Getting started with the mFilter package
This package provides some tools for decomposing time series into trend (smooth) and cyclical (irregular) components. The package implements come commonly used filters such as the Hodrick-Prescott, Baxter-King and Christiano-Fitzgerald filter.
For loading the package, type:
library(mFilter)
A good place to start learning the package usage is to examine examples for
the mFilter function. At the R prompt, write:
example("mFilter")
For a full list of functions exported by the package, type:
ls("package:mFilter")
Each exported function has a corresponding man page (some man pages are common to more functions). Display it by typing
help(functionName).
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 finite sample approximation to the ideal bandpass filter uses the alternative filter $$y_t = \hat{B}(L)x_t=\sum^{n_2}_{j=-n_1}\hat{B}_{t,j} x_{t+j}$$ 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\(=(\theta_1, \theta_2, \dots)\) then
the series is assumed to be
$$x_t = \mu + 1_{root} x_{t-1} + \theta_1 e_t + \theta_2 e_{t-1} + \dots$$
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.
The Baxter-King filter is a finite data approximation to the ideal bandpass filter with following moving average weights $$y_t = \hat{B}(L)x_t=\sum^{n}_{j=-n}\hat{B}_{j} x_{t+j}=\hat{B}_0 x_t + \sum^{n}_{j=1} \hat{B}_j (x_{t-j}+x_{t+j})$$ where $$\hat{B}_j=B_j-\frac{1}{2n+1}\sum^{n}_{j=-n} B_{j}$$
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 $$\hat{B}_j=\frac{1}{2\pi}\int^{\pi}_{-\pi} \frac{4\lambda(1-\cos(\omega))^2}{1+4\lambda(1-\cos(\omega))^2}e^{i \omega j} d \omega$$
The digital version of the Butterworth highpass filter is described by the rational polynomial expression (the filter's z-transform) $$\frac{\lambda(1-z)^n(1-z^{-1})^n}{(1+z)^n(1+z^{-1})^n+\lambda(1-z)^n(1-z^{-1})^n}$$ The time domain version can be obtained by substituting \(z\) for the lag operator \(L\).
Pollock (2000) 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 $$y_t=s_t+c_t=\frac{(1+L)^n}{(1-L)^d}\nu_t+(1-L)^{n-d}\varepsilon_t$$ where \(\nu_t \sim N(0,\sigma_\nu^2)\) and \(\varepsilon_t \sim N(0,\sigma_\varepsilon^2)\).
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 $${y}_t=\sum^{n_1}_{j=n_2}\left\{ a_j \cos(\omega_j t) + b_j \sin(\omega_j t) \right\}$$ where \(a_j\) and \(b_j\) are the coefficients obtained by regressing \(x_t\) on the indicated sine and cosine functions. Specifically,
\(a_j=\frac{T}{2}\sum^{T}_{t=1}\cos(\omega_j t) x_t,\ \ \ \) for \(j=1,\dots,T/2-1\)
\(a_j=\frac{T}{2}\sum^{T}_{t=1}\cos(\pi t) x_t,\ \ \ \) for \(j=T/2\)
and
\(b_j=\frac{T}{2}\sum^{T}_{t=1}\sin(\omega_j t) x_t,\ \ \ \) for \(j=1,\dots,T/2-1\)
\(b_j=\frac{T}{2}\sum^{T}_{t=1}\sin(\pi t) x_t,\ \ \ \) 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 \(t=1,2,\dots,T\). 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
$$\tilde{x}_{t}=x_t-t\left(\frac{x_{T}-x_{1}}{T-1}\right), \ \ t=0,1,\dots,T-1$$
where \(\tilde{x}_{t}\) is the undrifted series.
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-methods for listing all currently
available mFilter methods. For help on common interface function
"mFilter", mFilter. For individual filter function
usage, bwfilter, bkfilter,
cffilter, hpfilter, trfilter.