Measuring Spot Variance Spillovers when (Co)variances are Time‐varying – The Case of Multivariate GARCH Models
| Published date | 01 February 2018 |
| Author | Helmut Herwartz,Matthias R. Fengler |
| DOI | http://doi.org/10.1111/obes.12191 |
| Date | 01 February 2018 |
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©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 80, 1 (2018) 0305–9049
doi: 10.1111/obes.12191
Measuring Spot Variance Spillovers when
(Co)variances areTime-varying – The Case of
Multivariate GARCH Models*
Matthias R. Fengler† and Helmut Herwartz‡
†School of Economics and Political Science, University of St. Gallen, Bodanstrasse 6
CH–9000, St. Gallen, Switzerland (e-mail: matthias.fengler@unisg.ch)
‡Department of Economics, Georg-August-University G¨ottingen, Platz der G¨ottinger Sieben
5 D–37073, G¨ottingen, Germany (e-mail: hherwartz@uni-goettingen.de)
Abstract
We propose global and disaggregated spillover indices that allow us to assess variance
and covariance spillovers, locally in time and conditionally on time-tinformation. Key to
our approach is the vector moving average representation of the half-vectorized ‘squared’
multivariate GARCH process of the popular BEKK model. In an empirical application to a
four-dimensional system of broad asset classes (equity,fixed income, foreign exchange and
commodities), we illustrate the new spillover indices at various levels of (dis)aggregation.
Moreover, we demonstrate that they are informative of the value-at-risk violations of port-
folios composed of the considered asset classes.
I. Introduction
In highly integrated markets, shocks spread at a fast pace and bedevil risk management
and optimal asset allocation because of disappearing diversification benefits and cascade
effects. Awareness of this fact has risen especially during the financial crisis of 2008
and over the subsequent years of economic fragility. Consequently, much effort has been
devoted to developing quantitative measures of economic interdependence. Examples
include the systemic expected shortfall of Acharya et al. (2010), the conditional value-at-
risk of Adrian and Brunnermeier (2016), and the spillover indices of Diebold andYilmaz
(2009, 2012, 2014).
Among these, the spillover indices of Diebold and Yilmaz (2009, 2012, 2014) have
garnered much attention, because in contrast to other measures, they allow one to track
JEL Classification numbers: C32, C58, F3, G1.
*The authors gratefully acknowledge helpful comments and suggestions from anonymous referees and theAsso-
ciate Editor Heino Bohn Nielsen, from J¨org Breitung, Katja Gisler, Roman Liesenfeld, Ostap Okhrin, Kamil Yilmaz,
as well as from seminar participants at the Universit¨at zu K ¨oln,TU Dresden, and the CFE 2015, London.
Financial support by the Swiss National Science Foundation is gratefully acknowledged (Grant No. 144033).
Helmut Herwartz gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (Grant No.
HE 2188/8-1).
136 Bulletin
the associations between individual variables and the system as a whole at all levels, from
pairwise to system-wide, in a mutually consistent way. The notion of a spillover is that of a
forecast error variance share derived from the forecast error variance decomposition of an
underlying vector autoregressive model (VAR). For example, to study variance spillovers,
one simply estimates a VAR on measures of realized variance (see Yilmaz, 2013; Fengler
and Gisler, 2015; Barun´ık, Koˇcenda and V´acha, 2016).
Because the indices are based on the forecast error variance decomposition of a single
VAR, they produce static and average spillover information. While this is undoubtedly
valuable, it would be of greater use to have up-to-date spillover information, especially
when thinking of variance spillovers. There is ample evidence that conditional variances
are time-varying, and it is natural to expect that spillovers are as well. Diebold and Yilmaz
(2009, 2012) therefore suggest computing the indices from VAR models that are estimated
on rolling subsamples. In this way, one obtains an impression of the time-varying patterns
of spillovers. But as with all rolling window approaches, the estimates reflect only the
average information of the respective estimation window. Because the subsamples must
be of sufficient length to provide reasonably accurate parameter estimates, the rolling
window indices are probably more useful for a retrospective analysis than for the timely
monitoring of spillovers. For this purpose, one would need a time-tconditional spillover
index.
In this paper, we propose such an approach. We adopt the ideas of Diebold and Yilmaz
(2009, 2012) and construct variance spillover indices that are updated with time-tinforma-
tion. To this end, we build on multivariate GARCH (MGARCH) models of the BEKK-type
(Baba et al., 1990; Engle and Kroner, 1995) and calculate the indices from the forecast
error variance decomposition that is derived from the vector moving average (VMA) rep-
resentation of the ‘squared’and vectorized retur n process.This process is driven by serially
uncorrelated heteroskedastic innovations and, as we show here, its conditional covariance
matrix can be derived analytically. This allows us to absorb the regimedependence into the
parameters of the VMA representation.The variance spillover indices that are based on this
time-varying VMA representation therefore take full advantage of the time-tconditional
information of the prevailing variance regime. In contrast to rolling window estimates,
they convey on-the-spot variance spillover information. In our empirical applications, we
show not only that the time-tconditional variance spillover indices allow a study of the
prompt impact of major economic or political events, but also that they are informative of
the likelihood of value-at-risk violations.
Aside from the value of well-timed spillover information, our approach differs in
methodological terms from the extant literature in that we derive the spillover indices
from a full-fledged model of variance and covariance dynamics.This has advantages that
are more than conceptual. First and most importantly, the variance spillover indices take
full advantage of the informational content embedded in covariances. It appears common
sense to expect covariance dynamics to play a decisive role in the mechanisms of variance
spillovers. In Fenglerand Gisler (2015), a first step towards incorporating covariance infor-
mation into variance spillover indices is made, but the authors follow the traditional route
of applying a VAR to vectorized realized covariance matrices estimated from intra-day
data. Thus, they cannot ensure positive definiteness of the dynamic covariance matrices.
While the realized variance literature proposes such models, it does so at the expense of
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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