Vine-copula GARCH model with dynamic conditional dependence

Computational Statistics and Data Analysis ( ) –

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Computational Statistics and Data Analysis

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Vine-copula GARCH model with dynamicconditional dependenceMike K.P. So ∗, Cherry Y.T. YeungDepartment of Information Systems, Business Statistics and Operations Management, Hong Kong University of Science and Technology,Hong Kong

a r t i c l e i n f o

Article history:Received 13 March 2013Received in revised form 11 August 2013Accepted 12 August 2013Available online xxxx

Keywords:CopulaGARCHTime varying dependenceVine decomposition

a b s t r a c t

Constructingmultivariate conditional distributions for non-Gaussian return series has beenamajor research agenda recently. CopulaGARCHmodels combine the use ofGARCHmodelsand a copula function to allow flexibility on the choice of marginal distributions anddependence structures. However, it is non-trivial to define multivariate copula densitiesthat allow dynamic dependent structures in returns. The vine-copula method has beengaining attention recently in that a multi-dimensional density can be decomposed into aproduct of conditional bivariate copulas andmarginal densities. The dependence structureis interpreted individually in each copula pair. Yet, most studies have only focused on timevarying correlation. A vine-copula GARCH model with dynamic conditional dependenceis proposed. A generic approach to specifying dynamic conditional dependence usingany dependence measures is developed. The characterization also induces multivariateconditional dependence dynamically through vine decomposition. The main idea is toincorporate dynamic conditional dependence, such as Kendall’s tau and rank correlation,not tomention linear correlation, in each bivariate copula pair. The estimation is conductedthrough a sequential approach. Simulation experiments are performed and five Hong Kongblue chip stock data from January 2004 to December 2011 are studied. Using t and twoArchimedean copulas, it is revealed that Kendall’s tau and linear correlation of the stockreturns vary over time, indicating the presence of time varying properties in dependence.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

A large literature has contributed to modeling conditional dependence for multivariate financial time series. Popularestimationmethods of conditional correlationweremultivariateGARCH (MGARCH)models, such as the constant conditionalcorrelation (CCC)-GARCHmodels in Bollerslev (1990), the VECmodel of Bollerslev et al. (1988), the BEKKmodel of Engle andKroner (1995) and the dynamic correlation (DCC)-GARCHmodels of Engle (2002), Tse and Tsui (2002) and Asai andMcAleer(2009). These MGARCH methods estimate conditional dependence via a correlation matrix or covariance matrix. Yet, theassumptions on the distributions for each return series are often limited to using either normal distribution, t distributionor other elliptical distributions such that the joint distribution can be explicitly defined.

Sklar (1959) introduced the copula function, a joint distribution with arguments from uniform distributions. It describesdependence between random variables. Joe (1997) and Nelson (2006) discussed different types of copulas and theirproperties in detail. Aas and Berg (2009), Austin and Lopes (2010), Dias and Embrechts (2010), Longin and Solnik (2001)and Patton (2006a) introduced copula GARCH models, where a joint density function is modeled separately for marginal

∗ Corresponding author. Tel.: +852 23587726.E-mail addresses: [email protected] (M.K.P. So), [email protected] (C.Y.T. Yeung).

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2 M.K.P. So, C.Y.T. Yeung / Computational Statistics and Data Analysis ( ) –

time series, and amulti-dimensional copula density. Embrechts et al. (2003) and Joe (1997) discussed dependence structuresfrommultivariate Archimedean copulas. Frey andMcNeil (2003) applied latent variablemodels as Bernoulli mixturemodelsand described the dependence using exchangeable Archimedean copula. A survey on copula GARCHmodels was conductedby Fischer et al. (2009). Some papers have incorporated time-varying dependence structures into copula-GARCH models:see the work of Austin and Lopes (2010), Dias and Embrechts (2010), Jondeau and Rockinger (2006) and Patton (2006b).Although copula-GARCHmodels allow flexibility in the selection of types of copula functions, there is doubt about whetherone or two copula parameters in the multi-dimensional copula density are accurate enough to interpret all dependence foreach pair of financial time series.

Vine-copula GARCHmodels are gaining increased attention. Vines were first proposed by Joe (1997). Bedford and Cooke(2002) explored vines as graphical models and presented the general construction of regular vines. Vines are among thegraphical models in which conditional dependence exists for dependent random variables. Vine decomposition is crucialin understanding the dependence of each pair of return series, in terms of bivariate conditional copulas. Kurowicka andCooke (2006) proposed Gaussian vines. Aas et al. (2009) gave the general construction of a vine-copula GARCH, includingthe simulation algorithm, model selection and empirical study of tail dependence for canonical vines and D vines. Czadoand Min (2010) used the MCMC method to find confidence intervals for parameters in pair copula construction andmodel the tail dependence of each pair of copulas. For the dynamic of correlation, Patton (2006b) used ten lags of pastobservations while Dias and Embrechts (2010) used only one lag of past observations with Fisher transformation, ensuringthat the correlation was within the range of −1 and 1. Other examples of vine-copulas models were considered byKim et al. (2011), Nikoloulopoulos et al. (2010) and Smith et al. (2010). Most of the literature has studied correlation,assuming it to be either constant or time varying. However, copulas have to be elliptical such that there is an explicitcorrelation parameter. This may restrict the choice of copula functions. If time varying dependence other than correlationis allowed, properties such as nonlinear relationships and tail dependence over time can be observed. Besides, moststudies have focused on bivariate applications. In financial markets, risk managers may want to look at a few stocks oreven portfolios of higher dimensions simultaneously. Thus, an extension of modeling to multivariate application would beideal.

In short, for an adequate model to construct dependence, two criteria must be satisfied.

1. Dependence is time varying and not limited to linear correlation.2. The application should work other than bivariate cases.

This paper makes two main contributions to the literature. First, it develops a generic approach to specifying dynamicconditional dependence using any dependence measures. Second, multivariate conditional dependence is induceddynamically through vine decomposition. In other words, we build time varying conditional dependence, other thanjust linear correlation, in a structured way for multi-dimensional problems through vine-copula GARCH models. Almeidaand Czado (2012) proposed time varying dependence for Kendall’s tau using latent variables and the inverse Fishertransformation of Kendall’s tau for bivariate cases. However, due to the observation-driven characteristics in the likelihoodfunction of our vine-copula GARCH models, stepwise estimation may be more computationally feasible than the MCMCmethods used in the model of Almeida and Czado (2012). Flexible copula functions, besides Gaussian copulas and tcopulas, are applicable to highlight properties such as asymmetry and tail dependence using vine decomposition, etc..The estimation of dynamic conditional dependence also contributes to studying the relationship of stocks at differenttimes. The dependence measures focused in this paper are conditional linear correlation, rank correlation and Kendall’stau.

The reminder of this paper is presented as follows. Section 2 presents the vine-copula GARCH model with considerationof time varying conditional dependence. Section 3 covers estimation inference and simulation studies. Section 4 presentsreal financial data estimation using five blue chip stocks in Hong Kong. Section 5 concludes the paper.

2. Vine-copula GARCHmodel with dynamic dependence

2.1. Model form

Suppose a collection of p financial returns is expressed by a multivariate vector rt = {r1t , . . . , rpt} for t = 1, . . . , T .Let rxt , ryt and rzt be vector variable sets and F [t](rxt |rzt) be a conditional distribution of rxt given the conditioning set rztand information up to time t − 1, Ft−1, with marginal distribution parameters θx|z . If the conditioning set rzt is empty,F [t](rxt |rzt) simply represents the distribution of rxt given Ft−1. In line with Sklar (1959), the multivariate distribution of rxtand ryt conditional on Ft−1 can be expressed by a copula function, denoted by C [t]

xy (F[t](rxt), F [t](ryt)). Using the same idea,

the conditional distribution of rxt and ryt given rzt and Ft−1 is specified by a copula function C [t]xy|z(F

[t](rxt |rzt), F [t](ryt |rzt))where F [t](rxt |rzt) and F [t](ryt |rzt) are conditional distributions of rxt and ryt , respectively, given the conditioning set rztand Ft−1, with copula parameters ϑxy|z,t . All copula distributions and marginal distributions considered in this paperare assumed to be continuous at any time such that their density functions c[t]

xy|z and f [t] corresponding to C [t]xy|z and F [t]

exist.

M.K.P. So, C.Y.T. Yeung / Computational Statistics and Data Analysis ( ) – 3

Fig. 1. Tree decomposition showing canonical vines (left) and D vines (right) in five dimensions.

Sklar (1959) expressed the conditional joint cumulative distribution function for rt conditioning on Ft−1 in terms of ap-dimensional copula, which is

F [t](r1t , . . . , rpt) = C [t]1,...,p(F

[t]1 (r1t), . . . , F

[t]p (rpt)).

The joint density for rt is

f [t](r1t , . . . , rpt) = c[t]1,...,p(F

[t]1 (r1t), . . . , F

[t]p (rpt))

pi=1

f [t]i (rit), (1)

where C [t]1,...,p is the copula function of r1t , . . . , rpt givenFt−1 and c[t]

1,...,p is its copula density. A vine structure decomposes thecopula density in (1) into a product of conditional copulas with lower dimensions. Among the set of regular vines, canonicalvines and D vines are the most common choices. Bedford and Cooke (2001) defined regular vines on p variables. In regularvines, there are (p − 1) tree levels. In the first tree level, nodes are connected by edges among the p variables. For the i-thtree level, with i = 2 . . . , p− 1, new nodes are obtained from the edge set in (i− 1)-th tree level. New edges are formed onthe i-th level by connecting the nodes. The resulting dimension of the conditioning sets of the edges is one higher than thedimension of the nodes for i = 2, . . . , p − 1. The p-dimensional density function using canonical vines is

f [t](r1t , . . . , rpt) =

p−1i=1

p−ij=1

c[t]i,i+j|1,...,i−1(F

[t](rit |r1t , . . . , ri−1,t), F [t](ri+j,t |r1,t , . . . , ri−1,t))

pi=1

f [t]i (rit), (2)

and the p-dimensional density function using D vines is

f [t](r1t , . . . , rpt) =

p−1i=1

p−ij=1

c[t]j,i+j|j+1,...,i+j−1(F

[t](rjt |rj+1,t , . . . , ri+j−1,t), F [t](ri+j,t |rj+1,t , . . . , ri+j−1,t))

pi=1

f [t]i (rit). (3)

In (2) and (3), f [t]i is the marginal distribution of rit conditional on Ft−1. A key contribution of this paper is to define the

conditional copula c[t]1,...,p with dynamic dependence, which is presented in Section 2.2. The construction of c[t]

1,...,p is basedon vine decomposition. Fig. 1 illustrates the tree decomposition for canonical vines and D vines. The difference in the treestructure explains the major dissimilarity in the two vines. Canonical vines highlight one variable that tends to dominatethe others, while D vines treat every variable equally. Eqs. (2) and (3) do not show any difference for dimensions lowerthan or equal to three as there is only one possible decomposition. For dimensions higher than three, the conditioning setsbetween canonical vines and D vines are different. We propose methods to define dynamic conditional pairwise copulas,e.g. c[t]

i,i+j|1,...,i−1 in (2) and c[t]j,i+j|j+1,...,i+j−1 in (3), and their constructions will be discussed below.

2.2. Dynamic conditional dependence

Most discussions on dynamic conditional dependence such as those of Austin and Lopes (2010) and Jondeau andRockinger (2006) have focused on conditional correlation in elliptical copulas. Archimedean copulas and other families ofcopulas, which do not have correlation parameters, cannot be extended easily to allow time-varying linear correlations.If there exists a monotonic function between copula parameters and any conditional dependence, making the copulaparameters time varying will induce time-varying conditional dependence.


The flexibility of copulas is considered to measure dynamic conditional rank correlation and Kendall’s tau. Rankcorrelation examines dependence between the rank of variables, while Kendall’s tau concerns concordant and discordantpairs. For convenience, denotew(ϕ[t]

xy|z) as the conditional dependence at time t as a function of the dependence parameterϕ

[t]xy|z , which is one of the copula parameters in ϑxy|z,t . Table 1 summarizes the dynamic conditional dependence for various

copula functions. Conditional linear correlation is a function of the copula parameter ϕ[t]xy|z in the two elliptical copulas:

the Gaussian copula and t copula. The conditional Kendall’s tau has closed forms for common Archimedean copulas suchas Gumbel copulas and Clayton copulas. For example, copula parameters in conditional t copulas contain a correlationparameter, so the conditional linear correlation for t copulas,w(ϕ[t]

xy|z), does not need a transformation, i.e. it is given by

w(ϕ[t]xy|z) = ϕ

[t]xy|z . (4)

As the copula parameter ϕ[t]xy|z is not the conditional Kendall’s tau in Clayton copulas, a transformation is required. The

conditional Kendall’s tau, denoted byw(ϕ[t]xy|z), can be expressed as

w(ϕ[t]xy|z) =

ϕ[t]xy|z

ϕ[t]xy|z + 2

. (5)

Similarly, the conditional Kendall’s tau for Gumbel copulas, denoted byw(ϕ[t]xy|z), is given by

w(ϕ[t]xy|z) =

ϕ[t]xy|z − 1

ϕ[t]xy|z

. (6)

One advantage of conditional rank correlation andKendall’s tau over linear correlation is that they are invariant under strictlyincreasing component-wise transformations. As measures of concordance, this propertymakes conditional rank correlationand Kendall’s tau more useful.

Inspired by the DCC-GARCH models of Tse and Tsui (2002) and Engle (2002), a time-varying property is incorporatedinto the conditional dependence. We propose dynamic conditional dependence w(ϕ[t]

xy|z) between return variables rxt andryt given rzt and Ft−1 as

w(ϕ[t]xy|z) = (1 − axy|z − bxy|z)w(ϕxy|z)+ axy|zξxy|z,t−1 + bxy|zw(ϕ

[t−1]xy|z ), (7)

where w(ϕxy|z) is the long run dependence, ξxy|z,t is a sample conditional dependence and axy|z and bxy|z are coefficientsassociated with the sample conditional dependence ξxy|z,t−1 and conditional dependencew(ϕ[t−1]

xy|z ), respectively. Followingthe stationarity condition forDCC-GARCHmodels, the conditions 0 ≤ axy|z, bxy|z ≤ 1, axy|z+bxy|z ≤ 1 and−1 ≤ w(ϕ

[t]xy|z) ≤ 1

are applied.Note that ξxy|z,t is the sample conditional dependence between variable rxt and ryt given the pastm-period data and rzt .We

consider three dependence measures: namely linear correlation, rank correlation and Kendall’s tau. Conditional marginaldistribution F [t](rxt |rzt) and F [t](ryt |rzt) are valued between 0 and 1. For linear correlation, we transform F [t](rxt |rzt) andF [t](ryt |rzt) to a value in [−∞,∞], represented by rx|z,t and ry|z,t . This is obtained by finding the inverse of the correspondingcopulas for the pair of variables. From Table 1, conditional linear correlation is only applicable to elliptical copulas. Ifconditional t copula is used, the transformation function is rx|z,t = t−1

vxy|z(F [t](rxt |rzt)) and ry|z,t = t−1

vxy|z(F [t](ryt |rzt)), where

vxy|z is the degrees of freedom from the t copula c[t]xy|z . As rank correlation and Kendall’s tau are invariant under monotonic

transformation, conditionalmarginal distributions F [t](rxt |rzt) and F [t](ryt |rzt)donot need any transformation. In these cases,rx|z,t and ry|z,t are simply defined as F [t](rxt |rzt) and F [t](ryt |rzt) for conditional rank correlation and Kendall’s tau. This isapplicable to all copulas as long as there is a closed form expression for the conditional rank correlation or Kendall’s tauw(ϕ[t]). Sample conditional dependence ξxy|z,t−1 is summarized as follows.1. Dynamic conditional linear correlation

The sample conditional linear correlation betweenm pairs of variables rx|z,t−i and ry|z,t−i is defined as

ξxy|z,t−1 =

mi=1

rx|z,t−i ry|z,t−imi=1

r2x|z,t−i

mj=1

r2y|z,t−j

, (8)

where m is a positive integer greater than 1. Eq. (4) is motivated by the dynamic correlation formulation of Tse and Tsui(2002). It generalizes Austin and Lopes (2010) to handle a dimension greater than two, i.e. p > 2, and apply it to conditionalcopulas.


Table 1Summary of dynamic conditional dependence for different types of copulas and dependence from Sklar (1959). The table includes dependence functionsin terms of copula parameters for elliptical copulas, Archimedean copulas and different copula families. The corresponding copula cumulative distributionfunctions are provided. Denote ux|z,t = F [t](rxt |rzt ) and uy|z,t = F [t](ryt |rzt ). M is the upper Fréchet–Hoeffding bound. Π is the independence copula andW is the lower Fréchet–Hoeffding bound.

Copula Transformation functionw(ϕ[t]xy|z) Conditional copula function C [t]

x,y|z(ux|z,t , uy|z,t ) Dependence parameter ϕ[t]xy|z

Conditional linear correlation

Normal copulas ϕ[t]xy|z Φ(Φ−1(ux|z,t ),Φ

−1(uy|z,t )) 0 ≤ ϕ[t]xy|z ≤ 1

T -copulas ϕ[t]xy|z tv(t−1

v (ux|z,t ), t−1v (uy|z,t )) 0 ≤ ϕ

[t]xy|z ≤ 1

Conditional rank correlation

Fréchet family ofcopulas

α[t]xy|z − β

[t]xy|z α

[t]xy|zM + (1 − α

[t]xy|z − β

[t]xy|z)Π + β

[t]xy|zW ϕ

[t]xy|z = {α

[t]xy|z , β

[t]xy|z},

α[t]xy|z , β

[t]xy|z ∈ [0, 1],

α[t]xy|z + β

[t]xy|z < 1

Farlie–GumbelMorgensternfamily of copulas

ϕ[t]xy|z/3 ux|z,tuy|z,t + ϕ

[t]xy|zux|z,tuy|z,t

(1 − ux|z,t )(1 − uy|z,t )

−1 ≤ ϕ[t]xy|z ≤ 1

Marshall–Olkinfamily of copulas

(3α[t]xy|zβ

[t]xy|z)/(2α

[t]xy|z − α

[t]xy|zβ

[t]xy|z + 2β [t]

xy|z) min(u1−α[t]

xy|zx|z,t uy|z,t , ux|z,tu

1−β[t]xy|z

y|z,t ) ϕ[t]xy|z = {α

[t]xy|z , β

[t]xy|z},

0 < α[t]xy|z , β

[t]xy|z < 1

Raftery family ofcopulas

(ϕ[t]xy|z(4 − 3ϕ[t]

xy|z))/(2 − ϕ[t]xy|z)

2 min(ux|z,tuy|z,t )+1−ϕ[t]

xy|z

1+ϕ[t]xy|z(ux|z,tuy|z,t )

11−ϕ[t]

xy|z

(1 − (max(ux|z,t , uy|z,t ))−(1+ϕ[t]

xy|z )/(1−ϕ[t]xy|z ))

0 ≤ ϕ[t]xy|z ≤ 1

Conditional Kendall’s tau

Gumbel copulas (ϕ[t]xy|z − 1)/ϕ[t]

xy|z exp(−((− ln ux|z,t )ϕ

[t]xy|z

+ (− ln uy|z,t )ϕ

[t]xy|z )

1/ϕ[t]xy|z )

ϕ[t]xy|z ≥ 1

Clayton copulas ϕ[t]xy|z/(ϕ

[t]xy|z + 2) max((u

−ϕ[t]xy|z

x|z,t + u−ϕ

[t]xy|z

y|z,t − 1)−1/ϕ[t]xy|z , 0) ϕ

[t]xy|z > 0

Fréchet family ofcopulas

(α[t]xy|z − β

[t]xy|z)(α

[t]xy|z − β

[t]xy|z + 1)/3 α

[t]xy|zM + (1 − α

[t]xy|z − β

[t]xy|z)Π + β

[t]xy|zW ϕ

[t]xy|z = {α

[t]xy|z , β

[t]xy|z},

α[t]xy|z , β

[t]xy|z ∈ [0, 1],

α[t]xy|z + β

[t]xy|z < 1

Marshall–Olkinfamily of copulas

α[t]xy|zβ

[t]xy|z/(α

[t]xy|z − α

[t]xy|zβ

[t]xy|z + β

[t]xy|z) min(u

1−α[t]xy|z

x|z,t uy|z,t , ux|z,tu1−β[t]

xy|zy|z,t ) ϕ

[t]xy|z = {α

[t]xy|z , β

[t]xy|z},

α[t]xy|z , β

[t]xy|z ∈ (0, 1)

Raftery family ofcopulas

2ϕ[t]xy|z/(3 − ϕ

[t]xy|z) min(ux|z,tuy|z,t )+

1−ϕ[t]xy|z

1+ϕ[t]xy|z(ux|z,tuy|z,t )

11−ϕ[t]

xy|z

(1 − (max(ux|z,t , uy|z,t ))−(1+ϕ[t]

xy|z )/(1−ϕ[t]xy|z ))

0 ≤ ϕ[t]xy|z ≤ 1

2. Dynamic conditional rank correlationThe sample conditional rank correlation at time t − 1 is measured using m pairs of ranked variables of rx|z,t−i and ry|z,t−i,denoted by r∗

x|z,t−i and r∗

y|z,t−i. The ranked variables are integers ranging from 1 up to m, with i = 1, . . . ,m, m ≥ 1. It isgiven by

ξxy|z,t−1 =

mi=1

r∗

x|z,t−ir∗

y|z,t−imi=1

r∗

x|z,t−i2

mj=1

r∗

y|z,t−j2

, (9)

where m is a positive integer greater than 1.3. Dynamic conditional Kendall’s tauThe sample Kendall’s tau is related to the difference in the number of concordant pairs and the number of discordant pairsbetween m pairs of variables, rx|z,t−i and ry|z,t−i with i = 1, . . . ,m. The indicator function for concordant and discordantpairs (rx|z,t−i, ry|z,t−i) and (rx|z,t−j, ry|z,t−j), where i = j, are given respectively by

ICt−i,t−j =

1 if (rx|z,t−i − rx|z,t−j)(ry|z,t−i − ry|z,t−j) > 0,0 otherwise,

and

IDt−i,t−j =

1 if (rx|z,t−i − rx|z,t−j)(ry|z,t−i − ry|z,t−j) < 0,0 otherwise.


The sample conditional Kendall’s tau ξxy|z,t−1 at time t − 1 is given by

ξxy|z,t−1 =

mi=1

mj=1

ICt−i,t−j −mi=1

mj=1

IDt−i,t−j

m(m − 1)/2, (10)

where m is a positive integer greater than 1. The novelty of the formulation of w(ϕ[t]xy|z) in (7) is that it can be applied to

dependence measures other than linear correlation. As (7) is built under vine decomposition, the dynamic structure can beadopted in high-dimensional time series rather than limiting to p = 2 in many of the existing works on dynamic copulas.

3. Estimation and simulation study

3.1. Computational issues and inference

The conditional distribution function F [t](xt |vt)plays a crucial role in estimation inference and is associatedwith bivariateconditional copulas (see, Joe (1996, p. 125)) as follows:

F [t](xt |vt) =

∂C [t]x,vj|v−j

(F [t](xt |v−j,t), F [t](vj,t |v−j,t))

∂F [t](vj,t |v−j,t), (11)

where vt is a random vector at time t and v−j,t is a vector of vt without the j-th variable. For example, the conditionaldistribution in canonical vines is

F [t](rj,t |r1t , . . . , ri−1,t) =∂C [t]

j,i−1|1,...,i−2(F[t](rj,t |r1t , . . . , ri−2,t), F [t](ri−1,t |r1t , . . . , ri−2,t))

∂F [t](ri−1,t |r1t , . . . , ri−2,t),

where j = i, i + 1, . . . , p. The conditioning set {r1t , . . . , ri−2,t} contains one less variable than {r1t , . . . , ri−1,t} in the targetdistribution. This indicates that conditional distributions can always be calculated with conditional copulas from lower treelevels. The derivative on the right-hand side of (11) is usually expressed as a h-function, h[t]

xvj|v−j, i.e.

h[t]xvj|v−j

(F [t](xt |v−j,t), F [t](vj,t |v−j,t)) = F [t](xt |vt). (12)

The h-function for canonical vines is

h[t]j,i−1|1,...,i−2(F

[t](rj,t |r1t , . . . , ri−2,t), F [t](ri−1,t |r1t , . . . , ri−2,t)) = F [t](rj,t |r1t , . . . , ri−1,t),

where j = i, i + 1, . . . , p. Denote h[t]−1xvj|v−j

as the inverse of the h-function at time t with respect to F [t](xt |v−j,t) in (12). Ifwe are given F [t](xt |vt) and F [t](vj,t |v−j,t), then F [t](xt |v−j,t) can be obtained using the inverse of the h-function by

F [t](xt |v−j,t) = h[t]−1xvj|v−j

(F [t](xt |vt), F [t](vj,t |v−j,t)).

We observe that a characteristic of the inverse of the h-function is the reduction of the dimension of the conditioning setof F [t](xt |vt) by one. In other words, we calculate F [t](xt |v−j,t) from F [t](xt |vt) by h[t]−1

xvj|v−j. Observed in the conditional

distribution in (11), the h-function h[t]xvj|v−j

and its inverse h[t]−1xvj|v−j

are associated with the conditional copulas C [t]x,vj|v−j

withcopula parameters ϑx,vj|v−j,t . Table 2 gives the copula distribution functions, their density functions, h-functions and theinverse of h-functions of Gaussian copulas, t-copulas, Gumbel copulas and Clayton copulas for variable r1t and r2t . In thetable, u1t and u2t represent the cumulative distributions such that u1t = F [t]

1 (r1t) and u2t = F [t]2 (r2t). The inverse of the

h-function for Gumbel copulas is not explicit, but it can be obtained numerically by the Newton–Raphson method.The likelihood function in (2) and (3) or other vine structures comprises two components with one carrying marginal

distributions and the other carrying bivariate conditional copulas. The marginal distributions are estimated individuallyfor each return series. The estimation algorithm for copula parameters is summarized below. For illustration purposes, weassume that canonical vine decomposition is used. A sequential maximum likelihood estimation algorithm is as follows.

1. Consider the tree level of i = 1; the conditional (unconditional for the first tree level) distributionsF [t](rit |r1t , . . . , ri−1,t), . . . , F [t](rpt |r1t , . . . , ri−1,t) are given.

2. For the tree node of j = 1, we want to maximize the log likelihood function of the conditional copula densityc[t]i,i+j|1,...,i−1(F

[t](rit |r1t , . . . , ri−1,t), F [t](ri+j,t |r1t , . . . , ri−1,t))with the copula parameter ϑi,i+j|1,...,i−1,t in (2) given by

li,i+j =

Tt=1

log c[t]i,i+j|1,...,i−1(F

[t](rit |r1t , . . . , ri−1,t), F [t](ri+j,t |r1t , . . . , ri−1,t)). (13)


Table 2Summary of the copula distributions, density functions, h-function and inverse of the h-function forGaussian copula, t copulas, Gumbel copulas and Clayton copulas with u1t = F [t]

1 (r1t ) and u2t = F [t]2 (r2t ).

Gaussian copulas, ϑ12,t = {ϕ[t]12 }, −1 ≤ ϕ

[t]12 ≤ 1

C [t]12 (u1t , u2t ) = Φ(Φ−1(u1t ),Φ

−1(u2t ))

c[t]12 (u1t , u2t ) =

11−ϕ[t]

122exp(− ϕ

[t]12

2((Φ−1(u1t ))2+(Φ−1(u2t ))2)−2ϕ[t]

12Φ−1(u1t )Φ−1(u2t )

2(1−ϕ[t]12

2)

)

h[t]12(u1t , u2t ) = Φ(

Φ−1(u1t )−ϕ[t]12Φ

−1(u2t )1−ϕ[t]

122

)

h[t]−112 (u1t , u2t ) = Φ(Φ−1(u1t )

1 − ϕ

[t]12

2+ ϕ

[t]12Φ

−1(u2t ))

t copulas, ϑ12,t = {ϕ[t]12 , v}, −1 ≤ ϕ

[t]12 ≤ 1

C [t]12 (u1t , u2t ) = tv(t−1

v (u1t ), t−1v (u2t ))

c[t]12 (u1t , u2t ) =

Γ ( v+22 )Γ ( v2 )

(Γ ( v+12 ))2

2i=1(1+

t−1v (uit )2

v )−v+12

1−ϕ[t]

122(1 +

t−1v (u1t )2+t−1

v (u2t )2−2ϕ[t]12 t

−1v (u1t )t

−1v (u2t )

v(1−ϕ[t]12

2)

)−v+22

h[t]12(u1t , u2t ) = tv+1(

t−1v (u1t )−ϕ

[t]12 t

−1v (u2t )

(v+(t−1v (u2t ))2)(1−ϕ

[t]12

2)

v+1

)

h[t]−112 (u1t , u2t ) = tv(t−1

v+1(u1t )

(v+t−1

v (u2t )2)(1−ϕ[t]12

2)

v+1 + ϕ[t]12 t

−1v (u2t ))

Gumbel copulas (Venter, 2001), ϑ12,t = {ϕ[t]12 }, ϕ[t]

12 ≥ 1

C [t]12 (u1t , u2t ) = exp(−((− ln u1t )

ϕ[t]12 + (− ln u2t )

ϕ[t]12 )1/ϕ

[t]12 )

c[t]12 (u1t , u2t ) = C [t]

1,2(u1t , u2t )(log u1t log u2t )

ϕ[t]12 −1

u1t u2t((− log u1t )

ϕ12,t

+ (− log u2t )ϕ

[t]12 )−2+2/ϕ[t]

12 (1 + (ϕ[t]12 − 1)((− log u1t )

ϕ[t]12 + (− log u2t )

ϕ[t]12 )−1/ϕ[t]

12 )

h[t]12(u1t , u2t ) = C [t]

1,2(u1t , u2t )1u2t(− log u2t )

ϕ[t]12 −1((− log u1t )

ϕ[t]12 + (− log u2t )

ϕ[t]12 )1/ϕ

[t]12 −1

Clayton copulas, ϑ12,t = {ϕ[t]12 }, ϕ[t]

12 ≥ 0

C [t]12 (u1t , u2t ) = (u

−ϕ[t]12

1t + u−ϕ

[t]12

2t − 1)−1/ϕ[t]12

c[t]12 (u1t , u2t ) =

1+ϕ[t]12

(u1t u2t )1+ϕ[t]

12(u

−ϕ[t]12

1t + u−ϕ

[t]12

2t − 1)−1/ϕ[t]12 −2

h[t]12(u1t , u2t ) = u

−ϕ[t]12 −1

2t (u−ϕ

[t]12

1t + u−ϕ

[t]12

2t − 1)−1/ϕ[t]12 −1

h[t]−112 (u1t , u2t ) = (1 − u

−ϕ[t]12

2t + (u1tuϕ

[t]12 +1

2t )−

ϕ[t]12

ϕ[t]12 +1 )−1/ϕ[t]

12

The corresponding dependence parametersϕ[t]i,i+j|1,...,i−1 must be obtained first. For each time point t , calculate the sample

conditional dependence ξi,i+j|1,...,i−1,t−1 using either (8), (9) or (10), depending on the type of dependence of interest. Theconditional dependence w(ϕ[t]

i,i+j|1,...,i−1) is calculated using ξi,i+j|1,...,i−1,t−1 and w(ϕ[t−1]i,i+j|1,...,i−1) in (7). The dependence

parameter ϕ[t]i,i+j|1,...,i−1 at time t is then determined fromw(ϕ

[t]i,i+j|1,...,i−1) and so the copula parameter ϑi,i+j|1,...,i−1,t (see

Table 2 for some examples) can be substituted into the log likelihood function (13).3. Again from (2), repeat step 2 with j = 2 until j = p − i. As long as the tree level has not reached p − 1, consider the tree

node j = i + 1 and calculate the conditional distribution F [t](rj,t |r1t , . . . , rit) which is used in the next tree level and iscalculated using the h-function in (12), i.e.

F [t](rj,t |r1t , . . . , rit) = h[t]j,i|1,...,i−1(F

[t](rj,t |r1t , . . . , ri−1,t), F [t](ri,t |r1t , . . . , ri−1,t)).

This h-function is associated with the conditional copulas in the current tree level i, which is c[t]j,i|1,...,i−1,t , with the copula

parameter ϑi,i+j|1,...,i−1,t . Repeat this step with j = i + 1 until j = p.4. After obtaining the conditional distributions F [t](ri+1,t |r1t , . . . , rit), . . . , F [t](rp,t |r1t , . . . , rit) from step 3, go back to step

1 for i = 2 to i = p − 1.

If other vine decompositions are used, the log likelihood function in step 2 will be revised accordingly. As a result, theconditional distributions in step 3, which are to be used in the next tree level, will change.

3.2. Approximation of time varying pairwise dependence

The dynamic dependence is conditional on conditioning sets rzt andFt−1. In addition, computations are needed to convertthe conditional dependence into pairwise dependence, which is conditional on Ft−1 only. We discuss three methods in thefollowing.


Method 1: partial correlation

Proposed by Kurowicka and Cooke (2006), partial correlation is applied to approximate pairwise correlations for ellipticalcopulas. Assume that the size of the conditioning set rzt is i. The conditional correlation ϕ

[t]xy|z , which describes the correlation

between rxt and ryt given rzt , has a relationship with the conditional correlation ϕ[t]xy|z−j

, where z−j is the vector of z without

the j-th component and has a size of i − 1. Rummel (1976) gave the conditional correlation ϕ[t]xy|z−j

as

ϕ[t]xy|z−j

= ϕ[t]x,y|z

(1 − ϕ

[t]xzj|z−j

2)(1 − ϕ

[t]yzj|z−j

2)+ ϕ

[t]xzj|z−j

ϕ[t]yzj|z−j

, (14)

where zj is the j-th component of z. By recursively reducing the dimension of z by one using (14), pairwise correlation ϕ[t]xy

can be obtained. However, this method is not applicable to non-elliptical copulas such as Gumbel or Clayton copulas. It alsocannot be applied to Kendall’s tau or rank correlation.

Method 2: integration

The second method is based on integrating copula functions or densities. This method allows all copulas to have explicitcopula density functions in vine decomposition to find Kendall’s tau and rank correlation. Here, equations are derived undercanonical vine decomposition. For convenience, denote uit = F [t]

i (rit) for i = 1, . . . , p. Consider the pairwise rank correlationin the first tree level given by

ρ[t]1i = 12

[0,1]2

C [t]1i (u1t , uit)du1tduit − 3,

where i = 2, . . . , p. The pairwise rank correlation for the second tree level can be expressed by integrating copula densitiesas

ρ[t]2i = 12

1

0

1

0

u3t

0

u2t

0

1

0c[t]2i|1(h

[t]21(u

,2t , u

,1t), h

[t]i1 (u

,it , u

,1t))

c[t]12 (u

,1t , u

,2t)c

[t]1i (u

,1t , u

,it)du

,1tdu

,2tdu

,3tdu2tdu3t − 3, (15)

where i = 3, . . . , p. The proof of (15) is discussed in Appendix B. Similarly, the pairwise Kendall’s tau for the first tree levelis

τ[t]1i = 4

[0,1]2

C [t]1i (u1t , uit)c

[t]1i (u1t , uit)du1tduit − 1,

where i = 2, . . . , p. The pairwise Kendall’s tau for the second tree level is given by

τ[t]2i = 4

1

0

1

0

1

0

uit

0

u2t

0

1

0c[t]2i|1(h

[t]21(u

,2t , u

,1t), h

[t]i1 (u

,it , u

,1t))

c[t]12 (u

,1t , u

,2t)c

[t]1i (u

,1t , u

,it)c

[t]2i|1(h

[t]21(u2t , u1t), h

[t]i1 (uit , u1t))

c[t]12 (u1t , u2t)c

[t]1i (u1t , uit)du

,1tdu

,2tdu

,itdu1tdu2tduit − 1, (16)

where i = 3, . . . , p. The proof of (16) is presented in Appendix B. Because the dimension of the integrals increasessubstantially when the tree level moves up, this method is only applicable to obtaining ϕ[t]

xy , which involves low level trees.For high level trees associated with the dimension of z greater than two, we adopt the simulation method described below.

Method 3: simulation

This method can estimate any type of pairwise dependence. The main idea is to recursively simulate p-dimensional returnseries for t = 1, . . . , T with the estimated parameters. For a given time t , a sample of k independent return realizationsis simulated. In each return realization, p independent uniformly distributed series are generated first. Using the inverseof the h-function, p dependent uniformly distributed series are obtained sequentially from the independent series. Thedependent uniformly distributed series can then be transformed into return series with appropriate inverse distributions.Sample pairwise dependence can then be calculated for every pair of return series using the k returns at time t to estimateϕ

[t]xy|z . Details of the simulation procedure are presented in the next section using p = 5 as an example.The characteristics of the three approximation methods are summarized in Table 3. Partial correlation is not sufficient

to approximate dependence for non-elliptical copulas. This restricts us from understanding time varying dependence inArchimedean copulas or other copula families. Compared with partial correlation and integration, which have limitationsin the type of dependence for approximation, simulation does well in obtaining approximated dependence for the copulafunctions considered in this study.


Table 3This table compares the three approximation methods for pairwise dependence, including the type of copulaapplicable, type of dependence for approximation and number of pairwise dependences obtained.

Methods Type of copulasapplicable

Type of dependence No. of pairwisedependences obtained

Partial correlation Elliptical copulas Linear correlation AllIntegration All copulas Rank correlation, Kendall’s tau The first two tree levelsSimulation All copulas Any dependence All

3.3. Simulation study

In the simulation, five dimensional problems are considered. Conditional on Ft−1 the marginal density functionsf [t]1 (r1t), . . . , f

[t]5 (r5t) are derived from a GARCH(1, 1)-t innovationmodel. Assume that each financial return series hasmean

zero and conditional variance given by σ 2it , i.e.

rit = σitεit , (17)

σ 2it = wi + αir2it + βiσ

2i,t−1, (18)

where εit is the innovation for the i-th series and is distributed independently and identically as a standardized t distributionwith vi degrees of freedom. Thus, the marginal distribution parameters for the i-th return series are θi = {wi, αi, βi, vi}. Weimpose usual constraints for positive variance and covariance stationarity, αi > 0, βi > 0 and αi + βi < 1, i = 1, . . . , p.

To simulate p-dimensional time series with the joint density specified by the copula c[t]1,...,p in (1), independent

observations from Uniform[0,1] distribution are drawn. Each observation represents a realization of a variable rit giventhe conditioning set r1t , . . . , ri−1,t , i.e. ui|1,...,i−1,t = F [t](rit |r1t , . . . , ri−1,t), where i = 1, . . . , 5. Using the h-function in(12), dependent uniform F [t]

1 (r1t), . . . , F[t]1 (r5t) return series are obtained using the inverse of the h-function recursively to

reduce the conditioning set by one each time until it is empty. Using canonical vine decomposition as an illustration, fort = 1, . . . , T , we use the above scheme to determine F [t]

1 (r1t), . . . , F[t]5 (r5t):

F [t]1 (r1t) = u1t ,

F [t]2 (r2t) = h[t]−1

21 (u2|1,t , u1t),

F [t]3 (r3t) = h[t]−1

31 (h[t]−1

32|1(u3|12,t , u2|1,t), u1t),

F [t]4 (r4t) = h[t]−1

41 (h[t]−1

42|1(h[t]−1

43|12(u4|123,t , u3|12,t), u2|1,t), u1t),

F [t]5 (r5t) = h[t]−1

51 (h[t]−1

52|1(h[t]−1

53|12(h[t]−1

54|123(u5|1234,t , u4|123,t), u3|12,t), u2|1,t), u1t).

By applying the inverse of t distribution on uniform series F [t]1 (r1t), . . . , F

[t]5 (r5t), returns r1t , . . . , r5t are simulated. The

derivation of the above simulation procedure is given in Appendix C. In our simulation, two different types of dependence,conditional linear correlation and conditional Kendall’s tau, are tested. Three models are considered.

1. t marginal-t copula-linear correlation,2. t marginal-Clayton copula-Kendall’s tau,3. t marginal-Gumbel copula-Kendall’s tau.

All three models assume that the marginal distribution of returns is Student t , i.e. εt in (17) is t-distributed. The first modelis specified by having t copula as the conditional copula pair in c[t]

i,i+j|1,...,i−1 in (2) and tries to examine dynamic linearcorrelation through w(ϕ[t]). t copulas are chosen because they are commonly used in vine-copula GARCH models suchas that of Jondeau and Rockinger (2006). The definitions of the second and third models are similar. Clayton and Gumbelcopulas are twowidely discussed functions in Archimedean copulas. t copulas induce both upper and lower tail dependence.Clayton copulas exhibit lower tail dependence, while Gumbel copulas have upper tail dependence. Our true parametersare set based on estimates of a five dimensional real financial data set in the next section. We set the sample sizes to ben1 = · · · = n5 = 1500. We conduct 200 replications for each model setup. All of the models have marginal distributionparameters θi = {wi, αi, βi, vi} for the i-th marginal. For t marginal-t copula-linear correlation model, the parametersin the dynamic dependence specification of (7) are {ϕxy|z, axy|z, bxy|z, vxy|z}. The conditional dependence at time t , ϕ[t]

xy|z , isspecified in (4) and the sample conditional linear correlation with m = 2 in (8). There are 60 parameters to be estimatedin the simulation. For the remaining two models, the dynamic dependence parameters of (7) are {ϕxy|z, axy|z, bxy|z} with 50parameters being estimated. The conditional dependence ϕ[t]

xy|z for t marginal-Gumbel copula-Kendall’s tau and t marginal-Clayton–Kendall’s tau model are specified in (5) and (6), respectively.

As our interest is the copula parameters instead of themarginal distribution parameters, only true values of themarginaldistribution parameters are presented in Table 4. Simulation results for the marginals are available upon request. Thedynamic dependence parameter estimates for the three models are summarized in Tables 5 and 6. Most of the degrees


Table 4Summary of the true marginal distribution parameters in the three models in the simulation study.wi, αi and βi are from a GARCH(1, 1)-t innovation model for each time series in (18). vi specifies thedegree of freedom in t distribution, where i = 1, . . . , 5.

True parameter True parameter True parameter True parameter

w1 0.0890 α1 0.1400 β1 0.8500 v1 5.0000w2 0.0820 α2 0.1600 β2 0.8300 v2 8.0000w3 0.0120 α3 0.2000 β3 0.7900 v3 6.0000w4 0.0120 α4 0.1400 β4 0.8500 v4 7.0000w5 0.0930 α5 0.1100 β5 0.8800 v5 6.0000

Table 5Summary of the dynamic dependence parameters for the t marginal-t copula-linear correlationmodel in the simulation study. ϕ, a and b are from the time varying conditional correlation in (7).v is the degree of freedom from t copulas.

True parameter Mean estimate True parameter Mean estimate

v1,2 7.0000 6.9083 a1,2 0.0200 0.0225ϕ1,2 0.3800 0.3728 b1,2 0.9600 0.9218v1,3 7.2000 7.0229 a1,3 0.0300 0.0314ϕ1,3 0.4100 0.4123 b1,3 0.9400 0.9197v1,4 7.8000 6.9614 a1,4 0.0220 0.0244ϕ1,4 0.3790 0.3521 b1,4 0.9610 0.9228v1,5 7.4000 7.0627 a1,5 0.0100 0.0118ϕ1,5 0.4200 0.4200 b1,5 0.9710 0.9162

v2,3|1 13.0000 13.5172 a2,3|1 0.0100 0.0130ϕ2,3|1 0.3700 0.3670 b2,3|1 0.9600 0.9099v2,4|1 14.0000 13.5676 a2,4|1 0.0170 0.0178ϕ2,4|1 0.3100 0.3051 b2,4|1 0.9600 0.9193v2,5|1 13.5000 13.6512 a2,5|1 0.0100 0.0125ϕ2,5|1 0.3000 0.2975 b2,5|1 0.9500 0.8670

v3,4|1,2 18.0000 18.4557 a3,4|1,2 0.0140 0.0165ϕ3,4|1,2 0.2800 0.2784 b3,4|1,2 0.9300 0.8455v3,5|1,2 19.700 18.5525 a3,5|1,2 0.0190 0.0193ϕ3,5|1,2 0.2700 0.2682 b3,5|1,2 0.9400 0.8960

v4,5|1,2,3 20.3000 19.9438 a4,5|1,2,3 0.0180 0.0204ϕ4,5|1,2,3 0.2400 0.2381 b4,5|1,2,3 0.9610 0.9115

Table 6Summary of the dynamic dependence parameters for the t marginal-Clayton copula-Kendall’s tau andt marginal-Gumbel copula-Kendall’s tau models in the simulation study. ϕ, a and b are from the timevarying conditional Kendall’s tau in (7). The ‘‘Clayton’’ column is the mean estimate for the t marginal-Clayton copula-Kendall’s tau model, while the ‘‘Gumbel’’ column refers to the mean estimate for the tmarginal-Gumbel copula-Kendall’s tau model.

True parameter Clayton Gumbel True parameter Clayton Gumbel

ϕ1,2 0.3800 0.2760 0.3126 a1,2 0.0200 0.0160 0.0156b1,2 0.9600 0.9460 0.9317

ϕ1,3 0.4100 0.2700 0.3286 a1,3 0.0300 0.0230 0.0236b1,3 0.9400 0.9260 0.9087

ϕ1,4 0.3790 0.2650 0.3274 a1,4 0.0220 0.0180 0.0158b1,4 0.9610 0.9460 0.9211

ϕ1,5 0.4200 0.3460 0.3478 a1,5 0.0100 0.0080 0.0080b1,5 0.9710 0.9620 0.9612

ϕ2,3|1 0.3690 0.3010 0.3259 a2,3|1 0.0100 0.0110 0.0099b2,3|1 0.9600 0.9320 0.9326

ϕ2,4|1 0.3100 0.2510 0.2606 a2,4|1 0.0170 0.0130 0.0159b2,4|1 0.9600 0.9280 0.9333

ϕ2,5|1 0.2990 0.2520 0.2487 a2,5|1 0.0100 0.0110 0.0096b2,5|1 0.9500 0.8960 0.9464

ϕ3,4|1,2 0.2770 0.2250 0.2341 a3,4|1,2 0.0140 0.0150 0.0130b3,4|1,2 0.9300 0.8680 0.8440

ϕ3,5|1,2 0.2650 0.2240 0.2202 a3,5|1,2 0.0190 0.0160 0.0142b3,5|1,2 0.9400 0.8920 0.9342

ϕ4,5|1,2,3 0.2400 0.2020 0.2085 a4,5|1,2,3 0.0180 0.0200 0.0135b4,5|1,2,3 0.9610 0.9040 0.9472


Fig. 2. Tree decomposition shows the canonical vine (left) and the D vine (right) for the real data set.

of freedom vxy|z in t marginal-t copula-linear correlation have a bias of less than one. The values of axy|z in the t marginal-tcopulas-linear correlation are overestimated and that in the t marginal-Gumbel copula-Kendall’s tau are underestimated.Yet,with less than 0.01 error inmagnitude, the estimate of axy|z is considered to be accurate. The largest error of the estimatesof bxy|z is only 0.086 in magnitude. It can be concluded that the sequential estimation method in Section 3.1 is reliable.

4. Real data estimation

We use five daily return series of Hong Kong stocks from January 2004 to December 2011, resulting in a sample sizeof 1894 to demonstrate the proposed vine-copula GARCH model. The five chosen stocks are constituent stocks of the HangSeng index in Hong Kong: Cheung Kong (CK), CLP Holdings (CLP), HK and China Gas (HKCG),Wharf Holdings (WH) and HSBCHoldings (HSBC)with their return time series denoted by r1t , . . . , r5t respectively. The daily return of the i-th series is definedas rit = 100 × ln(Pit/Pi,t−1), where Pi,t is the price of the i-th stock series at time t collected from Bloomberg. CK and WHare from the conglomerate industry, CLP is an electricity company, HKCG is from the public utility sector and HSBC is fromthe banking industry. The three models adopted in the simulation study are fitted to this five-dimensional time series. AsAustin and Lopes (2010) discussed, the indexing of variables in canonical vines can be achieved by first selecting one pilotvariable that has the strongest dependence associated with all other variables. By listing the most dependence variableswith the pilot variable in decreasing order in dependence, the first tree level is generated. In D vines, indexing is achieved bychoosing and ranking themost dependent pairs. The pairs are then connected in the first tree. Dependence comparison is notrestricted to linear correlation, but it can also be Kendall’s tau or rank correlation. The tree decomposition of the canonicaland D vines are shown in Fig. 2, where the variable indexing is based on linear correlations. In the study, canonical vinedecomposition is used as an illustration. As in the simulation study in Section 3.3, three models are considered.

To facilitate our explanation, we denote CK as stock 1, WH as stock 2, HKCG as stock 3, HSBC as stock 4 and CLP asstock 5. The estimates of the three models in Table 7 are from marginal distributions, which are t distributed. They arethe same because marginal distributions are estimated separately from copula parameters. The sum of the estimates αand β is close to one. Table 8 summarizes copula estimates for the three models. Regarding the t marginal-t copula-linearcorrelation model, the degrees of freedom ν tends to increase and ψ tends to drop when the tree level increases (with moreconditioning variables). The above findings indicate that the conditional dependence, reflected from ψ , tends to be weakerwhen there are more conditioning variables. We also observe small a and large b with the persistence, a + b, close to one.For the t marginal-Clayton copula-Kendall’s tau and the t marginal-Gumbel copula-Kendall’s tau models, similar resultscan be retrieved from the table. The drops of ψ with the tree level are more pronounced in these two models than in thet marginal-t copula-linear correlation case. The conditional copula 23|1 in t marginal-Clayton copula-Kendall’s tau modelhas both a and b close to zero, suggesting that most of the weighting contributes to the long term conditional dependence,making the dependence invariant over time. Estimates of close to zero are observed in parameter a in some copulas, suchas 13 and 25|1 in all three models.

The pairwise linear correlation for the first tree level are estimated from the t marginal-t copula-linear correlationmodel.Two bivariate series, CK &WH and CK & HSBC, are shown in the top figure of Fig. 3. The series of all pairs are available uponrequest. Pairwise correlations for the two pairs in Fig. 3 are very close to each other and below 0.75. Among all pairs, CKhas the highest correlation with WH and HSBC clustered at around 0.6 due to their common business sector. An increasingtrend of correlation from 0.4 to 0.7 can be observed during 2007–2008 between CK and WH and CK and HSBC. It becomesstable afterward and increases again around 2011 to a level above 0.6. The upward trend of correlation before the economicrecession in 2008means that stocks tended to react together in the bear financialmarket, possibly triggering a chain reaction.The figures deliver two important messages. First, increasing correlation can induce chain reaction in the financial market,


Table 7Summary of the result for marginal distribution param-eters from the t marginal-t copula-linear correlation, tmarginal-Clayton copula-Kendall’s tau and t marginal-Gumbel copula-Kendall’s tau models. The parameters wi ,αi and βi are for the GARCH model in (18) and vi is the de-gree of freedom in the marginal t distribution. As sequen-tial estimation is used, the estimates are not affected by thechoice of copula model.

Estimate Estimate

w1 0.2300 β1 0.7592α1 0.2377 v1 3.9488w2 0.0095 β2 0.8563α2 0.1384 v2 6.9768w3 0.0130 β3 0.9114α3 0.0869 v3 4.1511w4 0.2103 β4 0.7411α4 0.2579 v4 6.6371w5 0.0146 β5 0.9069α5 0.0887 v5 4.2543

Table 8The copula parameters of the threemodels. For the t marginal-t copula-linear correlationmodeling conditional linear correlation, there are four parametersin each bivariate copulas, whilst there are only three parameters in t marginal-Clayton copula-Kendall’s tau and t marginal-Gumbel copula-Kendall’s taumodels. The parameters ϕ, a and b determine the time varying dependence in (7), and v is the degrees of freedom in the t copulas for conditional linearcorrelation.

t marginal-t copula-linear correlation t marginal-Clayton copula-Kendall’s tau tmarginal-Gumbel copula-Kendall’s tau

ϕ1,2 0.3916 a1,2 0.0584 ϕ1,2 0.0562 a1,2 0.0681 ϕ1,2 0.1079 a1,2 0.0435v1,2 5.0666 b1,2 0.8232 b1,2 0.7594 b1,2 0.7763ϕ1,3 0.4860 a1,3 0.0000 ϕ1,3 0.2718 a1,3 0.0000 ϕ1,3 0.3021 a1,3 0.0000v1,3 5.3082 b1,3 0.9479 b1,3 0.8580 b1,3 0.9574ϕ1,4 0.6470 a1,4 0.0345 ϕ1,4 0.2873 a1,4 0.0429 ϕ1,4 0.1579 a1,4 0.0478v1,4 4.7957 b1,4 0.8523 b1,4 0.7607 b1,4 0.8902ϕ1,5 0.6606 a1,5 0.0298 ϕ1,5 0.1807 a1,5 0.0388 ϕ1,5 0.1457 a1,5 0.0476v1,5 5.2215 b1,5 0.9249 b1,5 0.8640 b1,5 0.8902

ϕ2,3|1 0.3475 a2,3|1 0.0099 ϕ2,3|1 0.1222 a2,3|1 0.0059 ϕ2,3|1 0.0922 a2,3|1 0.0037v2,3|1 7.7815 b2,3|1 0.9824 b2,3|1 0.0066 b2,3|1 0.9699ϕ2,4|1 0.1693 a2,4|1 0.0211 ϕ2,4|1 0.0002 a2,4|1 0.0161 ϕ2,4|1 0.0001 a2,4|1 0.0187v2,4|1 7.0029 b2,4|1 0.9313 b2,4|1 0.8447 b2,4|1 0.8572ϕ2,5|1 0.1757 a2,5|1 0.0000 ϕ2,5|1 0.0695 a2,5|1 0.0000 ϕ2,5|1 0.0745 a2,5|1 0.0000v2,5|1 7.3937 b2,5|1 0.9548 b2,5|1 0.8890 b2,5|1 0.9659

ϕ34|12 0.1475 a34|12 0.0000 ϕ34|12 0.0093 a34|12 0.0036 ϕ3,4|1,2 0.0105 a3,4|1,2 0.0075v34|12 6.6624 b34|12 0.8364 b34|12 0.9661 b3,4|1,2 0.9176ϕ35|12 0.1542 a35|12 0.0011 ϕ35|12 0.0001 a35|12 0.0175 ϕ3,5|1,2 0.0693 a3,5|1,2 0.0000v35|12 18.5724 b35|12 0.9819 b35|12 0.8408 b3,5|1,2 0.9878

ϕ45|123 0.2004 a45|123 0.0035 ϕ45|123 0.0004 a45|123 0.0098 ϕ4,5|1,2,3 0.0005 a4,5|1,2,3 0.0041v45|123 24.4357 b45|123 0.9065 b45|123 0.9297 b4,5|1,2,3 0.9726

resulting in higher risk in portfolios. Second, the increasing correlation observed for CK, WH and HSBC around 2011 and2012 implies a possible increase in market risk in the foreseeable future.

For the t marginal-t copula-linear correlation model, the approximations of pairwise linear correlation using thesimulation method and the integration method are illustrated in Fig. 3. The pattern of linear correlation using simulationsis similar to the true linear correlation based on the dynamic structure specified by (4), (7) and (8). Linear correlation usingthe integration method gives a pattern similar to that of the true linear correlation (top figure) but with a lower level due tothe error inmultiple integration. As summarized in Table 9, theMAE and RMSE for the pairwise linear correlation in the firsttree is smaller in the simulation method than that in the integration method. The dimension of integrals in the integrationmethod affects the error. For the t marginal-Clayton copula-Kendall’s tau model, the approximates of pairwise Kendall’stau are obtained by the integration method and simulation using 10,000 iterations in Fig. 4. The true Kendall’s tau for thefirst tree level is calculated from the estimated copula parameters in the t marginal-Clayton copula-Kendall’s tau modelusing (7). When we compare the approximate Kendall’s tau from the two methods with the true Kendall’s tau for the firsttree level, MAE and RMSE are obtained in Table 10. Having a smaller MAE and RMSE, the integration method estimates arebetter than that obtained by simulation with 10,000 iterations. By observing the MAE and RMSE, it can be seen that theperformance of the integration method depends heavily on the dimension of integrals while simulation method has morestable performance as long as the number of iterations is large enough.


Table 9The MAE and RMSE of linear correlation estimates for thefirst tree level of t copulas for the integration and simulationmethods in the real data study.

Simulation of 10,000 iterations Integration

MAE 0.0110 0.0905RMSE 0.0144 0.0955

Table 10The MAE and RMSE of Kendall’s tau estimates for thefirst tree level of Clayton copulas for the integration andsimulation methods in the real data study.

Simulation of 10,000 iterations Integration

MAE 0.0051 0.0003RMSE 0.0064 0.0003

Fig. 3. Pairwise linear correlation in the t marginal-t copula-linear correlation model. Two series represent the dependence between Cheung Kong (CK)and Wharf Holdings (WH) and Cheung Kong (CK) and HSBC Holdings (HSBC). The true linear correlation based on the dynamic structure specified by (4),(7) and (8) (top figure), pairwise linear correlation estimates using simulation with 10,000 iterations (middle figure) and estimates using the integrationmethod (bottom figure) are plotted.

5. Conclusion

This paper develops a generic approach to specifying dynamic conditional dependence using any dependence measures.The characterization also induces multivariate conditional dependence dynamically through vine decomposition. The vinecopula GARCHmodelswith dynamic conditional dependence introduced contribute to building a unifiedmodeling approachof time varying dependence. Using vine decompositions, a joint density function can be decomposed into a product ofmarginal distributions and conditional copulas. Conditional dependence such as linear correlation, Kendall’s tau and rank


Fig. 4. Pairwise Kendall’s tau in t marginal-Clayton copula-Kendall’s taumodel. Two series represent the dependence between Cheung Kong (CK) &WharfHoldings (WH) and Cheung Kong (CK) & HSBC Holdings (HSBC). The true Kendall’s tau based on the dynamic structure specified by (5), (7) and (10) (topfigure), pairwise Kendall’s tau estimates using simulation with 10,000 iterations (middle figure) and estimates by using the integration method (bottomfigure) are plotted.

correlation become time varying via a transformation function of copula parameters. Using a five dimensional Hong Kongstock return data set, we estimate the parameters using three models: t marginal-t copula-linear correlation, t marginal-Clayton copula-Kendall’s tau and t marginal-Gumbel copula-Kendall’s tau. It has revealed in all the three models that mostconditional dependence among the stocks is highly persistent. While almost all conditional dependence is time-varying,there is still a minority of non-time-varying conditional dependence. Another highlight of this paper is the estimation ofpairwise dependence using the simulation of return series with estimated parameters and past return data. This method isof crucial importance to explaining the dependence levels, such as correlation, Kendall’s tau and rank correlation, betweenany two stocks. In 2008, the increase in correlation of many pairs of stocks can be attributed to the increased risk level ofthe stocks during financial turmoil. The increasing correlation around 2011 and 2012 suggests that high market risk maypersist.

Acknowledgments

This work is supported by an Hong Kong RGC General Research Fund (645111) and HKUST grant (SBI12BM07).

Appendix A. Conditional distributions in likelihood functions

In both canonical vines and D vines, the log likelihood function is the summation of logarithm of the joint densityfunction in (2) and (3) over t = 1, . . . , T . Given r1t , . . . , rpt , each copula density requires arguments from two conditionaldistributions. Referring to copula densities for canonical vines on the first tree level (i = 1) in (2), the arguments for copulac[t]1,1+j, where j = 1, . . . , p − 1, are F [t](r1t) and F [t](r1+j,t). These are calculated by mapping stock returns r1t , . . . , r1+j,t totheir cumulative distributions.


Looking at i = 2, the arguments for copula densities on the second tree level are F [t](r2,t |r1,t) and F [t](r2+j,t |r1,t), wherej = 1, . . . , p− 2. These two conditional distributions can be expressed in terms of conditional distributions on the first treelevel, which are given by

F [t](r2,t |r1,t) = h[t]21(F

[t]2 (r2t), F

[t]1 (r1t)),

F [t](r2+j,t |r1,t) = h[t]2+j,1(F

[t]2+j(r2+j,t), F

[t]1 (r1t)),

where j = 1, . . . , p − 2. It should be noted that F [t]1 (r1t), F

[t]2 (r2t) and F [t]

2+j(r2+j,t) for j = 1, . . . , p − 2 are the arguments forthe copula density on the first tree level.The arguments for copula densities for D vines in (3) essentially follow the same procedures as that for canonical vines withdifferent h-functions. Take i = 2 for example; arguments for copula densities on the second tree level are F [t](rj,t |rj+1,t) andF [t](rj+2,t |rj+1,t) for j = 1, . . . , p − 2. They are given by

F [t](rj,t |rj+1,t) = h[t]j,j+1(F

[t]j (rjt), F

[t]j+1(rj+1,t)),

F [t](rj+2,t |rj+1,t) = h[t]j+2,1(F

[t]j+2(rj+2,t), F

[t]j+1(rj+1,t)),

where j = 1, . . . , p − 2.

Appendix B. Pairwise rank correlation and Kendall’s tau using the integration method for the second tree level

First denote uit = F [t]i (rit) for i = 1, . . . , p. For rank correlation, the integration form for the second tree level is given by

ρ[t]2i = 12

[0,1]2

C [t]2i (u2t , uit)du2tduit − 3, (B.1)

where i = 3, . . . , p. Copula C [t]2i (u2t , uit) can be obtained using a three dimensional copula C [t]

1,2,i(u1t , u2t , uit) such that

C [t]2i (u2t , uit) = C [t]

1,2,i(1, u2t , uit) = F [t](∞, r2t , rit)

=

rit

−∞

r2t

−∞

∞

−∞

f [t](r ,2t , r,it |r

,1t)f

[t]1 (r ,1t)dr

,1tdr

,2tdr

,it . (B.2)

Now considering density f [t](r ,2t , r,it |r

,1t) in (B.2), we have

f [t](r ,2t , r,it |r

,1t) =

c[t]2i|1(F

[t](r ,2t |r,1t), F

[t](r ,it |r,1t))f

[t](r ,1t , r,2t)f

[t](r ,1t , r,it)

f [t]1 (r ,1t)

2 ,

and becausef [t](r ,1t , r

,jt) = c[t]

1j (F[t]1 (r

,1t), F

[t]j (r

,jt))f

[t]1 (r ,1t)f

[t]j (r ,jt),

where j = 2, i, we can obtain

f [t](r ,2t , r,it |r

,1t) = c[t]

2i|1(F[t](r ,2t |r

,1t), F

[t](r ,it |r,1t))c

[t]12 (F

[t]1 (r

,1t), F

[t]2 (r

,2t))c

[t]1i (F

[t]1 (r

,1t), F

[t]i (r

,it))f

[t]2 (r ,2t)f

[t]i (r ,it). (B.3)

Substituting the new equation of f [t](r ,2t , r,it |r

,1t) into Eq. (B.2), copula C [t]

2i (u2t , uit) becomes

C [t]2i (u2t , uit) = C [t]

1,2,i(1, u2t , uit)

=

u3t

0

u2t

0

1

0c[t]2i|1(h

[t]21(u

,2t |u

,1t), h

[t]i1 (u

,it |u

,1t))c

[t]12 (u

,1t , u

,2t)c

[t]1i (u

,1t , u

,it)du

,1tdu

,2tdu

,it . (B.4)

If copula C [t]2i (u2t , uit) in (B.4) is substituted back into (B.1), the resulting rank correlation for the second tree level is the same

as (15). Kendall’s tau for the second tree level is given by

τ[t]2i = 4

[0,1]2

C [t]2i (u2t , uit)c

[t]2i (u2t , uit)du2tduit − 1,

where i = 3, . . . , p. Copula C [t]2i (u2t , uit) is the same as (B.2) and there is a term for copula density c[t]

2i (F[t]2 (r2t), F

[t]i (rit)) in

the integration. The copula density is given by

c[t]2i (F

[t]2 (r2t), F

[t]i (rit)) =

f [t](r2t , rit)

f [t]2 (r2t)f

[t]i (rit)

=

∞

−∞f [t](r1t , r2t , rit)dr1t

f [t]2 (r2t)f

[t]i (rit)

=

∞

−∞f [t](r2t , rit |r1t)f

[t]1 (r1t)dr1t

f [t]2 (r2t)f

[t]i (rit)

,


and due to the fact given by (B.3) that

f [t](r2t , rit |r1t) = c[t]2i|1(F

[t](r2t |r1t), F [t](rit |r1t))c[t]12 (F

[t]1 (r1t), F

[t]2 (r2t))c

[t]1i (F

[t]1 (r1t), F

[t]i (rit))f

[t]2 (r2t)f

[t]i (rit),

the resulting copula density c[t]2i (F

[t]2 (r2t), F

[t]i (rit)) is

c[t]2i (u2t , uit) =

1

0c[t]2i|1(h

[t]21(u2t , u1t), h

[t]i1 (uit , u1t))c

[t]12 (u1t , u2t)c

[t]1i (u1t , uit)du1t . (B.5)

Substituting C [t]2i (u2t , uit) in (B.4) and c[t]

2i (u2t , uit) in (B.5) into integral form of pairwise Kendall’s tau for the second treelevel, the pairwise Kendall’s tau is the same as (16).

Appendix C. Derivation of simulation procedures

Denote ui|1,...,i−1,t as independent uniform random variables F [t](rit |r1t , . . . , ri−1,t) at time t for i = 1, . . . , p. For thereturn of the first stock r1t , we randomly draw a uniformly distributed number u1t = F [t]

1 (r1t). The return for the first stock

in this simulation iteration is simply r1t = F [t]1

−1(u1t). For the return of the second stock r2t , we draw uniformly distributed

number u2|1,t = F [t](r2t |r1t). From the h-function in (12), we know u2|1,t = h[t]21(F

[t]2 (r2t), F

[t]1 (r1t)). Thus, F

[t]2 (r2t), together

with the fact that u1t = F [t]1 (r1t), can be obtained by applying the inverse of h-function. It is

F [t]2 (r2t) = h[t]−1

21 (u2|1,t , u1t).

Then r2t is calculated by taking the inverse of its marginal distribution. For the return of the third stock r3t , draw u3|12,t =

F [t](r3t |r1t , r2t). This can be expressed by u3|12,t = F [t](r3t |r1t , r2t) = h[t]32|1(F

[t](r3t |r1t), F [t](r2t |r1t)). By applying the inverseof the h-functionwith argument u3|12,t and F [t](r2t |r1t), an expression for conditional distribution F [t](r3t |r1t), which containsone less conditioning variable than F [t](r3t |r1t , r2t), is formed. In addition, as u2|1,t = F [t](r2t |r1t), we have

F [t](r3t |r1t) = h[t]−132|1(u3|12,t , u2|1,t). (C.1)

Again, using the h-function on F [t](r3t |r1t) gives

F [t](r3t |r1t) = h[t]31(F

[t]3 (r3t), F

[t]1 (r1t)).

After using the inverse of the h-function and substituting (C.1), cumulative distribution F [t]3 (r3t) is given by

F [t]3 (r3t) = h[t]−1

31 (F[t](r3t |r1t), F

[t]1 (r1t)),

= h[t]−131 (h

[t]−132|1(u3|12,t , u2|1,t), u1t).

By taking the inverse of its marginal distribution, we obtain r3t . In a similar manner, we obtain all stock returns.

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