Identification of Economic Shocks by Inequality Constraints in Bayesian Structural Vector Autoregression
| DOI | http://doi.org/10.1111/obes.12338 |
| Date | 01 April 2020 |
| Author | Jani Luoto,Markku Lanne |
| Published date | 01 April 2020 |
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©2019 TheAuthors. OxfordBulletin of Economics and Statistics published by Oxford University and John Wiley & Sons Ltd.
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OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 82, 2 (2020) 0305–9049
doi: 10.1111/obes.12338
Identification of Economic Shocks by Inequality
Constraints in Bayesian Structural Vector
Autoregression*
Markku Lanne and Jani Luoto
Faculty of Social Sciences, University of Helsinki, Helsinki, Finland
Abstract
Theories often make predictions about the signs of the effects of economic shocks on
observable variables, thus implyinginequality constraints on the parameters of a str uctural
vector autoregression (SVAR). We introduce a new Bayesian procedure to evaluate the
probabilities of such constraints, and, hence, to validate the theoretically implied economic
shocks. We first estimate a SVAR, where the shocks are identified by statistical properties
of the data, and subsequently label these statistically identified shocks by the Bayes factors
calculated from their probabilities of satisfying given inequality constraints. In contrast to
the related sign restriction approach that also makes use of theoretically implied inequality
constraints, no restrictions are imposed. Hence, it is possible that only a subset or none of
the theoretically implied shocks can be labelled. In the latter case, we conclude that the
data do not lend support to the theory implying the signs of the effects in question. We
illustrate the method by empirical applications to the crude oil market, and U.S. monetary
policy.
I. Introduction
The structural vector autoregressive (SVAR) model is one of the prominent tools in em-
pirical macroeconomics. While the reduced-form VAR is useful for describing the joint
dynamics of a number of time series, it is only when some structure is imposed upon
it that interesting economic questions apart from forecasting can be addressed. Typically
SVAR analysis involves tracing out the dynamic effects (impulse responses) of economic
shocks on the variables included in the model, and these shocks are often identified by
restricting their effects in various ways (for a comprehensive survey on SVAR models,
see Kilian and L¨utkepohl, 2017). Recently, identification by sign restrictions, put forth
by Faust (1998); Canova and De Nicol´o (2002); Uhlig (2005), has become increasingly
JEL Classification numbers: C32, C5, E52, Q41.
*We thank Heino Bohn Nielsen (editor) and three anonymous referees for useful comments. Financial support
from theAcademy of Finland (grant 308628) is gratefully acknowledged. The first author also acknowledgesfinancial
support from CREATES(DNRF78) funded by the Danish National Research Foundation, while the second author is
grateful for financial support from the Research Funds of the University of Helsinki.
426 Bulletin
popular in the macroeconomic SVAR literature. Compared to most other approaches, sign
restrictions only constraining the signs of the effects of (some of) the shocks to accord
with economic theory or institutional knowledge, are less stringent, yet manage to convey
economic intuition. Therefore, they have a great appeal in empirical research.
In this paper, we propose a formal procedure to identify economic shocks without
actually imposing any restrictions on the parameters of the SVAR model, whilestill making
use of the signs of the effects of shocks. These signs can be easily expressed in the form
of inequality constraints on the parameters of the SVAR model.1Our starting point is the
SVAR model where, followingHyv ¨arinenet al. (2010); Lanne et al. (2017), identification is
achieved by means of statistical properties of the data.The statistically identified structural
shocks (errors) have no economic meaning as such, but for interpretation, they need to
be labelled using external information. To that end, sign constraints have been used in
the previous statistical identification literature (see, e.g. Herwartz and L¨utkepohl, 2014;
L¨utkepohl and Netˇsunajev, 2014; Lanne et al., 2017). The idea of this approach is to
visually check whether the impulse responses implied by the uniquely identified SVAR
model satisfy the constraints. If they are satisfied, the shocks can be labelled accordingly.
Our procedure formalizes this approach by quantifying the likelihood of the inequality
constraints. It also has the advantage that it uses all information from the joint (posterior)
distribution of the estimator of the impulse responses, while the previous approach is
based on their (pseudo) marginal sampling distributions. The latter approach is somewhat
deficient and unreliable, akin to a joint hypothesis testing using the usual tstatistics for
testing the restrictions one at a time.
Our analysis is based on Bayesian inference that facilitates straightforward assessment
of inequality constraints by posterior odds or Bayes factors (see, e.g. Koop, 2003, 39–
40). In particular, as shown in section III, each set of inequality constraints implies a
different model, whose posterior probability can be interpreted as the probability of the
constraints. This probability can then be transformed to the Bayes factor to weigh the
posterior evidence against the case where no constrains are imposed (see, e.g. Kass and
Raftery, 1995). Hence our approach facilitates the identification of the shocks that are
the likeliest to satisfy the constraints (i.e. are the likeliest to be the structural shocks of
interest). It may also turn out that only a subset or none of the inequality constraints are
supported by the data. It is then concluded that the constraints that the data do not lend
support to, are not useful in identifying the economic shocks in question. In this case, an
alternative set of constraints, potentially implied by a competing economic theory, could
be entertained, or the subsequent analysis may concentrate only on the subset of the shocks
that are identified.
A major difference between our approach and imposing sign restrictions is that the
latter only achieves set identification (see, e.g. Baumeister and Hamilton, 2015). There-
fore, assessing the plausibility of the given inequality constraints (sign restrictions) is
not straightforward, whereas, due to point identification, our approach facilitates direct
calculation of the Bayes factor for the constrained model against the unconstrained one.
1Inequality constraints may also be imposed on other functions of structural parameters, not just on impulse
responses, such as historical decompositions considered in Antol´ın-D´ıaz and Rubio-Ram´ırez (2018) (we thank an
anonymous referee for pointing this out). Our procedure obviously generalizes in a straightforward manner to such
cases.
©2019 The Authors. Oxford Bulletin of Economics and Statistics published by Oxford University and JohnWiley & Sons Ltd.
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