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International Journal of Forecasting 27 (2011) 347–364www.elsevier.com/locate/ijforecast

Multivariate semi-nonparametric distributions with dynamicconditional correlations

Esther B. Del Brioa, Trino-Manuel Nıguezb,∗, Javier Perotec

a Department of Business and Finance, University of Salamanca, 37007 Salamanca, Spainb Department of Economics and Quantitative Methods, Westminster Business School, University of Westminster, London NW1 5LS, UK

c Department of Economics, University of Salamanca, 37007 Salamanca, Spain

Available online 1 September 2010

Abstract

This paper generalizes the Dynamic Conditional Correlation (DCC) model of Engle (2002), incorporating a flexible non-Gaussian distribution based on Gram-Charlier expansions. The resulting semi-nonparametric-DCC (SNP-DCC) model allowsestimation in two stages and deals with the negativity problem which is inherent in truncated SNP densities. We test theperformance of a SNP-DCC model with respect to the (Gaussian)-DCC through an empirical application of density forecastingfor portfolio returns. Our results show that the proposed multivariate model provides a better in-sample fit and forecast of theportfolio returns distribution, and thus is useful for financial risk forecasting and evaluation.c⃝ 2010 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

Keywords: Density forecasts; Financial markets; GARCH models; Multivariate time series; Semi-nonparametric methods

1. Introduction

In recent decades, the forecasting of macroeco-nomic and financial variables in terms of intervals(particularly one-side interval forecasts, e.g. Value-at-Risk (VaR hereafter)) or densities has become a ma-jor topic of research, triggered by the developmentof powerful models for time-varying conditional vari-ances and densities and the need to satisfy practi-tioners’ demands for more accurate risk measures.

∗ Corresponding author.E-mail addresses: [email protected] (E.B. Del Brio),

[email protected] (T.-M. Nıguez), [email protected](J. Perote).

In particular, Diebold, Gunther, and Tay (1998) pro-vided techniques for density forecasting that havebeen applied to the distribution of both economicand financial variables (see Mitchell & Wallis, 2009;and Tay & Wallis, 2000, for comprehensive reviewson density forecast applications and evaluation meth-ods, respectively). Furthermore, extensions of densityforecasting to the multivariate case were consideredby Clements and Smith (2000) and Diebold, Hahn,and Tay (1999), and, more recently, a strand ofthe literature has focused on proposing density fore-cast evaluation methods based on different criteria,such as the Kullback-Leibler information criterion(Bao, Lee, & Saltoglu, 2007), weighted likelihoodratio tests (Amisano & Giacomini, 2007), Bayesian

0169-2070/$ - see front matter c⃝ 2010 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.ijforecast.2010.02.005


348 E.B. Del Brio et al. / International Journal of Forecasting 27 (2011) 347–364

approaches (Geweke & Amisano, 2010) and strictlyproper scoring rules (Gneiting & Raftery, 2007).

On the other hand, the literature on multivariatevolatility modeling has undergone steady developmentsince the 1980s, including the following contributions:the vech model (Bollerslev, Engle, & Wooldridge,1988), the Constant Conditional Correlation (CCC)model (Bollerslev, 1990), the BEKK model (Engle& Kroner, 1995), and, more recently, the DynamicConditional Correlation (DCC) model (Engle, 2002;Engle & Sheppard, 2001) and the Dynamic Equicor-relation (DECO) model (Engle & Kelly, 2007). In par-ticular, the DCC model was introduced to allow fortime-varying correlations, and, at the same time,deal with the well-known “dimensionality curse”of the multivariate context, by means of separatequasi-maximum likelihood estimation (QMLE) of theconditional variances and correlation dynamics.Unfortunately, the DCC model two-step estimationprocedure has so far been shown to be theoreticallyvalid only under normality. In spite of this, DCCprocesses have also been implemented in empiri-cal works using the Student-t distribution, i.e., us-ing the first stage QMLE of conditional variances,followed by the second stage MLE of the corre-lation processes and Student-t distribution parame-ters. Bauwens and Laurent (2005) and Jondeau andRockinger (2005) showed, by means of an em-pirical application, that although the decompositionproposed by Engle (2002) is not formally possi-ble for the Student-t distribution and a one-stepapproach should be adopted (i.e., a joint MLE of theconditional variance and covariance processes underthe Student-t), the estimation results from the one- andtwo-step approaches did not differ significantly.

In this paper we apply density forecasting tech-niques in a multivariate framework to test the per-formance of a generalization of Engle’s (2002) DCCmodel which can incorporate not only volatility clus-tering and time-varying correlations but also all ofthe non-normal stylized features of high frequencyfinancial variables, i.e. skewness, leptokurtosis, mul-timodality, etc. This new model, which we call semi-nonparametric-DCC (SNP-DCC hereafter), specifiesa general and flexible multivariate density based ontruncated Edgeworth and Gram-Charlier series. Theseseries, originally defined in the early 20th century(Edgeworth, 1907), have recently been the subject of

renewed interest in financial econometrics for deal-ing with the unsolved problem of fitting the heavy-tailed distribution of high-frequency asset returns —see for example Gallant and Tauchen (1989), Leon,Mencıa, and Sentana (2009) and Mauleon and Per-ote (2000). Nevertheless, as far as we know, this SNPframework has only been extended to the multivariatecontext for financial purposes by Perote (2004) andDel Brio, Nıguez, and Perote (2009). They showedthat multivariate SNP modeling produces flexible mul-tivariate densities that provide accurate fits to financialreturns data, but also that the densities implemented inthose papers assume constant correlations, which, al-though it eases their implementation for large portfo-lios, may be too restrictive. Furthermore, the so-calledmultivariate Edgeworth-Sargan (MES hereafter) dis-tribution introduced by Perote (2004) cannot strictlybe considered a probability density function (pdf here-after), since it is not positive for all values of the pa-rameters in the parametric space, and therefore its ap-plications, e.g. for forecasting, are limited.

In this article, we jointly analyze the aforemen-tioned problems, namely, the negativity probleminherent to truncated SNP distributions, and the multi-variate framework “curse of dimensionality”— specif-ically when the model accounts for time-varyingcorrelations — through a new family of multivari-ate SNP distributions. This family has the followingfeatures: (i) generality: it encompasses as marginalsnot only the Gaussian but also the different uni-variate Edgeworth and Gram-Charlier distributionsproposed in the literature; (ii) positiveness: it is pos-itive for all of its parameter values in the paramet-ric space; (iii) empirical tractability: it theoreticallyadmits the decomposition proposed by Engle (2002),allowing us to obtain consistent MLE (under a cor-rect specification), which eases the model implemen-tation; and (iv) it yields a reasonable out-of-sampleperformance for forecasting the density of portfolio re-turns, as measured using the probability integral trans-formation (PIT) paradigm (Diebold et al., 1999) andscoring rules for ranking density forecasting methods(Amisano & Giacomini, 2007).

Our SNP-DCC model is inspired by the fact that, tothe best of our knowledge, none of the aforementionednon-Gaussian multivariate approaches formally allowsthe two stage estimation proposed by Engle (2002).The solution provided by our approach in relation to


E.B. Del Brio et al. / International Journal of Forecasting 27 (2011) 347–364 349

the latter point, together with its other features, (i),(ii) and (iv) above, makes it a useful tool for financialeconometrics research and applications.

The remainder of the article is divided into thefollowing four sections. In Section 2, we present themethodology for defining multivariate SNP densities,and discuss their properties. Section 3 describes a par-ticular SNP-DCC model which is of interest for fi-nancial applications. Section 4 provides an empiricalapplication to two bivariate portfolios composed of USstock returns, an exchange rate index and a stock re-turns index, to test the in- and out-of-sample perfor-mances of the proposed model. Finally, Section 5 sum-marizes the main conclusions.

2. Multivariate SNP distributions

Let xt = (x1t , x2t , . . . , xnt )′

∈ Rn be a randomvector distributed with zero mean, uncorrelatedvariables and joint pdf Fζ (xt ; γ ),

Fζ (xt ; γ ) =1n

n∏

i=1

g(xi t )

n−

i=1

wζi qζ

i (xi t )

=(2π)−n/2

nexp

−

x′t xt

2

n−

i=1

wζi qζ

i (xi t )

(1)

qζi (xi t ) =

1 +

m−s=2

γis Hs(xi t ) if ζ = I ,

1 +

m−s=2

γ 2is H2

s (xi t ) if ζ = II,(2)

where g(·) denotes a standard Gaussian density, qζi (·)

represents the Hermite polynomial expansion for thei th variable (i = 1, 2, . . . , n), Hs(·) is the Hermitepolynomial of order s, m is the truncation order which,without loss of generality, is considered to be the samefor all i , and w

ζi are the constants which ensure that the

marginal densities integrate to one.This distribution type involves different specifica-

tions depending on the structure of the assumed Her-mite polynomials. Specifically, in this article we com-pare two different multivariate SNP alternatives, de-noted by the index ζ = I, II. These distributions,known as SNPI and SNPII , are inspired by the dis-tributions of Perote (2004) and Del Brio et al. (2009),respectively. However, in contrast with those distribu-tions (which consider a combination of a multivariate

Gaussian density with a non-diagonal covariance ma-trix and a term that incorporates the Hermitian expan-sion for every variable), SNPI and SNPII are initiallydefined in terms of uncorrelated variables (i.e., corre-lation is introduced by means of linear restrictions).Specifying the multivariate SNP density in these termsis the key to achieving the separability of the log-likelihood function, as proposed by Engle (2002), andallows us to formally implement two-step maximumlikelihood estimation, as is discussed in the next sec-tion.

The SNPI distribution has a simpler structure thanthe SNPII , but, unlike the latter, it does not guaranteepositivity for all values of the density parameters, γis ,∀i = 1, 2, . . . , n and ∀s = 2, . . . , m. The Hermitepolynomials, Hs(·), can be defined in terms of the sthorder derivative of the Gaussian pdf as

ds g(z)dzs = (−1)s g(z)Hs(z). (3)

Hs(·) satisfy, among others, the orthogonality proper-ties given in Eqs. (4)–(6) below (see Kendall & Stuart,1977, for further details on the properties of Hermitepolynomials):∫

Hs(z)g(z)dz = 0, ∀s > 0, (4)∫Hs(z)H j (z)g(z)dz =

0,

s!,∀s = j,∀s = j, (5)∫

Hs(z)2 H j (z)2g(z)dz = s! j !, ∀s = j. (6)

Under these orthogonality conditions, it is straightfor-ward to show that multivariate SNPζ (ζ = I, II) den-sities satisfy the following properties:

Property 1. The constants wζi weighting qζ

i (·) are (bydirect application of Eq. (5))

wζi =

[∫g(xi t )q

ζi (xi t )dxi t

]−1

=

1 if ζ = I ,

1 +

m−s=2

γ 2iss!

−1

if ζ = II.(7)

Property 2. Multivariate SNPζ (ζ = I, II) integrateto one (see Proof 1 in the Appendix).



Property 3. The marginal densities of the multivari-ate SNPI are the standard Gram-Charlier approx-imations, whilst the marginals of the multivariateSNPII are mixtures of univariate normal and PositiveEdgeworth-Sargan (PES hereafter) (Nıguez & Per-ote, 2004), as shown in Eq. (8) (see Proof 2 in theAppendix).

f ζi (xi t ) =

g(xi t )

1 +

m−s=2

γis

nHs(xi t )

,

if ζ = I ,

g(xi t )

[n − 1

n+

1nwII

i qIIi (xi t )

],

if ζ = II.

(8)

Property 4. All order moments of SNPI and SNPIIexist and can be obtained in terms of the standardnormal density moments (µr∀r ∈ N), as displayed inEq. (9),

Eζxr

i t

=

µr +

r−j=0

j !c jγi j

n, if ζ = I , ∀r ∈ N,

(n − 1)wIIi + 1

µr +

m∑s=2

γ 2iss!

r/2∑j=0

j !d j

nwII

i,

if ζ = II, ∀r ∈ N even,

(9)

where c j rj=0 is a sequence of constants such that

xri t =

∑rj=0 c j H j (xi t ), e.g.,

xri t =

H1(xi t ), if r = 1H2(xi t ) + 1, if r = 2H3(xi t ) + 3H1(xi t ), if r = 3,

(10)

and d j r/2j=0 is another sequence of constants such that

xri t =

∑r/2j=0 d j H j (xi t )

2, e.g.,

xri t =

H1(xi t )

2, if r = 2H2(xi t )

2+ 2H1(xi t )

2− 1, if r = 4

H3(xi t )2+ 6H2(xi t )

2+ 3H1(xi t )

2− 6, if r = 6.

(11)

Proof 3 in the Appendix proves this result by assuming(without loss of generality) that r < m.

Property 5. The SNP cumulative distribution func-tions (cdf hereafter) can easily be computed by meansof Eqs. (12) and (13) (see Proof 4 in the Appendix).

Pr[x1 ≤ a1, . . . , xn ≤ an]ζ

=1n

n−i=1

Ψ ζ (ai )

×

n∏

j=1, j=i

Φ(a j )

, ∀ζ = I, II, (12)

where Φ(·), Ψ I (·) and Ψ II(·) stand for the cdfs of thestandard normal, Gram-Charlier and PES univariatedistributions, respectively.

Ψζ (ai )=

Φ(ai ) − g(ai )m−

s=2γis Hs−1 (ai ) , if ζ = I ,

Φ(ai ) − wi g(ai )m−

s=2γ 2

is

×

s−1−l=0

s!Hs−l (ai )Hs−l−1(ai )

(s − l)!, if ζ = II.

(13)

Properties 1–5 support the multivariate SNP as avery flexible (i.e., it can capture skewness, leptokur-tosis, multimodality and most of the high frequencyfinancial features through its general parametric struc-ture) and easy to implement distribution (e.g., mo-ments, probabilities and quantiles can be computedin a straightforward manner). Nevertheless, in spiteof these interesting properties, this distribution wouldcertainly be more useful if correlations among vari-ables were incorporated. To do so, we transform thevector xt such that the transformed variable, ut =

Σ1/2t xt , has zero mean and variance-covariance ma-

trix Σ t , with Σ t = Σ1/2t Σ

1/2t being the symmet-

ric spectral decomposition, i.e. Σ1/2t = CtΛ

1/2t C′

t ,where Λt = diag λ1t , λ2t , . . . , λnt is the diagonalmatrix of the eigenvalues of Σ t , and Ct is the cor-responding orthogonal matrix of eigenvectors of Σ t .Note that this decomposition of the matrix Σ t , whichis always possible for symmetric and positive definitematrices, yields the product of two identical sym-metric matrices, unlike either the Cholesky decom-position (in terms of triangular matrices) or the “nonsymmetric” eigenvector decomposition (Σ 1/2

t = Ct

Λ1/2t ). Furthermore, note that if the variance-

covariance matrix of xt is given by k2t = diag

k2

1t ,

k22t , . . . , k2

nt, then the vector xt can be standard-

ized by the following transformation: x∗t = k−1

t xt . Ifthis is the case, the variance-covariance matrix of ut

= Σ1/2t x∗

t can be interpreted in terms of the ma-trix Σ t . However, if the focus of the empirical anal-ysis is on forecasting, the latter transformation isunnecessary.

Alternatively, we can write ut = Dt R1/2t xt , since

Σ t can be decomposed in terms of the diagonal



matrix of conditional standard deviations, Dt =

diagσ1t , . . . , σnt , and the correlation matrix, Rt , asgiven below:

Σ t = Dt Rt Dt = Dt R1/2t R1/2

t Dt . (14)

Then, ut is distributed according to a multivariate SNPdistribution, whose pdf is:

Fζ (ut ; γ ) = (2π)−n2 |Σ t |

−12 exp

−

12

u′tΣ

−1t ut

×

n−

i=1

wζi qζ

i

Σ

−12

t ut

1n, ∀ζ = I, II. (15)

Analogously, the SNP density can be re-written interms of εt = D−1

t ut as

Fζ (εt ; γ ) = (2π)−n2 |Rt |

−12 exp

−

12ε′

t R−1t εt

×

n−

i=1

wζi qζ

i

R−

12

t εt

1n, ∀ζ = I, II. (16)

In particular, for the bivariate case (i = 1, 2), theinverse transformation xt = R−1/2

t εt can easily bewritten as a function of the standardized variablesεi t =

ui tσi t

and the time-varying correlation ρt , as:

xi t =

12

1

√1 + ρt

+1

√1 − ρt

ε1t

+12

1

√1 + ρt

−1

√1 − ρt

ε2t , if i = 1,

12

1

√1 + ρt

−1

√1 − ρt

ε1t

+12

1

√1 + ρt

+1

√1 − ρt

ε2t , if i = 2.

(17)

3. The multivariate SNP-DCC model

Let rt be an n × 1 vector of asset returns condi-tionally distributed (on the information set Ω t−1) ac-cording to a multivariate SNP, with conditional firstand second moments defined by µt (φ) and Σ t (θ) =

Dt (α)Rt (ρ)Dt (α), respectively, where φ, θ , α and ρ

are parameter vectors. Following Engle’s (2002) DCCmodel specification, the matrices Dt (α) and Rt (ρ) aremodelled as displayed in Eqs. (21) to (24), with Eq.(23) being the MARCH family of Ding and Engle(2001), where S is the unconditional correlation ma-trix, ξ is a vector of ones, A, B and ξξ ′

− A − B

are positive definite matrices, and is the Hadamardproduct of two identically sized matrices (computedby element-by-element multiplication).

rt = µt (φ) + ut , (18)ut |Ωt−1 ∼ SN P(0,Σ t (θ)), (19)

Σ t (θ) = Dt (α)Rt (ρ)Dt (α)

= Dt (α)R1/2t (ρ)R1/2

t (ρ)Dt (α), (20)

D2t = diagαi0 + diagαi1 ut−1u′

t−1+ diagαi2 D2

t−1, (21)

εt = D−1t ut , (22)

Qt = S (ξξ ′− A − B) + A εt−1ε

′

t−1+ B Qt−1, (23)

Rt = diagQt −1/2Qt diagQ

−1/2. (24)

This model, named SNP-DCC, encompasses the(Gaussian)-DCC, and presents a log-likelihood which(like that of the (Gaussian)-DCC) can be split intotwo different components: (i) the mean-volatility part,L MV (rt , η), where η =

φ′, α′

′, and (ii) the “stan-dardized” SNP component, Lζ

SN P (εt , η, ϕ), whereϕ = (ρ′, γ ′)′ includes the correlation and the shapeparameters (see Proof 5 in the Appendix). Specifically,these two terms can be written (after deleting the un-necessary constants) as

L MV (rt , η) = −12

n−i=1

[T ln(2π)

+

T−t=1

ln(σ 2

i t ) +

ri t − µi t

2

σ 2i t

], (25)

LζSN P (εt , η, ϕ) =

T−t=1

ln

n−

i=1

wζi qζ

i

R−1/2

t εt

−12

ln |Rt | + ε′

t R−1t εt

∀ζ = I, II. (26)

Thus, the SNP-DCC model can be estimated byML in two steps as follows: first, η is obtained bymaximizing L MVi (ri t , η) independently for each i =

1, 2, . . . , n (note that this step assumes normality); andsecond, the conditional correlation parameters and themultivariate SNP shaping parameters, ϕ, are jointlyestimated in the log-likelihood function concentratedwith respect to η = arg max L MV (rt , η), i.e.,



LζSN P (εt ,η, ϕ). As was pointed out by Engle

(2002), this two-step estimation problem can beinterpreted in terms of the joint Generalized Method ofMoments estimation framework discussed by Neweyand McFadden (1994, Section 6.1). In particular,if the conditions 1

T∑T

t=1 ∇ηL MV (rt , η) = 0 and1T

∑Tt=1 ∇ϕ Lζ

SN P (εt ,η, ϕ) = 0 are satisfied with a

probability approaching one, η P→ η0, ϕ P

→ ϕ0, and,under the regularity conditions (i)–(v) of Theorem3.4 of Newey and McFadden (1994), η and ϕ areasymptotically normal, and√

Tϕ − ϕ0

d→ N (0, V) , (27)

V =

E

∇ϕϕ Lζ

SN P

−1

× EHH′

E

∇ϕϕ Lζ

SN P

−1, (28)

H = ∇ϕ LζSN P − E

∇ϕηLζ

SN P

×

E

∇ηηL MV

−1∇ηL MV , (29)

with ∇η(·) and ∇ηϕ(·) denoting the gradient vectorof first derivatives with respect to the variables in η

and the Hessian matrix of the second derivatives withrespect to η and ϕ, respectively (see Theorem 6.1 ofNewey & McFadden, 1994, and its application to theDCC model by Engle, 2002, and Engle & Sheppard,2001, for further details). It must be noted that inthe context of the SNP-DCC, the first step is QMLE,but the second step is MLE, since the likelihoodcorresponds to the SNP density. Nevertheless, weargue that the second-step MLE is also consistentand asymptotically normal as n → ∞, sincethe multivariate (infinite) SNP expansion is almostsurely the “true” density (Gallant & Tauchen, 1989).Moreover, the asymptotic variances of the GARCHparameters are the robust variance and covariancematrix of Bollerslev and Wooldridge (1992), but theasymptotic variance and covariance matrix of theestimates of the second step parameters involvesa more complex structure (see Engle & Sheppard,2001). Furthermore, the estimates of the SNP-DCCparameters are not fully efficient, since they areestimated by Limited Information MLE, although theestimates of the SNP-DCC model are more efficientthan those of the DCC models, provided that γ = 0.

On the other hand, it must be noted that, inpractice, the consistency of the second step cannot be

guaranteed for the SNP-DCC, since the Gram-Charlierexpansions need to be truncated. Nonetheless, weargue that the density misspecification error can beminimized by selecting the truncation order of theSNP density according to Wald specification testsor information criteria, for instance. In any case,the two-step SNP-DCC is consistent, provided thatthe SNP distribution is correctly specified, unlikevarious other non-Gaussian distributions which arewidely used to model departures from normality, forwhich the consistency of the DCC two-step estimationcannot be formally proven, even under a correctdensity specification, since the log-likelihood is notseparable (see Bauwens & Laurent, 2005; Jondeau& Rockinger, 2005; Newey & Steigerwald, 1997). Afurther discussion of the estimation procedure for atruncated SNP-DCC model is provided in Section 4.

Finally, Fig. 1 plots the bivariate Gaussian densityand the allowable shapes of a bivariate symmetricSNPII density for different values of its parameters(as orientating illustrations, we display the cases of(γi4, γi6) = (0, 0), (γi4, γi6) = (0.1, 0.01) and(γi4, γi6) = (0.5, 0.05)). The figures in the leftcolumn highlight the tail shape of the densities. It isstriking to see the flexibility presented by the SNPdensity, which is an advantage in fitting any empiricaldistribution, especially in the tails (see also Del Brioet al., 2009, for further examples of the flexibility ofother SNP-related multivariate densities).

4. An empirical application to portfolio returns

4.1. Data, estimation and in-sample analysis

In this section, we investigate the empirical per-formances of the models discussed above through anin- and out-of-sample comparative analysis. To do so,we consider three types of distributions with eitherconstant or dynamic conditional correlation, whichformally allow two-step estimation procedures. Themodels that we use are the following: Gaussian (CCCand DCC), non-positive-SNP (SNPI -CCC and SNPI -DCC), and positive-SNP (SNPII-CCC and SNPII-DCC). The data used are (daily) percentage logreturns, computed from observed samples of (daily)asset prices, ri t = 100 ln (Pi t/Pi t−1), of two (repre-sentative) stocks from the Dow Jones index (AT&T



Figure 1.1: Gaussian Figure 1.2: Gaussian

0.08

0.06

0.04

0.02

0

0.08

0.06

0.04

0.02

0

0.08

0.06

0.04

0.02

0

55

0

0

0-5

-5

-5

0.1

-10-8

-6-4

-20

-10-8

-6-4

-20

-10-8

-6-4

-20

-12

0.06

0.05

0.04

0.03

0.02

0.01

0

0.04

0.03

0.02

0.01

0

10105

50

0

0-5

-5-5

-10

10105

500-5 -5

-10 -10

-10

0

-5

-10

-10

Figure 1.3: SNP, gamma(i4) = 0.1, gamma(i6) = 0.01 Figure 1.4: SNP, gamma(i4) = 0.1, gamma(i6) = 0.01

Figure 1.5: SNP, gamma(i4) = 0.5, gamma(i6) = 0.05 Figure 1.6: SNP, gamma(i4) = 0.5, gamma(i6) = 0.05

0.03

0.025

0.02

0.015

0.01

0.005

0

Fig. 1. Plots of bivariate Gaussian and SNP distributions.



–

–

Fig. 2. Plots of daily log returns.

and JP Morgan), the Euro (e)/British Pound (£) ex-change rate (FX e /£), and the Nasdaq index. Weconsider two bivariate portfolios: Portfolio AT&T-JPMorgan, and Portfolio FX e /£-Nasdaq, with returnsdenoted as rt = (r1t , r2t ). All series are sampled overthe period August 17, 1995, to December 17, 2007,for a total of 3218 observations. The data were ob-tained from Datastream. Fig. 2 displays the plots ofthe four return series for the full sample. The shadedareas in the plots correspond to the out-of-sample pe-riod (February 17, 2004, to December 17, 2007, for atotal of N = 1000 observations) used for the forecast-ing analysis in the next section.

The estimation is carried out in two stages by(Q)MLE techniques, using a moving in-sample win-dow of a constant size, T = 2217 observations. Inthe first stage, an AR(1) process for the conditionalmean (selected according to the Akaike InformationCriterion (AIC hereafter)) and a GARCH(1, 1) pro-

cess for the conditional variance are jointly estimated(under normality) independently for each asset. Then,in the second stage, the standardized residuals fromthe previous step are used to estimate the conditionalcorrelation equation and the rest of the density pa-rameters. The first stage yields QMLE, which is con-sistent and asymptotically normal, although not ef-ficient. Our second stage MLE is not a priori con-sistent, since we use a truncated SNP density, but itmay be more efficient than the second step QMLEof the Gaussian-DCC. Thus, we argue that the betterour truncated SNP density can approximate the “true”distribution, the more efficient our second stage MLEis. Finally, following Pagan (1986), a further Newton-Raphson iteration without line search for the jointmodel is performed from the two-stage (Q)MLE: theestimates do not change, but the information matrix isnow block diagonal, thus obtaining estimators whichare asymptotically equivalent to joint QMLE. Robust



standard errors were computed following Bollerslevand Wooldridge (1992). Taking into account the lim-itations in the estimation of the truncated SNP-DCCmodel, the model performance is mainly evaluatedin an out-of-sample forecasting framework, in whichpossible losses in efficiency and consistency are notcrucial (see Ruiz & Pascual, 2002).

We observe that the estimation of the SNP modelsis not very demanding computationally, provided thatthe starting values are chosen properly. As it is knownthat SNP densities may have multiple local modes, theoptimization is monitored using different starting val-ues, to ensure that the (Q)MLE we obtain are globaloptima.

Table 1 presents the estimation results of thebivariate models. αiτ , i = 1, 2 and τ = 0, 1, 2, denotethe parameters of the GARCH(1, 1) models used forthe conditional variances, φi0 and φi1 are the interceptand the slope of the AR(1) process for the conditionalmean, respectively, and ρ is the correlation parameterin CCC models. The other parameters displayed in thetable follow the notation used in previous sections.Robust standard errors are in parentheses next to theparameter estimates.

For the specification of the SNP models, westarted by considering densities truncated at the eighthmoment, then persistently (across windows) removednon-significant parameters. In Table 1 we presentthe final SNP specifications. For both portfolios,the estimated SNP densities are unconditionallysymmetric, since the odd parameters, γis (s = 3, 5, 7),are not significant at any reasonable significance level.For Portfolio AT&T-JP Morgan, the even parametersof the SNPI , γ12 and γ16, were not statisticallysignificant, although they were not removed fromthe model for the sake of clear comparisons acrossSNP models. The SNPII models have three significanteven parameters for the first asset (γ1s , s = 2, 4, 6)and two significant even parameters for the secondasset (γ2s , s = 4, 6). For Portfolio FX e /£-Nasdaq,the existing leptokurtosis in the distribution of thefirst data window is gathered mainly by γ14, γ24and γ26 in the SNPI models, and by γ14 and γ26in the SNPII models. It is interesting to note thatthe estimates γi4 and γi6 obtained for both portfoliosare smaller than those considered in Figs. 1.3 and1.4, and therefore the evidence of multimodality inour data is mild. Nonetheless, the heavy tails of

the returns distribution do not necessarily decreaseuniformly, as they would do under the commonlyassumed parametric distributions in most financialeconometrics applications.

According to the AIC, which is defined as AIC =

2(ϑ − ln L)/n (with ϑ being the number of parametersin the model), we observe that SNP models providea noticeably better goodness-of-fit than Gaussianmodels, and that dynamic conditional correlationhelps the models to fit the data. Furthermore, amongthe SNP models, SNPI are preferred to their SNPIIcounterparts. Regarding the conditional mean andvariance estimates, we observe the usual smallstructure in the conditional mean and high persistencein the conditional variance, such as is observed inthe conditional correlation (in line with the resultsof Engle, 2002, and Engle & Sheppard, 2001, forUS stock returns). We also observe that the estimatesobtained for the unconditional correlation coefficient,ρ, given by the CCC and SNP-CCC models, are closeto the sample correlations.

4.2. Density forecasting

In this section we test the performance of theSNPII-DCC model relative to that of the DCCmodel for full density forecasting. The forecasts areproduced using a rolling window of size N thatdiscards old observations. The recursive optimizationis monitored using the same starting value for allwindows, instead of using the usual optimum fromthe previous data window. This mechanism is usedin order to avoid getting trapped in successive localoptima. On the other hand, it is worth noting thatalthough the SNPI -DCC is, in principle, a candidatefor providing a good forecasting performance, it is nota suitable model to use in a forecasting experimentthat involves rolling windows of data over a longperiod of time, since a combination of parameterestimates during the optimization process for a givenwindow may lead to a negative value of the density.In this paper we have addressed the problem bydefining the SNPII distribution for which positivityis guaranteed through a reformulation of the densityfunction. Furthermore, we consider models with DCCstructures, given their better in-sample fit relativeto their CCC counterparts (as shown in Table 1).Thus, SNPII-DCC and (Gaussian)-DCC models are



Table 1Estimation results.

CCC DCC SNPI -CCC SNPI -DCC SNPII -CCC SNPII -DCC

Panel 1: Portfolio AT&T — JP Morgan

Stage 1φ10 0.0343 (1.21)φ11 −0.0494 (−2.23)α10 0.0362 (1.65)α11 0.0577 (3.08)α12 0.9360 (45.1)φ20 0.0990 (2.08)φ21 0.0343 (1.61)α20 0.0318 (1.66)α21 0.0551 (4.53)α22 0.9407 (69.5)

Stage 2γ12 0.0037 (0.12) 0.0010 (0.07) 0.0913 (1.63) 0.0888 (1.59)γ14 0.0912 (6.43) 0.0914 (6.55) 0.0281 (3.57) 0.0267 (3.15)γ16 0.0012 (0.54) 0.0013 (0.55) 0.0021 (2.03) 0.0020 (1.56)γ22 0.0001 (0.11) 0.0001 (0.24) 0.0001 (0.03) 0.0001 (0.73)γ24 0.0902 (6.71) 0.0904 (6.71) −0.0293 (−4.95) −0.0283 (−4.79)γ26 0.0062 (2.43) 0.0060 (2.31) 0.0031 (3.12) 0.0031 (3.13)ρ 0.2859 (13.5) 0.2904 (14.1) 0.3240 (14.4)δ1 0.0077 (4.19) 0.0077 (4.27) 0.0085 (4.65)δ2 0.9877 (329) 0.9880 (348) 0.9880 (365)AIC 1.9137 1.9043 0.4539 0.4446 0.4798 0.4727

Panel 2: Portfolio FX e /£— Nasdaq

Stage 1φ10 0.0072 (0.73)φ11 −0.0689 (−3.02)α10 0.0035 (1.09)α11 0.0382 (2.21)α12 0.9474 (33.1)φ20 0.1063 (2.34)φ21 −0.0387 (−1.83)α20 0.0571 (1.67)α21 0.0659 (3.37)α22 0.9246 (39.7)

Stage 2γ12 −0.0114 (−0.33) −0.0115 (−0.33) −0.0001 (−0.03) 0.0001 (0.13)γ14 0.0647 (4.43) 0.0649 (4.44) 0.0270 (4.58) 0.0269 (4.55)γ16 0.0037 (1.37) 0.0037 (1.37) 0.0011 (0.64) 0.0012 (0.71)γ22 −0.0229 (−0.73) −0.0234 (−0.77) −0.0001 (−0.05) 0.0001 (0.10)γ24 0.0235 (2.10) 0.0227 (2.02) 0.0059 (0.31) −0.0003 (−0.11)γ26 0.0039 (2.00) 0.0039 (1.99) 0.0029 (3.43) 0.0031 (4.07)



Table 1 (continued)

CCC DCC SNPI -CCC SNPI -DCC SNPII -CCC SNPII -DCC

ρ 0.0642 (2.78) 0.0591 (2.66) 0.0612 (2.54)δ1 0.0090 (1.93) 0.0075 (1.12) 0.0091 (1.63)δ2 0.9748 (68.3) 0.9802 (38.4) 0.9768 (53.1)AIC 1.9924 1.9901 0.5837 0.5810 0.5858 0.5833

Mean equation: ri t = φi0 + φi1ri,t−1 + ui t , ui t = εi t σi t , i = 1, 2.Variance equation: σ 2

i t = αi0 + αi1u2i,t−1 + αi2σ 2

i,t−1.Correlation equation: ρt = (1 − δ1 − δ2)ρ + δ1ε1t−1ε2t−1 + δ2ρt−1.Notes: The coefficients presented in this table are (Q)ML estimates of the CCC, DCC, SNP-CCC and SNP-DCC models, for the bivariateportfolios’ returns. φis (s = 0, 1) stand for the AR(1) parameters of the conditional means and αis (s = 0, 1, 2) for the GARCH(1, 1)parameters of the conditional variances. ρ denotes the unconditional correlation parameter in CCC models and δs (s = 1, 2) the conditionalcorrelation parameters in DCC models. γis (s = 2, 4, 6) denote the order s polynomial weighting parameter in the SNP models. t-statisticscalculated from robust standard errors are in parentheses. Values of the Akaike Information Criterion (AIC) are displayed in the last row.

compared with respect to their density forecastingperformances using the following criteria:

1. The PIT paradigm (Diebold et al., 1998, 1999).The PIT paradigm establishes that if

r∗

i tT +N

t=T +1is a series of realizations generated from a seriesof conditional densities, f (ri t |Ωt−1)

T +Nt=T +1 =

fi,t−1(ri t )T +N

t=T +1, and fi,t−1(ri t )

T +Nt=T +1 is a

series of one-step-ahead density forecasts, thenthe series of PIT of

r∗

i tT +N

t=T +1 with respect to fi,t−1(ri t )T +N

t=T +1 is i.i.d. U (0, 1), i.e.

pi t T +Nt=T +1 =

∫ r∗i t

−∞

fi,t−1(ri t )dri t

T +N

t=T +1i.i.d∼ U (0, 1), (30)

provided that the forecast densities match the actualdensities at each t . Diebold et al. (1999) showedthat the application of the PIT paradigm in amultivariate framework can be performed basedon the PIT of the realized series with respectto the marginal and conditional one-step-aheaddensity forecasts, as shown in Eqs. (31) and (32),respectively, for the bivariate case (i, j = 1, 2) ofthe SNPII ,

pi t =

∫ r∗i t

−∞

f IIi,t−1(ri t )dri t ,

∀t = T + 1, . . . , T + N , (31)

pi | j,t =

r∗i t

−∞

r∗j t

−∞F II

t−1(ri t , r j t )dri t dr j t r∗j t

−∞f II

j,t−1(r j t )dr j t

,

∀t = T + 1, . . . , T + N , (32)

where f IIi,t−1(·) and F II

t−1(·, ·) stand for the one-step-ahead marginal and joint density forecasts ofthe standardized SNPII-DCC distribution. Sincethe PIT series pi t

T +Nt=T +1 and

pi | j,t

T +Nt=T +1 are

also interpreted as the p-values corresponding tothe series of quantiles

r∗

i tT +N

t=T +1 of the fore-casted marginal and conditional densities, respec-tively, we use p-value discrepancy plots (i.e.plotting Ppi t

(yϱ) − yϱ against yϱ) (see Davidson& MacKinnon, 1998; Fiorentini, Sentana, & Cal-zolari, 2003) to test for correct model specifica-tion, where Ppi t

(yϱ) is the cdf of pi t , calculated asPpi t(yϱ) =

1N

∑T +Nt=T +1 1(pi t ≤ yϱ), with 1 · be-

ing an indicator function and yϱ an arbitrary gridof ϱ points. Therefore, under the correct modelspecification, the variable Ppi t

(yϱ) − yϱ must con-verge to zero. The pi | j,t sequences can be analyzedanalogously.

2. Scoring rules (Amisano & Giacomini, 2007),(Gneiting & Raftery, 2007). A scoring rule is aloss function Υ( f , r∗) whose arguments are thedensity forecast f and the realization r∗ of thefuture observation of the variable. In this paperwe use the logarithmic scoring rule Υ( f , r∗) =

− ln f (r∗) to compare marginal and conditionaldensity forecasts. This is a (strictly proper) scor-ing rule that rewards a density forecast which as-signs a high probability to the event that actuallyoccurred. Density forecast models can be rankedby comparing their average scores, Υ( f , r∗) =

−N−1 ∑T +Nt=T +1 ln ft−1(r∗

t ). We take the logarith-



Table 2Density forecasting performance.

Portfolio AT&T —JP Morgan

Portfolio FX e/£— Nasdaq

Υ(·, ·) (t-stat) Υ(·, ·) (t-stat)

SNP marginal 1 0.3987 0.4194DCC marginal 1 0.4247 (−4.55) 0.4256 (−3.95)SNP marginal 2 0.4271 0.4318DCC marginal 2 0.4379 (−3.82) 0.4382 (−2.38)SNP conditional 1 0.4278 0.4319DCC conditional 1 0.4324 (−1.74) 0.4391 (−2.30)SNP conditional 2 0.3995 0.4195DCC conditional 2 0.4117 (−3.36) 0.4258 (−3.55)

Notes: This table presents the results of the average logarithmicscores and statistical tests for one-step-ahead density forecastsfrom the SNP-DCC and DCC models. Υ(·, ·) denotes the averagelogarithmic scoring rule, defined as a negatively oriented penalty,so that the lower the score, the better the model. The table alsogathers the results of the Amisano and Giacomini (2007) test forthe significance of the difference between the average logarithmicscores: the entries are t-statistics (t-stat) for pairwise comparisonsbetween Υ(·, ·) from the SNP-DCC models and the DCC models;a negative t-statistic means that the SNP model produces a lowerΥ(·, ·) than its DCC counterpart.

mic scoring rule as a negatively oriented penalty;thus, we prefer model f if Υ( f , r∗) < Υ(g, r∗),and model g otherwise. The null hypothesis H0 :

EΥ( f , r∗) − Υ(g, r∗)

= 0 is tested using the

Amisano and Giacomini (2007) test.

The results in Fig. 3 and Table 2 show that, over-all, the bivariate SNPII distribution provides a betterperformance for forecasting the full density than thenormal distribution. Specifically, the plots in Fig. 3show a remarkably improved performance of the SNP-DCC over the DCC at the 5% lower tail, which ishighlighted (right column plots in Fig. 3) because ofits direct relationship with various widely used finan-cial risk measures such as VaR and short-fall prob-abilities. On the other hand, if we look at higherpercentiles, e.g., in the range [0.05, 0.4], the resultsare mixed but tend to favour the DCC model. Fur-thermore, Table 2 provides clear evidence in favourof the SNP-DCC for full density forecasting, as theaverage logarithmic scores from the SNP-DCC modelare significantly lower than those obtained from theDCC model. It is worth noting that the observed dif-ferences are due to the density specification, since theforecasted means and volatilities from the two distri-butions are exactly the same, and the differences in

forecasted conditional correlations are very small. Ourempirical results show evidence that the extra term inthe SNPII distribution with respect to the normal pro-vides the SNPII with enough flexibility to adapt to(a) higher and possibly irregular frequencies at thelower percentiles (see the columns on the right ofFig. 3); and (b) more density around its mean. Theseresults are not surprising, since leptokurtic modelstend to show significantly better performances, thelower the quantile of the forecasted density; in addi-tion, the results are in line with previous findings in theliterature: for example, Fiorentini et al. (2003) show,through the graphical procedures used in this section,that a Student-t distribution provides a better perfor-mance than a normal distribution for fitting the fulldensity of stock returns, and Nıguez and Perote (2004)provide evidence on gains in the PES density with re-spect to the Gaussian and Student-t distributions forfull density forecasting. Both papers consider empiri-cal applications to asset returns in a univariate frame-work. On the other hand, in the multivariate context,Perote (2004) shows that multivariate SNP distribu-tions involve more accurate in-sample fits of asset re-turn distributions than the Student-t or Gaussian dis-tributions, and Del Brio et al. (2009) show evidenceof the superior performance of multivariate SNP-CCCmodels with respect to Gaussian models for asset port-folio data. The results of our analysis show that SNP-DCC models perform a reasonable amount better than(Gaussian)-DCC models for forecasting the full andlower percentile of the density of asset portfolio re-turns.

5. Concluding remarks

There exists an abundant body of literature on mul-tivariate volatility models for the time-varying first andsecond conditional moments of the asset return distri-bution. However, not many papers have dealt with themodeling of the full distribution of financial variables.In multivariate analysis, it is usually found that the dis-tributions considered are either not flexible enough toincorporate the salient empirical regularities of finan-cial returns (such as heavy tails, possible multimodal-ity, skewness, etc), or analytically intractable if they doincorporate these features. The SNP densities, whichare based on Edgeworth and Gram-Charlier expan-sions, allow us to address these topics, since not only



Fig. 3. p-value discrepancy plots.



Fig. 3. (continued)



can they fit any target density through their general andflexible parametric structure, but also they present ananalytical specification that is simple and easy to im-plement, due to the orthogonal structure of Hermitepolynomials (i.e., marginals, moments or cdfs can becomputed in a straightforward manner). There havebeen several attempts made to generalize SNP densi-ties to a multivariate context, but the proposed specifi-cations have serious drawbacks, such as not ensuringpositivity for the full density domain or not achiev-ing tractability for large portfolios and incorporatingDCC structures. This article fills this gap in the lit-erature by proposing a new multivariate positive SNPdensity, which, despite a very simple structure, is capa-ble of encompassing as marginals the alternative uni-variate SNP used in the financial literature. Moreover,the proposed multivariate SNP distribution allows thedecomposition of the likelihood function proposedby Engle (2002). We note that this is not trivial fornon-Gaussian distributions; in fact, to the best of theauthors’ knowledge, these decompositions are not pos-sible to either other multivariate distributions of thistype, such as the MES, or other non-normal distribu-tions. The implementation of Engle’s (2002) two-stepestimation techniques for SNP distributions helps tosolve the known “dimensionality curse”, and allows usto incorporate dynamic conditional correlations, thusnesting the (Gaussian)-DCC model in the SNP-DCC.

We compare the empirical performances of twotypes of multivariate SNP specifications (positive andnon-positive) for modeling the distribution of portfolioreturns, in relation to the Gaussian distribution whichis used as the benchmark. The models were estimatedunder either the CCC or DCC hypotheses. Our re-sults show the superiority of the DCC models relativeto their CCC counterparts. We also find that the SNPspecifications considered outperform the normal, andthat the non-positive SNP version seems to provide abetter in-sample fit. Nevertheless, since a positive SNPis required in an out-of-sample context, we comparedthe out-of-sample performances of the (Gaussian)-DCC and the positive SNP-DCC by using graphicalprocedures and proper scoring rules, and find that thelatter provide a superior performance overall (particu-larly at the distribution tails, i.e. for VaR forecasting).

In summary, this paper proposes a new approach tomodeling the distribution of financial returns that gen-eralizes the univariate SNP distributions to a multivari-

ate framework which guarantees the positivity of thedensity for all values of its parameters. Furthermore,the proposed model avoids the known “dimensional-ity curse” of the multivariate context, by means imple-menting the DCC methodology. An empirical applica-tion to asset portfolio returns shows that, overall, theSNP-DCC model provides a better in-sample fit andout-of-sample performance for full density forecastingthan the (Gaussian)-DCC.

Acknowledgements

We would like to thank the co-editor, two anony-mous referees and participants at the MultivariateVolatility Models Conference in Faro, Portugal, Oc-tober 2007, the XXXIII Symposium of the SpanishEconomic Association in Zaragoza, Spain, December2008, and the 29th International Symposium on Fore-casting in Hong Kong, China, June 2009, for manyuseful comments that have substantially improved thepaper. Financial support from the Spanish Ministryof Education under grants SEJ2006-06104/ECON andSEJ2007-66592-C03-03 is gratefully acknowledged.A version of this paper was published as Fundacionde las Cajas de Ahorros (FUNCAS) Working Paper.The responsibility for the content remains solely withthe authors.

Appendix

This appendix includes the proofs of the proper-ties of the multivariate SNP densities presented inSection 2. Proof 1 shows that multivariate SNP den-sities integrate up to one; Proofs 2, 3 and 4 provideclosed forms for the marginal distributions, the mo-ments and the cdf, respectively; and Proof 5 showsthe separability of the log-likelihood for the SNP-DCCmodel.

Proof 1∫· · ·

∫Fζ (xt )dx1t . . . dxnt

=1n

∫· · ·

∫ n∏

i=1

g(xi t )

×

n−

i=1

wζi qζ

i (xi t )

dx1t . . . dxnt



=1n

n−i=1

[w

ζi

∫g(xi t )q

ζi (xi t )dxi t

×

n∏j=1, j=i

∫g

x j t

dx j t

]=

1n

n = 1, ∀ζ = I, II

Proof 2

f ζi (xi t )=

∫· · ·

∫Fζ (xt )dx1t . . . dxi−1,t dxi+1,t . . . dxnt

=1n

g(xi t )wζi qζ

i (xi t )

×

n∏j=1, j=i

∫g

x j t

dx j t +

1n

g(xi t )

×

n−j=1, j=i

n∏

l=1,l=i

wζl

∫g (xlt ) qζ

l (xlt ) dxlt

=1n

g(xi t )wζi qζ

i (xi t ) +n − 1

ng(xi t ),

∀ζ = I, II.

Note that if ζ = I , then f Ii (xi t ) = g(xi t )

(n − 1) + q Ii (xi t )

1n = g(xi t )

1 +

∑qs=2

γisn Hs(xi t )

.

Proof 3

E I xr

i t

=

∫xr

i t f Ii (xi t )dxi t =

∫xr

i t g(xi t )dxi t

+

m−s=2

γis

n

∫xr

i t Hs(xi t )g(xi t )dxi t

= µr +

m−s=2

γis

n

r−j=0

c j

∫H j (xi t )Hs(xi t )g(xi t )dxi t

= µr +

r−j=2

j !c jγi j

n

E II xr

i t

=

∫xr

i t f IIi (xi t )dxi t

=n − 1

n

∫xr

i t g(xi t )dxi t +1

nwIIi

∫xr

i t g(xi t )dxi t

+1

nwIIi

m−s=2

γ 2is

∫xr

i t Hs(xi t )2g(xi t )dxi t

=

[n − 1

n+

1nwi

]µr +

1nwII

i

m−s=2

γ 2is

×

r/2−j=0

d j

∫H j (xi t )

2 Hs(xi t )2g(xi t )dxi t

=

n − 1

n+

1nwII

i

µr +

1nwII

i

×

m−s=2

γ 2iss!

r/2−j=0

j !d j .

Proof 4

Pr[x1 ≤ a1, . . . , xn ≤ an]ζ

=1n

∫ a1

−∞

. . .

∫ an

−∞

n∏

i=1

g(xi t )

×

n−

i=1

wζi qζ

i (xi t )

dx1t . . . dxnt

=1n

n−i=1

∫ ai

−∞

wζi g(xi t )q

ζi (xi t )dxi t

×

n∏j=1, j=i

∫ a j

−∞

gx j t

dx j t

=1n

n−i=1

Ψ ζ (ai )

n∏j=1, j=i

Φ(a j ), ∀ζ = I, II.

Proof 5

LζSN P (Φ, ρ, γ )

= −12

T−t=1

ln |Σ t | + (rt − µt )

′Σ−1t (rt − µt )

−2 ln

n−

i=1

wζi qζ

i (Σ−1/2t (rt − µt ))

+ κ

= −12

T−t=1

ln |Dt Rt Dt | + u′

t D−1t R−1

t D−1t ut

−2 ln

n−

i=1

wζi qζ

i (R−12

t D−1t ut )

+ κ

= −12

T−t=1

2 ln |Dt | + ln |Rt | + ε′

t R−1t εt

−2 ln

n−

i=1

wζi qζ

i (R−12

t εt )

+ κ



= −12

T−t=1

2 ln |Dt | + u′

t D−2t ut − ε′

tεt

+ ln |Rt | + ε′t R

−1t εt

−2 ln

n−

i=1

wζi qζ

i (R−12

t εt )

+ κ

= −12

T−t=1

n−i=1

ln

σ 2

i t

+

ri t − µi t

2

σ 2i t

−12

T−t=1

[ln |Rt | + ε′

t R−1t εt

−2 ln n−

i=1

wζi qζ

i (R−12

t εt )

]+ κ∗

= L MV (Φ) + LζSN P (Φ, ϕ) + κ∗,

where κ = −12 T n ln(2π) − T ln(n) and κ∗

= κ +

12

∑Tt=1 ε′

tεt , ∀ζ = I, II.

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