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A dynamic double asymmetric copulageneralized autoregressive conditionalheteroskedasticity model: applicationto China's and US stock marketYan Fanga, Ling Liua & JinZhi Liub

a School of Finance, Shanghai University of International Businessand Economics, 1900 Wenxiang Road, Shanghai 201620, People'sRepublic of Chinab Department of Basic English Group, Changchun InformationTechnology College, Changchun 130012, People's Republic of ChinaPublished online: 20 Aug 2014.

To cite this article: Yan Fang, Ling Liu & JinZhi Liu (2014): A dynamic double asymmetric copulageneralized autoregressive conditional heteroskedasticity model: application to China's and USstock market, Journal of Applied Statistics, DOI: 10.1080/02664763.2014.949639

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Journal of Applied Statistics, 2014http://dx.doi.org/10.1080/02664763.2014.949639

A dynamic double asymmetric copulageneralized autoregressive conditional

heteroskedasticity model: application toChina’s and US stock market

Yan Fanga∗, Ling Liua and JinZhi Liub

aSchool of Finance, Shanghai University of International Business and Economics, 1900 Wenxiang Road,Shanghai 201620, People’s Republic of China; bDepartment of Basic English Group, Changchun

Information Technology College, Changchun 130012, People’s Republic of China

(Received 9 October 2013; accepted 25 July 2014)

Modeling the relationship between multiple financial markets has had a great deal of attention in bothliterature and real-life applications. One state-of-the-art technique is that the individual financial marketis modeled by generalized autoregressive conditional heteroskedasticity (GARCH) process, while marketdependence is modeled by copula, e.g. dynamic asymmetric copula-GARCH. As an extension, we proposea dynamic double asymmetric copula (DDAC)-GARCH model to allow for the joint asymmetry caused bythe negative shocks as well as by the copula model. Furthermore, our model adopts a more intuitive way ofconstructing the sample correlation matrix. Our new model yet satisfies the positive-definite condition asfound in dynamic conditional correlation-GARCH and constant conditional correlation-GARCH models.The simulation study shows the performance of the maximum likelihood estimate for DDAC-GARCHmodel. As a case study, we apply this model to examine the dependence between China and US stockmarkets since 1990s. We conduct a series of likelihood ratio test tests that demonstrate our extension(dynamic double joint asymmetry) is adequate in dynamic dependence modeling. Also, we propose asimulation method involving the DDAC-GARCH model to estimate value at risk (VaR) of a portfolio. Ourstudy shows that the proposed method depicts VaR much better than well-established variance–covariancemethod.

Keywords: volatility; multivariate GARCH; skewed-t copula; dynamic asymmetric correlation

1. Introduction

Modeling volatility of the returns in financial time series has had a great deal of attention sincethe introduction of the autoregressive conditional heteroskedasticity process [12] and generalizedautoregressive conditional heteroskedasticity (GARCH) process [5]. The properties of univariate

∗Corresponding author. Email: yiffanyfang@163.com

c© 2014 Taylor & Francis

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2 Y. Fang et al.

GARCH sequences are well studied and general statistical methods have been established towork with these sequences.

More recently, researchers begin to study the relationship among different financial markets.A good understanding of the correlation/dependence among the markets is in turn used in deci-sion making in various areas, e.g. asset pricing, portfolio selection, risk management and soon. A common line of research extends GARCH from univariate to multivariate case [7]. Forexample, Karolyi [19] applied multivariate GARCH model to examine the short-run dynamicsof returns and volatility for stocks traded on the New York and Toronto stock exchanges; Tse [25]introduced a Lagrange Multiplier Test for the constant correlation hypothesis in a multivariateGARCH model; Silvennoinen and Teräsvirta [24] proposed a multivariate GARCH model witha time-varying conditional correlation structure.

However, traditional multivariate GARCH models have its limitations, a major one being thatthey enforce a symmetric response of volatility to both positive and negative shocks (i.e. drasticups and downs of a stock market). Existing studies have shown that a negative shock in financialtime series is more likely to raise volatility than a positive shock of the same magnitude, i.e.the leverage effect [4]. Thus, a large number of GARCH extensions exploring the dynamicsof the covariance of assets have been developed, such as the constant conditional correlation(CCC)-GARCH model [6], the BEKK model of Baba, Engle, Kroner and Kraft [14], the dynamicconditional correlation (DCC)-GARCH model [13,26], the asymmetric DCC (ADCC)-GARCH[9] and so forth. Yet it is noteworthy that the assumption of normality is common to all thesemodels. What if the daily returns are not normally distributed?

To address the issue, copula theory is developed to model dependence structure between vari-ables. Using a copula function to link univariate models provides an alternative approach tomultivariate GARCH models. The approach, though not new, has recently attracted considerableattention, as they have been shown to offer a more flexible and accurate modeling when appliedto financial time series. Moreover, copula has the advantage of allowing more flexible marginaldynamics, and handling the dynamical dependence parameter more elegantly. The ‘GARCH-copula framework’, where the marginals are characterized by GARCH process while the jointdistributions are characterized by copula functions, has become popular recently, e.g. Fang andMadsen [15] introduced a modified Gaussian copula with GARCH(1,1) marginals to study thedependence structure with US treasury bonds data.

Lately, Christoffersen et al. [11] developed a novel dynamic asymmetric copula (DAC)-GARCH model, which captures long-run as well as short-run dependence, multivariate non-normality and asymmetries in large cross-section. In order to tell whether the asymmetry is fromthe marginal distribution or the joint distribution, we will later in this study adopt the joint asym-metry simply representing the asymmetry from the joint distribution and the asymmetry denotingthe asymmetry from the univariate marginal distribution. However, Cappiello et al. [9] reveal thatwhen the correlation and volatilities are simultaneously considered, the joint asymmetry becomesstriking with a huge increase for joint negative shocks and little change even for larger positiveshocks; while Christoffersen et al. [11] neglect the joint asymmetry caused by negative shocks.Moreover, to satisfy the positive definite condition, DAC-GARCH model no longer retains theintuition and interpretation of the univariate GARCH model [26].

In this paper, we present an extension to the DAC-GARCH model. The most important featureof our new model is that it considers asymmetries caused by not only the skewness from themultivariate non-normalities, but also the positive/negative shocks in financial market series.Further, we adopt a sampling scheme [26] to obtain a positive-definite correlation matrix.

We refer our model as dynamic double asymmetric copula (DDAC)-GARCH model. OurDDAC-GARCH model is a two-stage process, we first estimate the univariate GARCH for eachasset series, and then, we estimate a DDAC correlation for the standardized residuals resultingfrom the first stage.

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Journal of Applied Statistics 3

Another line of research characterizing financial return volatility is value at risk (VaR). Nowa-days, there have been literatures using copula-GARCH to fit the financial data and compute theVaR. Jondeau and Rockinger [18] applied normal GARCH-based copula for the VaR estimationof a portfolio composed of international equity indices; Huang et al. [17] presented an appli-cation of the copula-GARCH model in estimation of a portfolio’s VaR. It is possible for us toapply the DDAC-GARCH model to compute a reasonable and efficient VaR of portfolios, sincethe DDAC-GARCH model fully considers the skewness, dynamics and asymmetries from thefinancial markets. In contrast to the traditional VaR estimation using the constant weight, we willallow for the dynamic weighting in our VaR estimation.

To assess our proposed modeling methodology, we run a small simulation study to check thequality of the model estimation.

As a case study, we apply our new model in exploring the relationship between the dailyreturns of China’s stock market and US stock market.

This paper is organized as follows. Section 2 introduces the model for both the marginal distri-bution and the joint distribution (i.e. the skewed-t copula with dynamic dependence). We discussthe estimation procedure of VaR in Section 3. Section 4 presents the quick results of the simu-lation, while Section 5 elaborates on our case study of the relationship of the two stock indices.We conclude with some remarks in Section 6.

2. DDAC model

This section provides an outline of our DDAC-GARCH model.

2.1 Marginal model

When modeling the marginal distributions, we need to take both the dynamic volatility and theasymmetry into account. Standard GARCH model assumes that positive and negative error termshave a symmetric effect on volatility. Yet, in reality, this assumption is frequently violated, par-ticularly in stock returns where a negative shock was more likely to cause volatility to risethan a positive shock of the same magnitude. This asymmetry is usually modeled through theleverage effect. For instance, Glosten et al. [16] introduced the Glosten–Jagannathan–Runkle(GJR)-GARCH model to fully model the leverage effect.

However, existing studies repeatedly show that the model residuals still reveal the evidenceof remaining skewness and fat tails even after considering the leverage effect. Accordingly, weallow for an asymmetry in the marginal return 1 distribution by modeling a leverage effect as wellas by using an asymmetric marginal distribution for the return innovation. Particularly, we useGJR-GARCH with the standard innovation distributed from the skewed-t distribution of Azzaliniand Genton [2], i.e. GJR-GARCH-t, for each individual markets.

Let the returns of a given asset i be {ri,t}, where t = 1, . . . , T . For simplicity, we only considerGJR-GARCH-t(1,1) which is denoted as follows:

ri,t = ui,t + ai,t,

μi,t = μi + βTX + ϕiri,t−1 − θiai,t−1,

ai,t = σi,tzi,t,

σ 2i,t = ωi + (αi + γiIi,t−1)a

2i,t−1 + βiσ

2i,t−1,

zi,t ∼ SKT(0, 1, νi, λi),

(1)

where X is the k-dimensional exogenous co-variate, β is the k-dimensional regression coeffi-cients; the indicator function Ii,t−1(·) = 1 if ai,t−1 < 0, and 0 otherwise; γi > 0, αi ≥ 0, βi ≥

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4 Y. Fang et al.

0, βi + γi ≥ 0 and αi + βi + 12γi < 1. Here, μi,t = E[E(ri,t | Fi,t−1)] and σ 2

i,t = Var(ri,t | Fi,t−1)

are the conditional mean and variance of series return of i, respectively, where Fi,t−1 is theinformation set at t − 1.

Good news ai,t−1 > 0 and bad news ai,t−1 < 0 have different effects on the conditional vari-ance in these models, that is to say, good news has an impact of αi, while bad news has animpact of αi + γi. If γi > 0 we say that the leverage effect exists, or if γi �= 0 the news impactis asymmetric. Besides the asymmetry from the volatility, another source of asymmetry comesfrom the skewness of the standard residual, zi,t, which is driven by the skewed-t distribution withthe skewness parameter λi and the degree of kurtosis parameter νi.

Recall that Z is a standard skewed-t distribution variable, i.e. Z ∼ SKT(0, 1, λ, ν), then, asstated by Azzalini and Genton [2], the density function of the skewed-t variable, Z, with locationparameter 0, scale parameter 1, shape parameter λ ∈ R and degrees of freedom ν can be denotedas

fST(z; λ, ν) = 2fT (z; ν)FT

(λz

√ν + 1

ν + z2; ν + 1

), z ∈ R, (2)

where fT (·) is the probability density function (pdf) for univariate standard Student’s t, Z, withdegree of freedom ν; and FT (·) stands for the cumulative distribution function (cdf) for theunivariate standard Student’s t variable with degrees of freedom ν + 1. Parameter λ controls theskewness, while ν controls the tail proportions. The notable degenerative cases of the densityfunction (2) include:

• if λ = 0, then it reduces to the standard Student’s t distribution;• if ν → ∞, then it reduces to the standard skewed normal distribution;• if both λ = 0 and ν → ∞, then it reduces to the standard normal distribution.

Function fT (z; ν) and FT (z; ν) for the univariate Student’s t variable Z with the degree offreedom ν can be specifically expressed as

fT (z; ν) = �((ν + 1)/2)√πν�(ν/2)

(1 + z2

ν

)−(ν+1)/2

(3)

and

FT (z; ν) =∫ z

−∞fT (y; ν) dy =

∫ z

−∞

�((ν + 1)/2)√πν�(ν/2)

(1 + y2

ν

)−(ν+1)/2

dy. (4)

Note that the distribution for the individual return shock is invariant to time, however, the distri-bution for the individual return does vary through time because the return mean and variance aredynamic.

2.2 Copula modeling

Having addressed the asymmetries for the marginal distributions in the first stage, we proceed tostudy the time-varying trend, the dynamic asymmetries as well as the tail dependence existingin the dependence structure of two different equity returns in the second stage. The proposedDDAC model, which extends DAC model introduced by Christoffersen et al. [11], is used tocapture the multivariate dynamics as well as the asymmetries.

In order to construct the density function for a d-dimensional copula, which evidently hasa dynamic double asymmetric correlation, we need first to portray the correlation matrix forDDAC model. In DDAC model, the correlation matrix regresses on its own lag term, t−1; the

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Journal of Applied Statistics 5

intrinsically constant term, ; the time-varying trend, ϒt; and the lagged negative shock, �t−1.In general terms, the correlation matrix in DDAC model, t, is specified as

t = (1 − β − α − γ )[(1 − ϕ ) + ϕ ϒt] + α �t−1 + β t−1 + γ �t−1, (5)

where α ∈ (0, 1), β ∈ (0, 1), γ ∈ (0, 1), α + β + γ < 1 and ϕ ∈ (0, 1). Appendix 1demonstrates that t remains a positive semi-definite correlation matrix. It is worth noting thatmatrix �t−1 models the leverage effect. If the return shock is positive, then the asymmetries areonly from matrix �t−1, otherwise, the asymmetries are derived from both matrix �t−1 and �t−1.

The pair (i, j) and (j, i) entries of the constant correlation matrix � is defined as the uncondi-tional correlation2 between z∗

i and z∗j , which takes no account of the temporal ordering. Usually,

matrix ϒt contains information about time trend as well as the explanatory variables. Here, how-ever, we will restrict out attention to the time trend only. With reference to Christoffersen et al.[11], all off-diagonal elements can be set to be δ log(t)/(1 + δ log(t)). Furthermore, the correla-tion estimated at a particular point in time is remarkably robust to the used sample period andthe upward trending in correlation has nothing to do with the sample period used. With referenceto Tse and Tsui [26], both correlation matrix �t−1 and �t−1 are obtained by using the samplecorrelation matrix of the shocks, i.e. {z∗

1, . . . , z∗t−1}. Generally, the off-diagonal entries of the

correlation matrix �t−1, denoted as ρi,j,t−1, and those of �t−1, denoted as �i,j,t−1, are defined as

ρi,j,t−1 =∑M

h=1 z∗i,t−hz∗

j,t−h√∑Mh=1(z

∗i,t−h)

2∑M

h=1(z∗j,t−h)

2

and

�i,j,t−1 =∑L

h=1 ni,j,t−h√∑Lh=1 n2

i,t−hIi,j,t−h∑L

h=1 n2j,t−hIi,j,t−h

Ii,j,t−1,

where i, j ∈ {1, . . . , d}; d is the dimension of variable z∗t ; ni,j,t−h = ni,t−h × nj,t−h; the indica-

tor function Ii,j,t−h being 1 if both z∗i,t−h < 0 and z∗

j,t−h < 0, and 0 otherwise. Variable nk,t−h =z∗

k,t−hIk,t−h, where k = 1, . . . , d; h = 1, . . . , L; the indicator function Ii,t−h being 1 if z∗i,t−h < 0,

and 0 otherwise. Note that subset (ni,j,t−1, . . . , ni,j,t−L) must contain M non-zero elements. Forsimplicity, in this paper, we set M = d . Variable z∗

i,t is

z∗i,t = F−1

ST (ηi,t; λ∗, ν∗),

ηi,t = FST(zi,t; λi, νi),

where ηi,t ∈ (0, 1); λ∗ and ν∗ are the shape parameter and the degree of freedom, respectively;FST(·) and F−1

ST (·) are the cdf of the standard univariate skewed-t distribution and the standardskewed-t quantile function, respectively. Obviously, z∗

i,t is still the standardized variable withmean 0 and variance 1. Function FST(·) can be expressed as

FST(z; λ, ν) =∫ z

−∞2fT (y; ν)FT

(λy

√ν + 1

ν + y2; ν + 1

)dy, z ∈ R. (6)

Accordingly, the special cases of the correlation matrix defined in Equation (5) are

• DAC correlation matrix of Christoffersen et al. [11], when γ = 0;• ADCC correlation matrix of Engle [13], when ϕ = 0;• DCC correlation matrix, when φ = γ = 0;• CCC correlation matrix, when φ = α = γ = 0.

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6 Y. Fang et al.

When the correlation is constructed, a d-dimensional dynamic asymmetric skewed-t copula’spdf can be written as

ct(ηt; t, λ∗, ν∗) = fMST(z∗

t ; t, λ∗, ν∗)∏di=1 fST(z∗

i,t; λ∗, ν∗)

, (7)

where ηt = (η1,t, . . . , ηd,t); z∗t = (z∗

1,t, . . . , z∗d,t); t is a d × d dynamic asymmetric correlation

matrix; ν∗ and λ∗ are the shape parameter and the degree of freedom parameter for the dynamicasymmetric skewed-t copula’s distribution; fMST(·) is the standard multivariate skewed-t pdf withthe dynamic asymmetric correlation matrix t. In view of Azzalini and Capitanio [1], functionfMST(·) can be simply denoted as

fMST(vt; t, λ∗, ν∗) = 2fMT(vt; t, ν

∗, d)FT

(ξT

t vt

√ν∗ + d

vTt −1

t vt + ν∗ ; ν∗ + d

), (8)

where fMT(·) is the standard multivariate Student’s t pdf; ξt = −1t ς with ς = (λ∗, . . . , λ∗) being

a d × 1 skewness controlling parameter. Function fMT(·) [27] is simplified as

fMT(vt; t, ν∗, d) = �((ν∗ + d)/2)√| t|(πν∗)d/2�(ν∗/2)

(1 + vT

t −1t vt

ν∗

)−(ν∗+d)/2

, vt ∈ Rd . (9)

In order to develop DDAC-GARCH model, we follow the assumptions set forth on DCC-GARCH model in Engle [13], which are subsequently adopted by ADCC-GARCH [9] and DAC-GARCH models [11]. The assumptions can be summarized as follows:

(1) All the roots of the characteristic equation lie outside the unit circle. E[rt | Ft−1] possessesan Ft measurable second-order stationary solution.

(2) The matrix is finite and ρ( ) has a positive lower bound over the parameter space.(3) The parameters α, β, γ and ϕ are non-negative.

Under these conditions, {rt, at, t} are strictly stationary and ergodic by Theorem 1 of Ling andMcAleer [21].

2.3 Parameter estimation

This paper uses the maximum likelihood estimation (MLE)3 and the inference function for mar-gins (IFM) method, where the estimation method applies to models in which the univariatemargins are computed independently of the dependence structure. Hence, the parameters of IFMare estimated in two separate stages, i.e. applying MLE to the marginal distribution as well asthe copula:

First: Estimating the marginal parameters �̂1 of the univariate marginal distributions:

�̂1,i = argmax�1,i

T∑t=1

lnf (ri,t; �1,i), (10)

where f (ri,t; �1,i) is the marginal distribution for returns at time t for equity market i, and i =1, . . . , d.

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Journal of Applied Statistics 7

Second: Given (�̂1,1, . . . , �̂1,d), we estimate the copula parameter �̂2, namely,

�̂2 = argmax�2

T∑t=1

lnct(ηt; t, λ∗, ν∗), (11)

where ηt is obtained with respect to the given value of (�̂1,1, . . . , �̂1,d).The IFM estimator is defined as

�̂MLE = (�̂1,1, . . . , �̂1,d , �̂2).

Under some regularity conditions, �̂MLE is consistent and asymptotically efficient, which alsocorroborates the properties of asymptotic normality [23]:

√T(�̂MLE − �0) → N(0, V−1(�0)),

where V−1(�0) is the Godambe information matrix [10].

2.4 Validate for constant correlation

In order to validate the specification of the dynamic asymmetric correlation, we will evaluate thefollowing three versions of the model:

• The full model, which allows for both correlation dynamics and the asymmetries in copuladependence, namely, correlation defined in Equation (5);

• The time-varying trend is ignored, i.e. ϕ = 0;• The third is that neither the effect from the negative shock nor the effect from the time-varying

trend is considered, that is, ϕ = γ = 0.

The three models are listed in the level of reductions; a conventional likelihood ratio test (LRT)will apply to test the adequacy of the reduction levels. The form of the test function is suggestedby its name, to wit, the ratio of two likelihood functions:

LRT = −2(log(Ls) − log(Lg)) ∼ χ2ν , (12)

where log(Ls) and log(Lg) are the likelihood function for the simpler model (s) and the generalmodel (g), respectively. The simpler model has fewer parameters than the more general model.Asymptotically, the test statistic is distributed as a chi-squared random variable, with degrees offreedom ν equal to the difference in the number of parameters of the two models.

3. Estimation of VaR

Generally, VaR is a concept developed in the field of risk management in finance. It is a measureof the maximum loss expected with a given confidence level over a specific period of time (fromtime t − 1 to t). With a confidence level α ∈ (0, 1), VaR is defined as

VaRt(α) = inf{s : Fp,t(s) ≥ α}, (13)

where Fp,t(·) is the distribution of the portfolio return Xp,t at time t. Equivalently, we haveP(Xp,t ≤ VaRt(α) | Ft−1) = α at time t [22], where Ft−1 is the information set at time t − 1.This means that we are 100(1 − α)% confident that the loss in period from time t − 1 to t willnot be larger than VaR.

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8 Y. Fang et al.

The portfolio return Xp,t considered here is composed of two equity returns denoted as r1,t andr2,t given by

Xp,t = �1,tr1,t + �2,tr2,t, (14)

where �1,t ∈ (0, 1), �2,t ∈ (0, 1), �1,t + �2,t = 1, �1,t and �2,t are the dynamic allocation ofinvestment weights for asset 1 and asset 2 at time t, respectively. The traditional allocation ofweights are based on constant weight, that is, the weights are unaffected by economic conditions.In general, the effect of economic conditions is changing over time. Constructing the dynamicweighting will produce superior risk and expected return trade-off relative to standard portfolios[3]. Therefore, we will use the inverse dynamic variance weighting. Specifically, the weights ofasset i at time t are defined as

�1,t = σ 22,t

σ 21,t + σ 2

i,t

and �2,t = σ 21,t

σ 21,t + σ 2

i,t

, (15)

where σ 21,t and σ 2

2,t are the dynamic variance for assets 1 and 2 at time t, respectively. Then, theportfolio return is expressed as

P(Xp,t ≤ VaRt | Ft−1) =∫ +∞

−∞

∫ (1/�t)VaRt−((1−�t)/�t)r2,t

−∞f (r1,t, r2,t | Ft−1) dr1,t dr2,t, (16)

where the specific form for function f (r1,t, r2,t | Ft−1), i.e. the dynamic joint probability functionfor assets 1 and 2, is given in Appendix 2.

The computation of the multivariate skewed-t quantile involved in the VaR estimation doesnot have a closed-form solution. As such, we will use the following simulation methods:

For each t ∈ (1, . . . , T)

• Generate N data points, i.e. sample (ε̂1,1,t, ε̂1,2,t), . . . , (ε̂N ,1,t, ε̂N ,2,t), from the bivariate skewed-tdistribution fMST(vt; t, λ∗, ν∗) defined in Equation (8) with d = 2. Note that (ε̂1,t, ε̂2,t) are thestandardized variables, that is, means are 0 and variances are 1.

• For i = 1, . . . , N , we convert them to the daily returns (r̂i,1,t, r̂i,2,t) by using r̂i,1,t =F−1

ST (ε̂i,1,t; λ1, ν1) and r̂i,2,t = F−1ST (ε̂i,2,t; λ2, ν2).

• Using Equation (14), we obtain an independently simulated sample with sample size N , i.e.(X̂1,p,t, . . . , X̂N ,p,t), for the portfolio Xp,t at time t.

• Corresponding to the given probability α, the VaR value at time t is approximate by the samplequantiles (X̂1,p,t, . . . , X̂N ,p,t).

Finally, we estimate the VaR by using the variance–covariance method: the VaR estimationformula is defined as

σ 2p,t = [

�t 1 − �t]×

[σ 2

1,t σ1,2,t

σ1,2,t σ 22,t

]×[

�t

1 − �t

]= ��t�

T,

VaRt(α) = σp,t × Zα + μp,t,

(17)

where μp,t and σ 2p,t are the mean and the variance of the portfolio return at time t, respectively; �t

and 1 − �t are portfolio weights of asset 1 and asset 2; Zα is the standardized normal inverse withprobability α. The variance–covariance method is fairly simple, albeit subject to the constraintsof normality assumption.

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4. Simulation study

An interesting issue for empirical applications concerns the properties of the MLE of the con-ditional heteroscedasticity models in small and moderate samples. In this section, we conducta small simulation study on samples of moderate size to examine the properties of the MLE ofDDAC-GARCH model. Instead of providing a comprehensive Monte Carlo experiment, we willonly focus on the bias and mean squared error.

Consider the bivariate DDAC-GARCH model. The conditional correlation equation is givenby (

z∗1,t

z∗2,t

)∼ MST( t, λ

∗, ν∗). (18)

The off-diagonal entry of t is

ρt = (1 − βρ − αρ− γρ)[(1 − ϕρ)ρ + ϕρϒ̃t] + αρ�̃t−1 + βρρt−1 + γρ�̃t−1, (19)

where �̃t−1 and �̃t−1 are defined as

�̃t−1 =∑2

h=1 z∗1,t−hz∗

2,t−h√∑2h=1(z

∗1,t−h)

2∑2

h=1(z∗2,t−h)

2

Table 1. Bias and MSE of DDAC-GARCH model’s MLE (λ = 0.1, ν = 5).

Experiment I where ρ = 0.7 Experiment II where ρ = 0.2

True Sample True SampleParameters value size Bias MSE value size Bias MSE

αρ 0.1 100 4.61e−6 6.74e−11 0.1 100 3.97e−6 6.15e−11500 1.78e−6 4.61e−12 500 1.60e−6 4.30e−12

1000 1.06e−6 1.35e−12 1000 9.85e−7 1.28e−12

βρ 0.8 100 −6.87e−6 8.61e−11 0.8 100 −7.17e−6 9.04e−11500 −5.94e−7 2.18e−12 500 −6.17e−7 2.26e−12

1000 −1.25e−7 4.07e−13 1000 −9.38e−8 3.90e−13

γρ 0.05 100 3.65e−6 5.32e−12 0.05 100 3.10e−6 4.67e−11500 9.37e−7 2.69e−12 500 9.25e−7 2.68e−12

1000 4.77e−7 6.87e−13 1000 6.04e−7 8.50e−13

δρ 0.1 100 6.43e−6 7.30e−11 0.1 100 1.90e−6 2.51e−11500 2.23e−6 5.53e−12 500 2.48e−7 1.05e−12

1000 1.23e−6 1.54e−12 1000 1.05e−7 2.63e−13

φρ 0.2 100 −6.28e−7 8.52e−12 0.2 100 −8.89e−7 7.60e−12500 6.85e−7 1.66e−12 500 −1.32e−7 6.54e−13

1000 7.73e−7 8.55e−13 1000 −5.39e−8 2.10e−13

ν 5 100 6.88e−5 4.75e−9 5 100 6.97e−5 4.87e−9500 1.38e−5 1.91e−10 500 1.40e−5 1.97e−10

1000 6.94e−6 4.83e−11 1000 7.02e−6 4.93e−11

λ 0.1 100 −0.097 0.0094 0.1 100 −0.097 0.0094500 −0.097 0.0095 500 −0.098 0.0095

1000 −0.098 0.0096 1000 −0.098 0.0096

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Table 2. Bias and MSE of DDAC-GARCH model’s MLE (λ = 0.1, ν = 5).

Experiment III where ρ = 0.7 Experiment IV where ρ = 0.2

True Sample True SampleParameters value size Bias MSE value size Bias MSE

αρ 0.2 100 5.23e−6 7.23e−11 0.2 100 3.92e−6 6.29e−11500 1.89e−6 4.34e−12 500 1.65e−6 4.03e−12

1000 1.05e−6 1.18e−12 1000 9.65e−7 1.13e−12

βρ 0.6 100 −1.58e−6 4.83e−11 0.6 100 −2.12e−6 5.13e−11500 5.38e−7 1.51e−12 500 3.85e−7 1.65e−12

1000 4.09e−7 3.42e−13 1000 3.34e−7 3.74e−13

γρ 0.1 100 4.26e−6 6.24e−11 0.1 100 3.03e−6 4.89e−11500 1.06e−6 3.01e−12 500 1.00e−6 2.93e−12

1000 5.91e−7 8.20e−13 1000 5.38e−7 7.73e−13

δρ 0.1 100 7.01e−6 8.03e−11 0.1 100 1.73e−6 2.45e−11500 2.39e−6 5.91e−12 500 3.04e−7 1.20e−12

1000 1.24e−6 1.54e−12 1000 1.51e−7 3.14e−13

φρ 0.2 100 −3.01e−8 1.21e−11 0.2 100 −6.83e−7 8.78e−12500 8.66e−7 2.11e−12 500 −6.66e−8 7.87e−13

1000 7.62e−7 8.71e−13 1000 −8.69e−9 2.48e−13

ν 5 100 6.86e−5 4.72e−9 5 100 6.95e−5 4.84e−9500 1.38e−5 1.90e−10 500 1.40e−5 1.97e−10

1000 6.89e−6 4.75e−11 1000 7.01e−6 4.92e−11

λ 0.1 100 −0.097 0.0094 0.1 100 −0.097 0.0094500 −0.097 0.0094 500 −0.098 0.0096

1000 −0.097 0.0095 1000 −0.098 0.0096

and

�̃t−1 =∑2

h=1 z∗1,t−hI1,t−hz∗

2,t−hIi,j,t−h√∑2h=1(z

∗1,t−h)

2∑2

h=1(z∗2,t−h)

2,

respectively.In the study, we use three sample sizes, two ρ’s, and two sets of values for

(αρ , βρ , γρ , δρ , ϕρ , ν, λ) to generate our simulation data. The MLEs are calculated for each gen-erated sample. Using Monte Carlo samples of 1000 runs, we compute the bias and mean squareerror (MSE) of the MLE. The true generating parameter values, together with the two MLEerrors, are reported in Tables 1 and 2.

It can be seen from the simulation results that the biases and MSE of the MLE estimators aregenerally very small, with only one exception of λ. This indicates that our proposed estimatingprocedures work well with our models.

In the next section, we illustrate the application of the DDAC-GARCH model with real dataset.

5. Empirical application

As an application of previously proposed methodology, this paper explores the relationshipbetween the Shanghai stock market4 and the New York stock market.5 We work with data from

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OLS−based CUSUM test

Time

Em

piric

al flu

ctu

atio

n p

roce

ss

1990 1995 2000 2005 2010 2015

−1.0

0.0

0.5

1.0

1.5

Figure 1. Ordinary least squares (OLS)-based CUSUM process for SHECI.

1 January 1991 through 31 May 2013, almost the entire history of Chinese stock market. Allstock price indices were obtained from the Wind Info (http://www.wind.com.cn/en/), a leadingprovider of financial data in China.

One problem with the date correspondence is that the holidays of the two countries seldomcoincide: one country’s holiday is usually the other’s trading day. Presumably people would havetraded differently in holidays, so generating holidays’ data by interpolating adjacent work daysmight not be very accurate; we simply discard unpaired dates, following Jondeau and Rockinger[18]. Thus, we end up with 5318 paired daily observations as our data set.

For time series this long, it is customary to perform ‘structural change’ tests and divideit into coherent chunks as necessary. We performed the test by computing the cumulativesums (CUSUM) fluctuations, which contain CUSUM of standardized residuals [8] of SHECI’smonthly closing prices. From Figure 1, we can see there are two prominent break points, namely,mid-2001 and late-2008.6 Accordingly, before analyzing the linkages between SHECI and DJI,we divide the entire time span into three periods:

• January 1991 to June 2001: This is the infancy decade of SHECI. The stock market fluctuatedsometimes wildly (2528 observations).

• July 2001 to October 2008: In this period, SHECI has experienced the rapid development andgradually stabilized. However, an important policy (the ‘Securities Act’) was introduced inthe end of 2005, which brought the new opportunities for the development of SHECI. Accord-ingly, we will add a factor before and after the end of 2005 to the marginal distribution ofSHECI (1718 observations).

• November 2008 to May 2013: In November 2008, China announced a stimulus package as anattempt to minimize the impact of the global financial crisis on its domestic economy. Sincethen, the turnover for SHECI began to slowly return to a more active state (1702 observations).

Table 3 provides mean, skewness and kurtosis on market returns for the entire time span andthe three periods. The values inside the parentheses are the unconditional correlation values.As a whole, we discover that DJI has a negative skewness (−0.21) while SHECI has a positiveskewness (5.32). Besides, both SHECI and DJI apparently exist significant excess kurtosis, whichmeans the empirical return observations display fatter tails than normal distributions. The uncon-ditional correlation among SHECI and DJI is 0.011, which is rather low. However, as explainedearlier, we will discuss the linkages between SHECI and DJI in three periods. Both stocks exhibitnegative skew for all periods, except for the first period of SHECI. Moreover, they all exhibit fat

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Table 3. Mean, skewness and kurtosis for daily returns series.

All data (0.011) Period 1 (−0.028) Period 2 (0.054) Period 3 (0.076)5318 2528 1718 1702

n SHECI DJI SHECI DJI SHECI DJI SHECI DJI

Mean 0.054 0.033 0.113 0.055 −0.015 −0.007 0.027 0.046Skewness 5.32 −0.21 5.58 −0.24 −0.17 −0.14 −0.28 −0.23Kurtosis 141.41 9.93 116.08 5.35 4.91 13.18 2.53 4.96

tail: all kurtosis are significant. Finally, the kurtosis of SHECI from all three periods vary wildly,which also justifies our period division.

Since the empirical return series exhibits extreme excess kurtosis and volatility clusteringin each period, it is necessary to consider the marginal distribution for adjusting the empiricalreturn distribution. Generally, heteroskedastic returns lead to skewness as well as fat tails, there-fore, returns standardized by their estimated conditional standard deviation may be (or close to)

Table 4. Parameters estimate and log likelihood (LL) values for the univariate GJR-GARCH(1,1) modelsfrom the different periods: SHECI vs. DJI.

Period 1 Period 2 Period 3

China USA China USA China USA

μ 0.12(0.15) 0.05(0.02)∗ −0.15(0.04)∗∗ 0.009(0.02) −0.006(0.05) 0.06(0.02)∗∗ϕ 0.97(0.02)∗∗ −0.29(0.26) −0.013(0.06) 0.42(0.53) −0.06(0.22) 0.19(0.40)θ −0.87(0.04)∗∗ 0.32(0.26) 0.009(0.04) −0.49(0.51) 0.03(0.22) −0.27(0.40)B1 0.13(0.22) 0.41(0.18)∗ 0.44(0.08)∗∗ 0.10(0.16) −1.11(0.37)∗ −0.20(0.09)∗B2 1.14 (0.60)∗ω 0.05(0.05) 0.01(0.004)∗ 0.08(0.03)∗ 0.01(0.004)∗ 0.01(0.01) 0.02(0.007)∗∗α 0.16(0.04)∗∗ 0.014(0.008) 0.05(0.02)∗ 2.76 × 10−8 0.03(0.01 )∗ 2.29 × 10−9

(7.47 × 10−5) (4.30 × 10−5 )β 0.78(0.06)∗∗ 0.94(0.01)∗∗ 0.88(0.03)∗∗ 0.93(0.01)∗∗ 0.96(0.01)∗∗ 0.88(0.02)∗∗γ 0.11(0.04)∗∗ 0.07(0.02)∗∗ 0.11(0.04)∗ 0.12(0.03)∗∗ −0.003(0.01) 0.21(0.05)∗∗λ 0.98(0.03)∗∗ 0.96(0.03)∗∗ 0.97(0.03)∗∗ 0.91(0.03)∗∗ 0.93(0.04)∗∗ 0.88(0.04)∗∗ν 4.21(0.28)∗∗ 5.97(0.65)∗∗ 4.30(0.52)∗∗ 7.85(1.48)∗∗ 5.38(0.70)∗∗ 6.93(1.42)∗∗LL −5007.0 −3154.7 −3173.871 −2337.908 −1528.6 −1865.1

Ljung-box testLag p-Value p-Value p-Value p-Value p-Value p-Value1 0.2729 0.1830 0.5142 0.6703 0.8424 0.94563 0.6710 0.0601 0.1255 0.7866 0.4580 0.99035 0.8161 0.1291 0.1037 0.8081 0.6915 0.24747 0.2810 0.2813 0.1270 0.6790 0.1777 0.3783

Augmented Dickey–Fuller test: Res. and S.R. indicates residuals and square of residuals, respectivelyRes. 0.01 0.01 0.01 0.01 0.01 0.01S.R. 0.01 0.01 0.01 0.01 0.01 0.01

Notes: The value inside parentheses is the standard deviation (sd).∗Significant at 5% level.∗∗Strongly significant (p-value close to 0) at 5% level.

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normally distributed. In order to investigate the properties of innovations, we standardized theresiduals by the GJR-GARCH(1,1) model (see Section 2.1). Meanwhile, we note that the holi-days in the two countries have a strong effect on the mean model of GJR-GARCH(1,1), thereforewe add the holiday effect as the exogenous regressors. Besides, there is additional effect in period2 of SHECI (the Securities Act).

Table 4 presents the results of the parameters we estimate, the maximum LL values, theLjung-box test and the augmented Dickey–Fuller test for model adequacy. In the mean model,we consider the exogenous variables, i.e. the holiday effect. Chinese Spring Festival (the mostimportant traditional Chinese holiday) and American Christmas are considered as the holidayeffect for China and USA, respectively. Interestingly, the holiday have the significant negativeeffect in period 3 for both SHECI and DJI and the positive effect in both periods 1 and 2. In termsof DJI, we find that the holiday effect decreased from 0.41 at the first period to −0.2 at the thirdperiod. As for SHECI, the holiday effect dropped from 0.44 to −1.11. Furthermore, the ChineseSecurities Act introduced in 2005 significantly improved the return by 1.14 with p-value 5.7%.

Table 4 also reveals that the standard innovation term are robustly significant skewed-t distri-bution for both SHECI and DJI from all periods. All the p-values for both the shape parametersλ and the degree of freedom are far less than 0.05. Moreover, the asymmetry parameter, γ , inthe GJR-GARCH process are also significant, all γ ’s are significantly away from 0 except forthe third period of SHECI. Therefore, univariate GJR-GARCH(1,1) completely fits DJI: it notonly fully characterizes the leverage effect but also thoroughly describes the fat tail from theinnovation shock.

Moreover, Table 4 shows that all the Ljung-box test applied to the standardized residuals of theGJR-GARCH(1,1) models do reject the null hypothesis of autocorrelations at lags 1, 3, 5 and 7at the 5% significant level, which indicates the stationarity of the standardized residuals obtainedfrom GJR-GARCH(1,1) for both assets. Additionally, the standardized residual series as well asthe square of the standardized residual series tested by the augmented Dickey–Fuller test stronglyreject the null hypothesis of the unit root at the 5% significant level, which also indicates thatthe standardized residuals of the GJR-GARCH(1,1) model are stationary. Therefore, we believethat the GJR-GARCH(1,1) adequately models all the data sets and successfully deliver the white

Table 5. Parameters estimates, LL and LRT values for thedynamic asymmetric skewed-t copula: SHECI and DJI.

Period 1 Period 2 Period 3

α 0.08(0.04)∗ 0.0006(0.03) 0.003(0.08)β 0.002(0.46) 0.14(0.81) 0.21(2.52)γ 0.11(0.14) 0.06(0.08) 0.21(0.11)∗δ 0.28(0.05)∗ 0.17(0.08)∗ 0.21(0.06)∗ϕ 0.48(0.05)∗ 0.69(0.18)∗ 0.93(0.80)ν∗ 5.09(0.32)∗ 6.06(0.68)∗ 6.15(0.92)∗λ∗ 1.62(0.008)∗ 1.59(0.02)∗ 1.53(0.02)∗LL 946.51 623.95 377.771

LRT with null hypothesis: ϕ = 0p-Value Close to 0

LRT with null hypothesis: ϕ = γ = 0p-Value 0.04 Close to 0 0.2

Notes: The values inside the parentheses are the sd.∗Significant at 5% level.

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0 500 1000 1500 2000 2500

−0.2

0.0

0.2

0.4

0.6

Dynamic Asymmetric Correlation: 1st period

Observations

Dyn

am

ic A

sym

me

tric

Co

rre

latio

n

0 500 1000 1500

−0

.20

.00

.20

.40

.6

Dynamic Asymmetric Correlation: 2nd period

Observations

Dyn

am

ic A

sym

me

tric

Co

rre

latio

n

0 200 400 600 800 1000

−0

.20

.00

.20

.40

.6

Dynamic Asymmetric Correlation: 3rd period

Observations

Dyn

am

ic A

sym

me

tric

Co

rre

latio

n

Figure 2. The series of the dynamic asymmetric correlations. The dark dot indicates the negative shocks,and the grey dot indicates the general shocks.

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noise residuals required to obtain the unbiased estimators of the dynamic asymmetric skewed-tcopula.

The parameter estimation for the dynamic asymmetric skewed-t copula is shown in Table 5.Neither parameter α (except for the first period) nor parameter β is significant at the 5% level,which probably implies that the correlation does not significantly depend on its own lag term, i.e.the dependence from time t − 1, and the intrinsically constant dependence. The negative shockis not a statistically significant effect of the link of returns between SHECI and DJI on the firstand second period, since all the p-values are greater than 5%; while it is sort of significant duringperiod 3 with p-value 0.056. Therefore, we should not ignore the impact caused by the negativeshocks when modeling the dynamic correlation.

Not surprisingly, the effects of all the time trends (δ) are significantly positive at the 5% level,which implies the DDAC correlation increases over time. In addition, the weighting parameters,ϕ , for the time-varying correlation matrix are significant for both periods 1 and 2. Besides, theresults of the conventional LRT show that:

• First, the introduction of the time-varying correlation matrices results in a statistically signifi-cant improvement in model fit for all three periods. Therefore, the correlation between SHECIand DJI is time-varying, and indeed the dependence is becoming more and more strong.

• Second, adding the effects of both the negative shocks and the time-varying trend resultsin a statistically significant improvement in model fitting for periods 1 and 2 (p-value closeto 0), but not for period 3 (p-value = 0.2). The insignificance result for the third period islargely because, although the parameter of the time-varying trend is significant, the parameterof time-varying matrix is not.

Besides, according to Table 5, the bivariate skewed-t copula fits nicely the joint asymmetry aswell as the fat tail for the residuals from the GJR-GARCH margins for all periods. All p-valuesfor both parameters λ∗ and parameters of the degree of freedom are much smaller than 5%. Inother words, there exists a strong tail dependence between SHECI and DJI. Even after consider-ing the fat tail individually for both markets, the joint tail dependence still cannot be ignored.

Figure 2 presents the DDAC correlations between SHECI and DJI for the three periods respec-tively. The most striking feature from these plots is that the two markets appear to be positively

Table 6. Number of violations of VaR estimation w.r.t. 5% and 1% levels.

First period 2528 Second period 1718 Third period 1072Trading daysα 5% 1% 5% 1% 5% 1%Expected no.of violations ω 126 25 86 17 54 11 Mean error

SM 0.5 81 15 52 9 56 12 16.7

1/σ 2i,t

1/σ 21,t + 1/σ 2

2,t

129 34 93 26 68 21 8.5

VC 0.5 56 11 30 9 31 6 29.3

1/σ 2i,t

1/σ 21,t + 1/σ 2

2,t

101 34 61 18 44 15 12.3

Note: VC means the variance–covariance method, while SM means the simulation method.

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0 500 1000 1500 2000 2500

−6

−4

−2

02

4

Estimated VaR: 1st Period

Observation

Po

rtfo

lio r

etu

rn /V

aR

(%

)

Portfolio Return

VaR at 5%

VaR at 1%

0 500 1000 1500

−6

−4

−2

02

4

Estimated VaR: 2nd Period

Observation

Po

rtfo

lio r

etu

rn /V

aR

(%

)

Portfolio ReturnVaR at 5%VaR at 1%

0 200 400 600 800 1000

−6

−4

−2

02

4

Estimated VaR: 3rd Period

Observation

Port

folio

retu

rn /V

aR

(%

)

Portfolio ReturnVaR at 5%VaR at 1%

Figure 3. Estimate VaR using the dynamic asymmetric correlation skewed-t copula with GJR-GARCH(1,1)as marginal distributions.

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correlated at all times. Moreover, negative shocks often lead to big correlation changes, which isvery obvious in first and third period. However, the correlation curve has gradually flattened out.

In order to check how well the DDAC-GARCH approach models the dynamic correlationbetween the two markets, we try to use the DDAC-GARCH model to evaluate VaR for a portfolioassets, which is composed of stocks from both SHECI and DJI. We estimate the portfolio return’sVaR with the dynamic weighting by assuming the skewed Student’s t margins.

The number of violations in Table 6 are the number of sample observations located out ofthe critical value and are calculated by two methods (i.e. the simulation method and tradi-tional variance–covariance method), and two weighting schemes (i.e. the equal weight and theinverse of the dynamic variance). The mean error shows the average absolute discrepancy permodel between the observed and expected number of violations for both the simulation methoddescribed in Section 3 and the traditional variance–covariance method. The smaller the meanerror, the better the model. From Table 6, using the simulation method with the inverse of vari-ance as the dynamic weights gives the smallest mean error (8.5) with α = 0.05 and α = 0.01.In other words, the return distribution of the portfolio is adequately portrayed by the model. Theconventional variance–covariance method does worse than the empirical quantile method, sincethe variance–covariance is limited to the normality assumption, and Table 3 has already shownthat neither market return meets the normality assumption. In addition, weighting is anotherimportant factor to estimate VaR adequately: the naïve equal weight appears overly simple, andthe inverse-variance weight accounts well for the data.

Finally, Figure 3 displays the VaR plot we estimate using the dynamic asymmetric skewed-tcopula with the GJR-GARCH(1,1) margins at α = 0.05 (dark grey line) and α = 0.01 (light greyline). Due to the limitation of space, VaR values are only calculated by using the inverse of theweighting dynamic variance. VaR is an estimate of the portfolio loss in the worst case scenariowith a relatively high level of confidence. In this figure VaR values of the portfolio are locatedalmost always below the portfolio returns, and it describes the expectation of investment losswell.

6. Conclusion

As an extension to the DAC-GARCH model, we propose the DDAC-GARCH model to allowfor the joint asymmetry caused by the negative shocks. And we explore constructing the samplecorrelation matrix for the shocks. The DDAC model permits modeling both dynamics and asym-metries in financial returns, and simultaneously characterizes the long-term trend in correlationfunction. We use this methodology to examine the dependence structure between China and USdaily stock market return over the period 1991–2013.

We demonstrate that the univariate distribution of daily returns may be well characterizedby the skewed Student’s t with time-varying volatility, and the joint distribution of dynamicvolatilities is characterized by the skewed-t copula with dynamic asymmetric correlations. Thedynamics of the dependency structure is parameterized by a time-varying correlation matrix, aconstant correlation, a lag term, as well as the negative return shock. Moreover, with the helpof the copula model, we model the tail dependence between SHECI and DJI. The empiricalevidence reveals that the dependency between China’s and US stock market trends up and thenlevels off. The dynamic correlation over time follow the same pattern very well.

Two simple tests are presented to test the adequacy of the DDAC-GARCH model. One is test-ing the existence of the negative shock on the dynamic asymmetric conditional correlation, theother is testing the existence of the time-varying effect in the dynamic correlation. The test resultsreveal the significance of dynamic double joint asymmetry. Thus, the DDAC-GARCH over-comes the shortcomings of the DAC-GARCH model, which ignores the impact of the negativeshock.

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In order to check how well the DDAC-GARCH model will work for the real data set, weapply the DDAC-GARCH model to estimate portfolio VaR. In this paper, the portfolio is com-posed of SHECI and DJI. Traditional risk measurement models mostly require the normalityassumption and commonly adopt constant weighting. However, empirical evidence shows thatthe true marginal distributions of financial assets commonly displays fat tail and skewness. Wethus employ the DDAC-GARCH model by using the simulation method with dynamic weightingto compute the portfolio VaR.

The comparison of the performance of the DDAC-GARCH model to that of the traditionalmethod shows that our model depicts VaR much better than the well-established variance–covariance method. Besides, the VaR obtained with dynamic weighting is much more accuratethan constant weighting. Therefore, the improved flexibility of DDAC-GARCH model has beendemonstrated that it is appropriate in studying highly volatile financial markets.

Whereas we have shown the DDAC-GARCH model is flexible and effective, it is currentlyrestricted to the specification of skew-t marginals and joint skew-t distributions. It would beinteresting and also more applicable to investigate the efficacy of our model to the more generalsettings by allowing different forms of marginals and joint distributions, as long as they arecompatible, as for example in [20].

Acknowledgements

This paper is supported by Shanghai University of International Business and Economics under grant YC-XK-13108 and085 Leading Academic Discipline Project and is supported by the Shanghai Pujiang Talent Plan (14PJ1404100) as well.

The author would like to thank the reviewers for providing valuable feedback.

Notes

1. In the sense of asset returns, hereinafter for conciseness.2. The unconditional correlation coefficient of two normalized sequences {xt} and {yt} is given by corr(x, y) =∑

(xt − E[x])(yt − E[y]).3. The computation procedure is robust and reliable. In addition, the model is identifiable when the longer time series

of returns are considered [11].4. Shanghai Stock Exchange Composite Index (SHECI).5. Dow Jones Industrial Average Index (DJI).6. Obviously, the timing is no surprising on a global scale: the dot-com bubble and the 2008 financial crisis.

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Appendix 1. �t is a correlation matrix

We will prove t defined in Equation (5) to be a correlation matrix.If a matrix is a correlation matrix, it should satisfy the following properties:

(1) symmetric;(2) having non-negative eigenvalues (or positive semi-definite).

We first demonstrate (1 − ϕ )� + ϕ ϒt , i.e. weighted average of correlation matrices, is still a correlation matrix.Since both � and ϒt are correlation matrices, they are both positive semi-definite. Therefore, for any vector z ∈ R

d ,zT�z ≥ 0 and zTϒtz ≥ 0 ; ϕ ∈ (0, 1), thus,

(1 − ϕ )zT�z ≥ 0 and ϕ zTϒtz ≥ 0, ∀ z ∈ Rd .

Then,

zT{(1 − ϕ )� + ϕ ϒt}z = {(1 − ϕ )zT�z + ϕ zTϒt}z ≥ 0,

hence (1 − ϕ )� + ϕ ϒt is positive semi-definite. Also, symmetry is preserved under the weighted average:

{(1 − ϕ )� + ϕ ϒt}T = (1 − ϕ )�T + ϕ ϒTt = (1 − ϕ )� + ϕ ϒt .

Therefore (1 − ϕ )� + ϕ ϒt is a correlation matrix.Likewise, the entire r.h.s. of Equation (5), a weighted average of correlation matrices, is also a correlation matrix.

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20 Y. Fang et al.

Appendix 2. The dynamic joint pdf

The dynamic joint pdf for the portfolio returns is

f (r1,t , r2,t | �t−1) = ct(u; t , λ∗, ν∗)

2∏i=1

∂ηi

∂εi

2∏i=1

∂εi

∂ri

= ct(u; t , λ∗, ν∗)

2∏i=1

fST(εi,t; λi, νi)

2∏i=1

1

σi,t

=2fMT(v; t , ν∗, 2)FT (λ∗ −1

t v√

(ν∗ + 2)/(vT −1t v + ν∗); ν∗ + 2)∏2

i=1 fST(F−1ST (ηi,t; λ∗, ν∗); λ∗, ν∗)

×2∏

i=1

fST(εi,t; λi, νi)

2∏i=1

1

σi,t. (A1)

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