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Dynamic Conditional Correlation: on Properties andEstimationGian Piero AielliAccepted author version posted online: 01 Feb 2013.

To cite this article: Gian Piero Aielli (2013): Dynamic Conditional Correlation: on Properties and Estimation, Journal ofBusiness & Economic Statistics, DOI:10.1080/07350015.2013.771027

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First draft: November, 2009; This version: December, 2012

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Dynamic Conditional Correlation: on Properties andEstimation

Gian Piero Aielli∗

Abstract

In this paper some issues that arise with the Dynamic Conditional Correlation (DCC) model are

addressed. It is proven that the DCC large system estimator can be inconsistent, and that the tra-

ditional interpretation of the DCC correlation parameters can lead to misleading conclusions. A

more tractable dynamic conditional correlation model, calledcDCC model, is then suggested.

ThecDCC model allows for a large system estimator which is heuristically proven to be con-

sistent. Sufficient stationarity conditions forcDCC processes of interest are established. The

empirical performances of the DCC andcDCC large system estimators are compared via simu-

lations and applications to real data.

KEYWORDS: Multivariate GARCH Model, Quasi-Maximum-Likelihood, Two-step Estima-

tion, Integrated Correlation, Generalized Profile Likelihood.

1 INTRODUCTION

During the last decade, the focus on the variance/correlation decomposition of the asset con-

ditional covariance matrix has become one of the most popular approaches to the modeling of

multivariate volatility. Seminal works in this area are the Constant Conditional Correlation (CCC)

model of Bollerslev (1990), the Dynamic Conditional Correlation (DCC) model of Engle (2002),

and the Varying Correlation (VC) model of Tse and Tsui (2002). Extensions of the DCC model

have been proposed, among others, by Cappiello, Engle, and Sheppard (2006), Billio, Caporin,

and Gobbo (2006), Pesaran and Pesaran (2007), and Franses and Hafner (2009). Related examples∗e-mail: gian.piero.aielli@gmail.com

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are the models of Silvennoinen and Terasvirta (2009), Pelletier (2006), McAleer, Chan, Hoti, and

Lieberman (2008), and Kwan, Li, and Ng (2010).

In the DCC model the conditional variances follow univariate GARCH models. The conditional

correlations are then modeled as peculiar functions of the past GARCH standardized returns. In

its original intention, such a modeling approach should have been capable of providing two major

advantages. Firstly, due to the modular structure of the DCC conditional covariance matrix, a con-

sistent three-step estimator for large systems (DCC estimator) should have been easily available.

Secondly, due to the explicit parametrization of the conditional correlation process, testing for cer-

tain correlation hypotheses, such as whether or not the correlation process is integrated, should

have been more immediate than with other data-driven volatility models. Aielli (2006) pointed out

that the DCC model is less tractable than expected. For example, the conjecture on the consistency

of the second step of the DCC estimator, in which the location correlation parameter is estimated,

turns out to be unsubstantiated. As a consequence of this, the third step of the DCC estimator, in

which the dynamic correlation parameters are estimated, could be inconsistent in turn. The author

then suggested reformulating the DCC correlation driving process as a linear multivariate gen-

eralized autoregressive conditional heteroskedasticity (MGARCH) process. The resulting model,

calledcDCC model, allows for an intuitive three-step estimator (cDCC estimator) which is feasible

with large systems. Compared with the DCC estimator, thecDCC estimator requires only a minor

additional computational effort.

This paper extends Aielli (2006) with new theoretical and empirical results. Regarding the

DCC model, it is proven that the second step of the DCC estimator can be inconsistent. It is

also shown that the traditional interpretation of the dynamic correlation parameters can lead to

misleading conclusions. For example, in the presence of a unit root, which is commonly tested to

check for integrated correlations (see, e.g., Engle and Sheppard 2001; Pesaran and Pesaran 2007),

the correlation driving process is weakly stationary.

Regarding thecDCC model, sufficient stationarity conditions are established as a modular

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stationarity principle which is capable of a wide range of applications. In simple terms, if the

correlation/standardized return process is stationary, checking for the stationarity of the covari-

ance/return process reduces checking for the stationarity of the univariate GARCH variances. As

for the stationarity of the correlation/standardized return process, explicit conditions are derived

from a recent result of Boussama, Fuchs, and Stelzer (2011) on the stationarity of linear MGARCH

processes. As for the stationarity of the univariate GARCH variances, relatively easy-to-check

conditions are available for many specifications adopted in practice (e.g., GARCH, EGARCH, and

TARCH; see Francq and Zakoıan 2010).

The behavior of thecDCC integrated correlation process is investigated via simulation. It is

proven that, in the presence of a unit root in the dynamic correlation parameters, the correlation

processes degenerate either to 1 or to -1 in the long run. Otherwise, the correlation processes are

mean reverting. Thus, testing for integrated correlation has an unambiguous meaning with the

cDCC model.

ThecDCC estimator is shown to be a generalized profile quasi–log–likelihood estimator (Sev-

erini 1998). The dynamic correlation parameters are the parameters of interest, and the variance

parameters and the location correlation parameters are the nuisance parameters. The estimator of

the nuisance parameters conditional on the true value of the parameter of interest is proven to be

consistent. Relying on this property, a heuristic proof of the consistency of thecDCC estimator is

provided.

The finite sample performances of thecDCC and DCC estimators are compared by means of

applications to simulated and real data. It is shown that, under the hypothesis of correctly spec-

ified model, for parameter values that are common in financial applications the bias of the DCC

estimator of the location correlation parameter is negligible. For less common parameter values, it

can be substantial. In general, the bias is an increasing function of the persistence of the correla-

tion process and of the impact of the innovations. ThecDCC estimator of the location correlation

parameter, instead, appears to be practically unbiased. Some simulation experiments under mis-

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specification are also discussed, in which the DCC andcDCC estimators prove to perform very

similarly. Regarding the applications to the real data, a small dataset of ten equity indices and a

large dataset of 100 equities are considered. With both datasets, thecDCC correlation forecasts

are proven to perform as well as, or significantly better than, the corresponding DCC correlation

forecasts.

The remainder of the paper is organized as follows: section 2 illustrates a theoretical compar-

ison of the DCC andcDCC models; section 3 discusses the DCC andcDCC estimators; section

4 compares the empirical performances of the two estimators, and section 5 concludes the paper.

The proofs of the propositions are compiled in the Appendix.

2 STRUCTURAL PROPERTIES

2.1 The DCC model

Let yt ≡ [y1,t, . . . , yN,t]′ denote the vector of excess returns at timet = 0,±1,±2, . . . It is assumed

thatyt is a martingale difference, orEt−1[yt] = 0, whereEt−1[ ∙ ] denotes expectations conditional

onyt−1,yt−2, . . . The asset conditional covariance matrix,Ht ≡ Et−1[yty′t ], can be written as

Ht = D1/2t Rt D

1/2t , (1)

whereRt ≡ [ρi j,t] is the asset conditional correlation matrix, andDt ≡ diag(h1,t, . . . , hN,t) is a

diagonal matrix with the asset conditional variances as diagonal elements. By construction,Rt is

the conditional covariance matrix of the vector of the standardized returns,εt ≡ [ε1,t, ε2,t, . . . , εN,t]′,

whereεi,t ≡ yi,t/√

hi,t. In the DCC model, the diagonal elements ofDt are modeled as univariate

GARCH models, that is,

hi,t = hi(θi; yi,t−1, yi,t−2, . . .), (2)

wherehi( ∙ ; ∙ , ∙ , . . .) is a known function, andθi is a vector of parameters,i = 1,2, . . . ,N. The con-

ditional correlation matrix is then modeled as a function of the past standardized returns, namely,

Rt = Q∗−1/2t Qt Q

∗−1/2t , (3)

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where

Qt = (1− α − β)S + αεt−1ε′t−1 + βQt−1, (4)

whereQt ≡ [qi j,t], Q∗t ≡ diag(q11,t, . . . , qNN,t), S ≡ [si j ], andα andβ are scalars. ForRt to be

positive definite (pd) it suffices thatQt is pd, which is the case ifα ≥ 0, β ≥ 0, α + β < 1, andS

is pd. Relying on the conjecture thatS is the second moment ofεt (see below),S is commonly

assumed to be unit–diagonal.

Some simulated series of DCC conditional correlations are plotted in Fig. (1). The persistence

of the correlation process is an increasing function ofα + β. The higher the impact of the innova-

tions, as measured byα, the higher the variance of the correlation process. The parametersi j is a

location parameter.

More general DCC models can be obtained setting

Qt = (ιι′ −A −B) � S + A � εt−1ε′t−1 + B �Qt−1, (5)

(Engle 2002, eq. (24)), whereA ≡ [αi j ], B ≡ [βi j ], ι denotes theN × 1 vector with unit entries,

and� denotes the element-wise (Hadamard) matrix product. ForQt to be pd, it suffices thatA

andB are positive semi definite (psd), and (ιι′ −A −B) � S is pd (Ding and Engle 2001). The

DCC correlation driving process,Qt, is often treated as a linear MGARCH process (see, e.g.,

Engle 2002, eq. (18)). The role of the parametersα, β, andS, is then interpreted accordingly.

Indeed, since the conditional covariance matrix ofεt is Rt (not Qt), the processQt is not a linear

MGARCH. A consequence of this is that the traditional interpretation ofα, β, andS can lead to

misleading conclusions. Consider for example the location correlation parameter,S. Applying

a standard result on linear MGARCH processes (Engle and Kroner 1995),S is thought of as the

second moment ofεt, or,

S = E[εtε′t ] (6)

(Engle 2002, eq. (18), (23) and (24)). Relying on this, during the fitting of large systems,S is

replaced by the sample second moment of the estimated standardized returns as a feasible estimator

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(Engle and Sheppard 2001; Cappiello et al. 2006; Billio et al. 2006; Pesaran and Pesaran 2007;

Franses and Hafner 2009). Unfortunately, as proven by the following proposition, eq. (6) does not

hold in general.

Proposition 2.1. For the model in (5), suppose that N= 2, α11 > 0, α22 = α12 = β11 = β22 = β12 =

0, and0 < s12 <√

1− α11. Then,Rt is pd, and E[ε1,t ε2,t] > s12.

Proof. All proofs are reported in the Appendix.

The only case in which eq. (6) holds seems to be the case of constant conditional correlations

(α = β = 0). Note in fact that, from eq. (4), under stationarity it follows thatS = E[εtε′t ] if and

only if E[εtε′t ] = E[Qt]. But E[εtε

′t ] = E[Et−1[εtε

′t ]] = E[Rt] = E[Q∗−1/2

t Qt Q∗−1/2t ], where, if

Qt is dynamic, it holds that a.s.Q∗−1/2t Qt Q

∗−1/2t , Qt.

Proposition 2.2. Suppose thatα + β < 1 and that E[Qt] and E[εtε′t ] are independent of t. Then,

S =1− β

1− α − βE[Q∗1/2t εt ε

′t Q∗1/2t ] −

α

1− α − βE[εt ε

′t ]. (7)

Eq. (7) shows that the expression ofS in terms of process moments is more complicated than

(6). Specifically, replacingS with the sample second moment ofεt proves not to be an obvious

estimation device.

Again relying on the conjecture thatQt follows a linear MGARCH process, if there is a unit

root in the dynamic correlation parameters (that is, forα + β = 1), the elements ofQt are thought

of as integrated GARCH processes (Engle 2002, eq. (17)). By analogy, the hypothesisα + β = 1

is often tested as a test of integrated correlation (see, e.g., Engle and Sheppard 2001; Pesaran and

Pesaran 2007). Indeed, as proven by the following arguments, forα + β = 1 the elements ofQt

are weakly stationary. Recalling thatS is unit-diagonal, forα + β = 1 the diagonal elements ofQt

can be written as

qii ,t = (1− β) + βqii ,t−1 + α(ε2it−1 − 1). (8)

Sinceα(ε2it−1 − 1) is a martingale difference,qii ,t follows an AR(1) process whose autoregressive

parameter isβ. Forα+ β = 1 andα > 0, the process is weakly stationary. Forα+ β = 1 andα = 0,

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the error term is zero andqii ,t is constant. SinceQt is pd, it holds that|qi j,t| ≤√

qii ,t qj j,t. Therefore,

for α + β = 1, the second moment ofqi j,t is finite, in which case, ifqi j,t is strictly stationary, it is

weakly stationary.

After recognizing thatQt is not a linear MGARCH, finding a tractable representation ofQt

proves to be difficult. A consequence of this is that the dynamic properties ofRt turn out to be

essentially unknown. Similar problems also arise with a major competitor of the DCC model, the

VC model of Tse and Tsui (2002). In this model the conditional correlation process is defined as

Rt = (1− α − β)S + αRM,t−1 + βRt−1, (9)

whereM ≥ N is fixed arbitrary andRM,t is the sample correlation matrix ofεt, εt−1, . . . , εt−M+1.

SinceRt is generally not the conditional expectation ofRM,t, the processRt is not standard. Taking

the expectations of both members of (9), under stationarity yields

S =1− β

1− α − βE[εtε

′t ] −

α

1− α − βE[RM,t]. (10)

As with the DCC model,S turns out to be neither easy to interpret nor simple to estimate by means

of a moment estimator.

2.2 ThecDCC model

The tractability of the DCC model can be substantially improved by reformulating the correlation

driving process as

Qt = (1− α − β) S + α{Q∗1/2t−1 εt−1ε

′t−1Q

∗1/2t−1

}+ βQt−1 (11)

(Aielli 2006). The resulting model is calledcDCC model, wherec stands forcorrected. Some

simulated series ofcDCC conditional correlations generated starting with the same seed of the

corresponding DCC series are plotted in the panel of Fig. (1). It is shown that, for typical parameter

values (that is, for smallα and largeα + β) the DCC andcDCC data generating processes are very

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similar. Pre and post multiplying the right hand side and the left hand side of (11) by an arbitrary

pd diagonal matrix,Z, yields

Qt = (1− α − β) S + α{Q∗1/2t−1 εt−1ε

′t−1Q

∗1/2t−1

}+ β Qt−1, (12)

whereS ≡ ZSZ, Qt ≡ ZQtZ, andQ∗1/2t ≡ Q∗1/2t Z. Noting thatQ∗1/2t Z = ZQ∗1/2t , it follows

that Q∗−1/2t Qt Q

∗−1/2t = Q∗−1/2

t Qt Q∗−1/2t = Rt. Thus,Qt and Qt deliver the same conditional

correlation processes orS is not identifiable. As a natural identification condition it is assumed

that S is unit-diagonal. Note that with the DCC model this assumption is an overidentifying

restriction.

One might argue that thecDCC model is not really a correlation model in thatRt is modeled

only implicitly, as a byproduct of the model ofQt. A more explicit representation ofρi j,t can

be obtained as follows. Let us divide the numerator and denominator of the right hand side of

ρi j,t = qi j,t/√

qii ,tqj j,t by√

qii ,t−1qj j,t−1. This yields

ρi j,t =ωi j,t−1 + α εi,t−1ε j,t−1 + βρi j,t−1

√{ωii ,t−1 + α ε

2i,t−1 + βρii ,t−1

}{ω j j,t−1 + α ε

2j,t−1 + βρ j j,t−1

} , (13)

whereωi j,t ≡ (1− α − β)si j/√

qii ,t qj j,t. The above equation shows that with thecDCC formula of

ρi j,t the relevant innovations and past correlations are combined into a correlation-like ratio. The

denominator of the time-varying parameters,ωi j,t, ωii ,t, andω j j,t, can be thought of as anad hoc

correction required for purposes of tractability. Let us now consider the DCC analog of (13), which

turns out to be

ρi j,t =ωi j,t−1 + αt−1εi,t−1ε j,t−1 + βρi j,t−1

√{ωii ,t−1 + αt−1ε

2i,t−1 + βρii ,t−1

}{ω j j,t−1 + αt−1ε

2j,t−1 + βρ j j,t−1

} ,

whereωi j,t ≡ (1− α − β) si j/√

qii ,t qj j,t andαt ≡ α/√

qii ,t qj j,t. Compared with thecDCC formula,

the DCC formula involves more time-varying parameters, none of them being supported by any

apparent motivation. As for the VC model (see (9)), the correlation process is modeled explicitly,

but at the cost of anad hocinnovation term,RM,t, which makes the model difficult to study (see

section 2.1).

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With thecDCC model,qii ,t follows a GARCH(1,1) process (Bollerslev 1986). Forα + β = 1,

the intercept ofqii ,t is zero, in which caseqii ,t −→ 0 for t −→ ∞ (Nelson 1990, Proposition 1).

This can cause numerical problems if, forα + β = 1 and larget, one computes the conditional

correlations asρi j,t = qi j,t/√

qii ,tqj j,t. Such problems disappear by computingρi j,t as in eq. (13),

which, forα + β = 1, does not depend onQt.

Settingε∗t ≡ Q∗1/2t εt , it follows thatEt−1[ε∗t ] = 0 andEt−1[ε

∗t ε∗′t ] = Qt, whereQt = (1− α −

β) S + α ε∗t−1ε∗′t−1 + βQt−1. Therefore,ε∗t follows a linear MGARCH process (Engle and Kroner

1995). Specifically,ε∗t follows a MARCH model (Ding and Engle 2001). A more generalcDCC

model can be obtained by assuming a more general linear MGARCH model forε∗t . Assumingε∗t

as a Baba-Engle-Kraft-Kroner (BEKK) model (Engle and Kroner 1995), after replacingε∗t with

Q∗1/2t εt yields

Qt = C +

Q∑

q=1

K∑

k=1

Aq,k {Q∗1/2t−q εt−qε

′t−qQ

∗1/2t−q } A

′q,k +

P∑

p=1

K∑

k=1

Bp,k Qt−p B′p,k, (14)

where

C ≡ S −Q∑

q=1

K∑

k=1

Aq,k S A′q,k −P∑

p=1

K∑

k=1

Bp,k S B′p,k. (15)

With this model,Qt is pd provided thatC is pd. A special case of the BEKK model is the

Diagonal Vech model (Bollerslev, Engle, and Wooldridge 1988), which, after replacingε∗t with

Q∗1/2t εt , yields

Qt =

ιι′ −

Q∑

q=1

Aq −P∑

p=1

Bp

� S +

Q∑

q=1

Aq � {Q∗1/2t−q εt−q ε′t−qQ

∗1/2t−q } +

P∑

p=1

Bp �Qt−p. (16)

With this model,Qt is pd if Aq ≡ [αi j,q] is psd for q= 1,2, . . . ,Q, Bp ≡ [βi j,p] is psd for p=

1,2, . . . ,P, and the intercept is pd (Ding and Engle 2001). SettingAq = αq ιι′ for q = 1,2, . . . ,Q,

andBp = βp ιι′ for p = 1,2, . . . ,P, provides the ScalarcDCC model (Ding and Engle 2001),

which, for P= Q = 1, is thecDCC model in eq. (11).

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2.3 cDCC stationarity principle

Relying on the modular structure of thecDCC model, thecDCC stationarity conditions can be

formulated as a flexible stationarity principle (cDCC stationarity principle). In simple terms, if the

correlation/standardized return process, (Rt, εt), is stationary, checking for the stationarity of the

covariance/return process, (Ht,yt), reduces checking for the stationarity of the univariate GARCH

processes,hi,t, i = 1,2, . . . ,N. Given that relatively easy–to–check stationarity conditions are

available for many univariate GARCH specifications adopted in practice (see Francq and Zakoıan,

2010), thecDCC stationarity principle is capable of a wide range of applications. A similar princi-

ple holds for the DCC and VC models as well, but with thecDCC model it takes on a practical in-

terest thanks to a recent result of Boussama et al. (2011), relying on which explicit stationarity con-

ditions for thecDCC specification of (Rt, εt) can be derived. Let us now introduce some notations.

For the BEKKcDCC model in (14–15), setAq ≡ U {∑K

k=1 Aq,k ⊗ Aq,k}W ′ for q = 1,2, . . . , Q,

andBp ≡ U {∑K

k=1 Bp,k ⊗ Bp,k}W ′ for p = 1,2, . . . , P, where⊗ denotes the Kronecker product.

The matricesU andW are the uniqueN(N + 1)/2× N2 matrices such that vech(Z) = Uvec(Z),

vec(Z) = W ′vech(Z), andUW ′ = IN(N+1)/2 for anyN-dimensional symmetric matrixZ, where

IM is theM-dimensional identity matrix and vec and vech, respectively, denote the operators stack-

ing the columns and the lower triangular part of the matrix argument. The existence and unique-

ness ofU andW hold by linearity of the vech and vec operators. Lethi,t = Hi(θi; εi,t−1, εi,t−2, . . .)

denote the representation ofhi,t in terms of standardized innovations, obtained by recursively sub-

stituting backward for laggedyi,t = εi,t√

hi,t into hi,t = hi(θi; yi,t−1, yi,t−2, . . .). It is assumed that

εt = R1/2t ηt , whereηt is iid such thatE[ηt] = 0 andE [ηtη

′t ] = IN, andR1/2

t is the unique psd

matrix such thatR1/2t R1/2

t = Rt (the matrixR1/2t is computed from the spectral decomposition of

Rt).

Proposition 2.3(cDCC stationarity principle). For the BEKKcDCC model in (14), suppose that:

H1) the density ofηt is absolutely continuous with respect to the Lebesgue measure, positive in

a neighborhood of the origin;

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H2) C is pd;

H3) the in modulus largest eigenvalue of∑Q

q=1Aq +∑P

p=1Bp is less than one.

Then, (i) the process[vech(Rt)′, ε′t ]

′ admits a non-anticipative, strictly and weakly stationary, and

ergodic solution.

In addition to H1-H3, suppose that:

H4) Hi(θi; εi,t−1, εi,t−2, . . .) is measurable for i= 1,2, . . . ,N.

Then, (ii) the process[vech(Ht)′,y′t , vech(Rt)′, ε′t ]

′ admits a non-anticipative, strictly stationary,

and ergodic solution.

In addition to H1-H4, suppose that:

H5) E[y2i,t] < ∞ for i = 1,2, . . . ,N.

Then, (iii) the processyt admits a weakly stationary solution.

Assumption H1 is typically fulfilled in any practical application. As for assumptions H2–H3,

they are relatively easy to check. As for the variance assumptions, H4–H5, their explicit form

(if any) will depend on the model ofhi,t, i = 1,2, . . . ,N. It is worth noting that H4 requires the

measurability ofHi,t under strictly stationaryεi,t, which is a less stringent condition than the usual

one of iidεi,t.

As an illustrative application of prop. 2.3, consider the GaussiancDCC model with GARCH(1,1)

conditional variances. Sinceεt is Gaussian,ηt is Gaussian, which implies that H1 holds. The in-

tercept ofQt satisfiesC = (1 − α − β)S; therefore, ifS is pd andα + β < 1, assumption H2

holds. The matrix∑Q

q=1Aq +∑P

p=1Bp turns out to be scalar and equal to (α + β)IN(N+1)/2. All the

eigenvalues of (α + β)IN(N+1)/2 are equal toα + β; therefore, ifα + β < 1, assumption H3 holds.

The GARCH(1,1) model is defined ashi,t = ci + ai y2i,t−1 + bi hi,t−1, whereci > 0, ai ≥ 0, and

bi ≥ 0. If εi,t is strictly stationary, assumption H4 holds provided thatE[log(ai ε2i,t + bi )] < 0 and

E[max{0, log(ai ε2i,t + bi )}] < ∞ (Brandt 1986 and Bougerol and Picard 1992). Ifεi,t is iid, as in the

GaussiancDCC model, for H4 to hold it suffices thatE[log(ai ε2i,t + bi )] < 0 (Nelson 1990). The

latter condition is less stringent thanai + bi < 1, in which case assumption H5 holds (Bollerslev

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1986). In summary, ifS is pd andα + β < 1, the correlation/standardized return process is strictly

and weakly stationary; if, in addition,ai + bi < 1 for i = 1,2, . . . ,N, the covariance/return process

is strictly stationary, and the return process is weakly stationary.

According to a result of Engle and Kroner (1995),S is the sample second moment ofε∗t .

Equivalently,S is the sample second moment ofQ∗1/2t εt .

Proposition 2.4. For the BEKKcDCC model in (14-15), suppose that assumptions H1-H3 of prop.

2.3 hold. Then,Q∗1/2t εt is covariance stationary, and

S = E[Q∗1/2t εt ε′t Q∗1/2t ]. (17)

Eq. (17) shows that, as with the DCC and VC models (see eq. (7) and eq. (10)), also with

thecDCC modelS is generally not the second moment ofεt. However, differing from the DCC

and VC models, with thecDCC model the sample counterpart ofS is by construction pd. Rely-

ing on this property, a pd generalized profile quasi-log-likelihoodcDCC estimator for large sys-

tems can easily be constructed (see sec. 3.2). Note also thatE[Q∗1/2t εtε′tQ∗1/2t ] = E[ε∗t ε

∗′t ] =

E[Et−1[ε∗t ε∗′t ]] = E[Qt], which, jointly with (17), yieldsS = E[Qt]; therefore,S is the expecta-

tion of the correlation driving process.

2.4 cDCC integrated correlations

The case ofα + β = 1 (integrated correlations) is not covered by prop. 2.3. In fact, in this

case,C = (1 − α − β)S = 0, which implies that H2 does not hold. The behavior of thecDCC

conditional correlations forα + β = 1 is illustrated in this section by means of simulations. It will

be proven that, forα + β = 1, the correlation processes degenerate either to 1 or to -1 in the long

run. Thus, recalling that, forα + β < 1, the correlation processes are mean reverting (see prop.

2.3), it follows that testing for integrated correlations has a clear meaning with thecDCC model.

The simulation experiment is planned as follows. Forα + β = 1, whereα = .01, .04, .16, .64, a

set of 250 bivariate ScalarcDCC series is generated, stopping each series at timet + M if either

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ρ12,t+m > 1 − ε, or ρ12,t+m < −1 + ε, for m = 1,2, . . . ,M, whereM = 100,000 andε = 10−3.

The occurrence of such a stopping rule is assumed as providing evidence that, in the long run,

ρ12,t degenerates either to 1 or to−1. As conditional distributions for [ε1,t, ε2,t]′ some elliptical

distributions are considered (McNeil, Frey, and Embrechts 2005), namely, the multivariate student

t distribution with degrees of freedom (d.f.)ν = 3,6,12, and the multivariate exponential power

(MEP) distribution with kurtosis parameterκ = 1,1.25,2,8. Such distributions are rescaled to get

unit variances and covariance equal toρ12,t. The total number of experiments is 28 (fourα–values

times seven distributional assumptions). The marginals of the MEP distribution are generalized

error distributions (Nelson 1991). Their kurtosis is a decreasing function ofκ. For κ = 1 they are

double exponential and forκ = 2 they are Gaussian. By the properties of the elliptical distributions

and the structure of thecDCC model, it can be easily shown that the dynamic properties ofρ12,t

in a bivariate model are the same as the dynamic properties ofρi j,t, i , j, in a model withN > 2

assets, provided that the conditional distribution of the two models is elliptical of the same kind.

For eachα–value, Fig. (2) reports the empirical distribution of the 250 absorbing times,t, under

student t innovations. It is shown that all correlation series degenerate. The smallerα, the longer

the absorbing time, which is coherent with the fact that the variance of the correlation process is an

increasing function ofα (see Fig. (1)). For typical parameter values (e.g.,α ≤ .04), the absorbing

time is a decreasing function of the kurtosis. Analogous results arise under MEP innovations (the

corresponding plots are not reported here for reasons of space). The amount of series such that the

absorbing state,ρ12,t, is positive (resp. negative) is an estimate of the probability that the assets be-

come perfectly positively (resp. negatively) correlated in the long run. None of the 28 experiments

provides an estimated probability which is significantly different from 1/2. This is due to having

setρ12,0 = 0 as a starting point of the correlation process. From another experiment run with

Gaussian innovations, whereα = .04 andρ12,0 = −.95,−.90,−.60,−.30,+.30,+.60,+.90,+.95,

the estimated probabilities turned out to be equal, respectively, to.10, .15, .32, .40, .59, .70, .84,

.91. As expected, the probability of perfectly positive correlation in the long run is an increasing

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function of the initial correlation.

3 LARGE SYSTEM ESTIMATION

3.1 The DCC estimator

Let LT(θ,S,φ) =∑T

t=1 lt(θ,S,φ) denote the (log) factorization of the DCC quasi-log-likelihood

(QLL), whereθ ≡ [θ′1,θ′2, . . . , θ

′N]′, φ ≡ [α, β]′, lt(θ,S,φ) = −(1/2)

{N log(2π) + log |Ht | + y′t H

−1t yt

},

andHt denotes the DCC conditional covariance matrix (see (1-4)) evaluated at (θ,S,φ) (here-

after, the symbol will denote series evaluated at given parameters values). The matrixS includes

N(N−1)/2 distinct parameters to estimate; therefore, even for knownθ, the joint quasi-maximum-

likelihood (QML) estimation of the DCC model is infeasible for largeN. As a feasible estimator,

Engle (2002) suggested a three-step procedure called DCC estimator. Before introducing it, let us

define the following estimator ofS conditional onθ.

Definition 3.1. (DCC conditional estimator ofS). For fixedθ, setSθ ≡ T−1 ∑Tt=1 εt ε

′t , where

εt ≡ [ε1,t, . . . , εN,t]′, εi,t ≡ yi,t/

√hi,t, andhi,t ≡ hi,t(θi; yi,t−1, yi,t−2, . . .).

The estimatorSθ is the sample second moment of the standardized returns evaluated atθ.

Alternatively,Sθ can be defined as the sample covariance matrix of ˜εt, or as the sample (centered

or uncentered) correlation matrix of ˜εt (Engle and Sheppard 2001). In the latter case,Sθ is unit-

diagonal, likeS.

Definition 3.2. (DCC estimator).

Step (1): setθ ≡ [θ′1, θ′2, . . . , θ

′N]′, where θi is the univariate QML estimator ofθi , i =

1,2, . . . ,N;

Step (2): setS ≡ Sθ ;

Step (3): setφ ≡ argmaxφ LT(θ, S,φ) subject toα ≥ 0, β ≥ 0, andα + β < 1.

Step 1 consists of a set ofN univariate fittings; step 2 is easy to compute after step 1; finally, step

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3 is a maximization in two variables. Therefore, the DCC estimator is feasible for largeN. A fur-

ther reduction of computing time can be obtained settingLT(θ,S,φ) ≡∑N

i=2 LT,i,i−1(θi ,θi−1, si,i−1,φ),

whereLT,i, j(θi ,θ j , si, j ,φ) is the QLL of the bivariate DCC submodel of (yi,t, yj,t). The function

LT(θ,S, φ) in this case is called bivariate composite QLL. Engle, Shephard, and Sheppard (2008)

proved that using a bivariate composite QLL in place of the full QLL can result in a dramatic

reduction of the bias ofφ.

Let us denote the true parameter values with a superscript zero. Relying on the conjecture that

S0 = E[εtε′t ] (see section 2.1),S is thought of as a consistent estimator (Engle and Sheppard 2001;

Cappiello et al. 2006; Billio et al. 2006; Pesaran and Pesaran 2007; Franses and Hafner 2009). As

proven in section 2.1, the equalityS0 = E[εtε′t ] does not hold in general. Under consistency ofθ,

if S0 , E[εtε′t ] and plimSθ is finite in a neighborhood ofθ0, it follows that

plim S = plim Sθ , S0

(Wooldridge 1994, Lemma A.1). IfS is inconsistent,φ is inconsistent in turn unlessS andφ are

orthogonal (Newey and McFadden 1986, sec. 6.2, p. 2179). The inconsistency ofS could be a

source of inconsistency also for the adjusted standard errors ofφ computed relying on the formulas

in Newey and McFadden (1986, Theorem 6.1; see Engle 2002, p. 342, eq. 33) or in White (1996;

see Engle and Sheppard 2001). In fact, the proof of the consistency of the adjusted standard errors

requires a consistent estimator of (θ,S).

3.2 ThecDCC estimator

As a large system estimator for thecDCC model, Aielli (2006) suggested a three-step estimator

called cDCC estimator. In the following,LT(θ,S,φ) =∑T

t=1 lt(θ,S,φ) will denote the (log)

factorization of the QLL of thecDCC model. The parameters (θ,S,φ) enter the QLL through the

cDCC defining equations (1-3) and (11). Before introducing thecDCC estimator, let us define the

following estimator ofS conditional on (θ,φ).

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Definition 3.3. (cDCC conditional estimator ofS). For fixed(θ,φ), setSθ ,φ ≡ T−1 ∑Tt=1 Q∗1/2t εt ε

′t Q∗1/2t ,

whereQ∗t ≡ diag(q11,t, . . . , qNN,t), where

qii ,t = (1− α − β) + α ε2i,t−1 qii ,t−1 + β qii ,t−1. (18)

The estimatorSθ ,φ is the sample counterpart ofS evaluated at (θ,φ) (see (17)). By construc-

tion, Sθ ,φ is pd. Alternatively,Sθ ,φ can be defined as the sample covariance matrix, or as the

sample correlation (centered or uncentered) matrix, ofQ∗1/2t εt . In the latter case,Sθ ,φ is unit-

diagonal, likeS.

Definition 3.4. (cDCC estimator).

Step (1): setθ as in def. 3.2;

Step (2): setφ ≡ argmaxφ LT(θ, Sθ ,φ ,φ) subject toα ≥ 0, β ≥ 0, andα + β < 1;

Step (3): setS ≡ Sθ ,φ .

Step 1 is the same as step 1 of the DCC estimator; step 2 is a maximization in two variables;

finally, step 3 does not actually need to be calculated, in thatS is nothing but the value ofSθ ,φ

at the end of step 2. Note that with the DCC estimator the estimator ofS is computed only once,

before estimatingφ, whereas, with thecDCC estimator, it is recomputed at each evaluation of the

objective fuction ofφ. This however does not require a sensible additional computational effort,

as the calculations required bySθ ,φ basically reduce running theN univariate GARCH recursions

providing qii ,t for fixed φ andθ = θ, i = 1,2, . . . ,N (see def. 3.3). Under appropriate conditions

(Newey and McFadden 1986, Theorem 3.5), a Newton-Raphson one-step iteration from thecDCC

estimation output will deliver an estimator with the same asymptotic efficiency as the joint QML

estimator. As with the DCC estimator, a bivariate composite version of thecDCC estimator can be

obtained replacing the fullcDCC QLL with the bivariate compositecDCC QLL.

Recalling thatS0 is the second moment ofQ∗1/2t εt , it follows thatS(θ ,φ ) is consistent if evalu-

ated at the true values of (θ,φ).

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Proposition 3.1. For thecDCC model in (11), suppose that assumptions H1-H3 of prop. 2.3 hold.

Then,plim Sθ 0,φ0 = S0.

SinceS0 is unit-diagonal andQ∗1/2t εt is zero-mean, prop. 3.1 holds whetherS(θ ,φ ) is computed

as a sample second moment, or as a sample covariance matrix, or as a sample (centered or un-

centered) correlation matrix, ofQ∗1/2t εt . Relying on prop. 3.1 a heuristic proof of the consistency

of the cDCC estimator can be provided as follows(see Fig. (3) for a graphical illustration of it).

Denote asΦ ≡ {(α, β) : α ≥ 0, β ≥ 0, α + β < 1} denote the constraint onφ. ThecDCC estimator,

(θ, S, φ), can be defined as the maximizer of thecDCC QLL subject to{θ = θ,S = Sθ ,φ ,φ ∈ Φ}.

If θ is consistent and plimSθ ,φ is finite for all (θ,φ), it follows that

plim {θ = θ,S = Sθ ,φ ,φ ∈ Φ} = {θ = θ0,S = plim Sθ 0,φ ,φ ∈ Φ}

(Wooldridge 1994, Lemma A.1). Recalling that plimSθ 0,φ0 = S0 (see prop. 3.1), from the above

equation it follows that the limit in probability of thecDCC constraint is a correctly specified

constraint. Therefore, if plimT−1LT(θ,S,φ) is finite for all (θ,S,φ), and uniquely maximized

at (θ0,S0,φ0), which is a common assumption in QML settings (see Bollerslev and Wooldridge

1992), plimT−1LT(θ, Sθ ,φ ,φ) is uniquely maximized atφ0 (Wooldridge 1994, Lemma A.1). If, in

addition, plimT−1LT(θ,S,φ) converges uniformly to its limit,φ is consistent (Newey and McFad-

den 1986, Theorem 2.1). Regarding the consistency ofS, if plim Sθ ,φ is finite for all (θ,φ), and

plim(θ, φ) = (θ0,φ0), then plimS = plim Sθ ,φ = S0 (Wooldridge 1994, Lemma A.1). Finally,

regarding the consistency ofθ, appropriate conditions can be found for several univariate GARCH

specifications adopted in practice (see Francq and Zakoıan 2010). Note that the above arguments

on the consistency of thecDCC estimator of (S,φ) do not apply to the DCC estimator of (S,φ)

because the DCC conditional estimator ofS is not generally consistent if evaluated at the true

value ofθ.

ThecDCC estimator ofφ can be thought of as a generalized profile QLL estimator (Severini

1998), in whichφ is the parameter of interest and (θ,S) is the nuisance parameter. The estimator of

the nuisance parameter conditional onφ is (θ, Sθ ,φ ). The functionLT(θ, Sθ ,φ ,φ) in step 2 of def.

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3.4 is the generalized profile QLL. A generalized profile QLL estimator of the DCC model, totally

analogous to thecDCC estimator, can be obtained definingS(θ ,φ ) as the sample counterpart of (7)

for fixed (θ,φ). Unfortunately, such an estimator is not guaranteed to be pd. Similar problems of

positive definiteness would also arise with the VC model (see eq. 10).

A large system estimator for the Diagonal VechcDCC model, (16), can be obtained as a direct

extension of thecDCC estimator. The parameterφ is defined as the vector collecting the distinct

elements of the dynamic parameter matrices,Aq ≡ [αi j,q] for q = 1,2, . . . ,Q, andBp ≡ [βi j,p] for

p = 1,2, . . . ,P. The conditional estimator ofS is computed replacing (18) with

qii ,t =

1−

Q∑

q=1

αii ,q −P∑

p=1

βii ,p

sii +

Q∑

q=1

αii ,q ε2i,t−q qii ,t−q +

P∑

p=1

βii ,p qii ,t−p,

wheresii = 1. The proof of the consistency ofS(θ0,φ0) is totally analogous to the proof of prop.

3.1. For suitably restrictedφ, the estimator is feasible (see, e.g., the block partition of Billio et al.

2006). The generalized profile QLL must be maximized over the set of the vectorsφ such that the

dynamic parameter matrices are psd, and the conditional estimator of the intercept, namely,ιι′ −

Q∑

q=1

Aq −P∑

p=1

Bp

� S(θ ,φ ),

is pd. Ensuring that the latter constraint holds can be a difficult task.

3.3 Inferences fromcDCC estimations

The approach in Newey and McFadden (1986) to the estimation of the asymptotic covariance ma-

trix of two-step estimators, can be adopted to compute inferences based on the (possibly Diagonal

Vech) cDCC estimator. Lets, s(θ,φ) and λt(θ,φ) denote the vectors stacking the lower off-

diagonal elements ofS, Sθ ,φ andQ∗1/2t εt ε′t Q∗1/2t , respectively. Letγ ≡ [θ′,φ′, s′]′, lt(θ, s,φ),

andli,t(θi) ≡ −(1/2){log(2π)+ log hi,t +y2i,t/hi,t}, wherei = 1,2, . . . ,N, denote thecDCC parameter,

the individualcDCC QLL at timet, and thei-th individual GARCH QLL at timet. ThecDCC

estimator, denoted as ˆγ ≡ [θ′, φ′, s′]′, is a solution of the estimating equationsT−1 ∑Tt=1 gt(γ) = 0,

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where the individual score is defined asgt(γ) ≡ [{ut(θ)}′, {vt(θ,φ)}′, {pt(θ,φ, s)}′]′, whereut(θ)

= [{u1,t(θ1)}′, . . . , {uN,t(θN)}′]′, whereui,t = (∂/∂θi)l i,t(θi), vt(θ,φ) ≡ (∂/∂φ)lt(θ, s(θ,φ),φ) and

pt(θ,φ, s) ≡ λt(θ,φ) − s. Under the conditions in Newey and McFadden (1986, Theorem 3.2,

with W replaced by the identity), it follows that√

T(γ − γ0)A≈ N(0, {C0}−1D0{C0′}−1), where

C0 ≡ plim

T−1T∑

t=1

∂

∂γ′gt(γ)

∣∣∣∣∣∣∣γ=γ 0

andD0 ≡ limT−→∞

VAR

T−1/2

T∑

t=1

gt(γ0)

.

The matrixC0 is block lower triangular with block partition determined by the partition ofγ intoθ,

φ, ands. It can be proven that the asymptotic covariance matrix ofθi coincides with its univariate

QML asymptotic covariance matrix. The asymptotic covariance matrix ofφ does not depend on

derivatives with respect to theO(N2) terms. Estimates of the asymptotic covariance matrices of

θ, φ and, if needed, ˆs, can be computed by replacing the relevant blocks ofC0 andD0 with the

sample counterparts evaluated at ˆγ. ForD0, a heteroskedastic and autocorrelation (HAC)–robust

estimator is required (see, e.g., Newey and West 1987). IfSθ ,φ is computed as a sample (centered

or uncentered) correlation matrix, the individual score ofs, namely,pt(θ,φ, s), must be designed

accordingly.

4 EMPIRICAL APPLICATIONS

In this section the empirical performances of the DCC andcDCC estimators are compared.

The calculations are run using a MATLAB code based on a sequential quadratic programming

optimizer. In the simulations under the hypothesis of correctly specified model the true value of

the parameters is used as a starting point for the maximization algorithms. In the simulations

under misspecification and in the applications to the real data the starting points are computed as

the maximizers over a grid of the related objective functions. The constraintsε ≤ α ≤ 1 − ε,

ε ≤ β ≤ 1−ε, andα+β ≤ 1−ε, whereε = 10−5, are imposed to achive more stable maximizations.

The conditional estimators ofS are computed as centered correlations.

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4.1 Simulations under correctly specified model

In this section the simulation performances of the DCC andcDCC estimators under correctly

specified model are compared. Fors ≡ s12 = 0,±.3,±.6,±.9, α = .01, .04, .16, andα + β =

.800, .990, .998, a set ofM = 500 independent DCC bivariate Gaussian series of lengthT = 1750

are generated. A burning period of 500 observations for each series is discarded to alleviate the

effect of the initial values. Since the focus of the simulation study is on the correlation fitting

performances, the DCC estimator is computed assuming that the standardized returns are known.

An analogous experiment is run in which thecDCC estimator is computed fromcDCC bivariate

Gaussian generated series. The results of the simulations are reported in the panels of Fig. (4)–

(7). Regarding the DCC estimator, the estimator ofs exhibits a positive bias fors < 0 and a

negative bias fors> 0 (see Fig. (4)). Such a shrinkage effect increases with the persistence of the

correlation process as measured byα + β and with the impact of the innovations as measured by

α. For varyings the bias moves according to a sinusoidal pattern. This is particularly evident for

α + β = .998 andα = .16. For typical values of the dynamic parameters (that is, forα + β ≥ .990

andα ≤ .04) the bias is negligible. Fors = 0, the estimator appears to be unbiased irrespective

of the value of the dynamic parameters. The panel of Fig. (5) reports the box plots of the relative

estimation error of ˆα. Forα + β = .998 andα = .16, the DCC box plots are symmetric and well

centered around zero. The same holds for the DCC box plots of the relative estimation error ofβ

(see Fig. (6)). This seems to suggest that the bias of the DCC estimator ofs does not seriously

affect the performances of the DCC estimator of (α, β). Fig. (7) reports the box plots of the mean

absolute error of the conditional correlation estimator, computed asM−1T−1 ∑Mm=1

∑Tt=1

∣∣∣ρt − ρt

∣∣∣ ,

whereρt ≡ ρ12,t andρt ≡ ρ12,t. Again, in spite of the large bias affecting the DCC estimator ofs for

α + β = .998 andα = .16, the performances of the DCC estimator turns out to be relatively good.

This can be explained as follows. By backward substitutions, for larget the DCC equation ofq12,t

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can be approximated as

q12,t ≈1− α − β

1− βs+

α

1− β

(1− β)

t−1∑

n=1

βn−1ε1,t−n ε2,t−n

.

The right hand side of the above equation is a weighted mean ofs and of the term in brackets. If

bothα + β andα are large (such as forα + β = .998 andα = .16) the weight ofs is small. In this

case, the effect on the estimates ofα, β, andρt due to replacings with a biased estimator will be

small in turn.

Regarding thecDCC estimator, the estimator ofs does not exhibit any apparent bias (see Fig.

(4)). All the related box plots are symmetric and well centered around zero. As forα, β, andρt

(see Fig. (5)–(7)), apart forα + β ≥ .990 andα = .16 the performances of thecDCC estimator

are practically identical to those of the DCC estimator. Forα + β ≥ .990 andα = .16, instead, the

cDCC estimator underperforms the DCC estimator, especially in terms of correlation mean abso-

lute error (see Fig. (7)). For such parameter values, however, a comparison of the two estimators

under correctly specified model is not appropriate because the data generating processes of the two

models are rather different. An illustration of this is reported in Fig. (8), in which the autocorre-

lation function (ACF) ofρt for the two models are depicted. Forα + β ≥ .99 andα = .16, the

cDCC conditional correlation process exhibits less memory than the DCC conditional correlation

process.

4.2 Simulations under misspecification

To compare the performances of the DCC andcDCC estimators under misspecification, a set of

500 bivariate Gaussian return series of lengthT = 1000 are generated in which the conditional

variances follow GARCH processes and the correlation process is fixed arbitrary. For each series

the DCC andcDCC estimators are computed, and a test of correctly specified model from the two

estimation outputs is carried out. The percentage of rejections at a given level is then considered

as a measure of the estimator performances under misspecification. The best estimator is the one

providing the smallest percentage of rejections. The conditional variances are GARCH(1,1), set

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ash1,t = .01+ .05y21,t−1 + .94h1,t−1 andh2,t = .30+ .20y2

2,t−1 + .50h2,t−1. One variance process is

highly persistent and the other is not. The orders of the GARCH variances (not the parameters)

are assumed as known. Following Engle (2002), the conditional correlation processes are set as

CONS T ANT≡ {ρt = .9}, S T EP≡ {ρt = .9− .5(t ≥ 500)}, S INE≡ {ρt = .5+ .4. cos(2πt/200)},

FAS TS INE≡ {ρt = .5 + .4. cos(2πt/20)}, andRAMP≡ {ρt = mod(t/200)/200}. The choice of

such processes ensures a variety of correlation dynamics, such as rapid changes, gradual changes,

and periods of constancy (see Engle 2002). As tests of correctly specified model, the following

three regression-based tests computed from the return series of a portfoliow′yt, wherew is the

vector of the portfolio weights, are considered.

• Engle-Colacito regression.The Engle-Colacito (E&C) regression (Engle and Colacito 2006) is

defined as{(w′yt )2 /(w′Ht w )} − 1 = λ + ξt, whereξt is an innovation term. Recalling that the

conditional variance ofw′yt is w′Htw, if the model is correctly specified it follows thatλ = 0.

The E&C test is a test of the null hypothesis thatλ = 0. An HAC-robust estimator of the standard

error ofξt is required.

• Dynamic Quantile test.Denote theτ×100%-quantile of the conditional distribution ofw′yt

asVaRt(τ) (whereVaRstands forValue at Risk). For fixedτ, setHITt ≡ 1 if w′yt < VaRt(τ), and

HITt ≡ 0 otherwise. If the model is correctly specified,{HITt − τ} is zero-mean iid. The Dynamic

Quantile (DQ) test of Engle and Manganelli (2004) is anF-test of the null hypothesis that all

coefficients, as well as the intercept, are zero in a regression of{HIT t − τ} on past values,VaRt(τ)

and any other variables. In this paper it is setVaRt(τ) = −1.96√

w′Htw , which corresponds to

the 2.5% estimated quantile under Gaussianity. Five lags and the current estimated VaR are used

as regressors.

• LM test of ARCH effects. If the model is correctly specified, the series (w′yt)2/(w′Htw)

does not exhibit serial correlation. The LM test of ARCH effects (Engle 1982) is a test of the null

hypothesis that (w′yt)2/(w′Htw) is serially uncorrelated. In this paper five lags are used.

Under the null hypothesis, the considered tests are asymptotically normal. In practice, because

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of the replacement of true quantities with estimated quantities, the asymptotic size of the tests can

be different from the nominal size. Each test is computed from three return portfolios, namely,

the equally weighted portfolio (EW), the minimum variance portfolio with short selling (MV), and

the minimum variance portfolio without short selling (∗MV). The vector of EW weights isι/N.

The MV and∗MV weights, which are functions of the true conditional covariance matrix,Ht, are

computed from the estimated conditional covariance matrix,Ht. For each estimator, 5× 3× 3 =

45 percentages of rejections at a 5% level are computed, where 5 is the number of correlation

specifications, 3 is the number of regression-based tests, and 3 is the number of portfolios. The null

hypothesis of equal percentage of rejections for the two estimators is not rejected at any standard

levels for any of the 45 cases. Thus, at least under the considered correlation misspecifications, the

DCC andcDCC estimators provide very similar performances.

4.3 Applications to real data

To compare the performances of the DCC andcDCC estimators on real data two datasets are con-

sidered. One dataset includes the S&P 500 composite index and the related nine S&P Depositary

Receipts (SPDR) sector indices for a total ofN = 10 assets. The sample period is May 2003

- February 2010, which results inT = 1750 daily returns. Another dataset includesN = 100

randomly selected equities from the S&P 1500 index industrial and consumer goods components.

The sample period is June 2003 - March 2010, again for a total ofT = 1750 daily returns. Both

datasets are extracted from Datastream. The two estimators are computed in their bivariate com-

posite version (see sec. 3.1 and 3.2) assuming GARCH(1,1) variances. As performance measures,

three sets of out-of-sample forecast criteria are considered: (i) Equal Predictive Ability (EPA) tests

of one-step-ahead forecasts, (ii) regression-based tests computed from one-step-ahead forecasts,

and (iii) EPA tests of multi-step-ahead correlation forecasts.

• EPA tests of one-step-ahead forecasts.Let dt anddct denote the loss due to replacing the

true one-step-ahead asset conditional covariance matrix,Ht, with the related DCC andcDCC

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estimates, denoted asHt|t−1 andHct|t−1, respectively, computed from a rolling window ofT < T

estimated excess returns, ˆyt− j, j = 1,2, . . . , T. The number of predictions isT − T. The null

hypothesisH0 : E[dc − d

]= 0, whered anddc are the average losses, denotes equal predictive

ability for the two estimators. Under appropriate conditions (Diebold and Mariano 1995), it holds

that

EPA≡

√T − T(dc − d )

√VAR[

√T − T(dc − d )]

A∼ N(0,1), (19)

whereVAR[√

T − T(dc − d )] is an HAC-robust estimate of the variance of√

T − T(dc − d ). Neg-

ative (resp. positive) values ofEPAprovide evidence in favor ofcDCC (resp. DCC) forecasts. The

length of the rolling window is set toT = 1250, for a total ofT − T = 500 forecasts. The esti-

mated excess returns are computed as demeaned observed returns using the mean of the data in the

rolling window. As loss functions, two mean square error (MSE) based losses (Diebold and Mari-

ano 1995), and two score based losses (Amisano and Giacomini 2007), are considered. Regarding

the DCC estimator, the MSE based losses are defined as

dt ≡ EWMS Et ≡((

w′yt|t−1)2 −w′ Ht|t−1w

)2, (20)

wherew is the vector of the EW weights and ˆyi,t|t−1 is the one-step-ahead estimated excess return,

and

dt ≡ CORRMS Et ≡1

N(N − 1)/2

∑

i< j=2,...,N

(εi,t|t−1 ε j,t|t−1 − ρi j,t|t−1

)2, (21)

whereρi j,t|t−1 ≡ hi j,t|t−1/√

hii ,t|t−1hj j,t|t−1 andεi,t|t−1 ≡ yi,t|t−1/

√hi,t|t−1, wherehi j,t|t−1 is thei j -th element

of Ht|t−1. The score based losses are defined as

dt ≡ EWSCOREt ≡ log(w′ Ht|t−1w) + (w′yt|t−1)2/(w′ Ht|t−1w), (22)

anddt ≡ CORRSCOREt ≡ log |Rt|t−1| + ε′t|t−1{Rt|t−1}

−1εt|t−1 (23)

where ˆεt|t−1 ≡ [ε1,t|t−1, . . . , εN,t|t−1]′ andRt|t−1 is the correlation matrix associated toHt|t−1. The

losses (20) and (22) focus on the performances of the EW conditional variance forecast, whereas

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the losses (21) and (23) focus on the performances of the asset correlation forecast. The losses

(20) and (22) are also computed for the MV and∗MV portfolios, in which case the related port-

folio weights are computed fromHt|t−1. The formulas of the corresponding losses for thecDCC

estimator are obtained replacingHt|t−1 with Hct|t−1 in (20–23). It is worth noting that the only model

dependent estimation error entering the considered losses is that due to the correlation estimator.

Therefore, the related EPA tests are essentially tests of equal predictive ability of correlation esti-

mators. The resulting test statistics are reported in Table (1). With the small dataset all tests are

insignificant at a 5% level. With the large dataset the message is in favor of thecDCC estimator,

as shown by the fact that the sign of the EPA tests is always negative, and that four tests of eight

are significant at a 5% level (one among them is significant at a 1% level).

• Regression-based tests computed from one-step-ahead forecasts.The regression-based

tests of section 4.2 can be used to assess the prediction performances of the DCC (resp.cDCC)

estimator after replacingyt andHt with yt|t−1 andHt|t−1 (resp.Hct|t−1). If the estimated forecasts

coincide with the true one-step-ahead quantities, the resulting test statistics are asymptotically

normal. Table (2) reports the test statistics. The performances of the DCC andcDCC estimators

are similar. With the small dataset thecDCC estimator turns out to be slightly better than the

DCC estimator, in that sixcDCC test statistics of nine are smaller than the corresponding DCC

test statistics. • EPA tests of multi-step-ahead correlation forecasts.The forecast ofρi j,t+m at

time t is defined asEt[ρi j,t+m], m ≥ 1. Apart fromm = 1, in which caseEt[ρi j,t+1] = ρi j,t+1, with

both the DCC model and thecDCC modelEt[ρi j,t+m] is infeasible. For the DCC model, Engle and

Sheppard (2001) suggest to approximateEt[ρi j,t+m] either with

ρi j,t+m|t ≡qi j,t+m|t

√qii ,t+m|t qj j,t+m|t

, (24)

where

qi j,t+m|t ≡ si j (1− α − β)m−2∑

n=0

(α + β)n + qi j,t+1(α + β)m−1, (25)

or with

%i j,t+m|t ≡ %i j (1− α − β)m−2∑

n=0

(α + β)n + ρi j,t+1(α + β)m−1, (26)

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where %i j ≡ E[εi,t ε j,t]. In practice, the unknown quantities appearing in (25–26) are replaced

by their estimates based on the rolling window ˆyt−1, yt−2, . . . , yt−T . Specifically, ˉ%i j is replaced

by the sample correlation of the estimated standardized returns. Analogous correlation forecasts

approximations can be defined for thecDCC model. A MSE based loss for ˉρi j,t+m|t can be defined

as

dt ≡1

N(N − 1)/2

∑

i< j=2,...,N

{εi,t+m|t+m−1 εi,t+m|t+m−1 − ρi j,t+m|t

}2, (27)

wheret = T + 1, T + 2, . . . ,T − m, andm ≥ 2. The rolling window one-step-ahead estimated

standardized returns appearing in the formula are preferred to the estimates of the standardized

returns based on the whole sample to alleviate the bias due to possible stractural breaks in the

univariate variance processes. Replacing ˉρi j,t+m|t with %i j,t+m|t in (27) will provide a MSE based loss

for %i j,t+m|t. The considered loss functions can be used to compare via EPA tests the performances of

the correlation forecasts ˉρi j,t+m|t and%i j,t+m|t for a given estimator, and the performances of the DCC

andcDCC estimators for a given correlation forecast. The related EPA test statistics, computed

for the forecast horizonsm = 2,3,5,8,13,21,34,55,89,144,233, are reported in Table (3). The

selected forecast horizons are the Fibonacci numbers greater than 1 and less than 250, where 250

is about the number of daily returns per year. Of course, for varying the forecast horizons and the

correlation forecasts the EPA tests are not independent. Let us consider the comparison of ˉρ and%

for a fixed estimator. Starting from the small dataset, with thecDCC estimator the preference is

for ρ (see the first row of the table, in which all test statistics are negative). Specifically, form≥ 55

the EPA test is significant at a 5% level. With the DCC estimator the message is less conclusive

(see the second row of the table). Moving to the large dataset, the preference is for ˉρ with both

estimators (see the fifth and sixth rows of the table). Let us now consider the comparison of the

DCC andcDCC estimators for a fixed correlation forecast. With the small dataset the test statistic

is always negative, which is in favor ofcDCC forecasts (see the third and fourth row of the table).

Specifically, for long forecast horizons (m ≥ 34) most tests are significant at a 5% level. With

the large dataset, apart form = 233 when ˉ% is used, in all cases the test is either not significant

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or significant and in favor ofcDCC forecasts (see the seventh and eighth rows of the table). In

summary, at least for the considered datasets, there is some evidence that ˉρ outperforms ˉ%, and that

thecDCC correlation forecasts outperform the DCC correlation forecasts.

5 CONCLUSIONS

This paper addressed some problems that arise with the DCC model. It has been proven that

the second step of the DCC large system estimator can be inconsistent, and that the traditional

GARCH-like interpretation of the DCC dynamic correlation parameters can lead to misleading

conclusions. A more tractable alternative to the DCC model, calledcDCC model, has been de-

scribed. A large system estimator for thecDCC model has been discussed and heuristically proven

to be consistent. Sufficient conditions for the stationarity ofcDCC processes of interest have also

been derived. The empirical performances of the DCC andcDCC large system estimators have

been compared by means of applications to simulated and real data. It has been shown that, un-

der correctly specified model, if the persistence of the correlation process and the impact of the

innovations are high, the DCC estimator of the location correlation parameter can be seriously bi-

ased. The correspondingcDCC estimator resulted practically uniformly unbiased. Regarding the

applications to the real data, thecDCC correlation forecasts have been proven to perform equally

or significantly better than the DCC correlation forecasts.

ACKNOWLEDGMENTS

I would like to thank Farid Boussama, Christian T. Brownlees, Giorgio Calzolari, Massimiliano

Caporin, Gabriele Fiorentini, Giampiero M. Gallo, Matteo Guastini, Davide Perilli, Pietro Rigo,

Paolo Santucci De Magistris, Nicola Sartori, Enrique Sentana and two anonymous referees for

their careful and useful suggestions. I would also like to thank my brother Gian Luca and Katerina

Zamit for their precious support.

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APPENDIX: PROOFS

Proof of prop. 2.1. It can be easily verified thatA is psd,B is psd, and (ιι′ − A − B) � S

is pd, which ensures thatRt is pd. Recalling thats11 = s22 = 1, the elements ofQt satisfy

q11,t = (1 − α11) + α11ε21,t−1, q22,t = 1, andq12,t = s12. By Jensen’s inequality andE[ε21,t] = 1, it

follows that

E[ε1,t ε2,t] = E[ρ12,t] = E[q12,t/√

q11,tq22,t]

= E[s12/

√(1− α11) + α11ε

21,t−1

]

= s12 E[1/

√(1− α11) + α11ε

21,t−1

]

> s12/√

(1− α11) + α11E[ε21,t−1] = s12. �

Proof of prop. 2.2. Taking the expectations of both members of (4) and rearranging, yieldsS =

{(1−β)E[Qt]−αE[εtε′t ]}/(1−α−β). Noting thatE[Qt] = E[Q∗1/2t Rt Q

∗1/2t ] = E[Q∗1/2t Et−1[εtε

′t ]Q

∗1/2t ] =

E[Et−1[Q∗1/2t εtε

′tQ∗1/2t ]] = E[Q∗1/2t εtε

′tQ∗1/2t ], proves the proposition.�

Proof of prop. 2.3. Fromε∗t = Q∗1/2t εt , whereεt = R1/2t ηt, it follows thatε∗t = Q1/2

t ηt, where

Q1/2t ≡ Q∗1/2t R1/2

t is the unique psd matrix such thatQ1/2t Q1/2

t = Qt. This fact, together with

H1-H3, ensures that the BEKK process, [vech(Qt)′, ε∗′t ]′, admits a non-anticipative, strictly sta-

tionary, and ergodic solution (Boussama et al. 2011, Theorem 2.3). Therefore, any time-invariant

function of [vech(Qt)′, ε∗′t ]′, such as [vech(Rt)′, ε′t ]′, admits a non-anticipative, strictly station-

ary, and ergodic solution (Billingsley 1995, Theorem 36.4). Since the elements of [vech(Rt)′, ε′t ]′

have finite variance, by Cauchy-Schwartz inequality the second moment of [vech(Rt)′, ε′t ]′ ex-

ists finite, which completes the proof of point (i). To prove point (ii), it suffices to note that

[vech(Ht)′,y′t , vech(Rt)′, ε′t ]′ is a time-invariant function of [h1,t, . . . , hN,t, vech(Rt)′, ε′t ]

′, which,

under H1-H4, is a measurable function of the non-anticipative, strictly stationary, and ergodic

process [vech(Rt)′, ε′t ]′. Under H1-H5, point (iii) follows by strict stationarity ofyt and Cauchy-

Schwartz inequality.�

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Proof of prop. 2.4. Under H1-H3 of prop. 2.3,ε∗t is weakly stationarity (Boussama et al. 2011,

Theorem 2.3). Ifε∗t is weakly stationary, the second moment ofε∗t is the matrixS in eq. (15)

(Engle and Kroner 1995). Sinceε∗t = Q∗1/2t εt , the proposition is proven.�

Proof of prop. 3.1. Under H1-H3 of prop. 2.3,ε∗t is weakly stationary and ergodic with station-

ary second momentS0 (see prop. 2.4). Hence, under H1-H3 of prop. 2.3, the matrix process

ε∗t ε∗′t , which is a time-invariant functions ofε∗t , is ergodic with finite first momentS0. This yields

plim T−1 ∑Tt=1 ε∗t ε

∗′t = S0, which proves the proposition in that, for (θ,φ) = (θ0,φ0), we have that

Q∗1/2t εt = ε∗t . �

Figure 1. DCC andcDCC Conditional Correlations. Simulated series ofρi j,t. The DGP parameter values are

reported at the top of the panel forα + β and on the right hand side of the panel forα. The location parameter is set as

si j = .3. DCC in straight line andcDCC in dashed line.

Figure 2. cDCC Integrated Conditional Correlations. Empirical distribution function of the absorbing time,t,

under student t innovations (x-axis in logarithmic scale). The straight, dashed and dotted lines correspond, respectively,

to d.f. equal to 3, 6, and 12.

Figure 3. Performance of thecDCC Estimator for IncreasingT. The figure provides an example of the behavior of

thecDCC estimator for increasing sample sizes. It is assumed thatθ andβ are known. The DGP is bivariate Gaussian

with parameters set as (s, α) = (.3, .4), wheres ≡ s12. The contour plots refer to the scaledcDCC QLL, which in

this case can be written asT−1LT(s, α). The dashed line refers to the constraint under which the scaledcDCC QLL

is maximized by thecDCC estimator (see section 3.2). Within the considered model, the constraint is a curve of the

plane, namely,{s = sα, α ∈ [0,1)}. For varyingα, the scaled objective function ofα, that is,T−1LT(sα, α), describes

the value of the scaledcDCC QLL along the constraint (the curve of the plane). ThecDCC estimator, ( ˆα, s), is denoted

with a bullet and the true value with a cross. The more observations, the more the scaledcDCC QLL centers on the

true value of the parameters, the more thecDCC constraint approaches a correctly specified constraint.

Figure 4. Estimation Error of ˆs. Box plots ofs− s. The DGP parameter values are reported at the top of the panel

for α + β, on the right hand side of the panel forα, and along thex-axis of each plot fors. For each box plot, the line

inside the box plot and the bullet denote, respectively, the median and the average of the estimates. The maximum

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length of the whiskers is set to the interquartile range. The percentages at the end of the whiskers refer to the outliers.

Figure 5. Estimation Error of ˆα. Box plots of (α − α)/α (for the layout of the panel and the construction of the

box plots, see the caption of Fig. 4). If there are estimates on the upper boundary, the related percentage is reported at

the top of the plot. In the presence of estimates on the upper boundary the end of the upper whisker can coincide with

the boundary, in which case the percentage of outliers is zero. Symmetric definitions hold for the percentages reported

below the lower whisker.

Figure 6. Estimation Error ofβ. Box plots of (β − β)/β (for the layout of the panel, the construction of the box

plots, and the percentages appearing in each plot, see the caption of Fig. 5).

Figure 7. Mean Absolute Error of ˆρt. (for the layout of the panel, the construction of the box plots, and the

percentages appearing in each plot, see the caption of Fig. 5).

Figure 8. Autocorrelation Function ofρi j,t. ACF computed as the mean of the sample ACF’s of 100 series of

lengthT = 100,000. The DGP parameter values are reported at the top of the panel forα + β, and on the right hand

side of the panel forα. The location parameter is set assi j = .3. DCC in straight line andcDCC in dashed line.

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Newey, W. K. and McFadden, D. (1986), “Large Sample Estimation and Hypothesis Testing,” inHandbook of Econo-

metrics, eds. Engle, R. F. and McFadden, D., New York: Elsevier Science, vol. 4, chap. 36, pp. 2111–2245.

Newey, W. K. and West, K. D. (1987), “A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation

Consistent Covariance Matrix,”Econometrica, 55, 703–708.

Pelletier, D. (2006), “Regime Switching for Dynamic Correlations,”Journal of Econometrics, 131, 445–473.

Pesaran, B. and Pesaran, M. (2007), “Modelling Volatilities and Conditional Correlations in Futures Markets with a

Multivariate t Distribution,” Cambridge Working Papers in Economics 0734, Faculty of Economics, University of

Cambridge.

Severini, T. A. (1998), “Likelihood Functions for Inference in the Presence of a Nuisance Parameter,” 85, 507–522.

Silvennoinen, A. and Terasvirta, T. (2009), “Modeling Multivariate Autoregressive Conditional Heteroskedasticity

with the Double Smooth Transition Conditional Correlation GARCH Model,”Journal of Financial Econometrics,

7, 373–411.

Tse, Y. K. and Tsui, A. K. C. (2002), “A Multivariate Generalized Autoregressive Conditional Heteroscedasticity

Model with Time-Varying Correlations,”Journal of Business& Economic Statistics, 20, 351–362.

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Engle, R. and McFadden, D., New York: Elsevier Science, vol. 4, pp. 2639–2738.

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Table 1: EPA Tests of One-step-ahead Forecasts

Small dataset LargedatasetLoss type CORR EW MV ∗MV CORR EW MV ∗MV

MSE -1.44 -0.17 -0.65 0.26 -2.41 -2.50 -1.82 -1.11SCORE -1.88 -1.31 0.28 0.12 -2.68 -1.43 -0.93 -2.46

NOTE: Negative (resp. positive) values are in favor of thecDCC (resp. DCC) estimator.Numbers in boldface denote significance at 5% level.

Table 2: Regression-based Tests Computed From One-step-ahead Forecasts

E & C Test DQ Test ARCH TestDataset Estimator EW MV ∗MV EW MV ∗MV EW MV ∗MV

Small cDCC 3.77 19.55 6.19 -0.67 -0.47 -0.50 0.59 3.72 3.53DCC 3.90 25.39 6.31 -0.67 -0.48 -0.50 0.68 4.63 3.53

Large cDCC 34.63 83.23 72.11 -0.25 -0.87 -0.77 4.28 2.84 1.32DCC 33.83 85.66 73.94 -0.23 -0.91 -0.75 3.92 2.24 1.37

NOTE: Numbers in boldface denote significance at 5% level.

Table 3: EPA Tests of Multi-step-ahead Correlation Forecasts

ForecasthorizonDataset EPA Test 2 3 5 8 13 21 34 55 89 144233

Small ρc vs. %c -1.31 -1.39 -1.45 -1.30 -0.88 -0.77 -0.69-2.47 -2.72 -2.61 -2.15ρ vs. % -1.41 -1.61 -1.71 -1.51 -0.51 0.48 2.41 0.46 -1.43 0.02 -1.92ρc vs. ρ -1.49 -1.51 -1.50 -1.39 -1.33 -1.36 -1.76-2.69 -2.64 -2.60 -2.15%c vs. % -1.51 -1.54 -1.56 -1.49 -1.52 -1.58-2.11 -2.72 -2.02 -2.11 -1.56

Large ρc vs. %c -2.19 -2.84 -2.88 -2.84 -2.45 -2.18-1.51 -1.17 0.42 1.14 1.44ρ vs. % -1.86 -2.69 -2.93 -2.99 -2.93 -2.91 -2.39 -2.29-0.60 -0.77 1.58ρc vs. ρ -2.31 -2.21 -2.16 -1.68 -1.71 -0.90 0.19 -0.07 0.79 0.98 1.63%c vs. % -2.25 -2.07 -1.93 -1.31 -1.43 -0.47 0.62 0.58 1.09 0.192.10

NOTE: Negative (resp. positive) values of the EPA test, “X vs. Y”, are in favor ofX (resp.Y). Numbers in boldface denotesignificance at 5% level. A superscriptc denotescDCC forecasts.

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0 200 400 600−.9

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.6

.9

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.3

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.9

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.3

.6

.9

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.3

.6

.9

t0 200 400 600

−.9

−.6

−.3

0

.3

.6

.9

t

α + β = .800 α + β = .990 α + β = .998α

=.01

α=

.04α

=.16

Figure 1: DCC andcDCC Conditional Correlations

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5,000 25,000 125,000 625,000

.2

.4

.6

.8

1α = .01

500 2,500 12,500 62,500

.2

.4

.6

.8

1α = .04

50 250 1,250 6,250

.2

.4

.6

.8

1α = .16

5 25 125 625

.2

.4

.6

.8

1α = .64

Figure 2:cDCC Integrated Correlation: student t innovations

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α

s

T = 2,000

.2 .25 .3 .35 .4 .45 .5.1

.15

.2

.25

.3

.35

.4

α

s

T = 4,000

.2 .25 .3 .35 .4 .45 .5.1

.15

.2

.25

.3

.35

.4

α

s

T = 8,000

.2 .25 .3 .35 .4 .45 .5.1

.15

.2

.25

.3

.35

.4

α

s

T = 16,000

.2 .25 .3 .35 .4 .45 .5.1

.15

.2

.25

.3

.35

.4

Figure 3:cDCC Estimator for IncreasingT

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−.9

−.6

−.3

0 .3 .6 .9

0%6%

.2%

3.6%

.6%

2%

.8%

.2%

2.8%

.8%

4%.6

%

5%0%

DCC

−.9

−.6

−.3

0 .3 .6 .9

.2%

5%

1.2%

3%

1%2.

4%

1.4%

.4%

1.8%

.8%

3.6%

1.2%

5.2%

.4%

cDCC

s

−.4

−.3

−.2

−.1

0

.1

.2

.3

.4

−.9

−.6

−.3

0 .3 .6 .9

.6%

3.2%

.4%

3.2%

1.2%

2%

1%1.

4%

3%2.

8%

1.6%

1.4%

3.4%

.2%

DCC

−.9

−.6

−.3

0 .3 .6 .9

1%2.

8%

1%3%

1%1.

6%

1.4%

1.4%

2.6%

2.2%

2.4%

1.8%

3.2%

.6%

cDCC

s

−.2

−.15

−.1

−.05

0

.05

.1

.15

.2

−.9

−.6

−.3

0 .3 .6 .9

2.6%

3.2%

1.8%

3.2%

2.6%

2%

3.4%

1.6%

1.6%

3%

2%1.

4%

3.8%

3%

DCC

−.9

−.6

−.3

0 .3 .6 .9

2.4%

2.8%

1.4%

2.8%

3.4%

2%

2.8%

1.8%

1.6%

2.8%

1.8%

1.6%

3.4%

2.6%

cDCC

s

−.1

−.05

0

.05

−.9

−.6

−.3

0 .3 .6 .9

0%12

.2%

0%4%

0%.4

%

.2%

.8%

2.2%

0%

5.2%

0%

11%

0%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%10

.6%

0%6.

4%

0%4.

2%

.8%

1.8%

5%0%

5.6%

0%

8.8%

0%

cDCC

s

−1

−.5

0

.5

1

−.9

−.6

−.3

0 .3 .6 .9

0%7.

8%

.2%

5.4%

.4%

3.6%

1.8%

.6%

3%.6

%

3.8%

0%

5.8%

0%

DCC

−.9

−.6

−.3

0 .3 .6 .9

1.2%

6.8%

.4%

4.2%

2.8%

5.6%

2%2.

2%

3.2%

.8%

4%.6

%

6.8%

1%

cDCC

s

−.4

−.3

−.2

−.1

0

.1

.2

.3

.4

−.9

−.6

−.3

0 .3 .6 .9

1.6%

2.4%

2.4%

2.2%

1.8%

1%

2.2%

2%

2.4%

1.4%

.6%

1.8%

1.6%

1.8%

DCC

−.9

−.6

−.3

0 .3 .6 .9

1.6%

3%

2%1.

8%

1.6%

1.4%

1.6%

2%

2%1.

6%

.6%

2.4%

2.2%

1.8%

cDCC

s

−.1

−.05

0

.05

.1

−.9

−.6

−.3

0 .3 .6 .9

0%4.

8%

0%2.

6%

0%1%

.2%

.8%

1.4%

0%

4.4%

0%

3.6%

0%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%8.

4%

1%7.

6%

1%4.

6%

3.6%

4.6%

5%1.

4%

6.8%

.2%

8.8%

0%

cDCC

s

−1.5

−1

−.5

0

.5

1

1.5

−.9

−.6

−.3

0 .3 .6 .9

0%3.

8%

.4%

2%

1.2%

2.6%

1.4%

1.4%

1.4%

.4%

2.2%

.2%

4%0%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%6.

8%

.2%

6.2%

1.2%

4.4%

2.4%

1.6%

3.2%

1.6%

4.4%

.2%

5.8%

0%

cDCC

s

−.8

−.6

−.4

−.2

0

.2

.4

.6

.8

−.9

−.6

−.3

0 .3 .6 .9

2.4%

4.8%

1.2%

4.4%

1.8%

1.4%

2.2%

3.4%

2%2%

2.2%

1.4%

2.6%

1.4%

DCC

−.9

−.6

−.3

0 .3 .6 .9

2.2%

5.4%

.6%

4%

1.8%

1.6%

1.2%

2.8%

3%2.

2%

2.8%

1%

3.6%

1.2%

cDCC

s

−.15

−.1

−.05

0

.05

.1

.15

α + β = .800 α + β = .990 α + β = .998

α=

.01α

=.04

α=

.16

Figure 4: Estimation Error of ˆs

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6.4%

−.9

7.2%

−.6

10.4

%−.

314

.4%

0

9% .3

9% .6

4.8%

.9

6.8%

6.8%

0%9.

6%

0%9.

6%

0%12

.4%

0%9.

2%

0%9.

8%

5.2%

7.2%

DCC

7%−.

97.

4%−.

69.

8%−.

314

.6%

0

9.6%

.3

10.2

% .6

4% .9

7.4%

6.2%

0%9.

2%

0%9%

0%11

.6%

0%9.

6%

0%10

.2%

4.4%

7.2%

cDCC

s

−1.5

−1

−.5

0

.5

1

1.5

2

2.5

7.2%

−.9

11.4

%−.

614

.8%

−.3

12.6

%

013

.6%

.3

7.4%

.6

9.2%

.9

0%10

.2%

0%9.

8%

0%9.

4%

0%9%

0%10

.8%

0%9.

6%

0%9.

4%

DCC

6.2%

−.9

11.6

%−.

615

.4%

−.3

12.6

%

014

.2%

.3

7.6%

.6

7.4%

.9

0%8.

4%

0%9%

0%9.

6%

0%8.

8%

0%10

%

0%9.

2%

0%8.

6%

cDCC

s

−2

−1

0

1

2

3

19.2

%−.

922

.4%

−.6

22.8

%−.

323

.2%

0

20.2

% .3

22%

.6

21.6

% .9

0%5.

4%

0%4.

4%

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6%

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0%5.

6%

DCC

19.2

%−.

922

.4%

−.6

22.6

%−.

322

.6%

0

20.8

% .3

23%

.6

21.4

% .9

0%5.

2%

0%3.

6%

0%5.

6%

0%5.

6%

0%4.

8%

0%7.

6%

0%5.

8%

cDCC

s

−2

0

2

4

6

−.9

−.6

−.3

0 .3 .6 .9

1.6%

2.8%

1.8%

2.8%

2.2%

2.6%

3.6%

3.8%

2.8%

4.8%

3.6%

1.8%

1.6%

2.4%

DCC

−.9

−.6

−.3

0 .3 .6 .9

1.2%

3.6%

3%2.

2%

2.6%

3.8%

4%4.

6%

3.8%

4.4%

2.8%

1.6%

2.6%

4.2%

cDCC

s

−.6

−.4

−.2

0

.2

.4

.6

−.9

−.6

−.3

0

.2%

.3 .6 .9

1.4%

3%

1%3%

1.4%

3.4%

2%4.

8%

3.8%

5.2%

1.8%

3.8%

2.6%

2%

DCC

−.9

−.6

−.3

.2%

0

.2%

.3 .6 .9

1.2%

3.8%

2.6%

3.8%

1%5%

2.2%

5%

3.4%

5.6%

2%4.

6%

2%2.

6%

cDCC

s

−.8

−.6

−.4

−.2

0

.2

.4

.6

.8

.4%

−.9

1.2%

−.6

.6%

−.3

1.6%

0

.8%

.3

.4%

.6

.2%

.9

2.4%

3.8%

0%4.

4%

0%5%

0%2.

8%

0%3.

4%

0%4.

2%

1.2%

4.8%

DCC

.4%

−.9

1.2%

−.6

.4%

−.3

1.4%

0

1% .3

.6%

.6

.2%

.9

2.2%

3.6%

4%6.

2%

0%5.

8%

0%3.

6%

0%3.

2%

0%4.

4%

1.4%

5.2%

cDCC

s

−1.5

−1

−.5

0

.5

1

1.5

2

−.9

−.6

−.3

0 .3 .6 .9

2.2%

2%

2.6%

2.6%

2.4%

1.4%

2.8%

2.8%

2.2%

2.4%

2.4%

3.2%

1%1.

6%

DCC

−.9

−.6

−.3

0 .3 .6 .9

2.6%

2.4%

2.4%

2.6%

3.6%

4.2%

.8%

2.4%

2.6%

3.6%

1%1.

2%

2.6%

2.8%

cDCC

s

−.4

−.2

0

.2

.4

−.9

−.6

−.3

0 .3 .6 .9

2.2%

2.8%

1.6%

3.4%

.8%

2%

2%1.

6%

1.4%

2%

2.4%

3.6%

3%3.

2%

DCC

−.9

−.6

−.3

0 .3 .6 .9

1.8%

2.6%

1.8%

3.6%

1.6%

2.4%

1.4%

3%

1%1.

4%

2.4%

4.6%

1%2.

8%

cDCC

s

−.4

−.3

−.2

−.1

0

.1

.2

.3

.4

−.9

−.6

−.3

0 .3 .6 .9

1.6%

2.8%

1.4%

2.2%

1.2%

1.8%

.4%

1.8%

1.2%

2%

3.6%

4.2%

1.8%

3.4%

DCC

−.9

−.6

−.3

0 .3 .6 .9

2.2%

1.8%

.6%

2%

1.2%

2.4%

1%3%

1.4%

2.4%

2.2%

3.4%

2.8%

3.4%

cDCC

s

−.6

−.4

−.2

0

.2

.4

.6

α + β = .800 α + β = .990 α + β = .998

α=

.01α

=.04

α=

.16

Figure 5: Estimation Error of ˆα

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7.4%

−.9

8.8%

−.6

13%

−.3

18.6

%

013

.2%

.3

10.8

% .6

6.2%

.9

16.2

%0%

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0%

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%0%

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23.2

%0%

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%0%

16.2

%0%

DCC

8%−.

910

%−.

612

%−.

319

.6%

0

15.4

% .3

12.6

% .6

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.9

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%0%

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%0%

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%0%

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%0%

cDCC

s

−1.2

−1

−.8

−.6

−.4

−.2

0

.2

9.2%

−.9

12%

−.6

17.4

%−.

318

%

017

.4%

.3

9.6%

.6

8% .9

20%

0%

20.4

%0%

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19.8

%0%

19.4

%0%

20.8

%0%

DCC

9.2%

−.9

12%

−.6

17%

−.3

17.8

%

018

.2%

.3

9.6%

.6

8.6%

.9

20.8

%0%

20.4

%0%

0%0%

0%0%

0%0%

20.4

%0%

20%

0%

cDCC

s

−1.2

−1

−.8

−.6

−.4

−.2

0

.2

21.6

%−.

923

.2%

−.6

25.6

%−.

326

.6%

0

21.4

% .3

23.4

% .6

22.2

% .9

0%0%

0%0%

0%0%

0%0%

0%0%

0%0%

0%0%

DCC

22%

−.9

22.6

%−.

625

.2%

−.3

28%

0

21.4

% .3

23.6

% .6

23%

.9

0%0%

0%0%

0%0%

0%0%

0%0%

0%0%

0%0%

cDCC

s

−1

−.5

0

.5

−.9

−.6

−.3

.2%

0 .3 .6 .9

3.6%

1.4%

4.6%

1.6%

5.8%

2.4%

5%2.

4%

3.6%

1.4%

3.2%

2.6%

3%1.

2%

DCC

−.9

−.6

−.3

.2%

0

.2%

.3 .6 .9

5.4%

1.4%

5%2.

2%

5%2%

5.6%

2.4%

4.2%

1.8%

4%3.

4%

5.4%

2.2%

cDCC

s

−.04

−.03

−.02

−.01

0

.01

.02

.03

−.9

.2%

−.6

.2%

−.3

.4%

0

.2%

.3 .6 .9

5%1.

4%

6.4%

1.4%

4.8%

.6%

6.8%

.6%

7.2%

2.2%

6%.8

%

4%1.

8%

DCC

−.9

.2%

−.6

−.3

.8%

0

.2%

.3 .6 .9

4.8%

1.8%

7%2%

5.8%

.8%

8.4%

1%

8%1.

6%

6.4%

1.6%

5.2%

2%

cDCC

s

−.06

−.04

−.02

0

.02

.04

5.2%

−.9

8.4%

−.6

9.4%

−.3

11.2

%

08.

8% .3

7% .6

3.8%

.9

11%

0%

14.2

%0%

13.2

%0%

13.4

%0%

13.2

%0%

11.2

%0%

10.8

%0%

DCC

5.2%

−.9

8.4%

−.6

10.4

%−.

312

.2%

0

8.4%

.3

7.2%

.6

3.8%

.9

10.2

%0%

13.2

%0%

13.6

%0%

13.8

%0%

12.8

%0%

11.4

%0%

11%

0%

cDCC

s

−1

−.5

0

.5

−.9

−.6

−.3

0 .3 .6 .9

3.6%

2.4%

3%3.

2%

2%2%

3.4%

2.6%

1.6%

1.8%

3%2.

2%

2.2%

1.4%

DCC

−.9

−.6

−.3

0 .3 .6 .9

2%2.

4%

4.2%

3.6%

5.2%

3.6%

2.4%

1.4%

3.4%

1.8%

2.6%

3.6%

3%2.

2%

cDCC

s

−.08

−.06

−.04

−.02

0

.02

.04

.06

.08

−.9

−.6

−.3

0 .3 .6 .9

3.2%

3.6%

3.2%

.8%

1.8%

1.4%

3%2.

2%

2%1.

6%

3.4%

1.6%

3.2%

2.2%

DCC

−.9

−.6

−.3

0 .3 .6 .9

2.4%

2.6%

3.6%

2.8%

2.2%

1%

4%2.

8%

2.4%

1.6%

4%1.

8%

2%1%

cDCC

s

−.08

−.06

−.04

−.02

0

.02

.04

.06

.08

−.9

−.6

−.3

0 .3 .6 .9

5.4%

1.8%

3%1%

4.6%

1.6%

5.6%

.4%

5%.6

%

2.8%

1.2%

3.6%

2%

DCC

−.9

−.6

−.3

.2%

0 .3 .6 .9

4.4%

2%

2.6%

1%

3.8%

2%

6.2%

.4%

5.2%

1%

2%1.

6%

4%1.

4%

cDCC

s

−.4

−.3

−.2

−.1

0

.1

.2

.3

.4

α + β = .800 α + β = .990 α + β = .998

α=

.01α

=.04

α=

.16

Figure 6: Estimation Error ofβ

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ACCEPTED MANUSCRIPT

−.9

−.6

−.3

0 .3 .6 .9

0%9.

2%

0%3.

4%

0%2.

2%

.8%

1.6%

0%2%

0%2.

6%

0%5.

8%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%7.

8%

0%3.

8%

0%2.

2%

.2%

1.8%

0%1.

8%

0%2.

2%

0%5.

2%

cDCC

s

−.02

0

.02

.04

.06

.08

.1

−.9

−.6

−.3

0 .3 .6 .9

0%2.

6%

0%2.

8%

.4%

3%

0%1.

4%

.2%

3.6%

0%2.

4%

0%3.

6%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%2%

0%2.

6%

.8%

3.2%

.2%

2%

.4%

4.4%

0%2%

0%4.

2%

cDCC

s

−.02

0

.02

.04

.06

.08

−.9

−.6

−.3

0 .3 .6 .9

0%4.

4%

0%4.

6%

0%3.

6%

0%4.

2%

0%3.

8%

0%3.

2%

0%5.

6%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%5%

0%4.

4%

0%3.

4%

0%4.

2%

0%4%

0%2.

8%

0%6%

cDCC

s

−.02

0

.02

.04

.06

.08

−.9

−.6

−.3

0 .3 .6 .9

0%13

.8%

.2%

9%

0%5%

.4%

3.6%

.2%

3%

0%11

.2%

0%15

.2%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%11

.2%

0%8%

0%4.

2%

.4%

3.8%

0%5.

2%

0%9.

8%

0%12

.6%

cDCC

s

−.02

0

.02

.04

.06

.08

−.9

−.6

−.3

0 .3 .6 .9

0%9.

8%

.2%

7.4%

0%5.

4%

0%3.

6%

.2%

3.4%

0%5.

6%

.4%

9.2%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%7%

0%6%

0%4.

4%

0%4%

0%4.

6%

.2%

6.8%

0%6.

4%

cDCC

s

−.02

0

.02

.04

.06

.08

−.9

−.6

−.3

0 .3 .6 .9

0%5.

2%

0%3%

0%3.

2%

.2%

3.8%

.6%

3.6%

.2%

3.6%

0%4%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%5.

4%

0%3.

4%

0%3.

2%

.4%

4.2%

.4%

4%

0%3.

2%

0%4.

8%

cDCC

s

−.02

0

.02

.04

.06

.08

−.9

−.6

−.3

0 .3 .6 .9

0%5%

0%5.

8%

.2%

2.8%

1%2.

6%

.2%

4.8%

0%5%

0%5.

6%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%12

.8%

0%7%

0%5.

8%

0%6%

0%6.

6%

0%6.

4%

0%8.

4%

cDCC

s

−.02

0

.02

.04

.06

.08

−.9

−.6

−.3

0 .3 .6 .9

0%10

.2%

0%10

%

0%5.

8%

0%3.

2%

0%4.

4%

0%9%

0%8.

6%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%8.

6%

0%5.

8%

0%5.

8%

0%6.

4%

0%4.

8%

0%4.

6%

0%10

%

cDCC

s

−.02

0

.02

.04

.06

.08

.1

−.9

−.6

−.3

0 .3 .6 .9

0%5.

6%

0%5.

2%

.2%

3%

.2%

5.4%

0%1.

8%

0%4%

.2%

5.6%

DCC

−.9

−.6

−.3

0 .3 .6 .9

0%5.

6%

0%4.

8%

.6%

4.8%

.2%

6.2%

0%3.

8%

0%5.

2%

0%5.

4%

cDCC

s

−.02

0

.02

.04

.06

.08

α + β = .800 α + β = .990 α + β = .998

α=

.01α

=.04

α=

.16

Figure 7: Mean Absolute Error of ˆρt

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ACCEPTED MANUSCRIPT

200 400 600 800−.2

0

.2

.4

.6

.8

1

Lags100 200

−.2

0

.2

.4

.6

.8

1

Lags10 20

−.2

0

.2

.4

.6

.8

1

Lags

200 400 600 800−.2

0

.2

.4

.6

.8

1

Lags100 200

−.2

0

.2

.4

.6

.8

1

Lags10 20

−.2

0

.2

.4

.6

.8

1

Lags

200 400 600 800−.2

0

.2

.4

.6

.8

1

Lags100 200

−.2

0

.2

.4

.6

.8

1

Lags10 20

−.2

0

.2

.4

.6

.8

1

Lags

α + β = .800 α + β = .990 α + β = .998α

=.01

α=

.04α

=.16

Figure 8: Autocorrelation Function ofρi j,t

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