Package 'mFilter' reference manual

Title:	Miscellaneous Time Series Filters
Description:	The mFilter package implements several time series filters useful for smoothing and extracting trend and cyclical components of a time series. The routines are commonly used in economics and finance, however they should also be interest to other areas. Currently, Christiano-Fitzgerald, Baxter-King, Hodrick-Prescott, Butterworth, and trigonometric regression filters are included in the package.
Authors:	Mehmet Balcilar <[email protected]>
Maintainer:	Mehmet Balcilar <[email protected]>
License:	GPL (>= 2)
Version:	0.1-6
Built:	2025-02-07 03:11:39 UTC
Source:	https://github.com/mbalcilar/mfilter

Getting started with the mFilter package

Description

Getting started with the mFilter package

Details

This package provides some tools for decomposing time series into trend (smooth) and cyclical (irregular) components. The package implements come commonly used filters such as the Hodrick-Prescott, Baxter-King and Christiano-Fitzgerald filter.

For loading the package, type:

library(mFilter)

A good place to start learning the package usage is to examine examples for the mFilter function. At the R prompt, write:

example("mFilter")

For a full list of functions exported by the package, type:

ls("package:mFilter")

Each exported function has a corresponding man page (some man pages are common to more functions). Display it by typing

help(functionName).

Almost all filters in this package can be put into the following framework. Given a time series $\{x_t\}^T_{t=1}$ we are interested in isolating component of $x_t$ , denoted $y_t$ with period of oscillations between $p_l$ and $p_u$ , where $2 \le p_l < p_u < \infty$ .

Consider the following decomposition of the time series

$x_t = y_t + \bar{x}_t$

The component $y_t$ is assumed to have power only in the frequencies in the interval $\{(a,b) \cup (-a,-b)\} \in (-\pi, \pi)$ . $a$ and $b$ are related to $p_l$ and $p_u$ by

$a=\frac{2 \pi}{p_u}\ \ \ \ \ {b=\frac{2 \pi}{p_l}}$

If infinite amount of data is available, then we can use the ideal bandpass filter

$y_t = B(L)x_t$

where the filter, $B(L)$ , is given in terms of the lag operator $L$ and defined as

$B(L) = \sum^\infty_{j=-\infty} B_j L^j, \ \ \ L^k x_t = x_{t-k}$

The ideal bandpass filter weights are given by

$B_j = \frac{\sin(jb)-\sin(ja)}{\pi j}$

$B_0=\frac{b-a}{\pi}$

The finite sample approximation to the ideal bandpass filter uses the alternative filter

$y_t = \hat{B}(L)x_t=\sum^{n_2}_{j=-n_1}\hat{B}_{t,j} x_{t+j}$

Here the weights, $\hat{B}_{t,j}$ , of the approximation is a solution to

$\hat{B}_{t,j}= \arg \min E \{ (y_t-\hat{y}_t)^2 \}$

The Christiano-Fitzgerald filter is a finite data approximation to the ideal bandpass filter and minimizes the mean squared error defined in the above equation.

Several band-pass approximation strategies can be selected in the function cffilter. The default setting of cffilter returns the filtered data $\hat{y_t}$ associated with the unrestricted optimal filter assuming no unit root, no drift and an iid filter.

If theta is not equal to 1 the series is assumed to follow a moving average process. The moving average weights are given by theta. The default is theta=1 (iid series). If theta $=(\theta_1, \theta_2, \dots)$ then the series is assumed to be

$x_t = \mu + 1_{root} x_{t-1} + \theta_1 e_t + \theta_2 e_{t-1} + \dots$

where $1_{root}=1$ if the option root=1 and $1_{root}=0$ if the option root=0, and $e_t$ is a white noise.

The Baxter-King filter is a finite data approximation to the ideal bandpass filter with following moving average weights

$y_t = \hat{B}(L)x_t=\sum^{n}_{j=-n}\hat{B}_{j} x_{t+j}=\hat{B}_0 x_t + \sum^{n}_{j=1} \hat{B}_j (x_{t-j}+x_{t+j})$

where

$\hat{B}_j=B_j-\frac{1}{2n+1}\sum^{n}_{j=-n} B_{j}$

The Hodrick-Prescott filter obtains the filter weights $\hat{B}_j$ as a solution to

$\hat{B}_{j}= \arg \min E \{ (y_t-\hat{y}_t)^2 \} = \arg \min \left\{ \sum^{T}_{t=1}(y_t-\hat{y}_{t})^2 + \lambda\sum^{T-1}_{t=2}(\hat{y}_{t+1}-2\hat{y}_{t}+\hat{y}_{t-1})^2 \right\}$

The Hodrick-Prescott filter is a finite data approximation with following moving average weights

$\hat{B}_j=\frac{1}{2\pi}\int^{\pi}_{-\pi} \frac{4\lambda(1-\cos(\omega))^2}{1+4\lambda(1-\cos(\omega))^2}e^{i \omega j} d \omega$

The digital version of the Butterworth highpass filter is described by the rational polynomial expression (the filter's z-transform)

$\frac{\lambda(1-z)^n(1-z^{-1})^n}{(1+z)^n(1+z^{-1})^n+\lambda(1-z)^n(1-z^{-1})^n}$

The time domain version can be obtained by substituting $z$ for the lag operator $L$ .

Pollock (2000) derives a specialized finite-sample version of the Butterworth filter on the basis of signal extraction theory. Let $s_t$ be the trend and $c_t$ cyclical component of $y_t$ , then these components are extracted as

$y_t=s_t+c_t=\frac{(1+L)^n}{(1-L)^d}\nu_t+(1-L)^{n-d}\varepsilon_t$

where $\nu_t \sim N(0,\sigma_\nu^2)$ and $\varepsilon_t \sim N(0,\sigma_\varepsilon^2)$ .

Let $T$ be even and define $n_1=T/p_u$ and $n_2=T/p_l$ . The trigonometric regression filter is based on the following relation

${y}_t=\sum^{n_1}_{j=n_2}\left\{ a_j \cos(\omega_j t) + b_j \sin(\omega_j t) \right\}$

where $a_j$ and $b_j$ are the coefficients obtained by regressing $x_t$ on the indicated sine and cosine functions. Specifically,

$a_j=\frac{T}{2}\sum^{T}_{t=1}\cos(\omega_j t) x_t,\ \ \$ for $j=1,\dots,T/2-1$

$a_j=\frac{T}{2}\sum^{T}_{t=1}\cos(\pi t) x_t,\ \ \$ for $j=T/2$

and

$b_j=\frac{T}{2}\sum^{T}_{t=1}\sin(\omega_j t) x_t,\ \ \$ for $j=1,\dots,T/2-1$

$b_j=\frac{T}{2}\sum^{T}_{t=1}\sin(\pi t) x_t,\ \ \$ for $j=T/2$

Let $\hat{B}(L) x_t$ be the trigonometric regression filter. It can be showed that $\hat{B}(1)=0$ , so that $\hat{B}(L)$ has a unit root for $t=1,2,\dots,T$ . Also, when $\hat{B}(L)$ is symmetric, it has a second unit root in the middle of the data for $t$ . Therefore it is important to drift adjust data before it is filtered with a trigonometric regression filter.

If drift=TRUE the drift adjusted series is obtained as

$\tilde{x}_{t}=x_t-t\left(\frac{x_{T}-x_{1}}{T-1}\right), \ \ t=0,1,\dots,T-1$

where $\tilde{x}_{t}$ is the undrifted series.

How to cite this package

Whenever using this package, please cite as

Balcilar, M. (2019). mFilter: Miscellaneous Time Series Filters. R package version 1.5-0. https://github.com/mbalcilar/mFilter.

@Manual{Balcilar_2019,
 author       = {Mehmet Balcilar},
 title        = {{mFilter}: Miscellaneous Time Series Filters.},
 year         = {2019},
 note         = {R package version 1.5-0.},
 URL          = {https://github.com/mbalcilar/mFilter}}

License

The releases of this package is licensed under GPL version 3.

Author(s)

Mehmet Balcilar, [email protected]

References

M. Baxter and R.G. King. Measuring business cycles: Approximate bandpass filters. The Review of Economics and Statistics, 81(4):575-93, 1999.

L. Christiano and T.J. Fitzgerald. The bandpass filter. International Economic Review, 44(2):435-65, 2003.

J. D. Hamilton. Time series analysis. Princeton, 1994.

R.J. Hodrick and E.C. Prescott. Postwar US business cycles: an empirical investigation. Journal of Money, Credit, and Banking, 29(1):1-16, 1997.

R.G. King and S.T. Rebelo. Low frequency filtering and real business cycles. Journal of Economic Dynamics and Control, 17(1-2):207-31, 1993.

D.S.G. Pollock. Trend estimation and de-trending via rational square-wave filters. Journal of Econometrics, 99:317-334, 2000.

Baxter-King filter of a time series

Description

This function implements the Baxter-King approximation to the band pass filter for a time series. The function computes cyclical and trend components of the time series using band-pass approximation for fixed and variable length filters.

Usage

bkfilter(x,pl=NULL,pu=NULL,nfix=NULL,type=c("fixed","variable"),drift=FALSE)
bkfilter(x,pl=NULL,pu=NULL,nfix=NULL,type=c("fixed","variable"),drift=FALSE)

Arguments

`x`	a regular time series
`type`	character, indicating the filter type, `"fixed"`, for the fixed length Baxter-King filter (default), `"variable"`, for the variable length Baxter-King filter.
`pl`	integer. minimum period of oscillation of desired component (pl<=2).
`pu`	integer. maximum period of oscillation of desired component (2<=pl<pu<infinity).
`drift`	logical, `FALSE` if no drift in time series (default), `TRUE` if drift in time series.
`nfix`	sets fixed lead/lag length or order of the filter. The `nfix` option sets the order of the filter by 2nfix+1. The default is `frequency(x)3`.

Details

Consider the following decomposition of the time series

$x_t = y_t + \bar{x}_t$

The component $y_t$ is assumed to have power only in the frequencies in the interval $\{(a,b) \cup (-a,-b)\} \in (-\pi, \pi)$ . $a$ and $b$ are related to $p_l$ and $p_u$ by

$a=\frac{2 \pi}{p_u}\ \ \ \ \ {b=\frac{2 \pi}{p_l}}$

If infinite amount of data is available, then we can use the ideal bandpass filter

$y_t = B(L)x_t$

where the filter, $B(L)$ , is given in terms of the lag operator $L$ and defined as

$B(L) = \sum^\infty_{j=-\infty} B_j L^j, \ \ \ L^k x_t = x_{t-k}$

The ideal bandpass filter weights are given by

$B_j = \frac{\sin(jb)-\sin(ja)}{\pi j}$

$B_0=\frac{b-a}{\pi}$

The Baxter-King filter is a finite data approximation to the ideal bandpass filter with following moving average weights

$y_t = \hat{B}(L)x_t=\sum^{n}_{j=-n}\hat{B}_{j} x_{t+j}=\hat{B}_0 x_t + \sum^{n}_{j=1} \hat{B}_j (x_{t-j}+x_{t+j})$

where

$\hat{B}_j=B_j-\frac{1}{2n+1}\sum^{n}_{j=-n}B_{j}$

If drift=TRUE the drift adjusted series is obtained

$\tilde{x}_{t}=x_t-t\left(\frac{x_{T}-x_{1}}{T-1}\right), \ \ t=0,1,\dots,T-1$

where $\tilde{x}_{t}$ is the undrifted series.

Value

A "mFilter" object (see mFilter).

Author(s)

Mehmet Balcilar, [email protected]

References

M. Baxter and R.G. King. Measuring business cycles: Approximate bandpass filters. The Review of Economics and Statistics, 81(4):575-93, 1999.

L. Christiano and T.J. Fitzgerald. The bandpass filter. International Economic Review, 44(2):435-65, 2003.

J. D. Hamilton. Time series analysis. Princeton, 1994.

R.J. Hodrick and E.C. Prescott. Postwar US business cycles: an empirical investigation. Journal of Money, Credit, and Banking, 29(1):1-16, 1997.

R.G. King and S.T. Rebelo. Low frequency filtering and real business cycles. Journal of Economic Dynamics and Control, 17(1-2):207-31, 1993.

D.S.G. Pollock. Trend estimation and de-trending via rational square-wave filters. Journal of Econometrics, 99:317-334, 2000.

Examples

## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.bk <- bkfilter(unemp)
plot(unemp.bk)
unemp.bk1 <- bkfilter(unemp, drift=TRUE)
unemp.bk2 <- bkfilter(unemp, pl=8,pu=40,drift=TRUE)
unemp.bk3 <- bkfilter(unemp, pl=2,pu=60,drift=TRUE)
unemp.bk4 <- bkfilter(unemp, pl=2,pu=40,drift=TRUE)

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.bk1$x,
    main="Baxter-King filter of unemployment: Trend, drift=TRUE",
    col=1, ylab="")
lines(unemp.bk1$trend,col=2)
lines(unemp.bk2$trend,col=3)
lines(unemp.bk3$trend,col=4)
lines(unemp.bk4$trend,col=5)
legend("topleft",legend=c("series", "pl=2, pu=32", "pl=8, pu=40",
      "pl=2, pu=60", "pl=2, pu=40"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.bk1$cycle,
main="Baxter-King filter of unemployment: Cycle,drift=TRUE",
      col=2, ylab="", ylim=range(unemp.bk3$cycle,na.rm=TRUE))
lines(unemp.bk2$cycle,col=3)
lines(unemp.bk3$cycle,col=4)
lines(unemp.bk4$cycle,col=5)
## legend("topleft",legend=c("pl=2, pu=32", "pl=8, pu=40", "pl=2, pu=60",
## "pl=2, pu=40"), col=1:5, lty=rep(1,5), ncol=1)

par(opar)
## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.bk <- bkfilter(unemp)
plot(unemp.bk)
unemp.bk1 <- bkfilter(unemp, drift=TRUE)
unemp.bk2 <- bkfilter(unemp, pl=8,pu=40,drift=TRUE)
unemp.bk3 <- bkfilter(unemp, pl=2,pu=60,drift=TRUE)
unemp.bk4 <- bkfilter(unemp, pl=2,pu=40,drift=TRUE)

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.bk1$x,
    main="Baxter-King filter of unemployment: Trend, drift=TRUE",
    col=1, ylab="")
lines(unemp.bk1$trend,col=2)
lines(unemp.bk2$trend,col=3)
lines(unemp.bk3$trend,col=4)
lines(unemp.bk4$trend,col=5)
legend("topleft",legend=c("series", "pl=2, pu=32", "pl=8, pu=40",
      "pl=2, pu=60", "pl=2, pu=40"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.bk1$cycle,
main="Baxter-King filter of unemployment: Cycle,drift=TRUE",
      col=2, ylab="", ylim=range(unemp.bk3$cycle,na.rm=TRUE))
lines(unemp.bk2$cycle,col=3)
lines(unemp.bk3$cycle,col=4)
lines(unemp.bk4$cycle,col=5)
## legend("topleft",legend=c("pl=2, pu=32", "pl=8, pu=40", "pl=2, pu=60",
## "pl=2, pu=40"), col=1:5, lty=rep(1,5), ncol=1)

par(opar)

Butterworth filter of a time series

Description

Filters a time series using the Butterworth square-wave highpass filter described in Pollock (2000).

Usage

bwfilter(x,freq=NULL,nfix=NULL,drift=FALSE)
bwfilter(x,freq=NULL,nfix=NULL,drift=FALSE)

Arguments

`x`	a regular time series
`nfix`	sets the order of the filter. The default is `nfix=2`, when `nfix=NULL`.
`freq`	integer, the cut-off frequency of the Butterworth filter. The default is `trunc(2.5*frequency(x))`.
`drift`	logical, `FALSE` if no drift in time series (default), `TRUE` if drift in time series.

Details

Consider the following decomposition of the time series

$x_t = y_t + \bar{x}_t$

The component $y_t$ is assumed to have power only in the frequencies in the interval $\{(a,b) \cup (-a,-b)\} \in (-\pi, \pi)$ . $a$ and $b$ are related to $p_l$ and $p_u$ by

$a=\frac{2 \pi}{p_u}\ \ \ \ \ {b=\frac{2 \pi}{p_l}}$

If infinite amount of data is available, then we can use the ideal bandpass filter

$y_t = B(L)x_t$

where the filter, $B(L)$ , is given in terms of the lag operator $L$ and defined as

$B(L) = \sum^\infty_{j=-\infty} B_j L^j, \ \ \ L^k x_t = x_{t-k}$

The ideal bandpass filter weights are given by

$B_j = \frac{\sin(jb)-\sin(ja)}{\pi j}$

$B_0=\frac{b-a}{\pi}$

The digital version of the Butterworth highpass filter is described by the rational polynomial expression (the filter's z-transform)

$\frac{\lambda(1-z)^n(1-z^{-1})^n}{(1+z)^n(1+z^{-1})^n+\lambda(1-z)^n(1-z^{-1})^n}$

The time domain version can be obtained by substituting $z$ for the lag operator $L$ .

Pollock derives a specialized finite-sample version of the Butterworth filter on the basis of signal extraction theory. Let $s_t$ be the trend and $c_t$ cyclical component of $y_t$ , then these components are extracted as

$y_t=s_t+c_t=\frac{(1+L)^n}{(1-L)^d}\nu_t+(1-L)^{n-d}\varepsilon_t$

where $\nu_t \sim N(0,\sigma_\nu^2)$ and $\varepsilon_t \sim N(0,\sigma_\varepsilon^2)$ .

If drift=TRUE the drift adjusted series is obtained as

$\tilde{x}_{t}=x_t-t\left(\frac{x_{T}-x_{1}}{T-1}\right), \ \ t=0,1,\dots,T-1$

where $\tilde{x}_{t}$ is the undrifted series.

Value

A "mFilter" object (see mFilter).

Author(s)

Mehmet Balcilar, [email protected]

References

M. Baxter and R.G. King. Measuring business cycles: Approximate bandpass filters. The Review of Economics and Statistics, 81(4):575-93, 1999.

L. Christiano and T.J. Fitzgerald. The bandpass filter. International Economic Review, 44(2):435-65, 2003.

J. D. Hamilton. Time series analysis. Princeton, 1994.

R.J. Hodrick and E.C. Prescott. Postwar US business cycles: an empirical investigation. Journal of Money, Credit, and Banking, 29(1):1-16, 1997.

R.G. King and S.T. Rebelo. Low frequency filtering and real business cycles. Journal of Economic Dynamics and Control, 17(1-2):207-31, 1993.

D.S.G. Pollock. Trend estimation and de-trending via rational square-wave filters. Journal of Econometrics, 99:317-334, 2000.

Examples

## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.bw <- bwfilter(unemp)
plot(unemp.bw)
unemp.bw1 <- bwfilter(unemp, drift=TRUE)
unemp.bw2 <- bwfilter(unemp, freq=8,drift=TRUE)
unemp.bw3 <- bwfilter(unemp, freq=10, nfix=3, drift=TRUE)
unemp.bw4 <- bwfilter(unemp, freq=10, nfix=4, drift=TRUE)

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.bw1$x,
     main="Butterworth filter of unemployment: Trend,
     drift=TRUE",col=1, ylab="")
lines(unemp.bw1$trend,col=2)
lines(unemp.bw2$trend,col=3)
lines(unemp.bw3$trend,col=4)
lines(unemp.bw4$trend,col=5)
legend("topleft",legend=c("series", "freq=10, nfix=2",
       "freq=8, nfix=2", "freq=10, nfix=3", "freq=10, nfix=4"),
       col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.bw1$cycle,
     main="Butterworth filter of unemployment: Cycle,drift=TRUE",
     col=2, ylab="", ylim=range(unemp.bw3$cycle,na.rm=TRUE))
lines(unemp.bw2$cycle,col=3)
lines(unemp.bw3$cycle,col=4)
lines(unemp.bw4$cycle,col=5)
## legend("topleft",legend=c("series", "freq=10, nfix=2", "freq=8,
## nfix=2", "freq## =10, nfix=3", "freq=10, nfix=4"), col=1:5,
## lty=rep(1,5), ncol=1)

par(opar)
## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.bw <- bwfilter(unemp)
plot(unemp.bw)
unemp.bw1 <- bwfilter(unemp, drift=TRUE)
unemp.bw2 <- bwfilter(unemp, freq=8,drift=TRUE)
unemp.bw3 <- bwfilter(unemp, freq=10, nfix=3, drift=TRUE)
unemp.bw4 <- bwfilter(unemp, freq=10, nfix=4, drift=TRUE)

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.bw1$x,
     main="Butterworth filter of unemployment: Trend,
     drift=TRUE",col=1, ylab="")
lines(unemp.bw1$trend,col=2)
lines(unemp.bw2$trend,col=3)
lines(unemp.bw3$trend,col=4)
lines(unemp.bw4$trend,col=5)
legend("topleft",legend=c("series", "freq=10, nfix=2",
       "freq=8, nfix=2", "freq=10, nfix=3", "freq=10, nfix=4"),
       col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.bw1$cycle,
     main="Butterworth filter of unemployment: Cycle,drift=TRUE",
     col=2, ylab="", ylim=range(unemp.bw3$cycle,na.rm=TRUE))
lines(unemp.bw2$cycle,col=3)
lines(unemp.bw3$cycle,col=4)
lines(unemp.bw4$cycle,col=5)
## legend("topleft",legend=c("series", "freq=10, nfix=2", "freq=8,
## nfix=2", "freq## =10, nfix=3", "freq=10, nfix=4"), col=1:5,
## lty=rep(1,5), ncol=1)

par(opar)

Christiano-Fitzgerald filter of a time series

Description

This function implements the Christiano-Fitzgerald approximation to the ideal band pass filter for a time series. The function computes cyclical and trend components of the time series using several band-pass approximation strategies.

Usage

cffilter(x,pl=NULL,pu=NULL,root=FALSE,drift=FALSE,
         type=c("asymmetric","symmetric","fixed","baxter-king","trigonometric"),
	 nfix=NULL,theta=1)
cffilter(x,pl=NULL,pu=NULL,root=FALSE,drift=FALSE,
         type=c("asymmetric","symmetric","fixed","baxter-king","trigonometric"),
	 nfix=NULL,theta=1)

Arguments

`x`	a regular time series.
`type`	the filter type, `"asymmetric"`, asymmetric Christiano-Fitzgerald filter (default), `"symmetric"`, symmetric Christiano-Fitzgerald filter `"fixed"`, fixed length Christiano-Fitzgerald filter, `"baxter-king"`, Baxter-King fixed length symmetric filter, `"trigonometric"`, trigonometric regression filter.
`pl`	minimum period of oscillation of desired component (pl<=2).
`pu`	maximum period of oscillation of desired component (2<=pl<pu<infinity).
`root`	logical, `FALSE` if no unit root in time series (default), `TRUE` if unit root in time series. The `root` option has no effect if `type` is `"baxter-king"` or `"trigonometric"`.
`drift`	logical, `FALSE` if no drift in time series (default), `TRUE` if drift in time series.
`nfix`	sets fixed lead/lag length or order of the filter with `"baxter-king"` and `"fixed"`. The `nfix` option sets the order of the filter by 2*nfix+1. The default is `nfix=1`.
`theta`	moving average coefficients for time series model: x(t) = mu + rootx(t-1) + theta(1)e(t) + theta(2)*e(t-1) + ..., where e(t) is a white noise.

Details

Consider the following decomposition of the time series

$x_t = y_t + \bar{x}_t$

The component $y_t$ is assumed to have power only in the frequencies in the interval $\{(a,b) \cup (-a,-b)\} \in (-\pi, \pi)$ . $a$ and $b$ are related to $p_l$ and $p_u$ by

$a=\frac{2 \pi}{p_u}\ \ \ \ \ {b=\frac{2 \pi}{p_l}}$

If infinite amount of data is available, then we can use the ideal bandpass filter

$y_t = B(L)x_t$

where the filter, $B(L)$ , is given in terms of the lag operator $L$ and defined as

$B(L) = \sum^\infty_{j=-\infty} B_j L^j, \ \ \ L^k x_t = x_{t-k}$

The ideal bandpass filter weights are given by

$B_j = \frac{\sin(jb)-\sin(ja)}{\pi j}$

$B_0=\frac{b-a}{\pi}$

The finite sample approximation to the ideal bandpass filter uses the alternative filter

$y_t = \hat{B}(L)x_t=\sum^{n_2}_{j=-n_1}\hat{B}_{t,j} x_{t+j}$

Here the weights, $\hat{B}_{t,j}$ , of the approximation is a solution to

$\hat{B}_{t,j}= \arg \min E \{ (y_t-\hat{y}_t)^2 \}$

The Christiano-Fitzgerald filter is a finite data approximation to the ideal bandpass filter and minimizes the mean squared error defined in the above equation.

$x_t = \mu + 1_{root} x_{t-1} + \theta_1 e_t + \theta_2 e_{t-1} + \dots$

where $1_{root}=1$ if the option root=1 and $1_{root}=0$ if the option root=0, and $e_t$ is a white noise.

If drift=TRUE the drift adjusted series is obtained as

$\tilde{x}_{t}=x_t-t\left(\frac{x_{T}-x_{1}}{T-1}\right), \ \ t=0,1,\dots,T-1$

where $\tilde{x}_{t}$ is the undrifted series.

Value

A "mFilter" object (see mFilter).

Author(s)

Mehmet Balcilar, [email protected]

References

M. Baxter and R.G. King. Measuring business cycles: Approximate bandpass filters. The Review of Economics and Statistics, 81(4):575-93, 1999.

L. Christiano and T.J. Fitzgerald. The bandpass filter. International Economic Review, 44(2):435-65, 2003.

J. D. Hamilton. Time series analysis. Princeton, 1994.

R.J. Hodrick and E.C. Prescott. Postwar US business cycles: an empirical investigation. Journal of Money, Credit, and Banking, 29(1):1-16, 1997.

R.G. King and S.T. Rebelo. Low frequency filtering and real business cycles. Journal of Economic Dynamics and Control, 17(1-2):207-31, 1993.

D.S.G. Pollock. Trend estimation and de-trending via rational square-wave filters. Journal of Econometrics, 99:317-334, 2000.

Examples

## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.cf <- cffilter(unemp)
plot(unemp.cf)
unemp.cf1 <- cffilter(unemp, drift=TRUE, root=TRUE)
unemp.cf2 <- cffilter(unemp, pl=8,pu=40,drift=TRUE, root=TRUE)
unemp.cf3 <- cffilter(unemp, pl=2,pu=60,drift=TRUE, root=TRUE)
unemp.cf4 <- cffilter(unemp, pl=2,pu=40,drift=TRUE, root=TRUE,theta=c(.1,.4))

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.cf1$x,
main="Christiano-Fitzgerald filter of unemployment: Trend \n root=TRUE,drift=TRUE",
col=1, ylab="")
lines(unemp.cf1$trend,col=2)
lines(unemp.cf2$trend,col=3)
lines(unemp.cf3$trend,col=4)
lines(unemp.cf4$trend,col=5)
legend("topleft",legend=c("series", "pl=2, pu=32", "pl=8, pu=40", "pl=2, pu=60",
"pl=2, pu=40, theta=.1,.4"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.cf1$cycle,
main="Christiano-Fitzgerald filter of unemployment: Cycle \n root=TRUE,drift=TRUE",
col=2, ylab="", ylim=range(unemp.cf3$cycle))
lines(unemp.cf2$cycle,col=3)
lines(unemp.cf3$cycle,col=4)
lines(unemp.cf4$cycle,col=5)
## legend("topleft",legend=c("pl=2, pu=32", "pl=8, pu=40", "pl=2, pu=60",
## "pl=2, pu=40, theta=.1,.4"), col=2:5, lty=rep(1,4), ncol=2)

par(opar)
## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.cf <- cffilter(unemp)
plot(unemp.cf)
unemp.cf1 <- cffilter(unemp, drift=TRUE, root=TRUE)
unemp.cf2 <- cffilter(unemp, pl=8,pu=40,drift=TRUE, root=TRUE)
unemp.cf3 <- cffilter(unemp, pl=2,pu=60,drift=TRUE, root=TRUE)
unemp.cf4 <- cffilter(unemp, pl=2,pu=40,drift=TRUE, root=TRUE,theta=c(.1,.4))

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.cf1$x,
main="Christiano-Fitzgerald filter of unemployment: Trend \n root=TRUE,drift=TRUE",
col=1, ylab="")
lines(unemp.cf1$trend,col=2)
lines(unemp.cf2$trend,col=3)
lines(unemp.cf3$trend,col=4)
lines(unemp.cf4$trend,col=5)
legend("topleft",legend=c("series", "pl=2, pu=32", "pl=8, pu=40", "pl=2, pu=60",
"pl=2, pu=40, theta=.1,.4"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.cf1$cycle,
main="Christiano-Fitzgerald filter of unemployment: Cycle \n root=TRUE,drift=TRUE",
col=2, ylab="", ylim=range(unemp.cf3$cycle))
lines(unemp.cf2$cycle,col=3)
lines(unemp.cf3$cycle,col=4)
lines(unemp.cf4$cycle,col=5)
## legend("topleft",legend=c("pl=2, pu=32", "pl=8, pu=40", "pl=2, pu=60",
## "pl=2, pu=40, theta=.1,.4"), col=2:5, lty=rep(1,4), ncol=2)

par(opar)

Hodrick-Prescott filter of a time series

Description

This function implements the Hodrick-Prescott for estimating cyclical and trend component of a time series. The function computes cyclical and trend components of the time series using a frequency cut-off or smoothness parameter.

Usage

hpfilter(x,freq=NULL,type=c("lambda","frequency"),drift=FALSE)
hpfilter(x,freq=NULL,type=c("lambda","frequency"),drift=FALSE)

Arguments

`x`	a regular time series.
`type`	character, indicating the filter type, `"lambda"`, for the filter that uses smoothness penalty parameter of the Hodrick-Prescott filter (default), `"frequency"`, for the filter that uses a frequency cut-off type Hodrick-Prescott filter. These are related by $lambda = (2*sin(pi/frequency))^{-4}$ .
`freq`	integer, if `type="lambda"` then `freq` is the smoothing parameter (lambda) of the Hodrick-Prescott filter, if `type="frequency"` then `freq` is the cut-off frequency of the Hodrick-Prescott filter.
`drift`	logical, `FALSE` if no drift in time series (default), `TRUE` if drift in time series.

Details

Consider the following decomposition of the time series

$x_t = y_t + \bar{x}_t$

The component $y_t$ is assumed to have power only in the frequencies in the interval $\{(a,b) \cup (-a,-b)\} \in (-\pi, \pi)$ . $a$ and $b$ are related to $p_l$ and $p_u$ by

$a=\frac{2 \pi}{p_u}\ \ \ \ \ {b=\frac{2 \pi}{p_l}}$

If infinite amount of data is available, then we can use the ideal bandpass filter

$y_t = B(L)x_t$

where the filter, $B(L)$ , is given in terms of the lag operator $L$ and defined as

$B(L) = \sum^\infty_{j=-\infty} B_j L^j, \ \ \ L^k x_t = x_{t-k}$

The ideal bandpass filter weights are given by

$B_j = \frac{\sin(jb)-\sin(ja)}{\pi j}$

$B_0=\frac{b-a}{\pi}$

The Hodrick-Prescott filter obtains the filter weights $\hat{B}_j$ as a solution to

$\hat{B}_{j}= \arg \min E \{ (y_t-\hat{y}_t)^2 \} = \arg \min \left\{ \sum^{T}_{t=1}(y_t-\hat{y}_{t})^2 + \lambda\sum^{T-1}_{t=2}(\hat{y}_{t+1}-2\hat{y}_{t}+\hat{y}_{t-1})^2 \right\}$

The Hodrick-Prescott filter is a finite data approximation with following moving average weights

$\hat{B}_j=\frac{1}{2\pi}\int^{\pi}_{-\pi} \frac{4\lambda(1-\cos(\omega))^2}{1+4\lambda(1-\cos(\omega))^2}e^{i \omega j} d \omega$

If drift=TRUE the drift adjusted series is obtained as

$\tilde{x}_{t}=x_t-t\left(\frac{x_{T}-x_{1}}{T-1}\right), \ \ t=0,1,\dots,T-1$

where $\tilde{x}_{t}$ is the undrifted series.

Value

A "mFilter" object (see mFilter).

Author(s)

Mehmet Balcilar, [email protected]

References

M. Baxter and R.G. King. Measuring business cycles: Approximate bandpass filters. The Review of Economics and Statistics, 81(4):575-93, 1999.

L. Christiano and T.J. Fitzgerald. The bandpass filter. International Economic Review, 44(2):435-65, 2003.

J. D. Hamilton. Time series analysis. Princeton, 1994.

R.J. Hodrick and E.C. Prescott. Postwar US business cycles: an empirical investigation. Journal of Money, Credit, and Banking, 29(1):1-16, 1997.

R.G. King and S.T. Rebelo. Low frequency filtering and real business cycles. Journal of Economic Dynamics and Control, 17(1-2):207-31, 1993.

D.S.G. Pollock. Trend estimation and de-trending via rational square-wave filters. Journal of Econometrics, 99:317-334, 2000.

Examples

## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.hp <- hpfilter(unemp)
plot(unemp.hp)
unemp.hp1 <- hpfilter(unemp, drift=TRUE)
unemp.hp2 <- hpfilter(unemp, freq=800, drift=TRUE)
unemp.hp3 <- hpfilter(unemp, freq=12,type="frequency",drift=TRUE)
unemp.hp4 <- hpfilter(unemp, freq=52,type="frequency",drift=TRUE)

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.hp1$x,  ylim=c(2,13),
main="Hodrick-Prescott filter of unemployment: Trend, drift=TRUE",
     col=1, ylab="")
lines(unemp.hp1$trend,col=2)
lines(unemp.hp2$trend,col=3)
lines(unemp.hp3$trend,col=4)
lines(unemp.hp4$trend,col=5)
legend("topleft",legend=c("series", "lambda=1600", "lambda=800",
       "freq=12", "freq=52"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.hp1$cycle,
main="Hodrick-Prescott filter of unemployment: Cycle,drift=TRUE",
     col=2, ylab="", ylim=range(unemp.hp4$cycle,na.rm=TRUE))
lines(unemp.hp2$cycle,col=3)
lines(unemp.hp3$cycle,col=4)
lines(unemp.hp4$cycle,col=5)
## legend("topleft",legend=c("lambda=1600", "lambda=800",
## "freq=12", "freq=52"), col=1:5, lty=rep(1,5), ncol=1)

par(opar)
## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.hp <- hpfilter(unemp)
plot(unemp.hp)
unemp.hp1 <- hpfilter(unemp, drift=TRUE)
unemp.hp2 <- hpfilter(unemp, freq=800, drift=TRUE)
unemp.hp3 <- hpfilter(unemp, freq=12,type="frequency",drift=TRUE)
unemp.hp4 <- hpfilter(unemp, freq=52,type="frequency",drift=TRUE)

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.hp1$x,  ylim=c(2,13),
main="Hodrick-Prescott filter of unemployment: Trend, drift=TRUE",
     col=1, ylab="")
lines(unemp.hp1$trend,col=2)
lines(unemp.hp2$trend,col=3)
lines(unemp.hp3$trend,col=4)
lines(unemp.hp4$trend,col=5)
legend("topleft",legend=c("series", "lambda=1600", "lambda=800",
       "freq=12", "freq=52"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.hp1$cycle,
main="Hodrick-Prescott filter of unemployment: Cycle,drift=TRUE",
     col=2, ylab="", ylim=range(unemp.hp4$cycle,na.rm=TRUE))
lines(unemp.hp2$cycle,col=3)
lines(unemp.hp3$cycle,col=4)
lines(unemp.hp4$cycle,col=5)
## legend("topleft",legend=c("lambda=1600", "lambda=800",
## "freq=12", "freq=52"), col=1:5, lty=rep(1,5), ncol=1)

par(opar)

Decomposition of a time series into trend and cyclical components using various filters

Description

mFilter is a generic function for filtering time series data. The function invokes particular filters which depend on filter type specified via its argument filter. The filters implemented in the package mFilter package are useful for smoothing, and estimating tend and cyclical components. Some of these filters are commonly used in economics and finance for estimating cyclical component of time series.

The mFilter currently applies only to time series objects. However a default method is available and should work for any numeric or vector object.

Usage

  mFilter(x, ...)
  ## Default S3 method:
mFilter(x, ...)
  ## S3 method for class 'ts'
mFilter(x, filter=c("HP","BK","CF","BW","TR"), ...)
mFilter(x, ...)
  ## Default S3 method:
mFilter(x, ...)
  ## S3 method for class 'ts'
mFilter(x, filter=c("HP","BK","CF","BW","TR"), ...)

Arguments

`x`	a regular a time series.
`filter`	filter type, the filter types are `"HP"` (Hodrick-Prescott), `"BK"` (Baxter-King), `"CF"` (Christiano-Fitzgerald), `"BW"` (Butterworth), and `"TR"` (trigonometric regression).
`...`	Additional arguments to pass to the relevant filter functions. These are passed to `hpfilter`, `bkfilter`, `cffilter`, `bwfilter`, and `trfilter`, respectively for the `"HP"`, `"BK"`, `"CF"`, `"BW"`, and `"TR"` filters.

Details

The default behaviour is to apply the default filter to ts objects.

Value

An object of class "mFilter".

The function summary is used to obtain and print a summary of the results, while the function plot produces a plot of the original series, the trend, and the cyclical components. The function print is also available for displaying estimation results.

The generic accessor functions fitted and residuals extract estimated trend and cyclical componets of an "mFilter" object, respectively.

An object of class "mFilter" is a list containing at least the following elements:

`cycle`	Estimated cyclical (irregular) component of the series.
`trend`	Estimated trend (smooth) component of the series.
`fmatrix`	The filter matrix applied to original series.
`method`	The method, if available, for the filter type applied.
`type`	The filter type applied to the series.
`call`	Call to the function.
`title`	The title for displaying results.
`xname`	Name of the series passed to `mFilter` for filtering.
`x`	The original or drift adjusted, if `drift=TRUE`, time series passed to the `mFilter`.

Following additional elements may exists depending on the type of filter applied:

`nfix`	Length or order of the fixed length filters.
`pl`	Minimum period of oscillation of desired component (2<=pl).
`pu`	Maximum period of oscillation of desired component (2<=pl<pu<infinity).
`lambda`	Lambda (smoothness) parameter of the HP filter.
`root`	Whether time series has a unit root, TRUE or FALSE (default).
`drift`	Whether time series has drift, TRUE or FALSE (default).
`theta`	MA coefficients for time series model, used in `"CF"` filter.

Author(s)

Mehmet Balcilar, [email protected]

Examples

## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.hp <- mFilter(unemp,filter="HP")  # Hodrick-Prescott filter
print(unemp.hp)
summary(unemp.hp)
residuals(unemp.hp)
fitted(unemp.hp)
plot(unemp.hp)

unemp.bk <- mFilter(unemp,filter="BK")  # Baxter-King filter
unemp.cf <- mFilter(unemp,filter="CF")  # Christiano-Fitzgerald filter
unemp.bw <- mFilter(unemp,filter="BW")  # Butterworth filter
unemp.tr <- mFilter(unemp,filter="TR")  # Trigonometric regression filter

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp,main="Unemployment Series & Estimated Trend", col=1, ylab="")
lines(unemp.hp$trend,col=2)
lines(unemp.bk$trend,col=3)
lines(unemp.cf$trend,col=4)
lines(unemp.bw$trend,col=5)
lines(unemp.tr$trend,col=6)

legend("topleft",legend=c("series", "HP","BK","CF","BW","TR"),
    col=1:6,lty=rep(1,6),ncol=2)

plot(unemp.hp$cycle,main="Estimated Cyclical Component",
     ylim=c(-2,2.5),col=2,ylab="")
lines(unemp.bk$cycle,col=3)
lines(unemp.cf$cycle,col=4)
lines(unemp.bw$cycle,col=5)
lines(unemp.tr$cycle,col=6)
## legend("topleft",legend=c("HP","BK","CF","BW","TR"),
## col=2:6,lty=rep(1,5),ncol=2)

unemp.cf1 <- mFilter(unemp,filter="CF", drift=TRUE, root=TRUE)
unemp.cf2 <- mFilter(unemp,filter="CF", pl=8,pu=40,drift=TRUE, root=TRUE)
unemp.cf3 <- mFilter(unemp,filter="CF", pl=2,pu=60,drift=TRUE, root=TRUE)
unemp.cf4 <- mFilter(unemp,filter="CF", pl=2,pu=40,drift=TRUE,
             root=TRUE,theta=c(.1,.4))

plot(unemp,
main="Christiano-Fitzgerald filter of unemployment: Trend \n root=TRUE,drift=TRUE",
      col=1, ylab="")
lines(unemp.cf1$trend,col=2)
lines(unemp.cf2$trend,col=3)
lines(unemp.cf3$trend,col=4)
lines(unemp.cf4$trend,col=5)
legend("topleft",legend=c("series", "pl=2, pu=32", "pl=8, pu=40",
"pl=2, pu=60", "pl=2, pu=40, theta=.1,.4"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.cf1$cycle,
main="Christiano-Fitzgerald filter of unemployment: Cycle \n root=TRUE,drift=TRUE",
     col=2, ylab="", ylim=range(unemp.cf3$cycle))
lines(unemp.cf2$cycle,col=3)
lines(unemp.cf3$cycle,col=4)
lines(unemp.cf4$cycle,col=5)
## legend("topleft",legend=c("pl=2, pu=32", "pl=8, pu=40", "pl=2, pu=60",
## "pl=2, pu=40, theta=.1,.4"), col=2:5, lty=rep(1,4), ncol=2)

par(opar)
## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.hp <- mFilter(unemp,filter="HP")  # Hodrick-Prescott filter
print(unemp.hp)
summary(unemp.hp)
residuals(unemp.hp)
fitted(unemp.hp)
plot(unemp.hp)

unemp.bk <- mFilter(unemp,filter="BK")  # Baxter-King filter
unemp.cf <- mFilter(unemp,filter="CF")  # Christiano-Fitzgerald filter
unemp.bw <- mFilter(unemp,filter="BW")  # Butterworth filter
unemp.tr <- mFilter(unemp,filter="TR")  # Trigonometric regression filter

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp,main="Unemployment Series & Estimated Trend", col=1, ylab="")
lines(unemp.hp$trend,col=2)
lines(unemp.bk$trend,col=3)
lines(unemp.cf$trend,col=4)
lines(unemp.bw$trend,col=5)
lines(unemp.tr$trend,col=6)

legend("topleft",legend=c("series", "HP","BK","CF","BW","TR"),
    col=1:6,lty=rep(1,6),ncol=2)

plot(unemp.hp$cycle,main="Estimated Cyclical Component",
     ylim=c(-2,2.5),col=2,ylab="")
lines(unemp.bk$cycle,col=3)
lines(unemp.cf$cycle,col=4)
lines(unemp.bw$cycle,col=5)
lines(unemp.tr$cycle,col=6)
## legend("topleft",legend=c("HP","BK","CF","BW","TR"),
## col=2:6,lty=rep(1,5),ncol=2)

unemp.cf1 <- mFilter(unemp,filter="CF", drift=TRUE, root=TRUE)
unemp.cf2 <- mFilter(unemp,filter="CF", pl=8,pu=40,drift=TRUE, root=TRUE)
unemp.cf3 <- mFilter(unemp,filter="CF", pl=2,pu=60,drift=TRUE, root=TRUE)
unemp.cf4 <- mFilter(unemp,filter="CF", pl=2,pu=40,drift=TRUE,
             root=TRUE,theta=c(.1,.4))

plot(unemp,
main="Christiano-Fitzgerald filter of unemployment: Trend \n root=TRUE,drift=TRUE",
      col=1, ylab="")
lines(unemp.cf1$trend,col=2)
lines(unemp.cf2$trend,col=3)
lines(unemp.cf3$trend,col=4)
lines(unemp.cf4$trend,col=5)
legend("topleft",legend=c("series", "pl=2, pu=32", "pl=8, pu=40",
"pl=2, pu=60", "pl=2, pu=40, theta=.1,.4"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.cf1$cycle,
main="Christiano-Fitzgerald filter of unemployment: Cycle \n root=TRUE,drift=TRUE",
     col=2, ylab="", ylim=range(unemp.cf3$cycle))
lines(unemp.cf2$cycle,col=3)
lines(unemp.cf3$cycle,col=4)
lines(unemp.cf4$cycle,col=5)
## legend("topleft",legend=c("pl=2, pu=32", "pl=8, pu=40", "pl=2, pu=60",
## "pl=2, pu=40, theta=.1,.4"), col=2:5, lty=rep(1,4), ncol=2)

par(opar)

Methods for mFilter objects

Description

Common methods for all mFilter objects usually created by the mFilter function.

Usage

## S3 method for class 'mFilter'
residuals(object, ...)
## S3 method for class 'mFilter'
fitted(object, ...)
## S3 method for class 'mFilter'
print(x, digits = max(3, getOption("digits") - 3), ...)
## S3 method for class 'mFilter'
plot(x, reference.grid = TRUE, col = "steelblue", ask=interactive(), ...)
## S3 method for class 'mFilter'
summary(object, digits = max(3, getOption("digits") - 3), ...)
## S3 method for class 'mFilter'
residuals(object, ...)
## S3 method for class 'mFilter'
fitted(object, ...)
## S3 method for class 'mFilter'
print(x, digits = max(3, getOption("digits") - 3), ...)
## S3 method for class 'mFilter'
plot(x, reference.grid = TRUE, col = "steelblue", ask=interactive(), ...)
## S3 method for class 'mFilter'
summary(object, digits = max(3, getOption("digits") - 3), ...)

Arguments

`object`, `x`	an object of class `"mFilter"`; usually, a result of a call to `mFilter`.
`digits`	number of digits used for printing (see `print`).
`col`	color of the graph (see `plot`).
`ask`	logical. if `TRUE` the user is asked for input before a new graph drawn in an interactive session (see `interactive`).
`reference.grid`	logical. if true grid lines are drawn.
`...`	further arguments passed to or from other methods.

Value

for residuals and fitted a univariate time series; for plot, print, and summary the "mFilter" object.

Author(s)

Mehmet Balcilar, [email protected]

Examples

## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.hp <- mFilter(unemp,filter="HP")  # Hodrick-Prescott filter
print(unemp.hp)
summary(unemp.hp)
residuals(unemp.hp)
fitted(unemp.hp)
plot(unemp.hp)

par(opar)
## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.hp <- mFilter(unemp,filter="HP")  # Hodrick-Prescott filter
print(unemp.hp)
summary(unemp.hp)
residuals(unemp.hp)
fitted(unemp.hp)
plot(unemp.hp)

par(opar)

Trigonometric regression filter of a time series

Description

This function uses trigonometric regression filter for estimating cyclical and trend components of a time series. The function computes cyclical and trend components of the time series using a lower and upper cut-off frequency in the spirit of a band pass filter.

Usage

trfilter(x,pl=NULL,pu=NULL,drift=FALSE)
trfilter(x,pl=NULL,pu=NULL,drift=FALSE)

Arguments

`x`	a regular time series.
`pl`	integer. minimum period of oscillation of desired component (pl<=2).
`pu`	integer. maximum period of oscillation of desired component (2<=pl<pu<infinity).
`drift`	logical, `FALSE` if no drift in time series (default), `TRUE` if drift in time series.

Details

Consider the following decomposition of the time series

$x_t = y_t + \bar{x}_t$

The component $y_t$ is assumed to have power only in the frequencies in the interval $\{(a,b) \cup (-a,-b)\} \in (-\pi, \pi)$ . $a$ and $b$ are related to $p_l$ and $p_u$ by

$a=\frac{2 \pi}{p_u}\ \ \ \ \ {b=\frac{2 \pi}{p_l}}$

If infinite amount of data is available, then we can use the ideal bandpass filter

$y_t = B(L)x_t$

where the filter, $B(L)$ , is given in terms of the lag operator $L$ and defined as

$B(L) = \sum^\infty_{j=-\infty} B_j L^j, \ \ \ L^k x_t = x_{t-k}$

The ideal bandpass filter weights are given by

$B_j = \frac{\sin(jb)-\sin(ja)}{\pi j}$

$B_0=\frac{b-a}{\pi}$

Let $T$ be even and define $n_1=T/p_u$ and $n_2=T/p_l$ . The trigonometric regression filter is based on the following relation

${y}_t=\sum^{n_1}_{j=n_2}\left\{ a_j \cos(\omega_j t) + b_j \sin(\omega_j t) \right\}$

where $a_j$ and $b_j$ are the coefficients obtained by regressing $x_t$ on the indicated sine and cosine functions. Specifically,

$a_j=\frac{T}{2}\sum^{T}_{t=1}\cos(\omega_j t) x_t,\ \ \$ for $j=1,\dots,T/2-1$

$a_j=\frac{T}{2}\sum^{T}_{t=1}\cos(\pi t) x_t,\ \ \$ for $j=T/2$

and

$b_j=\frac{T}{2}\sum^{T}_{t=1}\sin(\omega_j t) x_t,\ \ \$ for $j=1,\dots,T/2-1$

$b_j=\frac{T}{2}\sum^{T}_{t=1}\sin(\pi t) x_t,\ \ \$ for $j=T/2$

If drift=TRUE the drift adjusted series is obtained as

$\tilde{x}_{t}=x_t-t\left(\frac{x_{T}-x_{1}}{T-1}\right), \ \ t=0,1,\dots,T-1$

where $\tilde{x}_{t}$ is the undrifted series.

Value

A "mFilter" object (see mFilter).

Author(s)

Mehmet Balcilar, [email protected]

References

M. Baxter and R.G. King. Measuring business cycles: Approximate bandpass filters. The Review of Economics and Statistics, 81(4):575-93, 1999.

L. Christiano and T.J. Fitzgerald. The bandpass filter. International Economic Review, 44(2):435-65, 2003.

J. D. Hamilton. Time series analysis. Princeton, 1994.

R.J. Hodrick and E.C. Prescott. Postwar US business cycles: an empirical investigation. Journal of Money, Credit, and Banking, 29(1):1-16, 1997.

R.G. King and S.T. Rebelo. Low frequency filtering and real business cycles. Journal of Economic Dynamics and Control, 17(1-2):207-31, 1993.

D.S.G. Pollock. Trend estimation and de-trending via rational square-wave filters. Journal of Econometrics, 99:317-334, 2000.

Examples

## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.tr <- trfilter(unemp, drift=TRUE)
plot(unemp.tr)
unemp.tr1 <- trfilter(unemp, drift=TRUE)
unemp.tr2 <- trfilter(unemp, pl=8,pu=40,drift=TRUE)
unemp.tr3 <- trfilter(unemp, pl=2,pu=60,drift=TRUE)
unemp.tr4 <- trfilter(unemp, pl=2,pu=40,drift=TRUE)

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.tr1$x,
main="Trigonometric regression filter of unemployment: Trend, drift=TRUE",
     col=1, ylab="")
lines(unemp.tr1$trend,col=2)
lines(unemp.tr2$trend,col=3)
lines(unemp.tr3$trend,col=4)
lines(unemp.tr4$trend,col=5)
legend("topleft",legend=c("series", "pl=2, pu=32", "pl=8, pu=40",
"pl=2, pu=60", "pl=2, pu=40"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.tr1$cycle,
main="Trigonometric regression filter of unemployment: Cycle,drift=TRUE",
     col=2, ylab="", ylim=range(unemp.tr3$cycle,na.rm=TRUE))
lines(unemp.tr2$cycle,col=3)
lines(unemp.tr3$cycle,col=4)
lines(unemp.tr4$cycle,col=5)
## legend("topleft",legend=c("pl=2, pu=32", "pl=8, pu=40", "pl=2, pu=60",
## "pl=2, pu=40"), col=1:5, lty=rep(1,5), ncol=1)

par(opar)
## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.tr <- trfilter(unemp, drift=TRUE)
plot(unemp.tr)
unemp.tr1 <- trfilter(unemp, drift=TRUE)
unemp.tr2 <- trfilter(unemp, pl=8,pu=40,drift=TRUE)
unemp.tr3 <- trfilter(unemp, pl=2,pu=60,drift=TRUE)
unemp.tr4 <- trfilter(unemp, pl=2,pu=40,drift=TRUE)

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.tr1$x,
main="Trigonometric regression filter of unemployment: Trend, drift=TRUE",
     col=1, ylab="")
lines(unemp.tr1$trend,col=2)
lines(unemp.tr2$trend,col=3)
lines(unemp.tr3$trend,col=4)
lines(unemp.tr4$trend,col=5)
legend("topleft",legend=c("series", "pl=2, pu=32", "pl=8, pu=40",
"pl=2, pu=60", "pl=2, pu=40"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.tr1$cycle,
main="Trigonometric regression filter of unemployment: Cycle,drift=TRUE",
     col=2, ylab="", ylim=range(unemp.tr3$cycle,na.rm=TRUE))
lines(unemp.tr2$cycle,col=3)
lines(unemp.tr3$cycle,col=4)
lines(unemp.tr4$cycle,col=5)
## legend("topleft",legend=c("pl=2, pu=32", "pl=8, pu=40", "pl=2, pu=60",
## "pl=2, pu=40"), col=1:5, lty=rep(1,5), ncol=1)

par(opar)

US Quarterly Unemployment Series

Description

Quarterly US unemployment series for 1959.1 to 2000.4.

number of observations : 168

observation : country

country : United States

Usage

data(unemp)
data(unemp)

Format

A time series containing :

unemp: unemployment rate (average of months in quarter)

Author(s)

Mehmet Balcilar, [email protected]

Source

Bureau of Labor Statistics, OECD, Federal Reserve.

References

Stock, James H. and Mark W. Watson (2003) Introduction to Econometrics, Addison-Wesley Educational Publishers, chapter 12 and 14.

Examples

## library(mFilter)

data(unemp)

unemp.hp <- mFilter(unemp,filter="HP")  # Hodrick-Prescott filter
unemp.bk <- mFilter(unemp,filter="BK")  # Baxter-King filter
unemp.cf <- mFilter(unemp,filter="CF")  # Christiano-Fitzgerald filter

opar <- par(no.readonly=TRUE)
par(mfrow=c(2,1),mar=c(3,3,2,1))
plot(unemp,main="Unemployment Series & Estimated Trend",col=1,ylab="")
lines(unemp.hp$trend,col=2)
lines(unemp.bk$trend,col=3)
lines(unemp.cf$trend,col=4)
legend("topleft",legend=c("series", "HP","BK","CF"),col=1:4,
       lty=rep(1,4),ncol=2)

plot(unemp.hp$cycle,main="Estimated Cyclical Component",col=2,
     ylim=c(-2,2),ylab="")
lines(unemp.bk$cycle,col=3)
lines(unemp.cf$cycle,col=4)
legend("topleft",legend=c("HP","BK","CF"),col=2:4,lty=rep(1,3),ncol=2)
par(opar)
## library(mFilter)

data(unemp)

unemp.hp <- mFilter(unemp,filter="HP")  # Hodrick-Prescott filter
unemp.bk <- mFilter(unemp,filter="BK")  # Baxter-King filter
unemp.cf <- mFilter(unemp,filter="CF")  # Christiano-Fitzgerald filter

opar <- par(no.readonly=TRUE)
par(mfrow=c(2,1),mar=c(3,3,2,1))
plot(unemp,main="Unemployment Series & Estimated Trend",col=1,ylab="")
lines(unemp.hp$trend,col=2)
lines(unemp.bk$trend,col=3)
lines(unemp.cf$trend,col=4)
legend("topleft",legend=c("series", "HP","BK","CF"),col=1:4,
       lty=rep(1,4),ncol=2)

plot(unemp.hp$cycle,main="Estimated Cyclical Component",col=2,
     ylim=c(-2,2),ylab="")
lines(unemp.bk$cycle,col=3)
lines(unemp.cf$cycle,col=4)
legend("topleft",legend=c("HP","BK","CF"),col=2:4,lty=rep(1,3),ncol=2)
par(opar)

Package 'mFilter'

Help Index

Getting started with the mFilter package

Description

Details

How to cite this package

License

Author(s)

References

See Also

Baxter-King filter of a time series

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Butterworth filter of a time series

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Christiano-Fitzgerald filter of a time series

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Hodrick-Prescott filter of a time series

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Decomposition of a time series into trend and cyclical components using various filters

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Methods for mFilter objects

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Trigonometric regression filter of a time series

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

US Quarterly Unemployment Series

Description

Usage