cffilter: 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

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.

minimum period of oscillation of desired component (pl<=2).

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 + root*x(t-1) + theta(1)*e(t) + theta(2)*e(t-1) + …, where e(t) is a white noise.

Value

A "mFilter" object (see mFilter).

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.

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

Run this code

# NOT RUN {
## 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)
# }

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