# bwfilter: Butterworth filter of a time series In mFilter: Miscellaneous Time Series Filters

## Description

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

## Usage

 1 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

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

\frac{λ(1-z)^n(1-z^{-1})^n}{(1+z)^n(1+z^{-1})^n+λ(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}ν_t+(1-L)^{n-d}\varepsilon_t

where ν_t \sim N(0,σ_ν^2) and \varepsilon_t \sim N(0,σ_\varepsilon^2).

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, 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.

mFilter, hpfilter, cffilter, bkfilter, trfilter

## Examples

  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 35 36 ## 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) 

### Example output




mFilter documentation built on June 5, 2019, 1:03 a.m.