# cffilter: Christiano-Fitzgerald filter of a time series In mbalcilar/mFilter: Miscellaneous Time Series Filters

## 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

 1 2 3 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

## 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 ≤ p_l < p_u < ∞.

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 (-π, π). a and b are related to p_l and p_u by

a=\frac{2 π}{p_u}\ \ \ \ \ {b=\frac{2 π}{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) = ∑^∞_{j=-∞} 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)}{π j}

B_0=\frac{b-a}{π}

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

y_t = \hat{B}(L)x_t=∑^{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=(θ_1, θ_2, …) then the series is assumed to be

x_t = μ + 1_{root} x_{t-1} + θ_1 e_t + θ_2 e_{t-1} + …

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≤ft(\frac{x_{T}-x_{1}}{T-1}\right), \ \ t=0,1,…,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.

mFilter, bwfilter, bkfilter, hpfilter, trfilter
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 ## 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)