bwfilter | R Documentation |
Filters a time series using the Butterworth square-wave highpass filter described in Pollock (2000).
bwfilter(x,freq=NULL,nfix=NULL,drift=FALSE)
x |
a regular time series |
nfix |
sets the order of the filter. The default is
|
freq |
integer, the cut-off frequency of the Butterworth
filter. The default is |
drift |
logical, |
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.
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
, hpfilter
, cffilter
,
bkfilter
, trfilter
## 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)
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