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

Description Usage Arguments Details Value Author(s) References See Also Examples

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.

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

`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.

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.

A "mFilter" object (see mFilter).

Mehmet Balcilar, mehmet@mbalcilar.net

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

## 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)