Applications of state space models in finance

5.3 Multivariate conditional heteroskedasticity 77
5.3.1.2 The diagonal vech model
The diagonal vech model is a first way to restrict Equation (5.47) and to reduce the
number of parameters. Bollerslev et al. (1988) restrict the matrices Γ ∗ and ∆ ∗ to be
diagonal such that the conditional covariance between ɛ1,t and ɛ2,t depends only on
lagged cross-products of the residuals and its own lagged value. In this specification,
each element of the conditional covariance matrix follows a univariate GARCH(1,1)
model:
hij,t = ωij + γijɛi,t−1ɛj,t−1 + δijhij,t−1, (5.49)
where ωij, γij and δij denote the ij-th element of the symmetric N ×N matrices Ω, Γ and
∆, respectively. The latter two matrices are implicitly defined by Γ ∗ = diag(vech(Γ))
and ∆ ∗ = diag(vech(∆)). 16 Following Ding and Engle (2001) the diagonal vech model
can be expressed in terms of the Hadamard product, 17 denoted by ⊙:
Ht = Ω + Γ ⊙ ɛt−1ɛ ′ t−1 + ∆ ⊙ Ht−1, (5.50)
The number of parameters is reduced to 3N(N + 1)/2. Using this representation, H t
can be shown to be positive definite for positive definite Ω and positive semi-definite Γ
and ∆ (cf. Franses and van Dijk 2000, ¢ 4.7).
5.3.1.3 The BEKK model
A more general representation is the BEKK model of Engle and Kroner (1995). It
includes all positive definite diagonal models and nearly all positive definite vech specifications.
The model is named after an earlier version of the paper, which was based
on the contributions of Baba, Engle, Kraft and Kroner. The BEKK model elegantly
imposes the restrictions of positive definiteness of H t by (i) decomposing the constant
matrix into Ω ′ Ω, and (ii) by using quadratic forms of Γ and ∆ instead of imposing
restrictions on these matrices. The model is given by
Ht = Ω ′ Ω + Γ ′ ɛt−1ɛ ′ t−1Γ + ∆ ′ Ht−1∆, (5.51)
where Ω, Γ and ∆ are symmetric N × N matrices with Ω being upper triangular. In
the bivariate case the BEKK model can be written as
� �′ �
� � �
γ11 γ12
ɛ1,t−1ɛ2,t−1 γ11 γ12
Ht = Ω ′ Ω ′ +
γ21 γ22
�′
�
δ11 δ12
+
δ21 δ22
ɛ 2 1,t−1
ɛ2,t−1ɛ1,t−1
�
δ11 δ12
Ht−1
δ21 δ22
ɛ 2 2,t−1
γ21 γ22
�
. (5.52)
The number of parameters equals N(5N + 1)/2. For N = 2, two more unknowns than
in the diagonal vech setting have to be estimated. The number of parameters can be
16 The operation diag(x) with x = [x1 · · · xN] ′ denotes the N × N diagonal matrix with the
diagonal elements given by x.
17 For two N × N matrices A and B, the Hadamard product A ⊙ B is the N × N matrix that
contains the element-by-element products [aijbij]i,j=1,...,N.