Forecasting with High‐Dimensional Panel VARs
| DOI | http://doi.org/10.1111/obes.12303 |
| Published date | 01 October 2019 |
| Author | Dimitris Korobilis,Gary Koop |
| Date | 01 October 2019 |
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©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 81, 5 (2019) 0305–9049
doi: 10.1111/obes.12303
Forecasting with High-Dimensional Panel VARs*
Gary Koop,†Dimitris Korobilis‡
†University of Strathclyde, Glasgow, G4 0GE, UK (e-mail: gary.koop@strath.ac.uk)
‡Essex Business School, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, UK
(e-mail: d.korobilis@essex.ac.uk)
Abstract
This paper develops methods for estimating and forecasting in Bayesian panel vector au-
toregressions of large dimensions with time-varying parameters and stochastic volatility.
We exploit a hierarchical prior that takes into account possible pooling restrictions in-
volving both VAR coefficients and the error covariance matrix, and propose a Bayesian
dynamic learning procedure that controls for various sources of model uncertainty. We
tackle computational concerns by means of a simulation-free algorithm that relies on an-
alytical approximations to the posterior. We use our methods to forecast inflation rates in
the eurozone and show that these forecasts are superior to alternative methods for large
vector autoregressions.
I. Introduction
As the economies of the worldbecome increasingly linked through trade and financial flows,
the need for multi-country econometric modelling has increased. Panel vector autoregres-
sions (PVARs), which jointly model manymacroeconomic variables in many countries, are
becoming a popular way of fulfilling this need. We use the general term PVAR for models
where the dependent variables for all countries are modelled jointly in a single VAR and,
thus, the VAR for each individual country is augmented with lagged dependent variables
from other countries. In this paper, wedevelop econometric methods for PVARsof possibly
large dimensions using a hierarchical prior which can help overcome the concerns about
over-parameterization that arise in these models.The novelties of our approach are that (i)
we tackle in one integrated setting all concerns pertaining to out-of-sample forecasting,
such as controlling for over-fitting and dealing with model uncertainty; (ii) we allow for
empirically relevant extensionsthat account for str uctural breaks and changing volatilities;
and (iii) we do so in a computationally efficient way, building on previous work by Koop
and Korobilis (2013) for single-country VARs.
JEL Classification numbers: C11, C32, C53, C55.
*Wewant to thank Ana Galvao, TonyGar ratt, GeorgeKapetanios, James Mitchell, Ivan Petrella, Rob Taylor and
participants at the 9th European Central Bank Workshopon Forecasting Techniques; the Bank of England workshop
on Time-Variation in Econometrics and Macroeconomics; and the 10th RCEA Bayesian EconometricWorkshop, for
helpful comments and discussions. All remaining errors are ours.
938 Bulletin
To explain the significance of the econometric contributions of this paper, we note a
major characteristic of the existing literature on multi-country VAR models lies in the
need to appropriately model linkages between countries. This literature includes Bayesian
multi-country VARs (Canova and Ciccarelli, 2009; Koop and Korobilis, 2016), Global
VARs (Dees et al., 2007; Feldkircher and Huber, 2016), multi-country factor models (Kose,
Otrok and Whiteman, 2003), and spatial VARs (Chudik and Pesaran, 2011). The common
ground of all these modelling approaches is the need to account for the panel structure in
the data, and explicitly model inter-dependencies and commonalities in the units (countries
or individuals). This is an important consideration as the dimension of panel VARs tends
to grow rapidly: the case with only five variables for 10 countries results in a model
with 50 endogenous variables and thousands of parameters. Therefore, any panel-specific
restrictions one can impose, such as clustering/pooling coefficients across units, could
help identify parsimonious models that are useful in forecasting or structural inference.
Alternative solutions to the overparametrization problem are offered in the literature on
large (single-country) VARs – such as those estimated by Ba´nbura, Giannone and Reichlin
(2010) and Koop and Korobilis (2013), that incorporate shrinkage estimators and efficient
computational algorithms. However, large VAR methods typically rely on shrinkage priors
that ignore the panel structure in the data and this loss of useful information might have
adverse effects in forecasting.1
Consequently, our motivation and first econometric contribution is to develop efficient
methods for VARs of large dimensions that feature panel-specific restrictions as well as
time-varying parameters and stochastic volatility, and fill this particular gap in the VAR
literature. Our starting point is the seminal contribution of Canova and Ciccarelli (2009)
who introduce a hierarchical shrinkage prior for multi-country VARs with time-varying
parameters and develop Markovchain Monte Carlo (MCMC) simulation methods to tackle
estimation.2We extend their methods to account for stochastic volatility in the panel VAR
error covariance matrix. We also propose a model formulation that allows us to introduce
their hierarchical shrinkage prior to the time-varying error covariance matrix. Both these
extensions are empirically relevant. There is ample evidence that volatility in empirical
macroeconomic models is extremely important for forecasting (see, among many others,
Clark and Ravazzolo, 2015 and Diebold, Schorfheide and Shin, 2017), in which case it
is imperative to relax the assumption of homoscedasticity used by Canova and Ciccarelli
(2009). However, given that time-varying covariancematrices are non-parsimonious, some
form of shrinkage is also needed for these parameters. Off-diagonal elements of the error
1In particular, popular applications of large VARs rely on the Minnesota prior (Doan, Litterman and Sims, 1984)
that places weak prior shrinkage on the intercepts and own autoregressive dynamics of each variable, but heavily
shrinks cross-terms and more distant lags. In the case of panel VARs, right-hand side lags include own lags of each
variable of a given country, but also (i) lags of the same variableof other countries; (ii) lags of other variables of the
same country; and (iii) lags of other variables of other countries.The Minnesota prior would simply place equal prior
weight on these three categories of right-hand side variables, which would result in discarding useful information
about interdependencies and homogeneities among countries. Similar arguments would hold for anyshrinkage prior
or penalized estimator that is not developed specifically for panel VARs, such as the popular lasso of Tibshirani
(1996).
2Aswe explain later in this paper, this hierarchical prior pools certain VARcoefficients by country or by variable,and
results in a lowerdimensional VAR, whichg reatlyreduces the computational burden. However, the use of simulation
methods means that the authors consider only structural analysis (not much more computationallyintensive recursive
forecasting), using 27-variable PVARs.
©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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