Testing for Granger Causality in Moments

AuthorYi‐Ting Chen
Published date01 April 2016
Date01 April 2016
DOIhttp://doi.org/10.1111/obes.12108
265
©2015 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 78, 2 (2016) 0305–9049
doi: 10.1111/obes.12108
Testing for Granger Causality in Moments*
Yi-Ting Chen
Institute of Economics, Academia Sinica, Taipei 115, Taiwan
(e-mail: ytchen@sinica.edu.tw)
Abstract
In this paper, we consider a generalized approach which is flexibly applicable to testing
Granger causality in various moments and in both the full-sample and out-of-sample con-
texts. We further use this approach to establish a class of cross-correlation tests for financial
time series analysis, and show the advantages of this class of tests in unifying and general-
izing Box–Pierce-type Granger causality tests. We also conduct a Monte Carlo simulation
to show the validity of our tests, and provide an empirical example to demonstrate the
flexibility of our tests in exploring various types of Granger causality.
I. Introduction
An economic variable is Granger-causing another variable if its history is indispensable
from the information set generated by the history of both variables in modeling or pre-
dicting the conditional distribution of the latter; see Granger (1969, 1980). Researchers
have proposed a class of residuals-based tests based on restricted univariate models, in-
cluding the Granger causality in mean test of Pierce (1977) and Pierce and Haugh (1977)
based on an autoregressive (AR) model, the Granger causality in variance tests of Cheung
and Ng (1996) and Hong (2001) based on a generalized autoregressive conditional het-
eroskedasticity (GARCH)-type model, and the Granger causality in risk test of Hong, Liu
and Wang (2009) based on a fully specified GARCH-type model or a conditional quantile
model.
Similar to the autocorrelation test statistic of Box and Pierce (1970), the test statistics
of Pierce (1977), Pierce and Haugh (1977) and Cheung and Ng (1996) are all expressed
as sums of squared sample cross-correlations multiplied by the sample size. Thus, we re-
fer to them as Box–Pierce-type (Granger causality) tests. Although these tests are simple
for implementation, they are designed for testing Granger causality in a particular mo-
ment, based on a particular model and estimation method and derived in the full-sample
context, and assume that the models of the variables being examined have sequences of
JEL Classification numbers: C12, C13, C52.
*The author is indebted to the editor (Anindya Banerjee) and two anonymous referees for their comments
and suggestions, and acknowledge the research support provided by the Ministry of Science and Technology
of Taiwan (NSC101-2410-H-001-009). A supplementary appendix of the paper is available at http://idv.sinica.
edu.tw/ytchen/GrangerCausalityTestAppendix.pdf or upon request.
266 Bulletin
independently and identically distributed (iid) error terms and the error-term sequences
are mutually independent.
In applications, researchers may be interested in exploring Granger causality not only
in mean, variance or quantiles but also in any moments with suitable statistical or eco-
nomic meanings, such as symmetry or various risk measures, separately or simultane-
ously. In addition, possibly influenced by the notion: Granger causality needs ‘evidence
of improved forecasts’, see, e.g. Granger (1980, p. 348), practitioners also conduct out-
of-sample analyses in related empirical studies. In this scenario, they need suitable out-
of-sample tests for Granger causality. Moreover, the aforementioned independence
assumption is indeed stronger than the hypothesis of Granger non-causality and not
ensured to be valid in empirical contexts. Motivated by these problems, we consider a
general approach which is applicable to testing Granger causality in a more flexible and
robust way.
Our test approach is established by combining the asymptotic method for full-sample
moment tests conducted by Newey (1985) andTauchen (1985), among many others, with
its out-of-sample counterpart proposed by West (1996) and West and McCracken (1998).
This combination has also been applied by Chen (2011) to density forecast evaluation.
In this paper, we apply this approach to establishing a generalized parametric test which
examines the hypothesisof Granger non-causality from the aspect of ‘cor rect specification’.
The generalized test is applicable to unifying existing parametric tests and to generating
new tests for Granger causality. In context of GARCH-type models, we use it to establish
a class of cross-correlation tests, and show how this class of tests extend the applicability
of the existing Box–Pierce-type tests to various moments and to the out-of-sample context
without imposing the independence assumption.
Although our frameworkis quite general for encompassing parametric tests for Granger
causality, it is byconstr uction unableto account for non-parametric tests, such as the spec-
tral tests of Hong (2001) and Hong et al. (2009) in checking Granger causality in variance
or in value-at-risk (VaR), the sup-Wald test applied by Chuang, Kuan and Lin (2009) for
checking Granger causality in a continuum of quantiles, and the model-free quantile test
of Jeong, H¨ardle and Song (2012). The out-of-sample predictive ability test of McCracken
(2007), designed for nested regressions, is also beyond our context. Furthermore, one
might choose between the full-sample test and the out-of-sample test by formally compar-
ing the tests’ power properties; see, e.g. Inoue and Kilian (2004), Clark and McCracken
(2005) and Chen (2005). Such a comparison may need to explore how the distribution of
a test statistic is influenced by the structural instability of model, and is not considered
here.
The remainder of this paper is organized as follows. In section II, we review Box–
Pierce-type tests, and introduce the generalized test. In section III, we establish a class of
cross-correlation tests, and compare them with existing tests. In section IV, we conduct
a simulation study to assess the performance of our cross-correlation tests. In section V,
we use an empirical example to showthe flexibility of our tests in detecting various types of
Granger causality. We conclude this paper in sectionVI. In addition, we provide a supple-
mentary appendix that includes further discussions of the tests, mathematical derivations,
simulation results and empirical tables. This appendix is not reported here for the sake of
space, but it is available from the author.
©2015 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd

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