trfilter: Trigonometric regression filter of a time series
In mFilter: Miscellaneous Time Series Filters

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

This function uses trigonometric regression filter for estimating cyclical and trend components of a time series. The function computes cyclical and trend components of the time series using a lower and upper cut-off frequency in the spirit of a band pass filter.

1	trfilter(x,pl=NULL,pu=NULL,drift=FALSE)

`x`	a regular time series.
`pl`	integer. minimum period of oscillation of desired component (pl<=2).
`pu`	integer. maximum period of oscillation of desired component (2<=pl<pu<infinity).
`drift`	logical, `FALSE` if no drift in time series (default), `TRUE` if drift in time series.

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}

Let T be even and define n_1=T/p_u and n_2=T/p_l. The trigonometric regression filter is based on the following relation

{y}_t=∑^{n_1}_{j=n_2}≤ft\{ a_j \cos(ω_j t) + b_j \sin(ω_j t) \right\}

where a_j and b_j are the coefficients obtained by regressing x_t on the indicated sine and cosine functions. Specifically,

a_j=\frac{T}{2}∑^{T}_{t=1}\cos(ω_j t) x_t,\ \ \ for j=1,…,T/2-1

a_j=\frac{T}{2}∑^{T}_{t=1}\cos(π t) x_t,\ \ \ for j=T/2

and

b_j=\frac{T}{2}∑^{T}_{t=1}\sin(ω_j t) x_t,\ \ \ for j=1,…,T/2-1

b_j=\frac{T}{2}∑^{T}_{t=1}\sin(π t) x_t,\ \ \ for j=T/2

Let \hat{B}(L) x_t be the trigonometric regression filter. It can be showed that \hat{B}(1)=0, so that \hat{B}(L) has a unit root for t=1,2,…,T. Also, when \hat{B}(L) is symmetric, it has a second unit root in the middle of the data for t. Therefore it is important to drift adjust data before it is filtered with a trigonometric regression filter.

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, bwfilter

## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.tr <- trfilter(unemp, drift=TRUE)
plot(unemp.tr)
unemp.tr1 <- trfilter(unemp, drift=TRUE)
unemp.tr2 <- trfilter(unemp, pl=8,pu=40,drift=TRUE)
unemp.tr3 <- trfilter(unemp, pl=2,pu=60,drift=TRUE)
unemp.tr4 <- trfilter(unemp, pl=2,pu=40,drift=TRUE)

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.tr1$x,
main="Trigonometric regression filter of unemployment: Trend, drift=TRUE",
     col=1, ylab="")
lines(unemp.tr1$trend,col=2)
lines(unemp.tr2$trend,col=3)
lines(unemp.tr3$trend,col=4)
lines(unemp.tr4$trend,col=5)
legend("topleft",legend=c("series", "pl=2, pu=32", "pl=8, pu=40",
"pl=2, pu=60", "pl=2, pu=40"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.tr1$cycle,
main="Trigonometric regression filter of unemployment: Cycle,drift=TRUE",
     col=2, ylab="", ylim=range(unemp.tr3$cycle,na.rm=TRUE))
lines(unemp.tr2$cycle,col=3)
lines(unemp.tr3$cycle,col=4)
lines(unemp.tr4$cycle,col=5)
## legend("topleft",legend=c("pl=2, pu=32", "pl=8, pu=40", "pl=2, pu=60",
## "pl=2, pu=40"), col=1:5, lty=rep(1,5), ncol=1)

par(opar)