaso and Alessandra Luati (Dec. 2008). “Real time estimation in local polynomial regression, with application to trend-cycle analysis”. In: Ann. Appl. Stat. 2.4, pp. 1523–1553. URL: https://doi.org/10.1214/08-AOAS195.
Wildi, Marc and Tucker McElroy (2019). “The trilemma between accuracy, timeliness and smoothness in real-time signal extraction”. In: International Journal of Forecasting 35.3, pp. 1072–1084. URL: https://EconPapers.repec.org/RePEc:eee:intfor:v:35:y:2019:i:3:p:1072-1084.
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