Real‐Time Signal Extraction with Regularized Multivariate Direct Filter Approach

• DOI: http://doi.org/10.1002/for.2352
• Author: Ginters Buss
• Published date: 01 April 2016
• Date: 01 April 2016

Journal of Forecasting,J. Forecast. 35, 206–216 (2016)

Published online 25 November 2015 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/for.2352

Real-Time Signal Extraction with Regularized Multivariate

Direct Filter Approach

GINTERS BUSS

ABSTRACT

The paper studies the regularized direct filter approach as a tool for real-time signal extraction using high-dimensional

datasets. It is shown that the filter is able to process high-dimensional datasets by controlling for effective degrees

of freedom through longitudinal and cross-sectional regularization. The paper illustrates the merit of the proposed

approach by tracking the medium- to long-run component in euro area gross domestic product growth. The created

real-time indicators outperform Eurocoin with respect to timeliness. Copyright © 2015 John Wiley & Sons, Ltd.

KEY WORDS high-dimensional filtering; real-time estimation; business cycle; coincident indicator; leading

indicator

INTRODUCTION

Nowadays, gathering rich datasets is relatively easy. A more difficult task is to use them effectively for a particular

problem at hand. This paper studies the regularized multivariate direct filter approach (Wildi, 2012) as a tool for

real-time signal extraction with many variables.

Wildi (2012) derives a regularized multivariate filter as the successor to his multivariate direct filter approach

(Wildi, 2011). The regularization, or shrinkage, is performed in three dimensions: (i) longitudinal shrinkage reduces

the number of free filter coefficients by forcing more distant filter coefficients to converge to zero; it is, in spirit,

similar to the ‘lag decay’ term in Minnessota prior (e.g. Doan et al., 1984) in Bayesian econometrics; (ii) cross-

sectional shrinkage makes the filter coefficients to behave similarly for similar input series, and, in the limit, forces

filter coefficients to be the same for all input series; (iii) smoothness restriction makes the filter coefficients change

smoothly along the longitudinal/time dimension.

Wildi (2012) does not apply the filter to the data. Therefore, the added value of the current paper is to study

how the regularized filter can be implemented in real-time signal extraction exercises, and in particular, in the

trend-cycle (in this paper, defined by a cycle of 1 year and longer) estimation using high-dimensional datasets. By

high-dimensionality here I mean at least several dozen of variables, since that many variables makes it difficult or

impossible to successfully apply traditional, unregularized methods in real-time signal extraction due to the many free

estimated coefficients. Trend-cycle estimation is typical in economics and finance, where there is often short-term

noise in the data that does not help in understanding the medium- to long-term trends. For example, Altissimo et al.

(2010) use the same definition of the target signal as in this paper.

The key notion I am using to make the multivariate filter perform decently out-of-sample is the effective degrees

of freedom (the trace of the smoother matrix; henceforth e.d.f.). My motivation is that, typically, methods with a

few degrees of freedom perform well out-of-sample. For example, in forecasting, such would be the autoregressive

moving average model, or the factor model with one or a few factors. Therefore, I propose to fix the desired e.d.f. to

a sufficiently small value, and use the above shrinkage terms to achieve the desired e.d.f. This paper finds that two

of Wildi’s proposed three shrinkage terms help in real-time signal extraction problems. The most effective is found

to be the longitudinal shrinkage but it cannot be applied excessively since too short a filter cannot discriminate well

between the frequencies. I find that it is desirable to allow for at least about half a year of the effective length of

the filter (the length of the filter where its coefficients are beyond the neighborhood of zero). The rest of the e.d.f.

are suppressed by the cross-sectional shrinkage. I find that the cross-sectional shrinkage is useful particularly if the

dataset is rather homogeneous.

I illustrate the design of the filter and its real-time output by applying the above method to 72 variables in order

to track the medium- to long-run component of the euro area (henceforth GDP) growth. The results show that the

filter output is robust, and that such a design can mimic and even outperform in terms of timeliness the established

Eurocoin indicator (Altissimo et al., 2010) that is based on a dynamic factor methodology (Forni et al., 2000, 2005). A

comparison of the proposed method to the dynamic factor methodology is instructive since both methods have much

in common but also feature clear-cut differences. While dynamic factor methodology shrinks the dimension of the

dataset to a few unobserved factors and thus has a few parameters to estimate, the regularized filter does not shrink the

Correspondence to: Ginters Buss, Latvijas Banka, K. Valdemara2A, Riga, LV-1050, Latvia. E-mail: ginters.buss@gmail.com

Copyright © 2015 John Wiley & Sons, Ltd