Table Of ContentTesting for Granger causality in large mixed-frequency
VARs
Citation for published version (APA):
Götz, T. B., & Hecq, A. W. (2014). Testing for Granger causality in large mixed-frequency VARs.
Maastricht University, Graduate School of Business and Economics. GSBE Research Memoranda No.
028 https://doi.org/10.26481/umagsb.2014028
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Thomas B. Götz, Alain Hecq,
Testing for Granger Causality
in Large Mixed-Frequency
VARs
RM/14/028
Testing for Granger Causality in Large
Mixed-Frequency VARs*
Thomas G¨otz(cid:132)and Alain Hecq(cid:133)
Maastricht University, SBE, Department of Quantitative Economics
July 7, 2014
Abstract
InthispaperweanalyzeGrangercausalitytestinginamixed-frequencyVAR,originally
proposed by Ghysels (2012), where the difference in sampling frequencies of the variables
is large. In particular, we investigate whether past information on a low-frequency vari-
able help in forecasting a high-frequency one and vice versa. Given a realistic sample
size, the number of high-frequency observations per low-frequency period leads to param-
eter proliferation problems in case we attempt to estimate the model unrestrictedly. We
propose two approaches to solve this problem, reduced rank restrictions and a Bayesian
mixed-frequency VAR. For the latter, we extend the approach in Banbura et al. (2010)
to a mixed-frequency setup, which presents an alternative to classical Bayesian estimation
techniques. We compare these methods to a common aggregated low-frequency model as
well as to the unrestricted VAR in terms of their Granger non-causality testing behavior
using Monte Carlo simulations. The techniques are illustrated in an empirical application
involving daily realized volatility and monthly business cycle fluctuations.
JEL Codes: C12, C32
JEL Keywords: Granger Causality, Mixed Frequency VAR, Bayesian VAR, Volatility,
MIDAS
*WewanttoparticularlythankEricGhyselsformanyfruitfuldiscussions,commentsandsuggestionsonearlier
versions of the paper. Moreover, we thank Daniela Osterrieder, Lenard Lieb, Stephan Smeekes, Jo¨rg Breitung,
RomanLiesenfeldandparticipantsoftheWorkshoponAdvancesinQuantitativeEconomicsinMaastricht2013,
theCEISSeminarinRome2013,the33rdInternationalSymposiumonForecastinginSeoul2013,the24th(EC)2
conferenceinCyprus2013andandthe22th SNDEAnnualSymposiuminNewYork2014forusefulsuggestions
and comments on earlier versions of the paper. The usual disclaimer applies.
(cid:132)Corresponding author: Thomas Go¨tz, Department of Quantitative Economics, School of Business
and Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands. Email:
[email protected]
(cid:133)DepartmentofQuantitativeEconomics,SchoolofBusinessandEconomics,MaastrichtUniversity,P.O.Box
616, 6200 MD Maastricht, The Netherlands. Email: [email protected]
1 Introduction
Economic time series are published at various frequencies. While higher frequency variables
used to be aggregated (e.g., Silvestrini and Veredas, 2008), it has become more and more
popular to consider models that take into account the difference in frequencies of the processes
under consideration. As argued extensively in the mixed-frequency literature (e.g., Ghysels
et al., 2007), working in a mixed-frequency setup instead of a common low-frequency one is
advantageous due to the potential loss of information in the latter scenario and feasibility of
the former through MI(xed) DA(ta) S(ampling) regressions (Ghysels et al., 2004), even in the
presence of many high-frequency variables compared to the number of observations.
Until recently, the MIDAS literature was limited to the single-equation framework, in which
one of the low-frequency variables is chosen as the dependent variable and the high-frequency
ones are in the regressors. Since the work of Ghysels (2012) for stationary series and the
extensionofG¨otzetal.(2013)andGhyselsandMiller(2013)forthenon-stationaryandpossibly
cointegrated case, we can analyze the link between high- and low-frequency series in a VAR
system treating all variables as endogenous. Ghysels et al. (2013) define causality in such
a mixed-frequency VAR and develop a corresponding test statistic. The authors, however,
make an implicit assumption on the variables involved, which does not hold in many economic
applications: Thenumberofhigh-frequencyobservationswithinalow-frequencyperiodisrather
small, e.g., as in a year/quarter- or quarter/month-example.
In this paper we analyze the finite sample behavior of Granger non-causality tests when
the number of high-frequency observations per low-frequency period is large, e.g., as in a
month/working day-example (like the application presented in this chapter). To potentially
avoid the proliferation of parameters we consider two parameter reduction techniques, reduced
rank restrictions and a Bayesian mixed-frequency VAR. With respect to the latter we show
how to properly extend the dummy observation approach of Banbura et al. (2010) to a mixed-
frequency setting.1 Importantly, due to stacking the high-frequency variables in the mixed-
frequency VAR (Ghysels, 2012), their approach cannot be applied directly such that a properly
adapted choice of auxiliary dummy variables corresponding to the prior moments is required.
The aforementioned approaches are compared to a common low-frequency VAR, obtained by
temporally aggregating the high-frequency variable (Breitung and Swanson, 2002), and the
unrestricted model, which is expected to suffer from a parameter proliferation problem.
The rest of the paper is organized as follows. In Section 2 notations are introduced, the
mixed-frequency VAR (MF-VAR hereafter) for our specific case at hand is presented and
Granger (non-)causality is defined formally. Section 3 discusses the approaches to reduce the
number of parameters to be estimated, whereby reduced rank restrictions (Section 3.1) as well
as Bayesian MF-VARs (Section 3.2) are presented in detail. The finite sample performances
of these tests are analyzed via a Monte Carlo experiment in Section 4. An empirical example
with U.S. data on the monthly industrial production index and daily volatility in Section 5
1Banburaetal.(2010)refertothesevariablesasdummyobservations. Toavoidconfusionbetweenhigh-and
low-frequency observations and auxiliary variables, we use the term ’auxiliary dummy variables’.
2
illustrates the results. Section 6 gives concluding remarks.
2 Causality in a Mixed-frequency VAR
2.1 Notation
Let us start from a two variable mixed-frequency system, where y , t = 1,...,T, is the low-
t
(m)
frequency variable and x are the high-frequency variables with m high-frequency observa-
t−i/m
tions per low-frequency period t. Contrary to Ghysels et al. (2013), we assume m to be rather
large as in a year/month- or month/working day-example. We also assume m to be constant
rather than varying with t.2 The value of i indicates the specific high-frequency observation
(m)
under consideration, ranging from the beginning of each t-period (x ) until the end
t−(m−1)/m
(m)
(x with i = 0). These notational conventions have become standard in the mixed-frequency
t
literature and are similar to the ones in G¨otz et al. (2014), Clements and Galv˜ao (2008, 2009)
or Miller (2012).
Furthermore, let W = (W(cid:48) , W(cid:48) , ..., W(cid:48) )(cid:48), i.e., it stacks the last p low-frequency lags
t t−1 t−2 t−p
of any process W. Finally, 0 (1 ) denotes an (i×j)-matrix of zeros (ones), I is an identity
i×j i×j i
matrix of dimension i, ⊗ represents the Kronecker product and vec corresponds to the operator
stacking the columns of a matrix.3
2.2 MF-VARs
Considering each high-frequency variable such that
X(m) = (x(m),x(m) ,...,x(m) )(cid:48),
t t t−1/m t−(m−1)/m
a dynamic structural equations model for the stationary multivariate process Z = (y ,X(m)(cid:48))(cid:48)
t t t
isgivenbyA Z = c+A Z +...+A Z +ε ,4 whereΣ isadiagonalcovariancematrixand
c t 1 t−1 p t−p t ε
A pertains to contemporaneous relationships between the series. Note that the parameters in
c
(m)
A are related to the ones in A due to stacking the high-frequency observations X in Z
c 1 t t
2As long as m is deterministic, even time-varying frequency discrepancies do not pose a problem on a theo-
retical level. However, the assumption of constant m simplifies the notation greatly (Ghysels, 2012).
3ExtensionstowardsrepresentationsofhigherdimensionalmultivariatesystemsasinGhyselsetal.(2013)can
beconsidered,butareleftforfurtherresearchhere. AnalyzingGrangercausalityamongmorethantwovariables
inherentlyleadstomulti-horizoncausality(seeLu¨tkepohl,1993amongothers). Thelatterimpliesthepotential
presence of a causal chain: For example, in a trivariate system, X may cause Y through an auxiliary variable
Z. To abstract from that scenario, Ghysels et al. (2013) often consider cases, in which high- and low-frequency
variables are grouped and causality patterns between these groups, viewed as a bivariate system, are analyzed.
They study the presence of a causal chain and multi-horizon causality in a Monte Carlo analysis though.
4Compared to Ghysels (2012) we put the low-frequency variable first.
3
(Ghysels, 2012). Explicitly, for a lag length of p = 1 the model reads as:
1 β β ... β yt
1 2 m
x(m)
δ1 1 −ρ1 ... −ρm−1 t
(m)
δ2 0 1 ... −ρm−2 xt−1/m =
... ... ... ... ... ...
δ 0 0 ... 1 (m)
m x
t−(m−1)/m
cc...12 + ψψρ...y23 ρρmθ...m1−1 ρ.θ....m2. ............ θ00...m x(t−mxy1)t(t...−−−m111)/m + (1)
cm+1 ψ ρ ρ ... ρ (m)
m+1 1 2 m x
t−1−(m−1)/m
ε
1t
ε2t
. .
.
.
ε
(m+1)t
The matrix A links contemporaneous values of y and x: β (cid:54)= 0 implies that y is affected
c j t
(m)
by incoming observations of X , whereas δ (cid:54)= 0 implies that the high-frequency observations
t j
are influenced by y (see G¨otz and Hecq, 2014). The latter becomes interesting for studying
t
policy analysis, where the high-frequency policy variable(s) may react to current low-frequency
conditions (see Ghysels, 2012 for details). Note that an AR(m) model is assumed for the high-
frequency process. If instead it is assumed to follow an AR(q) model with q < m, one should
set ρ = ... = ρ = 0 in (1).
q+1 m
Pre-multiplying (1) by A−1 we get to the mixed-frequency reduced-form VAR(p), i.e., the
c
aforementioned MF-VAR, model:
Z = A−1c+A−1A Z +...+A−1A Z +A−1ε
t c c 1 t−1 c p t−p c t
= µ+Γ Z +...+Γ Z +u
1 t−1 p t−p t
or
Z = µ+B(cid:48)Z +u , (2)
t t t
where B = (Γ ,Γ ,...,Γ )(cid:48). Equation (2) is easy to estimate on small systems. As an example,
1 2 p
4
let us explicitly write a VAR(1) as
y
t
x(m) µ1 γ1,1 γ1,2 ··· γ1,m+1
t
x(t−m..1)/m = µ...2 + γ2...,1 γ2...,3 ·.·.·. γ2,m...+1
.
(m) µm+1 γm+1,1 γm+1,2 ··· γm+1,m+1
x (cid:124) (cid:123)(cid:122) (cid:125)
t−(m−1)/m
Γ1 (3)
y
t−1
x(m) u1t
t−1
× x(t−m1)−1/m + u..2t ,
.. .
.
(m) u(m+1)t
x
t−1−(m−1)/m
σ σ ... σ
1,1 1,2 1,m+1
.
.
ut ∼ N(0((m+1)×1),Σu), Σu = σ2...,1 σ2...,2 ...... .... .
σ ... ... σ
m+1,1 m+1,m+1
2.3 Granger Causality in MF-VARs
Let the information set generated by the collection of sigma-fields F = σ(Z ,s ≤ t),t ≥ 0
t s
be denoted by Ω . Furthermore, let Z be adapted to that filtration such that Ω represents
t t t
the set of information available at moment t (Mikosch, 1998). ΩW denotes the information set
t
containing the information for all stochastic process except W up to moment t. P[X(m)|ΩW]
t+h t
is the best linear forecast of X(m) based on ΩW and likewise for P[y |ΩW]. Granger non-
t+h t t+h t
causality is defined as follows (Eichler and Didelez, 2009):
Definition 1 y does not Granger cause X(m) if
P[X(m)|Ωy] = P[X(m)|Ω ].
t+1 t t+1 t
Similarly, X(m) does not Granger cause y if
P[y |ΩX(m)] = P[y |Ω ].
t+1 t t+1 t
In other words, y does not Granger cause X(m) if past information of the low-frequency
variable do not help in predicting current (or future) values of the high-frequency variables
and vice versa (Granger, 1969). In terms of (3), testing for Granger non-causality implies the
following null (and alternative) hypotheses:
5
(cid:136) y does not Granger cause X(m)
H : γ = γ = ... = γ = 0,
0 2,1 3,1 m+1,1
H : γ (cid:54)= 0 for at least one i = 2,...,m+1.
A i,1
(cid:136) X(m) does not Granger cause y
H : γ = γ = ... = γ = 0,
0 1,2 1,3 1,m+1
H : γ (cid:54)= 0 for at least one i = 2,...,m+1.
A 1,i
Ghysels et al. (2013) implicitly assume m to be rather small (m = 3 or 4 in their case) in
order to estimate the MF-VAR and then test for Granger non-causality. With m large (say
m ≥ 12), we would need a much larger sample than is usually applicable for macroeconomic
data sets to estimate the parameters and test for causality properly. Additional lags (p > 1)
would further complicate the issue.
2.4 Remark on Nowcasting Causality
It becomes clear from Section 2.3 that Granger non-causality in both directions is defined in
terms of the low frequency index t. Given the mixed-frequency nature of the variables under
(m)
consideration, it may be of interest to analyze whether knowing the values of x ,i =
t−i/m
0,...,m−1, helps to predict y , which is referred to as instantaneous causality in the common-
t
frequency case (Renault et al., 1998, Lu¨tkepohl, 2005, p.42, Hamilton, 1994, p. 301, Gianetto
and Ra¨ıssi, 2012).
Go¨tzandHecq(2014)haveintroducedthetermnowcastingcausalityasthemixed-frequency
analoguetoinstantaneouscausality,becauseitamountstopredictingy usingvaluesofthehigh-
t
frequency variables within period t. In terms of testing for nowcasting non-causality, G¨otz and
(m)
Hecq (2014) argue that X is not nowcasting causal for y if and only if the corresponding
t t
errors of the data generating process of Z are uncorrelated (Lu¨tkepohl, 2005). Indeed, testing
t
for nowcasting non-causality corresponds to β = δ = 0 ∀j in (1). In that case, though, Σ
j j u
is block diagonal with σ = σ = 0, j = 2,...,m+1, because Σ = A−1Σ A−1(cid:48). Thus, we
1,j j,1 u c ε c
can test for nowcasting non-causality by using a Wald test on the (1,j)-elements of Σ(cid:99)u, i.e.,
σ = 1 (cid:80)T u u ,j = 2,...,m+1, where u corresponds to the residuals of equation j.5
(cid:98)1,j T t=1(cid:98)1t(cid:98)jt (cid:98)jt
3 Parameter Reduction
Thissectionpresentstechniquesthatwehaveconsidered, andevaluatedthroughaMonteCarlo
exercise, with the aim to reduce the amount of parameters to be estimated in the MF-VAR
model. Two approaches are discussed in detail, reduced rank restrictions and a Bayesian MF-
VAR.
5In this paper we do not include degree of freedom corrections when computing estimates of covariance
matrices due to their invariance asymptotically.
6
There are many alternative approaches to reduce the number of parameters among which
are principal components, Lasso (Tibshirani, 1996) or ridge regressions.6 The reason not to
consider principal components is that it does not necessarily preserve the dynamics of the
VAR under the null. To be more precise, nothing prevents, e.g., the first and only principal
component to be loading only on y implying that the remaining dynamics enter the error term.
In other words, principal components may and will most likely not preserve the autoregressive
matrices in (2). This affects, in a possibly drastic manner, the block of parameters we test on
for Granger non-causality.
3.1 Reduced Rank Restrictions
3.1.1 Reduced Rank Regression Model
In order to reduce the number of parameters to estimate in the MF-VAR, we propose the
followingreducedrankregressionmodel,inwhichweassumethatrk(B(cid:48) ) = r < m(Carriero
X(m)
et al., 2011) with B(cid:48) being obtained from B(cid:48) by excluding the first columns of Γ , Γ , ...,
X(m) 1 2
Γ . The model reads as
p
Z = µ+γ y +α(cid:80)p δ(cid:48)X(m)+ν
t ·,1 t i=1 i t−i t (4)
= µ+γ y +αδ(cid:48)X(m)+ν ,
·,1 t t t
where γ is the (m+1)×p matrix containing the first columns of Γ ,Γ ,...,Γ and α and
·,1 1 2 p
δ = (δ(cid:48),...,δ(cid:48))(cid:48) are (m+1)×r and pm×r matrices, respectively. Note that (4) can also be
1 p
written in terms of Z . Let us define Γ ≡ (γ(i),αδ(cid:48)), where γ(i),i = 1,...,p, corresponds to
t i ·,1 i ·,1
the ith column of γ . Then, (4) is equivalent to
·,1
Z = µ+B(cid:48)Z +ν , (5)
t t t
6CubaddaandGuardabascio(2012)analyzeaso-called’medium-N’approacharguingthatmanyoftheresults
intheliteraturefavoranumberofpredictors(N)thatisconsiderablylargerthaninusualsmall-scaleforecasting
problems,butnottoolargeforbeingforcedtorelyondouble(T andN)asymptoticmethods. Theauthorsfind,
using Monte Carlo simulations, that, under the so-called Helland and Almoy (1994) condition, both principal
components and partial least squares (to be discussed later) provide consistent estimates in such a medium-N
framework as only the sample size diverges.
7
where B = (Γ ,...,Γ )(cid:48). For p = 1 the models becomes
1 p
y
t x(m)
x(m) γ1,1 α1 t−1
x(t−m...1)/m = µ+ γ2...,1 yt−1+ α...2 δ1(cid:48) x(t−m1)...−1/m +νt
γ α (m)
(m) m+1,1 m+1 x
x t−1−(m−1)/m
t−(m−1)/m
y
t−1
γ1,1 α1 x(m)
t−1
= µ+ γ2..,1 , α..2 δ1(cid:48)× x(t−m1)−1/m +νt,
. . ..
.
γm+1,1 αm+1 (m)
(cid:124) (cid:123)(cid:122) (cid:125) x
t−1−(m−1)/m
Γ
1
where each α ,i = 1,...,m+1, is of dimension 1×r and where δ is an m×r matrix. Hence,
i 1
we could call δ(cid:48)X(m) a vector of r observable high-frequency factors. Note that r = m − s,
t
where s represents the rank reduction we are able to achieve within X(m).7
t
In terms of parameter reduction, the unrestricted VAR in (2) requires p(m+1)2 coefficients
tobeestimatedintheautoregressivematrices,whereastheVARunderreducedrankrestrictions
in (4) or (5) needs p(m+1)+r(m+1)+prm parameter estimates. As an example, assume
p = 1 and m = 20. Then, if r = 1,2 or 3, there are, respectively, 62,103 or 144 coefficients to
be estimated in Γ instead of 441 in Γ in (3). Note that we do not require y to be included
1 1 t−1
in the same transmission mechanism as the x variables.
3.1.2 Obtaining Observable Factors
We use three ways to obtain the observable factors, canonical correlation analysis (CCA here-
after), partial least squares (PLS hereafter) and heterogeneous autoregressive (HAR heferafter)
type restrictions.
CCA is based on analyzing the eigenvalues and corresponding eigenvectors of
Σˆ−1 Σˆ Σˆ−1Σˆ (6)
X˜(m)X˜(m) X˜(m)Z˜ Z˜Z˜ Z˜X˜(m)
or, similarly, of the symmetric matrix
Σˆ−1/2 Σˆ Σˆ−1Σˆ Σˆ−1/2 . (7)
X˜(m)X˜(m) X˜(m)Z˜ Z˜Z˜ Z˜X˜(m) X˜(m)X˜(m)
7There are two ways to justify the reduced rank feature of the autoregressive matrix. First, at the model
representationlevelwemayassumethatxfollowsanAR(q)processwithq<mandthatthelastm−qelements
ofeachX(m),i=1,...,p,haveazerocoefficientintheequationfory . Pluggingtheserestrictionsinto(2)results
t−i t
in a reduced rank of B(cid:48) . Second, at the empirical level we can interpret the MF-VAR as an approximation
X(m)
of the VARMA obtained after the block marginalization of a high-frequency VAR for each variable. In this
situation, reduced rank matrices may empirically not be rejected by the data.
8
Description:a mixed-frequency VAR and develop a corresponding test statistic. The authors, however, .. In terms of testing for nowcasting non-causality, Götz and.
