4.2.1
VECH model
In the class of Conditional covariance matrix, the first multivariate GARCH model was introduced by Bollerslev et al. [20], which is called VECH model. It is much general compared to the subsequent formulations. In the VECH model, every condi- tional variance and covariance is a function of all lagged conditional variances and covariances, as well as lagged squared residuals and cross-product of residuals and lagged values of the elements of Ht. The VECH model can be express as:
vech(Ht) = Ω + p
X
i=1
Aivech(ut−iu0t−i) + q
X
j=1
Bjvech(Ht−j) (4.2)
where Ht is the covariance matrix of the residuals, Ω is an (K(K + 1)/2) × 1 vector
of constants , Ai and Bj are K(K + 1)/2 × K(K + 1)/2 parameter matrices, ut−i
is an K × 1 vector and K is the number of variables. Note that, from equation 4.2, it can be seen that VECH(p,q) model is multivariate extension of the GARCH(p,q) model, that is VAR-MGARCH(p,q).
The representation in 4.2 is very general and flexible but there is a disadvantage that only a sufficient condition for Ht to be positive definite for all t is not restrictive.
Furthermore the number of parameters is (p+q)×(K(K +1)/2)2+K(K +1)/2 which is excessively large as N increases. For example, if p = q = 1 and K = 2, the number of parameters is 21, if K = 3 it is 78. This may cause computational difficulties, which implies that in practice the VECH model is used only in the bivariate case.
4.2.2
Diagonal VECH model
To improve parsimony Bollerslev et al. [20], proposed diagonal VECH (p,q) or DVECH (p,q) model. Where the coefficient matrices Ai, i = 1, ..., p and Bj, j = 1, .., q
are assumed diagonal. In this case, the conditional covariance between ui,t and uj,t,
that is hij,t depends only on lagged cross-products of the two shocks involved and
lagged values of the covariance itself. The DVECH(p,q) model is given by:
Ht= Ω + p X i=1 A∗i (ut−iu0t−i) + q X j=1 B∗j Ht−j (4.3)
where denotes the Hadamard product, that is, element-by-element product; Ω, is the K ×K symmetric matrix of constant, A∗i = diag(vech(Ai)), B∗j = diag(vech(Bj))
and Ai, Bj are K × K symmetric matrices with elements αij and βij respectively.
Each covariance equation in 4.3 can be specified in scalar form as: hij,t = ωij +
αijui,t−1uj,t−1+ βijhij,t−1.
The matrix of residual covariances Ht is positive definite for all t and the number
of parameters is reduced to (p + q + 1)(K(K + 1)/2), as no interaction is allowed between the different conditional variances and covariances. For example, if p = q = 1 and K = 2 then the number of parameters for DVECH(1,1) model is 9, and if K = 3 it is 18.
According to Bauwens et al. [6], Engle and Kroner [43] and Ledoit et al. [72], assuming that Ht is positive definite, the DVECH(p,q) model from equation 4.3
is ensured to be covariance stationary if wii > 0, for ∀i = 1, ..., K. All elements of
Ai, i = 1, ..., p and Bj, j = 1, ..., q, all eigenvalues of p P i=1 Ai+ q P j=1
Bj are less than one
in modulus.
4.2.3
BEKK model
According to Bauwens et al. [6], because it is difficult to guarantee the positivity of Ht in the VECH representation without imposing strong restrictions on the
parameters, based on earlier work by Baba et al. [4], Engle and Kroner [43], propose a new parametrization for Ht and call BEKK (p,q) model, therefore, the acronym
BEKK refers to the initial letter of the authors Baba, Engle, Kraft and Kroner, who developed the multivariate GARCH model in a preliminary version.
Ht= ΩΩ0+ p X i=1 K X k=1 A0kiut−iu0t−iAki + q X j=1 K X k=1 B0kjHt−jBkj (4.4)
where Ω is a lower triangular matrix with (K(K + 1)/2) parameters, Ai and Bj
denote K × K matrices with K2 parameters each. The BEKK model is covariance stationary if and only if the eigenvalues of
p P i=1 K P k=1 Aki⊗ Aki+ q P j=1 K P k=1 Bkj⊗ Bkj are
less than one in modulus, where ⊗ denotes the Kronecker product of two matrices. Even restricting to the model of the first order, empirical applications often involve the highly simplification version. For example, when both A and B are assumed to be diagonal matrices, the model is called Diagonal BEKK, proposed by Bollerslev et al. [20]. The main advantage in Diagonal BEKK model is that the number of parameters to be estimated decrease and the new parametrization guarantees that Ht be positive definite, if ΩΩ0 is a positive definite matrix.
4 . 3 . C O N D I T I O N A L C O R R E L AT I O N S M G A RC H M O D E L S
Note that for VECH, DVECH and BEKK models the conditional covariance matrices Htare imposed to be positive definite for each t. These requires complicated
restrictions on the off-diagonal elements.