Inference in Misspecified GARCH‐M Models*
| Published date | 01 April 2022 |
| Author | Aaron D. Smallwood |
| Date | 01 April 2022 |
| DOI | http://doi.org/10.1111/obes.12468 |
Inference in Misspecified GARCH-M Models*
AARON D. SMALLWOOD
Department of Economics, University of Texas Arlington, 701 S. West Street. (mailbox: 19479),
Arlington, Texas 76019, USA (e-mail: smallwood@uta.edu)
Abstract
The manuscript studies testing methods in systems with relationships between observed
variables and conditional variances drawn from popular multivariate GARCH models.
Although these methods have been extensively used to study the effects of uncertainty
proxied by GARCH variables, inferential results are absent under misspecification or
when using multi-step estimators. Concentrating on test statistics for the hypothesis of
no uncertainty impact, extensive Monte Carlo evidence is presented. Results show that
severe size distortion and low power can occur when using two-step procedures unless
existing heteroskedasticity is modelled at every stage. In contrast, under moderate
unconditional residual cross-correlation, joint estimation of all model parameters yields
test statistics with impressive relative power. In terms of misspecification, the
consequences of ignoring asymmetries in the conditional variance matrix are shown to
be potentially severe. Otherwise, estimation of DCC and diagonal BEKK models may
be preferred relative to extended DCC and full BEKK counterparts, even under weak
negative volatility spillovers. Issues are highlighted with an analysis of the
relationships between production growth, inflation and their volatilities.
I. Introduction
In economics, there has been intense interest in the effects of uncertainty, which has
become even more exacerbated after the recent financial crisis and global pandemic. In
yielding a measure of the variance of a variable’s unexplained component, GARCH
methods have been used extensively as researchers search for an appropriate proxy for
risk. Applications are virtually boundless and include tests for economic uncertainty’s
effects on variables ranging from fertility (Hondroyiannis, 2010) to output and inflation
(Grier and Perry, 2000).
In many applications, researchers first estimate a GARCH model to obtain an
uncertainty variable treated as known in subsequent analysis. Although it might be
recognized that a generated regressor problem exists, the more general consequences of
JEL Classification numbers:C32. C22. C53. E00.
*The author thanks the Texas Advanced Computing Center for providing HPC resources used to generate
simulation results (URL: http://www.tacc.utexas.edu). The author acknowledges outstanding comments from the
Editor and two anonymous reviewers and also thanks Paul Beaumont, William Crowder and Kevin Grier for
invaluable feedback on earlier drafts of this paper.
334
©2021 The Department of Economics, University of Oxford and John Wiley & Sons Ltd
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 84, 2 (2022) 0305-9049
doi: 10.1111/obes.12468
heteroskedasticity for the variable under study are often ignored. It is also easy to
overlook the fact that GARCH proxies are often obtained from the residuals of variables
in a system that could possess both conditional and unconditional cross-correlation.
Additionally, the measure of risk will depend on the precise GARCH method that was
used to obtain it. As such, GARCH proxies could be obtained under misspecfication.
Unfortunately, for researchers using GARCH-based uncertainty variables, inferential
results are largely absent from the literature. Conrad and Mammen (2016) do provide
asymptotic results for QMLE estimators for specific univariate GARCH-in-mean models.
However, inference has yet to be extended to multivariate methods, and there is a dearth in
the literature in providing any guidance under misspecification and the use of multi-step
procedures. This is especially unfortunate in GARCH-M systems that couple mean
equations with a multivariate GARCH method, as joint parameter estimation can be
computationally demanding and even infeasible. As it seems clear that exact analytical
expressions for test statisticswill be very difficult to obtain in this environment, this
manuscript fills the void within the literature by providing extensive Monte Carlo evidence.
To motivate the problem, the paper empirically analyses the impacts of
macroeconomic uncertainty on inflation and industrial production growth using the
most popular methods that have been analysed in this extensive literature. The paper
demonstrates that two-step methods can produce erratic results, depending on whether
heteroskedasticity is accounted for in all stages. In contrast, fairly consistent evidence
is presented to show that inflation uncertainty has harmful effects on growth when
joint parameter estimation is coupled with a multivariate GARCH model.
Aspects of the empirical results here and others from related studies are used to
generate experiments that yield insight absent from existing research. The design assumes
all residuals are heteroskedastic, where variables are possibly functions of the conditional
variance of others within a system. Numerous parametric models are assumed for the
conditional variance matrix, including various dynamic conditional correlation (DCC)
and BEKK GARCH specifications, where asymmetry is considered for both individual
volatilities and conditional covariances/correlations. The results show that multi-step
methods that account for GARCH in the second stage rarely suffer from size distortion
under standard inference. However, if second-stage heteroskedasticity is ignored when
using OLS, severe size distortion and substantially lower size adjusted power result,
where heteroskedastic-based corrections offer only mild relief. These results show that
simple OLS-based test statistics, including those based on Granger causality, should not
be used when GARCH effects exist in second stage errors.
In terms of detecting uncertainty impacts, two-step methods produce reasonable size-
adjusted power if residual cross-correlation is low, so long as GARCH equations are
estimated throughout. In contrast, large gains occur under stronger residual cross-
correlation, and it is thus recommended that joint parameter estimation be used when
residuals have higher cross-correlation. The results further show that it is vital to properly
model asymmetric behaviour, particularly under shock and volatility spillovers, where
misspecification can yield size-distortion rendering certain procedures uninformative. In
contrast, the consequences of unnecessarily modelling asymmetries appear to be
relatively minor. Remaining results otherwise show that unless there are strong spillover
effects, estimation of the DCC and diagonal BEKK models produce impressive relative
©2021 The Department of Economics, University of Oxford and John Wiley & Sons Ltd
Misspecified GARCH-M Models335
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