Sparse Bayesian vector autoregressions in huge dimensions

• Author: Gregor Kastner,Florian Huber
• Published date: 01 November 2020
• DOI: http://doi.org/10.1002/for.2680
• Date: 01 November 2020

Received: 1 November 2018 Revised: 3 December 2019 Accepted: 22 February 2020

DOI: 10.1002/for.2680

RESEARCH ARTICLE

Sparse Bayesian vector autoregressions in huge dimensions

Gregor Kastner1Florian Huber2

1Institute for Statistics and Mathematics,

WU Vienna University of Economics and

Business, Vienna, Austria

2Salzburg Centre of European Union

Studies (SCEUS),University of Salzburg,

Salzburg, Austria

Correspondence

Gregor Kastner, Institute for Statistics and

Mathematics, WU Vienna University of

Economics and Business,

Welthandelsplatz 1, 1020 Vienna, Austria.

Email: gregor.kastner@wu.ac.at

[Correction added on 18 June 2020, after

first online publication: The abstract in

the original version of this article

contained a typographical error (some

words appeared duplicated). This version

corrects the mistake.]

Abstract

We develop a Bayesian vector autoregressive (VAR) model with multivariate

stochastic volatility that is capable of handling vast dimensional information

sets. Three features are introduced to permit reliable estimation of the model.

First, we assume that the reduced-form errors in the VAR feature a factor

stochastic volatility structure, allowing for conditional equation-by-equation

estimation. Second, we apply recently developed global–local shrinkage priors

to the VAR coefficients to cure the curse of dimensionality. Third, we utilize

recent innovations to sample efficiently from high-dimensional multivariate

Gaussian distributions. This makes simulation-based fully Bayesian inference

feasible when the dimensionality is large but the time series length is mod-

erate. We demonstrate the merits of our approach in an extensive simulation

study and apply the model to US macroeconomic data to evaluate its forecasting

capabilities.

KEYWORDS

Dirichlet-Laplace prior, efficient MCMC, factor stochastic volatility, normal-Gamma prior,

shrinkage

1INTRODUCTION

Previous research has identified two important features

that macroeconometric models should possess: the abil-

ity to exploit high-dimensional information sets (Ba ´

nbura

et al., 2010; Koop et al., 2019; Rockova & McAlinn,

2017; Stock & Watson, 2011) and the possibility to cap-

ture nonlinear features of the underlying time series

(Bitto & Frühwirth-Schnatter, 2019; Clark, 2011; Clark &

Ravazzolo, 2015; Cogley & Sargent, 2001; Huber et al.,

2019; Primiceri, 2005). While the literature suggests sev-

eral paths to estimate large models, the majority of

such approaches imply that once nonlinearities are taken

into account analytical solutions are no longer avail-

able and the computational burden becomes prohibitive.

This implies that high-dimensional nonlinear models can

practically be estimated only under strong (and often

unrealistic) restrictions on the dynamics of the model.

However, especially in forecastingapplications or in struc-

tural analysis, successful models should generally be able

to exploit much information and also control for breaks

in the autoregressive parameters or, more importantly,

changes in the volatility of economic shocks (Koop et al.,

2009; Primiceri, 2005; Sims & Zha, 2006).

Two reasons limit the use of large (or even huge) non-

linear models. The first reason is statistical. Since the

number of parameters in a standard vector autoregression

rises quadratically with the number of time series included

and commonly used macroeconomic time series are rather

short, in-sample overfitting turns out to be a serious issue.

As a solution, the Bayesian literature on vector autoregres-

sive (VAR)modeling (e.g., Ankargren et al., 2019; Ba ´

nbura

et al., 2010; Clark, 2011; Clark & Ravazzolo, 2015; Doan

et al., 1984; Follett & Yu, 2019; George et al., 2008; Huber

& Feldkircher, 2019; Koop, 2013; Korobilis & Pettenuzzo,

This is an open access article under the terms of the Creative Commons AttributionLicense, which permits use, distribution and reproduction in any medium, providedthe

original work is properly cited.

© 2020 The Authors. Journal of Forecasting published by John Wiley & Sons, Ltd.

1142 wileyonlinelibrary.com/journal/for Journalof Forecasting. 2020;39:1142–1165.

KASTNER AND HUBER 1143

2019; Litterman, 1986; Sims & Zha, 1998) suggests shrink-

age priors that push the parameter space towards some

stylized prior model like a multivariate random walk. On

the other hand, Ahelegbey et al. (2016) suggest viewing

VARs as graphical models and perform model selection

drawing from the literature on sparse directed acyclic

graphs. This typically leads to much improved forecast-

ing properties and more meaningful structural inference.

Moreover, the majority of the literatureon Bayesian VARs

imposes conjugate priors on the autoregressive parame-

ters, allowing for analytical posterior solutions and thus

avoiding simulation-based techniques such as Markov

chain Monte Carlo (MCMC). Frequentist approaches often

consider multistep approaches (e.g., Davis et al., 2016).

The second reason is computational. Nonlinear

Bayesian models typically have to be estimated by means

of MCMC, and computational intensity increases vastly

when the number of component series becomes large.

This increase stems from the fact that standard algorithms

for multivariate regression models call for the inversion

of large covariance matrices. Especially for sizable sys-

tems, this can quickly turn prohibitive since the inverse

of the posterior variance–covariance matrix on the coeffi-

cients has to be computed for each sweep of the MCMC

algorithm. For natural conjugate models, this step can be

vastly simplified because the likelihood possesses a con-

venient Kronecker structure, implying that all equations

in the VAR feature the same set of explanatory vari-

ables. This speeds up computation by large margins but

restricts the flexibility of the model. Carriero et al. (2016),

for instance, exploit this fact and introduce a simplified

stochastic volatility specification. Another strand of the lit-

erature augments each equation of the VAR by including

the residuals of the preceding equations (Carriero et al.,

2019), which also provides significant improvements in

terms of computational speed. Finally, in a recent contri-

bution, Koop et al. (2019) reduce the dimensionality of

the problem at hand by randomly compressing the lagged

endogenous variables in the VAR.

All papers mentioned hitherto focus on capturing

cross-variable correlation in the conditional mean through

the VAR part, and the comovement in volatilities is cap-

tured by a rich specification of the error variance (Prim-

iceri, 2005) or by a single factor (Carriero et al., 2016).

Another strand of the literature, typically used in financial

econometrics, utilizes factor models to provide a parsi-

monious representation of a covariance matrix, focusing

exclusively on the second moment of the predictive den-

sity. For instance, Pitt and Shephard (1999) and Aguilar

and West (2000) assume that the variance–covariance

matrix of a broad panel of time series might be described

by a lower dimensional matrix of latent factors featuring

stochastic volatility and a variable-specific idiosyncratic

stochastic volatility process.1

The present paper combines the virtues of exploiting

large information sets and allowing for movements in the

error variance. The overfitting issue mentioned above is

solved as follows. First, we use a Dirichlet–Laplace (DL)

prior specification (see Bhattacharya et al., 2015) on the

VAR coefficients. This prior is a global–local shrinkage

prior in the spirit of Polson and Scott (2011) that enables us

to heavily shrink the parameter space but at the same time

provides enough flexibility to allow for nonzero regression

coefficients if necessary.Second, a factor stochastic volatil-

ity model on the VAR errors grants a parsimonious rep-

resentation of the time-varying error variance–covariance

matrix of the VAR. To deal with the computational com-

plexity, we exploit the fact that, conditionally on the

latent factors and their loadings, equation-by-equation

estimation becomes possible within each MCMC itera-

tion. Moreover, we apply recent advances for fast sam-

pling from high-dimensional multivariate Gaussian distri-

butions (Bhattacharya et al., 2016) that permit estimation

of models with hundreds of thousands of autoregressive

parameters and an error covariance matrix with tens of

thousands of nontrivial time-varying elements on a quar-

terly US data set in a reasonable amount of time. In a

careful analysis, we show to what extent our proposed

method improves upon a set of standard algorithms typi-

cally used to simulate from the joint posterior distribution

of large-dimensional Bayesian VARs.

We first assess the merits of our approach in an exten-

sive simulation study based on a range of different

data-generating processes (DGPs). Relative to a set of com-

peting benchmark specifications we show that, in terms

of point estimates, the proposed global–local shrinkage

prior yields precise parameter estimates and successfully

introduces shrinkage in the modeling framework, without

overshrinking significant signals.

In an empirical application, we adopt a modified ver-

sion of the quarterly data set proposed by Stock and Wat-

son (2011) and McCracken and Ng (2016). To illustrate

the out-of-sample performance of our model, we forecast

important economic indicators such as output, consumer

price inflation, and short-term interest rates, amongst oth-

ers. The proposed model is benchmarked against sev-

eral alternatives. Our findings suggest that it performs

well in terms of one-step-ahead predictive likelihoods. In

addition, investigating the time profile of the cumulative

log-predictive likelihood reveals that allowing for large

information sets in combination with the factor structure

especially pays off in times of economic stress.

1Two recent exceptionsare Koop and Korobilis (2013) and Carriero et al.

(2016).