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Computational Statistics and Data Analysis ( )

Contents lists available at ScienceDirect

Computational Statistics and Data Analysis

journal homepage: www.elsevier.com/locate/csda

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 Kendalls 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 Kendalls 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: immkpso@ust.hk (M.K.P. So), cherryyeung032003@gmail.com (C.Y.T. Yeung).

0167-9473/$ see front matter 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.csda.2013.08.008

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 Kendalls tau using latent variables and the inverse Fishertransformation of Kendalls 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 Kendallstau.

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, Ft1, with marginal distribution parameters x|z . If the conditioning set rzt is empty,F [t](rxt |rzt) simply represents the distribution of rxt given Ft1. In line with Sklar (1959), the multivariate distribution of rxtand ryt conditional on Ft1 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 Ft1 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 Ft1, with copula parameters xy|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 Ft1 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))p

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

where C [t]1,...,p is the copula function of r1t , . . . , rpt givenFt1 and c[t]1,...,p is its copula density. A vine structure decomposes the

copula 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) =p1i=1

pij=1

c[t]i,i+j|1,...,i1(F[t](rit |r1t , . . . , ri1,t), F [t](ri+j,t |r1,t , . . . , ri1,t))

pi=1

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

and the p-dimensional density function using D vines is

f [t](r1t , . . . , rpt) =p1i=1

pij=1

c[t]j,i+j|j+1,...,i+j1(F[t](rjt |rj+1,t , . . . , ri+j1,t), F [t](ri+j,t |rj+1,t , . . . , ri+j1,t))

pi=1

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

In (2) and (3), f [t]i is the marginal distribution of rit conditional on Ft1. A key contribution of this paper is to define theconditional copula c[t]1,...,p with dynamic dependence, which is presented in Section 2.2. The construction of c

[t]1,...,p is based

on 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,...,i1 in (2) and c

[t]j,i+j|j+1,...,i+j1 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.

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

The flexibility of copulas is considered to measure dynamic conditional rank correlation and Kendalls tau. Rankcorrelation examines dependence between the rank of variables, while Kendalls tau concerns concordant and discordantpairs. For convenience, denotew([t]xy|z) as the conditional dependence at time t as a function of the dependence parameter[t]xy|z , which is one of the copula parameters in xy|z,t . Table 1 summarizes the dynamic conditional dependence for various

copula functions. Conditional linear correlation is a function of the copula parameter [t]xy|z in the two elliptical copulas:the Gaussian copula and t copula. The conditional Kendalls 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([t]xy|z), does not need a transformation, i.e. it is given by

w([t]xy|z) = [t]xy|z . (4)

As the copula parameter [t]xy|z is not the conditional Kendalls tau in Clayton copulas, a transformation is required. Theconditional Kendalls tau, denoted byw([t]xy|z), can be expressed as

w([t]xy|z) =

[t]xy|z

[t]xy|z + 2

. (5)

Similarly, the conditional Kendalls tau for Gumbel copulas, denoted byw([t]xy|z), is given by

w([t]xy|z) =

[t]xy|z 1[t]xy|z

. (6)

One advantage of conditional rank correlation andKendalls tau over linear correlation is that they are invariant under strictlyincreasing component-wise transformations. As measures of concordance, this propertymakes conditional rank correlationand Kendalls 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([t]xy|z) between return variables rxt andryt given rzt and Ft1 as

w([t]xy|z) = (1 axy|z bxy|z)w(xy|z)+ axy|zxy|z,t1 + bxy|zw([t1]xy|z ), (7)

where w(xy|z)...

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