Following Silvennoinen and Terasvirta [108], the models in the class of Conditional Correlations (CC), are based on the decomposition of the conditional covariance matrix Ht, into a matrix of conditional correlations Rt, and a diagonal matrix of
conditional variances Dt, that is:
rt= µ + Φirt−i+ ut ; ut= Ht1/2zt where Ht= DtRtDt (4.5)
The class of CC-MGARCH models are considered to be the simple in terms of rep- resentation and estimation, as the number of parameters to be estimate increases gradually as the number of variables increases. In this category the covariance be- tween variables hij,t= ρij,t
q
hii,t× hjj, where the variances hii and hjj are obtained
by a univariate GARCH process discussed in chapter 2. The correlations ρij show
how the series of residuals move together. The CC - MGARCH models, include Constant Conditional Correlation (CCC), Dynamic Conditional Correlation (DCC) and Varying Conditional Correlation (VCC) models.
4.3.1
Constant conditional correlation
The Constant Conditional Correlation (CCC) GARCH model was proposed by Boller- slev [17], assuming that the conditional correlations between the elements of ut are
time-invariant. By definition a process ut is called CCC - GARCH(p,q) model if the
matrix of variances Dt= diag(h1/211,t, ..., h1/2kk,t), and the correlation matrix R = ρij is
symmetric positive definite with ρii = 1, for ∀i.
Ht= Ω + p X i=1 Aiu2t−i+ q X j=1 BjHt−j (4.6)
where Ω is a K × 1 vector with positive coefficients Ai and Bj are K × K matrices
with non-negative coefficients.
The individual conditional variances and conditional covariances are:
hii,t = ωii+ αiiu2ii,t−1+ βiihii,t−1 ; i = 1, ..., K (4.7)
hij,t = ρij
q
4.3.2
Dynamic conditional correlation
The assumption that the conditional correlations are constant may seem unrealis- tic in many empirical applications. Engle [40] and Tse and Tsui [117] propose a generalization of the CCC model of Bollerslev [17] by making the conditional cor- relation matrix time-dependent. The model is then called a Dynamic Conditional Correlation (DCC) model. Here we will just focus on DCC-GARCH model proposed by Engle [40], where the matrix of variances Dt are diagonal with variances hii
described by a univariate GARCH(1,1) process.
Dt= diag(h1/211,t, ..., h 1/2
kk,t) where hii,t= ωii+ αiiu
2
ii,t−1+ βiihii,t−1 (4.8)
From equation 4.5, where rt= µ + Φirt−i+ ut, ut= H
1/2
t zt and Ht= DtRtDt,
The process ut is called DCC - GARCH model if the conditional correlation matrix
Rt is defined as: Rt= diag(q11,t1/2, ..., q 1/2 kk,t)Qtdiag(q 1/2 11,t, ..., q 1/2 kk,t) (4.9)
and Qt= (qii,t) is the K × K symmetric positive definite matrix which has the form:
Qt= (1 − λ1− λ2)Q + λ1zt−1zt−10 + λ2Qt−1 (4.10)
note that zi,t is a standardized residuals, λ1 and λ2 are adjustment parameter, they
are non-negative and satisfy λ1+ λ2 < 1 and Q = Cov(ztzt0) is the unconditional
variance matrix of standardized residuals.
According to Bauwens et al. [6], the main drawback of the model is that all con- ditional correlations follow the same dynamic structure. The number of parameters to be estimated is given by (K + 1)(K + 4)/2, relatively smaller compared to the number obtained in the BEKK model for cases where K is small. When the number of variables increases to continue to have parsimonious results and decrease the complexity in estimation DCC parameters, the process is performed in two steps.
4.3.3
Varying Conditional Correlation
The Varying Conditional Correlation (VCC) model proposed by Tse and Tsui [117] uses a nonlinear combination of univariate GARCH model with time-varying cross equation weights to model the conditional covariance matrix of errors. The VCC - GARCH model decomposes the conditional variance and covariances Ht into
conditional variance matrix Dt and the conditional correlation matrix Rt, where
Dt= diag(h1/211,t, ..., h 1/2
kk,t). The variance hii,tfollows a univariate GARCH(1,1) model,
4 . 4 . E S T I M AT I O N O F M U LT I VA R I AT E G A RC H M O D E L S
process, hij,t = ρij,t
q
hii,thjj,t. Because the correlations ρij,t vary with time, this
model is known as the VCC - GARCH model.
The time varying conditional correlation matrix Rt is generated by the equation:
Rt= (1 − λ1− λ2)R + λ1Rt−1+ λ2Ψt−1 (4.11)
where R = {ρij} is a K × K positive definite parameter matrix with ρii = 1 and
Ψt−1 is a K × K matrix whose elements are functions of the lagged observations of
ut. The parameters λ1 and λ2 are assumed to be non-negative with the additional
constraint that λ1+ λ2< 1. Thus, Rt is a weighted average of R, Rt−1 and Ψt−1.
Hence, if Ψt−1 is a well-defined correlation matrix, that is, positive definite with
unit diagonal elements, Rt will also be a well-defined correlation matrix.
The sample correlation matrix of {ut−1, ..., ut−m}, denoted by Ψt = {ψij,t}, is
computed as follows: ψij,t= Pm k=1ui,t−kuj,t−k q (Pm k=1u2i,t−k)( Pm k=1u2j,t−k) ; 1 ≤ i < j ≤ K (4.12)
Note that, for DCC or VCC, if λ1= λ2= 0, the model reduces to CCC model.