Applications of state space models in finance

76 5 Conditional heteroskedasticity models
This section summarizes some of the most important multivariate conditional heteroskedasticity
models. For those topics and the many extensions that are not covered
here, the reader is referred to Bauwens et al. (2003) and Asai et al. (2006) who provide
surveys of multivariate GARCH and multivariate SV models, respectively.
5.3.1 Multivariate GARCH
A general multivariate GARCH model for an N-dimensional process ɛt|Ωt−1 is given by
ɛt = ztH 1/2
t , (5.45)
where zt is an N-dimensional IID process with zero mean and the identity matrix
IN as covariance matrix. These properties of zt together with Equation (5.45) imply
that E(ɛt|Ωt−1) = 0 and E(ɛtɛ ′ t|Ωt−1) = Ht. For illustrative purposes, only the case
with N = 2 will be considered in the following. In the bivariate case, the conditional
covariance matrix is given by
Ht =
� h11,t h12,t
h21,t h22,t
�
, (5.46)
where Ht depends on lagged errors ɛt−1 and on lagged conditional covariance matrices
Ht−1. The most influential parameterizations of H t can be summarized as follows.
5.3.1.1 The vech model
The most general representation of H t−1 is the vech model as proposed by Bollerslev
et al. (1988). By employing the vech( ) operator, which vertically stacks the matrix
elements on or below the principal diagonal and thus transforms an N × N matrix into
an N(N + 1)/2 × 1 vector, all non-redundant elements of H t are stacked into a column
vector:
vech(Ht) = ω ∗ + Γ ∗ vech(ɛt−1ɛ ′ t−1 ) + ∆∗ vech(Ht−1), (5.47)
where ω ∗ = vech(Ω) is a N(N + 1)/2 × 1 parameter vector and Γ ∗ and ∆ ∗ are N(N +
1)/2 × N(N + 1)/2 matrices. In the bivariate case, Equation (5.47) becomes
⎡
⎣
h11,t
h21,t
h22,t
⎤
⎡
⎦ = ⎣
ω ∗ 11
ω ∗ 21
ω ∗ 22
⎤
⎡
⎦ + ⎣
⎡
+ ⎣
γ ∗ 11 γ ∗ 12 γ ∗ 13
γ ∗ 21 γ ∗ 22 γ ∗ 23
γ ∗ 31 γ ∗ 32 γ ∗ 33
δ ∗ 11 δ ∗ 12 δ ∗ 13
δ ∗ 21 δ ∗ 22 δ ∗ 23
δ ∗ 31 δ ∗ 32 δ ∗ 33
⎤ ⎡
⎦ ⎣
⎤ ⎡
⎦ ⎣
ɛ 2 1,t−1
ɛ2,t−1ɛ1,t−1
ɛ2 2,t−1
h11,t−1
h21,t−1
h22,t−1
⎤
⎤
⎦
⎦ . (5.48)
Despite its flexibility, the vech model has two major drawbacks: to guarantee positive
definiteness of H t it is necessary to impose further constraints on Γ ∗ and ∆ ∗ ; see
Engle and Kroner (1995) for a discussion. Besides, overall N(N + 1)/2 + N 2 (N + 1) 2 /2
parameters have to be estimated. As this number grows at a polynomial rate with
increasing N, estimation of this general model may become quite cumbersome without
further restrictions.