Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive
Conditional Heteroskedasticity Models
Author(s): Robert Engle
Source: Journal of Business & Economic Statistics, Vol. 20, No. 3 (Jul., 2002), pp. 339-350
Published by: American Statistical Association
Stable URL: http://www.jstor.org/stable/1392121
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Dynamic
A Simple
Conditional
Class
of
Correlation:
Multivariate
Conditional
Autoregressive
Models
Heteroskedasticity
Generalized
Robert ENGLE
Departmentof Finance, New YorkUniversityLeonardN. Stern School of Business, New York,NY 10012
and Departmentof Economics, Universityof California,San Diego (rengle@stem.nyu.edu)
Time varying correlationsare often estimated with multivariategeneralized autoregressiveconditional
heteroskedasticity(GARCH) models that are linear in squares and cross products of the data. A new
class of multivariatemodels called dynamic conditional correlationmodels is proposed. These have
the flexibility of univariate GARCH models coupled with parsimonious parametricmodels for the
correlations.They are not linear but can often be estimated very simply with univariateor two-step
methods based on the likelihood function. It is shown that they performwell in a variety of situations
and provide sensible empiricalresults.
KEY WORDS: ARCH; Correlation;GARCH;MultivariateGARCH.
1. INTRODUCTION
need for bigger correlationmatrices.In most cases, the number
of parametersin large models is too big for easy optimization.
In this article, dynamic conditional correlation(DCC) estimators are proposed that have the flexibility of univariate
GARCH but not the complexity of conventionalmultivariate
GARCH. These models, which parameterizethe conditional
correlations directly, are naturally estimated in two stepsa series of univariate GARCH estimates and the correlation estimate. These methods have clear computationaladvantages over multivariateGARCH models in that the numberof
parametersto be estimated in the correlationprocess is independent of the numberof series to be correlated.Thus potentially very large correlationmatrices can be estimated.In this
article, the accuracyof the correlationsestimatedby a variety
of methods is compared in bivariate settings where many
methods are feasible. An analysis of the performanceof the
DCC methods for large covariance matrices was considered
by Engle and Sheppard(2001).
Section 2 gives a brief overview of various models
for estimating correlations. Section 3 introduces the new
method and compares it with some of the cited approaches.
Section 4 investigates some statistical properties of the
method. Section 5 describes a Monte Carlo experiment and
results are presentedin Section 6. Section 7 presents empirical results for several pairs of daily time series, and Section 8
concludes.
Correlationsare critical inputs for many of the common
tasks of financial management. Hedges require estimates of
the correlationbetween the returnsof the assets in the hedge.
If the correlationsand volatilities are changing, then the hedge
ratio should be adjustedto account for the most recent information. Similarly,structuredproductssuch as rainbowoptions
that are designed with more than one underlying asset have
prices that are sensitive to the correlationbetween the underlying returns.A forecast of future correlationsand volatilities
is the basis of any pricing formula.
Asset allocation and risk assessment also rely on correlations; however, in this case a large number of correlationsis
often required.Constructionof an optimal portfolio with a set
of constraintsrequires a forecast of the covariance matrix of
the returns. Similarly, the calculation of the standarddeviation of today's portfolio requires a covariance matrix of all
the assets in the portfolio. These functions entail estimation
and forecasting of large covariancematrices, potentially with
thousandsof assets.
The quest for reliable estimates of correlations between
financial variables has been the motivationfor countless academic articles and practitionerconferences and much Wall
Street research.Simple methods such as rolling historicalcorrelations and exponential smoothing are widely used. More
complex methods, such as varieties of multivariategeneralMODELS
2. CORRELATION
ized autoregressiveconditional heteroskedasticity(GARCH)
The conditional correlationbetween two random variables
or stochastic volatility, have been extensively investigatedin
have mean zero is defined to be
the econometricliteratureand are used by a few sophisticated r1 and r2 that each
practitioners. To see some interesting applications, examEt= (r1l,r2, t)
P12, t
ine the work of Bollerslev, Engle, and Wooldridge (1988),
IE,(r2 )E, (r2
Bollerslev (1990), Kroner and Claessens (1991), Engle and
Mezrich (1996), Engle, Ng, and Rothschild(1990), Bollerslev,
? 2002 American Statistical Association
Chou, and Kroner (1992), Bollerslev, Engle, and Nelson
Journal of Business & Economic Statistics
July 2002, Vol. 20, No. 3
(1994), and Ding and Engle (2001). In very few of these artiDOI 10.1198/073500102288618487
cles are more than five assets considered,despite the apparent
339
Journalof Business & EconomicStatistics,July 2002
340
In this definition,the conditionalcorrelationis based on information known the previous period; multiperiod forecasts of
the correlationcan be defined in the same way. By the laws of
probability,all correlationsdefined in this way must lie within
the interval [-1, 1]. The conditional correlationsatisfies this
constraintfor all possible realizationsof the past information
and for all linear combinationsof the variables.
To clarify the relationbetween conditionalcorrelationsand
conditional variances, it is convenient to write the returns
as the conditional standarddeviation times the standardized
disturbance:
h,t
= Et-1(i2t),
it
=
/hiti;t,
i= 1, 2;
(2)
e is a standardizeddisturbancethat has mean zero and variance one for each series. Substitutinginto (4) gives
An alternativesimple approachto estimating multivariate
models is the OrthogonalGARCH method or principle component GARCH method. This was advocated by Alexander
(1998, 2001). The procedureis simply to constructunconditionally uncorrelatedlinearcombinationsof the series r. Then
univariateGARCH models are estimated for some or all of
these, and the full covariancematrixis constructedby assuming the conditional correlationsare all zero. More precisely,
find A such that y, = Art, E(yty') V is diagonal. Univariate GARCH models are estimated for the elements of y and
combined into the diagonal matrix Vt. Making the additional
strong assumptionthat Et_-(yty') = Vt, then
Ht = A-' VtA-'.
(8)
In the bivariatecase, the matrix A can be chosen to be triangular and estimated by least squares where r1 is one compo(3)
(82
2
= E1(l2).
nent and the residuals from a regression of r, on r2 are the
lEtl12t)Et_1(2,t)
second. In this simple situation,a slightly better approachis
Thus, the conditionalcorrelationis also the conditionalcovari- to run this regressionas a GARCHregression,therebyobtainance between the standardizeddisturbances.
ing residualsthat are orthogonalin a generalizedleast squares
Many estimators have been proposed for conditional cor- (GLS) metric.
relations. The ever-popular rolling correlation estimator is
MultivariateGARCH models are naturalgeneralizationsof
defined for returnswith a zero mean as
this problem.Many specificationshave been considered;however, most have been formulatedso that the covariancesand
rl, sr2, s
Ss=t-n-1
variances are linear functions of the squares and cross prodV S=t-n-1,(4)
ucts of the data. The most general expression of this type is
called the vec model and was describedby Engle and Kroner
Substitutingfrom (4) it is clear thatthis is an attractiveestima- (1995). The vec model parameterizesthe vector of all covaritor only in very special circumstances.In particular,it gives ances and variances
expressed as vec(Ht). In the first-order
equal weight to all observations less than n periods in the case this is
given by
past and zero weight on older observations. The estimator
will always lie in the [-1, 1] interval,but it is unclear under
(9)
vec(Ht) = vec(f1) + A vec(rt,_ r,_) + B vec(Ht,_),
what assumptionsit consistentlyestimatesthe conditionalcorrelations. A version of this estimator with a 100-day winwhere A and B are n2x n2 matrices with much structure
dow, called MA100, will be comparedwith other correlation
following from the symmetry of H. Without furtherrestricestimators.
this model will not guaranteepositive definitenessof the
tions,
The exponentialsmootherused by RiskMetricsuses declinH.
matrix
ing weights based on a parameterA, which emphasizescurrent
Useful
restrictionsare derived from the BEKK representadata but has no fixed terminationpoint in the past where data
introduced
tion,
by Engle and Kroner (1995), which, in the
becomes uninformative.
first-order
can
be written as
case,
I
P2,t
P12,-
Et-1(e1,t82,_t)
I
=
2,t
(1,s
tslt
t-j-
I r, sr2,s
s=1
) Z
1
s)
t--2,
.
(5)
ri S)
It also will surely lie in [-1, 1]; however,there is no guidance
from the data for how to choose A. In a multivariatecontext,
the same A must be used for all assets to ensure a positive
definite correlationmatrix. RiskMetricsuses the value of .94
for A for all assets. In the comparisonemployed in this article,
this estimatoris called EX .06.
Defining the conditionalcovariancematrixof returnsas
H, = fl + A(rt,_Irt'1)A'+ BH,_IB'.
(10)
Variousspecial cases have been discussed in the literature,
startingfrom models where the A and B matrices are simply
a scalaror diagonalratherthan a whole matrixand continuing
to very complex, highly parameterizedmodels that still ensure
positive definiteness.See, for example, the work of Engle and
Kroner (1995), Bollerslev et al. (1994), Engle and Mezrich
(1996), Kroner and Ng (1998), and Engle and Ding (2001).
In this study the scalar BEKK and the diagonal BEKK are
(6) estimated.
E _1(rtr)
t,
As discussed by Engle and Mezrich (1996), these models
these estimators can be expressed in matrix notation respeccan be estimatedsubjectto the variancetargetingconstraintby
tively as
which the long run variance covariancematrix is the sample
covariancematrix. This constraintdiffers from the maximum
nt--- j=1 rtjr_) and Hn-A(rtIt
)?+(1- -A)Hn_. (7)
likelihood estimator(MLE) only in finite samples but reduces
Engle: DynamicConditionalCorrelation
341
the number of parametersand often gives improved perfor- followed by the q's will be integrated,
mance. In the general vec model of Equation(9), this can be
expressed as
qi, j,t = (1 - A)(Ei,
t_-l) + A(qij,t-1),
r-Ij,
vec(f)=
where
(I- A-B)vec(S),
(11)
S=-L(r,rt).
T
APi, t
qij, t
/q, tqjj, t
(17)
A natural alternative is suggested by the GARCH(1,1)
This expression simplifies in the scalar and diagonal BEKK model:
cases. For example, for the scalar BEKK the intercept is
simply
- Pi,j) +P(qi,j,- Pi,j) (18)
qi,j,t = Pi,j + a(Ei,-1j,,-1
l = (1- a- P)s.
3.
(12) where
i,j is the unconditionalcorrelationbetween Ei,, and
Rewriting
gives
Ej,t.
DCCs
qi,j,t - Pi,j
1 --
a
--
00S-
+ aE•s
(19)
i,
tssj,
This article introducesa new class of multivariateGARCH
s=1estimators that can best be viewed as a generalizationof the
Bollerslev (1990) constantconditionalcorrelation(CCC) esti- The averageof qi,j, t will be fi, j, and the averagevariancewill
be 1.
mator.In Bollerslev's model,
where =diagjh-,
Dt
Ht= DRD,
,
(13)
1-j3-p
qi,ji
(20)
Pi, j.
where R is a correlation matrix containing the conditional The correlationestimator
correlations,as can directly be seen from rewritingthis equat
Pi,qij,
tion as
E,_, (8•,•) = D-1HD,-
=R
since s, = Dt, r.
(14)
The expressions for h are typically thought of as univariate GARCH models; however, these models could certainly
include functions of the other variablesin the system as predeterminedvariablesor exogenous variables.A simple estimate
of R is the unconditionalcorrelationmatrix of the standardized residuals.
This article proposes the DCC estimator.The dynamic correlation model differs only in allowing R to be time varying:
H, = D, RtD,.
(15)
Parameterizationsof R have the same requirementsas H,
except thatthe conditionalvariancesmust be unity.The matrix
R, remains the correlationmatrix.
Kroner and Ng (1998) proposed an alternativegeneralization that lacks the computationaladvantages discussed here.
They proposed a covariance matrix that is a matrix weighted
average of the Bollerslev CCC model and a diagonal BEKK,
both of which are positive definite.
Probably the simplest specification for the correlation
matrixis the exponentialsmoother,which can be expressed as
will be positive definite as the covariancematrix, Qt with typical element qi, j,,. is a weighted average of a positive definite
and a positive semidefinite matrix. The unconditionalexpectation of the numeratorof (21) is fi, j and each term in the
denominatorhas expected value 1. This model is mean reverting as long as a + < 1, and when the sum is equal to 1 it
is just the model in (17). Matrix versions of these estimators
can be written as
Qt = (1 - A)(etl_•1)
ts 1 /L
i t-s
j't-s-
(16)
+
AQ_1
(22)
+
(23)
and
Q, = S(1 - a - 0) +
a(e,_,
Et_1)
eQtl,
where S is the unconditionalcorrelationmatrixof the epsilons.
Clearly more complex positive definite multivariate
GARCH models could be used for the correlationparameterization as long as the unconditionalmoments are set to the
sample correlationmatrix. For example, the MARCH family
of Ding and Engle (2001) can be expressed in first-orderform
as
Q, = So(t'-A=
Pi,j,t- (/s
(21)
B)+Ao
,_1E,_1+ B oQ,_,1,
(24)
where t is a vector of ones and o is the Hadamardproduct
of two identically sized matrices, which is computed simply
by element-by-elementmultiplication.They show that if A, B,
a geometrically weighted average of standardizedresiduals. and
(tt' - A - B) are positive semidefinite, then Q will be
Clearly these equations will produce a correlationmatrix at positive semidefinite. If any one of the matrices is
positive
each point in time. A simple way to construct this correla- definite, then Qwill also be. This
family includes both earlier
tion is throughexponentialsmoothing.In this case the process models as well as many generalizations.
[R,],j,
342
Journalof Business & EconomicStatistics,July 2002
4. ESTIMATION
The DCC model can be formulatedas the following statistical specification:
--
rt •t,
The volatility part of the likelihood is apparentlythe sum of
individualGARCH likelihoods
1
L(O) = -EE
2
N(O, ODtgtt),
n
(log(2r) + log(hi,)+
t i=)1
r2\
hi, t
,
(30)
D 2 = diag{wi}+ diag{K}i o rt-l'-1
+diag{Ai} oD2t-1•
t
which is jointly maximized by separately maximizing each
term.
(25)
E, = Dt, rt,
The second part of the likelihood is used to estimate the
correlation
parameters.Because the squaredresiduals are not
Q,= So (It' - A - B)+ Ao st-1 _1 + Bo Qt-1,
on
these parameters,they do not enter the firstdependent
R, = diag{Q,}-' , diag{Qt}-'.
order conditions and can be ignored. The resulting estimator
The assumptionof normalityin the first equationgives rise to is called DCC LL MR if the mean revertingformula (18) is
a likelihood function. Without this assumption,the estimator used and DCC LL INT with the integratedmodel in (17).
The two-step approachto maximizing the likelihood is to
will still have the Quasi-MaximumLikelihood (QML) interfind
pretation.The second equation simply expresses the assumption that each asset follows a univariate GARCH process.
0 = argmax{Lv(O)}
(31)
Nothing would change if this were generalized.
The log likelihood for this estimatorcan be expressed as
and then take this value as given in the second stage:
rt t,_,I
N(O, H,),
L=--1
max{Lc(0,4)}.
(32)
__(nlog(2r) +log IHtI+ rHt-'rt)
Under reasonable regularityconditions, consistency of the
first step will ensure consistency of the second step. The
maximum of the second step will be a function of the first2
step parameterestimates, so if the first step is consistent, the
1
second step will be consistent as long as the function is con+
2 t=
T
rtDt R;1 Dtl rt)
(26) tinuous in a
neighborhoodof the true parameters.
and
McFadden(1994), in Theorem 6.1, formulated
Newey
R
1 +
a
Method of Moments (GMM) probGeneralized
two-step
(nlog(2Rt+log ID,R,D,t)
lem that can be applied to this model. Consider the moment
condition corresponding to the first step as VL,(O) = 0.
moment condition correspondingto the second step is
The
+ r'Dt-1D-rt
- E(nlog(2r) + 2log
IODI
2 t=1
VLc(0, 4))= 0. Under standardregularityconditions, which
are given as conditions (i) to (v) in Theorem 3.4 of Newey
- ete, +log I+
Rt E'R,1t,),
and McFadden, the parameterestimates will be consistent,
which can simply be maximized over the parametersof the and asymptoticallynormal,with a covariancematrixof familmodel. However, one of the objectives of this formulationis iar form. This matrixis the productof two invertedHessians
to allow the model to be estimated more easily even when aroundan outer productof scores. In particular,the covariance
the covariance matrix is very large. In the next few para- matrix of the correlationparametersis
graphs several estimation methods are presented, giving simV(4) = [E(VOOLc)]-1
ple consistent but inefficient estimates of the parametersof
the model. Sufficient conditions are given for the consistency
x E({VLc - E(VoLc)[E(VoLv)]-'VoLv}
and asymptoticnormalityof these estimatorsfollowing Newey
and McFadden (1994). Let the parametersin D be denoted
x {VLc - E(VqLc)[E(VLv)]-I VL,}')
0 and the additionalparametersin R be denoted 4. The logx [E(V??Lc)1.
(33)
likelihood can be written as the sum of a volatility part and a
correlationpart:
Details of this proof can be found elsewhere (Engle and
Sheppard2001).
(27)
L(O, 4) = Lv(O)+Lc((O, 4).
Alternativeestimation approaches,which are again consistent but inefficient, can easily be devised. Rewrite (18) as
The volatility term is
L,(O) =
_(nlog(2ir)+ log [Dt2 + rDt-2rt),
(28)
a
ei,j, = Pi,j(1 - -
- •(ei1, jt-1
and the correlationcomponentis
1
tc(O, 2)= -
,
t,
)+ (a+
(logIRg + et
where e =,, t
't -88).
Ei,t.,t
)ei,j,t_1
-- qi, j,t<-1) ? (ei1, jt - qi, j,t),
(34)
This equation is an ARMA(1, 1)
(29) because the errorsare a Martingaledifferenceby construction.
(29)
The autoregressivecoefficient is slightly bigger if a is a small
Engle: DynamicConditionalCorrelation
343
positive number,which is the empirically relevant case. This
equation can therefore be estimated with conventional time
series software to recover consistent estimates of the parameters. The drawbackto this method is that ARMA with nearly
equal roots are numericallyunstable and tricky to estimate. A
furtherdrawbackis that in the multivariatesetting, there are
many such cross productsthat can be used for this estimation.
The problem is even easier if the model is (17) because then
the autoregressiveroot is assumed to be 1. The model is simply an integratedmoving average (IMA) with no intercept,
(e, j,t1 - qi,j,t-1) + (ei, j,t - qi,j,t),
Aei,j,t =
(35)
which is simply an exponential smootherwith parameterA =
p. This estimatoris called the DCC IMA.
3. DCC IMA-DCC with integratedmoving average estimation as in (35)
4. DCC LL INT-DCC by log-likelihood for integrated
process
5. DCC LL MR-DCC by log-likelihood with mean reverting model as in (18)
6. MA100-moving average of 100 days
7. EX .06-exponential smoothing with parameter= .06
8. OGARCH-orthogonal GARCH or principle components GARCH as in (8).
Three performancemeasures are used. The first is simply
the comparisonof the estimatedcorrelationswith the true correlations by mean absolute error.This is defined as
1
MAE =
OF ESTIMATORS
5. COMPARISON
In this section, several correlation estimators are compared in a setting where the true correlation structure is
known. A bivariateGARCH model is simulated200 times for
1,000 observationsor approximately4 years of daily data for
each correlationprocess. Alternativecorrelationestimatorsare
comparedin terms of simple goodness-of-fit statistics, multivariate GARCH diagnostic tests, and value-at-risktests.
The data-generating process consists of two Gaussian
GARCH models; one is highly persistentand the other is not.
= .01
h2,t
= .5+.2r,_t-1
(8',
82, t)
1N
-
+
+ .05rt-1
h,
(Pt
+.5h2,
tPt
I1
,
vt = H7t1/2rt,
(
(38)
which in this bivariatecase is implementedwith a triangular
squareroot defined as
'2, t
t-1,
(37)
and of course the smallest values are the best. A second measure is a test for autocorrelationof the squared standardized
residuals.For a multivariateproblem,the standardizedresiduals are defined as
Pl,t = ri,t/
.94h,_t-1,
Pt, - Pt,
=
Hil,t,
1
r2, t
rl,t
Pt
Pt((
(39)
22,t(1-)
(36) The test is
computed as an F test from the regression of v2
6
and v2, on five lags of the squares and cross productsof the
standardizedresidualsplus an intercept.The numberof rejections using a 5% critical value is a measure of the perforThe correlationsfollow several processes that are labeled as
mance of the estimatorbecause the more rejections,the more
follows:
evidence that the standardizedresiduals have remaining time
1. Constantp, = .9
varying volatilities. This test obviously can be used for real
2. Sine p, = .5 + .4 cos(27r t/200)
data.
3. Fast sine p, = .5 +.4cos(2r t/20)
The thirdperformancemeasureis an evaluationof the esti4. Step p, = .9 -.5(t > 500)
mator for calculating value at risk. For a portfolio with w
5. Ramp p, = mod (t/200)
invested in the first asset and (1 - w) in the second, the value
These processes were chosen because they exhibit rapid at risk, assuming normality,is
changes, gradualchanges, and periods of constancy.Some of
VaR,= 1.65(w2Hl,,+(1-w)2H22,t+2*w(1-w)tlt
22,t), (40)
the processes appear mean reverting and others have abrupt
changes. Various other experiments are done with different
and a dichotomousvariablecalled hit should be unpredictable
errordistributionsand differentdata-generatingparametersbut
based on the past where hit is defined as
the results are quite similar.
Eight different methods are used to estimate the
hit, = I(w*r,+ (1 - w)*r2,t< -VaR,)- .05.
(41)
correlations-two multivariate GARCH models, orthogonal
GARCH, two integratedDCC models, and one mean revert- The dynamicquantiletest introducedby
Engle and Manganelli
ing DCC plus the exponentialsmootherfrom Riskmetricsand (2001) is an F test of the hypothesisthat all coefficientsas well
the familiar 100-day moving average. The methods and their as the
interceptare zero in a regressionof this variableon its
descriptionsare as follows:
past, on currentVaR, and any other variables.In this case, five
rl,t
= hEt8lt,
r2,t =
h2,t2, t
1. Scalar BEKK-scalar version of (10) with variance targeting as in (12)
2. Diag BEKK--diagonal version of(10) with variancetargeting as in (11)
lags and the currentVaR are used. The numberof rejections
using a 5% critical value is a measureof model performance.
The reportedresults are for an equal weighted portfolio with
w = .5 and a hedge portfolio with weights 1, -1.
344
Journalof Business & Economic Statistics,July 2002
1.0
1.0
0.9
0.8 -
0.8 -
0.6
0.7 -
0.4 -
0.6 0.5 -
0.2-
0.4 -
0.0 -
1
2
-
0.3
--
RHO_SINE
1.0
1.0
0.8 -
0.8
0.6 -
0.6
0.4 -
0.4
0.2 -
0.2
0.0
2.
-...
I1
. 20
r
1
40,'
20
60.
r
80)
-
1000
0.0
20
SRHORAMP
. .
RHOSTEP
.
60
.. 40
.i
RHOFASTSINE
80
. 10'
Figure1. CorrelationExperiments.
6. RESULTS
log-likelihood; overall, it is also the best method, followed by
the mean revertingDCC and the IMA DCC.
The value-at-risktest based on the long-shortportfolio finds
that the diagonal BEKK is best for three of six, whereas the
DCC MR is best for two. Overall, the DCC MR is observed
to be the best.
From all these performancemeasures,the DCC methodsare
either the best or very near the best method. Choosing among
these models, the mean revertingmodel is the general winner,
althoughthe integratedversions are close behind and perform
best by some criteria.Generallythe log-likelihood estimation
method is superior to the IMA estimator for the integrated
DCC models.
The confidence with which these conclusions can be drawn
can also be investigated.One simple approachis to repeatthe
experimentwith different sets of randomnumbers.The entire
Monte Carlo was repeated two more times. The results are
very close with only one change in ranking that favors the
DCC LL MR over the DCC LL INT.
Table 1 presents the results for the mean absolute error
(MAE) for the eight estimatorsfor six experimentswith 200
replications.In four of the six cases the DCC mean reverting
model has the smallest MAE. When these errorsare summed
over all cases, this model is the best. Very close secondand third-placemodels are DCC integratedwith log-likelihood
estimation and scalar BEKK.
In Table 2 the second standardizedresidual is tested for
remaining autocorrelationin its square. This is the more
revealing test because it depends on the correlations;the test
for the first residual does not. Because all models are misspecified, the rejectionrates are typically well above 5%. For
three of six cases, the DCC mean revertingmodel is the best.
When summed over all cases it is a clear winner.
The test for autocorrelationin the first squaredstandardized
residual is presentedin Table 3. These test statistics are typically close to 5%, reflectingthe fact that many of these models are correctly specified and the rejectionrate should be the
size. Overallthe best model appearsto be the diagonalBEKK
with OGARCHand DCC close behind.
The VaR-baseddynamicquantiletest is presentedin Table4
7. EMPIRICAL
RESULTS
for a portfolio that is half investedin each asset and in Table 5
for a long-short portfolio with weights plus and minus one.
Empirical examples of these correlationestimates are preThe numberof 5% rejectionsfor many of the models is close sented for several interesting series. First the correlation
to the 5% nominal level despite misspecificationof the struc- between the Dow Jones IndustrialAverage and the NASDAQ
ture. In five of six cases, the minimumis the integratedDCC composite is examined for 10 years of daily data ending
Engle: DynamicConditionalCorrelation
345
Table1. MeanAbsoluteErrorof CorrelationEstimates
MODEL
SCALBEKK
DIAGBEKK
DCCLLMR
DCCLLINT
DCCIMA
EX.06
MA100
O-GARCH
FAST SINE
.2292
.2307
.2260
.2555
.2581
.2737
.2599
.2474
SINE
STEP
RAMP
CONST
T(4) SINE
.1422
.0859
.1610
.0273
.1595
.1451
.0931
.1631
.0276
.1668
.1381
.0709
.1546
.0070
.1478
.1455
.0686
.1596
.0067
.1583
.1678
.0672
.1768
.0105
.2199
.1541
.0810
.1601
.0276
.1599
.3038
.0652
.2828
.0185
.3016
.2245
.1566
.2277
.0449
.2423
in SquaredStandardizedSecond Residual
Table2. Fractionof 5% TestsFindingAutocorrelation
MODEL
FASTSINE
SINE
STEP
RAMP
CONST
T(4) SINE
SCALBEKK
DIAGBEKK
DCCLLMR
DCCLLINT
DCCIMA
EX.06
MA100
.3100
.5930
.8995
.5300
.9800
.2800
.0950
.2677
.6700
.2600
.3600
.1900
.1300
.1400
.2778
.2400
.0788
.2050
.3700
.1850
.3250
.5450
.0900
.2400
.3700
.3350
.6650
.7500
.1250
.1650
.7250
.7600
.8550
.7300
.9700
.3300
.9900
1.0000
.9950
1.0000
.9950
.8950
O-GARCH
.1100
.2650
.7600
.2200
.9350
.1600
in SquaredStandardizedFirstResidual
Table3. Fractionof 5% TestsFindingAutocorrelation
MODEL
FASTSINE
SINE
STEP
RAMP
CONST
T(4) SINE
SCALBEKK
DIAGBEKK
DCCLLMR
DCCLLINT
DCCIMA
EX.06
MA100
.2250
.0804
.0302
.0550
.0200
.0950
.0450
.0657
.0400
.0500
.0250
.0550
.0600
.0400
.0505
.0500
.0242
.0850
.0600
.0300
.0500
.0600
.0250
.0800
.0650
.0600
.0450
.0600
.0250
.0950
.0750
.0400
.0300
.0650
.0400
.0850
.6550
.6250
.6500
.7500
.6350
.4900
O-GARCH
.0600
.0400
.0250
.0400
.0150
.1050
Table4. Fractionof 5%DynamicQuantileTestsRejectingValueat Risk,EqualWeighted
MODEL
FASTSINE
SINE
STEP
RAMP
CONST
T(4) SINE
SCALBEKK
DIAGBEKK
DCCLLMR
DCCLLINT
DCCIMA
EX.06
MA100
.0300
.0452
.1759
.0750
.0600
.1250
.0450
.0556
.1650
.0650
.0800
.1150
.0350
.0250
.0758
.0500
.0667
.1000
.0300
.0350
.0650
.0400
.0550
.0850
.0450
.0350
.0800
.0450
.0550
.1200
.2450
.1600
.2450
.2000
.2600
.1950
.4350
.4100
.3950
.5300
.4800
.3950
O-GARCH
.1200
.3200
.6100
.2150
.2650
.2050
Table5. Fractionof 5%DynamicQuantileTestsRejectingValueat Risk,Long-Short
MODEL
FASTSINE
SINE
STEP
RAMP
CONST
T(4) SINE
SCALBEKK
DIAGBEKK
DCCLLMR
DCCLLINT
DCCIMA
EX.06
MA100
.1000
.0553
.1055
.0800
.1850
.1150
.0950
.0303
.0850
.0650
.0900
.0900
.0900
.0450
.0404
.0800
.0424
.1350
.2550
.0900
.0600
.1750
.0550
.1300
.2550
.1850
.1150
.2500
.0550
.2000
.5800
.2150
.1700
.3050
.3850
.2150
.4650
.9450
.4600
.9000
.5500
.8050
O-GARCH
.0850
.0650
.1250
.1000
.1050
.1050
346
Journalof Business & EconomicStatistics,July 2002
in March 2000. Then daily correlationsbetween stocks and
bonds, a central feature of asset allocation models, are
considered. Finally, the daily correlationbetween returnson
severalcurrenciesaroundmajorhistoricalevents includingthe
launch of the Euro is examined. Each of these datasets has
been used to estimate all the models describedpreviously.The
DCC parameterestimates for the integratedand mean reverting models are exhibited with consistent standarderrorsfrom
(33) in Appendix A. In that Appendix, the statisticreferredto
as likelihood ratio is the differencebetween the log-likelihood
of the second-stage estimates using the integratedmodel and
using the mean revertingmodel. Because these are not jointly
maximizedlikelihoods,the distributioncould be differentfrom
its conventional chi-squared asymptotic limit. Furthermore,
nonnormalityof the returns would also affect this limiting
distribution.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
90
91
92
93
94
95
96
97
98
99
DCC INTDJNQ
Figure3. TenYearsof Dow Jones-NASDAQCorrelations.
7.1 Dow Jones and NASDAQ
The dramatic rise in the NASDAQ over the last part of
the 1990s perplexed many portfolio managers and delighted
the new internet start-ups and day traders. A plot of the
GARCH volatilities of these series in Figure 2 reveals that the
NASDAQ has always been more volatile than the Dow but
that this gap widens at the end of the sample.
The correlationbetween the Dow and NASDAQ was estimated with the DCC integrated method, using the volatilities in Figure 2. The results, shown in Figure 3, are quite
interesting.
Whereas for most of the decade the correlations were
between .6 and .9, there were two notable drops. In 1993 the
correlationsaveraged .5 and droppedbelow .4, and in March
2000 they again dropped below .4. The episode in 2000 is
sometimes attributedto sector rotationbetween new economy
stocks and "brick and mortar"stocks. The drop at the end
of the sample period is more pronouncedfor some estimators
than for others. Looking at just the last year in Figure 4, it
60
can be seen that only the orthogonalGARCHcorrelationsfail
to decline and that the BEKK correlationsare most volatile.
7.2 Stocks and Bonds
The second empirical example is the correlationbetween
domestic stocks and bonds. Taking bond returnsto be minus
the change in the 30-year benchmarkyield to maturity,the
correlationbetween bond yields and the Dow and NASDAQ
are shown in Figure 5 for the integratedDCC for the last 10
years. The correlationsare generally positive in the range of
.4 except for the summerof 1998, when they become highly
negative, and the end of the sample, when they are about 0.
Although it is widely reportedin the press that the NASDAQ
does not seem to be sensitive to interestrates,the data suggests
that this is true only for some limited periods, including the
first quarterof 2000, and that this is also true for the Dow.
Throughoutthe decade it appearsthat the Dow is slightly more
correlatedwith bond prices than is the NASDAQ.
50
7.3 Exchange Rates
40
ti
;
30
20l
lI
90
91
-
92
93
94
95
96
97
98
99
VOL_DJ_GARCH----- VOL NO_GARCH
Figure2. TenYearsof Volatilities.
Currency correlations show dramatic evidence of nonstationarity.Thatis, there are very pronouncedapparentstructural
changes in the correlationprocess. In Figure 6, the breakdown
of the correlationsbetween the Deutschmarkand the pound
and lira in August of 1992 is very apparent.For the pound
this was a returnto a more normal correlation,while for the
lira it was a dramaticuncoupling.
Figure 7 shows currency correlations leading up to the
launch of the Euro in January1999. The lira has lower correlations with the Franc and Deutschmarkfrom 93 to 96, but
then they gradually approachone. As the Euro is launched,
the estimated correlation moves essentially to 1. In the last
year it drops below .95 only once for the Franc and lira and
not at all for the other two pairs.
Engle: Dynamic Conditional Correlation
347
.9
.9
.8
.8-
.7
.7
.6\
.6
.5 -
.5
.4 -
.4
.3
1999:07
1999:10
2000:01
i
.3
1999:04
1999:07
-
SDCCINTDJNQ
1999:10
2000:01
DCCMRDJNQ
.8
.9
.8-
.7.7.6 -
.6-
.5 -
.5-
.4.
.4
.3-.
-
.2
.3
1999:07
1999:10
2000:01
SDIAGBEKKDJNQ
1999:04
1999:07
OGARCHDJ
I
I
1999:10
2000:01
NQ
Figure4. CorrelationsfromMarch1999 to March2000.
.4_
"0."
O
0..8
,
.07-
.61.06
0.5
90
-
91
92
93
94
DCCINT_DJ_BOND
95
96
97
98
99
----- DCC_INTNQBOND
Figure5. TenYearsof Bond Correlations.
86878889909192939495
-
DCC_INT_RDEM_RGBP ---- DCC INTRDEMRITL
Figure6. TenYearsof CurrencyCorrelations.
348
Journalof Business & EconomicStatistics,July 2002
1.0
If a 1% test is used reflecting the larger sample size, then
the numberof rejectionsranges from 7 to 21. Again the MA
100 is the worst but now the EX .06 is the winner.The DCC
LL MR, DCC LL INT, and diagonal BEKK are all tied for
second with 9 rejections each.
The implications of this comparisonare mainly that a bigger and more systematiccomparisonis required.These results
suggest first that real data are more complicated than any of
these models. Second, it appears that the DCC models are
competitive with the other methods, some of which are difficult to generalize to large systems.
0.60.4
-
0.2-
0.0-0.2-
"1994 1995 1996 1997 1998
S
DCCINT_RDEM_RFRF--------- DCCINT_RDEM_RITL
8. CONCLUSIONS
DCC_INT_RFRF_RITL
Figure7. CurrencyCorrelations.
From the results in Appendix A, it is seen that this is the
only datasetfor which the integratedDCC cannot be rejected
against the mean revertingDCC. The nonstationarityin these
correlationspresumablyis responsible.It is somewhatsurprising that a similar result is not found for the prior currency
pairs.
7.4 Testingthe EmpiricalModels
For each of these datasets, the same set of tests that were
used in the Monte Carloexperimentcan be constructed.In this
case of course, the mean absolute errorscannot be observed,
but the tests for residualARCH can be computedand the tests
for value at risk can be computed.In the lattercase, the results
are subject to various interpretationsbecause the assumption
of normality is a potential source of rejection. In each case
the numberof observationsis largerthan in the Monte Carlo
experiment,ranging from 1,400 to 2,600.
The p-statistics for each of four tests are given in Appendix
B. The tests are the tests for residualautocorrelationin squares
and for accuracy of value at risk for two portfolios. The two
portfolios are an equally weighted portfolio and a long-short
portfolio. They presumably are sensitive to rather different
failures of correlation estimates. From the four tables, it is
immediatelyclear that most of the models are misspecifiedfor
most of the data sets. If a 5% test is done for all the datasets
on each of the criteria, then the expected number of rejections for each model would be just over 1 of 28 possibilities.
Across the models there are from 10 to 21 rejections at the
5% level!
Without exception, the worst performeron all of the tests
and datasetsis the moving averagemodel with 100 lags. From
counting the total numberof rejections, the best model is the
diagonal BEKK with 10 rejections. The DCC LL MR, scalar
BEKK, O_GARCH, and EX .06 all have 12 rejections, and
the DCC LL INT has 14. Probably,these differences are not
large enough to be convincing.
In this article a new family of multivariateGARCH models was proposed that can be simply estimated in two steps
from univariateGARCH estimates of each equation.A maximum likelihood estimatorwas proposed and several different
specificationswere suggested. The goal for this proposalis to
find specifications that potentially can estimate large covariance matrices. In this article, only bivariate systems were
estimated to establish the accuracy of this model for simpler structures.However, the procedurewas carefully defined
and should also work for large systems. A desirablepractical
feature of the DCC models is that multivariateand univariate volatility forecasts are consistent with each other. When
new variables are added to the system, the volatility forecasts of the original assets will be unchangedand correlations
may even remain unchanged,depending on how the model is
revised.
The main finding is that the bivariate version of this
model provides a very good approximationto a variety of
time-varyingcorrelationprocesses. The comparison of DCC
with simple multivariateGARCH and several other estimators
shows that the DCC is often the most accurate.This is true
whether the criterionis mean absolute error,diagnostic tests,
or tests based on value at risk calculations.
Empirical examples from typical financial applicationsare
quite encouragingbecause they reveal importanttime-varying
features that might otherwise be difficult to quantify.Statistical tests on real data indicate that all these models are misspecified but that the DCC models are competitive with the
multivariateGARCHspecificationsand are superiorto moving
average methods.
ACKNOWLEDGMENTS
This research was supportedby NSF grant SBR-9730062
and NBER AP group. The author thanks Kevin Sheppard
for research assistance and Pat Burns and John Geweke for
insightful comments. Thanks also to seminar participantsat
New York University;University of Californiaat San Diego;
Academica Sinica; Taiwan;CIRANO at Montreal;University
of Iowa; Journal of Applied Econometrics Lectures, Cambridge, England; CNRS Aussois; Brown University; Fields
InstituteUniversity of Toronto;and Riskmetrics.
Engle: DynamicConditionalCorrelation
349
APPENDIXA
Mean RevertingModel
Asset 1
NQ
alphaDCC
betaDCC
alphaDCC
betaDCC
alphaDCC
betaDCC
T-stat
.039029
.941958
6.916839405
92.72739572
Asset 1
RATE
Asset 2
DJ
alphaDCC
betaDCC
alphaDCC
betaDCC
Asset 1
NQ
Log-likelihood
18079.5857
Parameter
T-stat
Log-likelihood
.037372
.950269
2.745870787
44.42479805
13197.82499
Asset 1
NQ
Asset 2
RATE
Parameter
T-stat
Log-likelihood
.029972
.953244
2.652315309
46.61344925
12587.26244
lambdaDCC
LRTEST
lambdaDCC
LRTEST
lambdaDCC
LRTEST
Log-likelihood
.0991
.863885
3.953696951
21.32994852
21041.71874
lambdaDCC
LRTEST
Asset 2
GBP
Parameter
T-stat
.03264
.963504
1.315852908
37.57905053
Asset 1
rdem90
Asset 2
rfrf90
T-stat
Log-likelihood
.030255569
4.66248
18062.79651
33.57836423
Asset 1
RATE
Asset 2
DJ
T-stat
Log-likelihood
.02851073
3.675969
13188.63653
18.37690833
Asset 1
NQ
Asset 2
RATE
Parameter
T-stat
.019359061
2.127002
Asset 1
DM
T-stat
19508.6083
Parameter
T-stat
Log-likelihood
.059413
.934458
4.154987386
59.19216459
12426.89065
Asset 1
rdem90
Asset 2
ritl90
Parameter
T-stat
Log-likelihood
.056675
.943001
3.091462338
5.77614662
11443.23811
lambdaDCC
LRTEST
lambdaDCC
LRTEST
lambdaDCC
LRTEST
Log-likelihood
12578.06669
18.39149373
Asset 2
ITL
Parameter
T-stat
.052484321
4.243317
Asset 1
DM
Log-likelihood
Asset 2
DJ
Parameter
Parameter
Asset 2
ITL
Parameter
Asset 1
DM
alphaDCC
betaDCC
Asset 2
DJ
Parameter
Asset 1
DM
alphaDCC
betaDCC
IntegratedModel
Log-likelihood
20976.5062
13.4250734
Asset 2
GBP
Parameter
T-stat
.024731692
1.932782
Asset 1
rdem90
Asset 2
rfrf90
Parameter
T-stat
.047704833
2.880988
Asset 1
rdem90
Asset 2
ritl90
Log-likelihood
19480.21203
56.79255661
Log-likelihood
12416.84873
20.08382828
Parameter
T-stat
Log-likelihood
.053523717
2.971859
11442.50983
1.456541924
NOTE: Empirical results for bivariate DCC models. T-statistics are robust and consistent using (33). The estimates in the left column are DCC LLMR and the estimates in the right column are
DCC LLINT.The LR statistic is twice the difference between the log likelihoods of the second stage. The data are all logarithmic differences: NQ=Nasdaq composite, DJ=Dow Jones Industrial
Average, RATE=returnon 30 year US Treasury, all daily from 3/23/90 to 3/22/00. Furthermore: DM=Deutschmarks per dollar, ITL=ltalian Lira per dollar, GBP=British pounds per dollar, all from
1/1/85 to 2/13/95. Finally rdem90=Deutschmarks per dollar, rfrf90=French Francs per dollar, and ritl90=ltalian Lira per dollar, all from 1/1/93 to 1/15/99.
Journalof Business & EconomicStatistics,July 2002
350
APPENDIXB
P-StatisticsFromTestsof EmpiricalModels
ARCHin SquaredRESID1
MODEL
NASD&DJ
DJ&RATE
NQ&RATE
DM&ITL
DM&GBP
FF&DM90
DM&IT90
SCALBEKK
DIAGBEKK
DCCLLMR
DCCLLINT
EX.06
MA100
.0047
.0000
.0000
.4071
.4437
.2364
.1188
.0281
.0002
.0044
.3593
.4303
.2196
.3579
.3541
.0003
.0100
.2397
.4601
.1219
.0075
.3498
.0020
.0224
.1204
.3872
.1980
.0001
.3752
.0167
.0053
.5503
.4141
.3637
.0119
.0000
.0000
.0000
.0000
.0000
.0000
.0000
O-GARCH
.2748
.0001
.0090
.4534
.4213
.0225
.0010
ARCHin SquaredRESID2
MODEL
NASD&DJ
DJ&RATE
NQ&RATE
DM&ITL
DM&GBP
FF&DM90
DM&IT90
SCALBEKK
DIAGBEKK
DCCLLMR
DCCLLINT
EX.06
MA100
.0723
.7090
.0052
.0001
.0000
.0002
.0964
.0656
.7975
.0093
.0000
.0000
.0010
.1033
.0315
.8251
.0075
.0000
.0000
.0000
.0769
.0276
.6197
.0053
.0000
.0000
.0000
.1871
.0604
.8224
.0023
.0000
.1366
.0000
.0431
.0000
.0007
.0000
.0000
.0000
.0000
.0000
O-GARCH
.0201
.1570
.1249
.0000
.4650
.0018
.5384
DynamicQuantileTestVaR1
MODEL
NASD&DJ
DJ&RATE
NQ&RATE
DM&ITL
DM&GBP
FF&DM90
DM&IT90
SCALBEKK
DIAGBEKK
DCCLLMR
DCCLLINT
EX.06
MA100
.0001
.7245
.5923
.1605
.4335
.1972
.1867
.0000
.4493
.5237
.2426
.4348
.2269
.0852
.0000
.3353
.4248
.1245
.4260
.1377
.5154
.0000
.4521
.3203
.0001
.3093
.1375
.7406
.0002
.5977
.2980
.3892
.1468
.0652
.1048
.0000
.4643
.4918
.0036
.0026
.1972
.4724
O-GARCH
.0018
.2085
.8407
.0665
.1125
.2704
.0038
DynamicQuantileTestVaR2
MODEL
NASD&DJ
DJ&RATE
NQ&RATE
DM&ITL
DM&GBP
FF&DM90
DM&IT90
SCALBEKK
DIAGBEKK
DCCLLMR
DCCLLINT
EX.06
MA100
.0765
.0119
.0432
.0000
.0006
.4638
.2130
.1262
.6219
.4324
.0000
.0043
.6087
.4589
.0457
.6835
.4009
.0000
.0002
.7098
.2651
.0193
.4423
.6229
.0000
.0000
.0917
.0371
.0448
.0000
.0004
.0209
.1385
.4870
.3248
.0000
.1298
.4967
.0081
.0000
.1433
.0000
O-GARCH
.0005
.3560
.3610
.0000
.0003
.5990
.1454
NOTE: Dataare the same as in the previoustable and tests are based on the resultsin the previoustable.
[Received January2002. Revised January2002.]
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