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.

Author(s)

Mehmet Balcilar, mehmet@mbalcilar.net

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.

See Also

mFilter-methods for listing all currently available mFilter methods. For help on common interface function "mFilter", mFilter. For individual filter function usage, bwfilter, bkfilter, cffilter, hpfilter, trfilter.

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)

Arguments

Details

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 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, mehmet@mbalcilar.net

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.

See Also

mFilter, bwfilter, cffilter, hpfilter, trfilter

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)

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)

Arguments

Details

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 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, mehmet@mbalcilar.net

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.

See Also

mFilter, hpfilter, cffilter, bkfilter, trfilter

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)

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)

Arguments

Details

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.

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, mehmet@mbalcilar.net

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.

See Also

mFilter, bwfilter, bkfilter, hpfilter, trfilter

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)

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)

Arguments

Details

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 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, mehmet@mbalcilar.net

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.

See Also

mFilter, bwfilter, cffilter, bkfilter, trfilter

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)

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"), ...)

Arguments

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:

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

Author(s)

Mehmet Balcilar, mehmet@mbalcilar.net

See Also

Other functions which return objects of class "mFilter" are bkfilter, bwfilter, cffilter, bkfilter, trfilter. Following functions apply the relevant methods to an object of the "mFilter" class: print.mFilter, summary.mFilter, plot.mFilter, fitted.mFilter, residuals.mFilter.

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)

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), ...)

Arguments

Value

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

Author(s)

Mehmet Balcilar, mehmet@mbalcilar.net

See Also

mFilter for the function that returns an objects of class "mFilter". Other functions which return objects of class "mFilter" are bkfilter, bwfilter, cffilter, bkfilter, trfilter.

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)

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)

Arguments

Details

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}

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.

Value

A "mFilter" object (see mFilter).

Author(s)

Mehmet Balcilar, mehmet@mbalcilar.net

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.

See Also

mFilter, hpfilter, cffilter, bkfilter, bwfilter

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)

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)

Format

A time series containing :

unemp

unemployment rate (average of months in quarter)

Author(s)

Mehmet Balcilar, mehmet@mbalcilar.net

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)