mFilter-package: Getting started with the mFilter package

Description

Getting started with the mFilter package

Arguments

How to cite this package

Whenever using this package, please cite as

Balcilar, M. (2019). mFilter: Miscellaneous Time Series Filters. R package version 1.5-0. https://github.com/mbalcilar/mFilter.

@Manual{Balcilar_2019,
 author       = {Mehmet Balcilar},
 title        = {{mFilter}: Miscellaneous Time Series Filters.},
 year         = {2019},
 note         = {R package version 1.5-0.},
 URL          = {https://github.com/mbalcilar/mFilter}}

License

The releases of this package is licensed under GPL version 3.

Author

Mehmet Balcilar, mehmet@mbalcilar.net

Details

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.

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.