hpfilter {mFilter}

Hodrick-Prescott filter of a time series

Description

This function implements the Hodrick-Prescott for estimating cyclical and trend component of a time series. The function computes cyclical and trend components of the time series using a frequency cut-off or smoothness parameter.

Usage

hpfilter(x,freq=NULL,type=c("lambda","frequency"),drift=FALSE)

Arguments

x

a regular time series.

type

character, indicating the filter type, "lambda", for the filter that uses smoothness penalty parameter of the Hodrick-Prescott filter (default), "frequency", for the filter that uses a frequency cut-off type Hodrick-Prescott filter. These are related by lambda = (2*sin(pi/frequency))^{-4}.

freq

integer, if type="lambda" then freq is the smoothing parameter (lambda) of the Hodrick-Prescott filter, if type="frequency" then freq is the cut-off frequency of the Hodrick-Prescott filter.

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 \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 Hodrick-Prescott filter obtains the filter weights \hat{B}_j as a solution to

\hat{B}_{j}= \arg \min E \{ (y_t-\hat{y}_t)^2 \} = \arg \min \left\{ \sum^{T}_{t=1}(y_t-\hat{y}_{t})^2 + \lambda\sum^{T-1}_{t=2}(\hat{y}_{t+1}-2\hat{y}_{t}+\hat{y}_{t-1})^2 \right\}

The Hodrick-Prescott filter is a finite data approximation with following moving average weights

\hat{B}_j=\frac{1}{2\pi}\int^{\pi}_{-\pi} \frac{4\lambda(1-\cos(\omega))^2}{1+4\lambda(1-\cos(\omega))^2}e^{i \omega j} d \omega

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.

Value

A "mFilter" object (see mFilter).

Author(s)

Mehmet Balcilar, [e-mail]

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.

See Also

mFilter, bwfilter, cffilter, bkfilter, trfilter

Examples

## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.hp <- hpfilter(unemp)
plot(unemp.hp)
unemp.hp1 <- hpfilter(unemp, drift=TRUE)
unemp.hp2 <- hpfilter(unemp, freq=800, drift=TRUE)
unemp.hp3 <- hpfilter(unemp, freq=12,type="frequency",drift=TRUE)
unemp.hp4 <- hpfilter(unemp, freq=52,type="frequency",drift=TRUE)

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.hp1$x,  ylim=c(2,13),
main="Hodrick-Prescott filter of unemployment: Trend, drift=TRUE",
     col=1, ylab="")
lines(unemp.hp1$trend,col=2)
lines(unemp.hp2$trend,col=3)
lines(unemp.hp3$trend,col=4)
lines(unemp.hp4$trend,col=5)
legend("topleft",legend=c("series", "lambda=1600", "lambda=800",
       "freq=12", "freq=52"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.hp1$cycle,
main="Hodrick-Prescott filter of unemployment: Cycle,drift=TRUE",
     col=2, ylab="", ylim=range(unemp.hp4$cycle,na.rm=TRUE))
lines(unemp.hp2$cycle,col=3)
lines(unemp.hp3$cycle,col=4)
lines(unemp.hp4$cycle,col=5)
## legend("topleft",legend=c("lambda=1600", "lambda=800",
## "freq=12", "freq=52"), col=1:5, lty=rep(1,5), ncol=1)

par(opar)