Dimension Reduction for High‐Dimensional Vector Autoregressive Models*
| Published date | 01 October 2022 |
| Author | Gianluca Cubadda,Alain Hecq |
| Date | 01 October 2022 |
| DOI | http://doi.org/10.1111/obes.12506 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 84, 5 (2022) 0305-9049
doi: 10.1111/obes.12506
Dimension Reduction for High-Dimensional Vector
Autoregressive Models*
GIANLUCA CUBADDA† and ALAIN HECQ‡
†Dipartimento di Economia e Finanza, Universita’ di Roma ‘‘Tor Vergata’’, Via Columbia 2,
00133 Roma, Italy (e-mail: gianluca.cubadda@uniroma2.it)
‡Department of Quantitative Economics, Maastricht University, P.O.Box 616, 6200 MD
Maastricht, The Netherlands (e-mail: a.hecq@maastrichtuniversity.nl)
Abstract
This article aims to decompose a large dimensional vector autoregressive (VAR) model
into two components, the first one being generated by a small-scale VAR and the second
one being a white noise. Hence, a reduced number of common components generates
the entire dynamics of the large system through a VAR structure. This modelling, which
we label as the dimension-reducible VAR, extends the common feature approach to
high-dimensional systems, and it differs from the dynamic factor model in which the
idiosyncratic component can also embed a dynamic pattern. We show the conditions
under which this decomposition exists. We provide statistical tools to detect its presence
in the data and to estimate the parameters of the underlying small-scale VAR model.
Based on our methodology, we propose a novel approach to identify the shock that is
responsible for most of the common variability at the business cycle frequencies. We
evaluate the practical value of the proposed methods by simulations as well as by an
empirical application to a large set of US economic variables.
I. Introduction
For decades, the vector autoregressive (VAR) model is de facto a standard tool for
investigating multivariate time series data.1In macroeconometrics, VARs are routinely
used for forecasting, extracting co-movements such as the presence of cointegration,
JEL Classification numbers: C32, C55, E32.
*Various versions of this article were presented at the 31th (EC)2Conference on ‘‘High dimensional modelling
in time series’’, the 9th Italian Congress of Econometrics and Empirical Economics, the International Association
for Applied Econometrics 2021 Annual Conference, the workshop on ‘‘Dimensionality Reduction and Inference
in High-Dimensional Time Series’’ at Maastricht University, the 15th International Conference on Computational
and Financial Econometrics, the 3rd Italian Workshop of Econometrics and Empirical Economics: ‘‘High-
dimensional and Multivariate Econometrics: Theory and Practice’’, the workshop ‘‘Revisiting and Improving
Prediction Tools for Central Banks’’, and at a seminar at Sichuan University. We thank the participants, as well
as Massimiliano Marcellino, Sung K. Ahn and two anonymous referees, for helpful comments and suggestions.
The usual disclaimers apply.
1It is usual and correct to quote Sims (1980) for his fundamental contribution to the VAR literature. An earlier
reference is Quenouille (1957).
1123
©2022 Oxford University and John Wiley & Sons Ltd.
1124 Bulletin
testing for Granger causality as well as to perform structural analyses. However, the
number of VAR parameters to be estimated increases quadratically with the number of
variables and linearly with the number of lags. This quickly compromises the estimation
results outside the case of small-scale models.
There is currently a growing interest for jointly modelling many variables and
consequently to look at the feasibility to work with large dimensional VAR models.
Indeed, the increase of data availability in economics and finance2is associated with
the common belief that using more information with high-dimensional econometric
and statistical models will improve our understanding of the macroeconomy as well
as forecast accuracy (see Boivin and Ng, 2006 for counterexamples). Consequently,
as the increase of the number of time series jointly considered in VARs cannot be
too large compared with a given number of observations, different attempts have been
proposed in the literature to curb the curse of dimensionality problem. These methods
can be gathered in two categories: dimension reduction approaches on the one hand and
regularization techniques on the other hand. The former group of methods, to which
our paper wishes to contribute, includes reduced-rank techniques (Reinsel, 1983;Ahn
and Reinsel, 1988; Carriero, Kapetanios and Marcellino, 2011; Cubadda and Hecq, 2011;
Bernardini and Cubadda, 2015) and the huge literature on factor models (surveyed in Stock
and Watson, 2016, and Lippi, 2018,2019). We include in the latter group both Bayesian
methods (surveyed in Karlsson, 2013, Koop, 2017), although Bayesian techniques are
also used to estimate large reduced-rank VARs (see Carriero et al.,2011), and the
more recent booming contributions on penalized estimation of sparse VARs (Hsu, Hung
and Chang, 2008; Kock and Callot, 2015; Davis, Zang and Zheng, 2016; Wilms and
Croux, 2016; Nicholson et al., 2018; Smeekes and Wijler, 2018; Hecq, Margaritella and
Smeekes, 2021).
Differently from previous contributions on system dimension reduction, we provide a
framework where the whole dynamics of the system is due to an underlying small-scale
VAR model. Following Lam, Yao and Bathia (2011) and Lam and Yao (2012), we first
decompose the large multivariate time series Ytinto two parts: a linear function of a
small-scale dynamic component xtand a static component εtthat, and this is the key point
that makes our approach different from the usual dynamic factor model, is unpredictable
from the past. The capital Ytstresses that we start from a potentially high-dimensional time
series process, whereas the small xtstresses that a small number of factors are responsible
for the entire dynamics of the system. Then we provide the conditions under which
such a dynamic component xtis generated by a small-scale VAR model. Particularly,
we show that it is required that the large VAR model of series Ytis endowed with both
the serial correlation common feature (Engle and Kozicki, 1993) and an index structure
(Reinsel, 1983) in order to ensure that the dynamic component xtfollows a VAR model.
Hence, we provide a link between the factor modelling for high-dimensional time series
and the reduced-rank VAR approach, which, to the best of our knowledge, was not noted
before. This bridge allows us to unravel common cyclical features and to impose their
presence in large VARs.
2Both in terms of the number of series that are easily available and in the use of different sampling frequencies (e.g.
mixed frequency VARs, see Goetz, Hecq and Smeekes, 2016).
©2022 Oxford University and John Wiley & Sons Ltd.
Get this document and AI-powered insights with a free trial of vLex and Vincent AI
Get Started for FreeStart Your Free Trial of vLex and Vincent AI, Your Precision-Engineered Legal Assistant
-
Access comprehensive legal content with no limitations across vLex's unparalleled global legal database
-
Build stronger arguments with verified citations and CERT citator that tracks case history and precedential strength
-
Transform your legal research from hours to minutes with Vincent AI's intelligent search and analysis capabilities
-
Elevate your practice by focusing your expertise where it matters most while Vincent handles the heavy lifting
Start Your Free Trial of vLex and Vincent AI, Your Precision-Engineered Legal Assistant
-
Access comprehensive legal content with no limitations across vLex's unparalleled global legal database
-
Build stronger arguments with verified citations and CERT citator that tracks case history and precedential strength
-
Transform your legal research from hours to minutes with Vincent AI's intelligent search and analysis capabilities
-
Elevate your practice by focusing your expertise where it matters most while Vincent handles the heavy lifting
Start Your Free Trial of vLex and Vincent AI, Your Precision-Engineered Legal Assistant
-
Access comprehensive legal content with no limitations across vLex's unparalleled global legal database
-
Build stronger arguments with verified citations and CERT citator that tracks case history and precedential strength
-
Transform your legal research from hours to minutes with Vincent AI's intelligent search and analysis capabilities
-
Elevate your practice by focusing your expertise where it matters most while Vincent handles the heavy lifting
Start Your Free Trial of vLex and Vincent AI, Your Precision-Engineered Legal Assistant
-
Access comprehensive legal content with no limitations across vLex's unparalleled global legal database
-
Build stronger arguments with verified citations and CERT citator that tracks case history and precedential strength
-
Transform your legal research from hours to minutes with Vincent AI's intelligent search and analysis capabilities
-
Elevate your practice by focusing your expertise where it matters most while Vincent handles the heavy lifting
Start Your Free Trial of vLex and Vincent AI, Your Precision-Engineered Legal Assistant
-
Access comprehensive legal content with no limitations across vLex's unparalleled global legal database
-
Build stronger arguments with verified citations and CERT citator that tracks case history and precedential strength
-
Transform your legal research from hours to minutes with Vincent AI's intelligent search and analysis capabilities
-
Elevate your practice by focusing your expertise where it matters most while Vincent handles the heavy lifting
