Applications of state space models in finance

80 5 Conditional heteroskedasticity models
approximation using a first-order Taylor series expansion of the correlation coefficient:
with
ρij,t+l ≈
¯qij
(¯qii ¯qjj) 1/2 + qij,t+l − ¯qij
(¯qii ¯qjj) 1/2
− 1
2
¯qij
(¯qii ¯qjj) 1/2
� qii,t+l − ¯qii
¯qii
+ qjj,t+l
�
− ¯qjj
, (5.64)
¯qjj
Et(qij,t+l) = ¯ρ(1 − α − β) + αEt(˜ɛi,t+l−1˜ɛj,t+l−1) + βEt(qij,t+l−1), (5.65)
where Et(˜ɛi,t+l−1˜ɛj,t+l−1) = 1 for i = j, and Et(˜ɛi,t+l−1˜ɛj,t+l−1) = Et(ρij,t+l−1) for
i �= j.
The most important feature of the DCC model is that it can be estimated in two
steps. This makes the method as feasible for the estimation of large variance-covariance
matrices as the CCC model. Besides, the DCC model allows for the incorporation of
nonlinear extensions and non-Gaussian conditional densities through the modeling of
the univariate conditional variances, as introduced in ¢ 5.1.2 and ¢ 5.1.3, respectively. A
drawback is that the dynamics of all conditional correlations are modeled to be the same
as they all depend on the scalars α and β. For a review of the recent literature coping
with this problem, see Bauwens et al. (2003, ¢ 2.3).
Even though the DCC model represents a more general approach to multivariate
GARCH models, the assumption of constant correlations has been shown to be reasonable
in many empirical applications; see, for example, Baillie and Bollerslev (1990) and
Schwert and Seguin (1990). Hence, for the empirical analyses in this thesis the CCC
model will be employed. As the multivariate SV model to be outlined below will also
be based on the assumption of constant conditional correlations, the results obtained
from the chosen multivariate conditional heteroskedasticity models can be compared in
a straightforward fashion. In the absence of distractions caused by different ways of
modeling conditional correlations, both chosen multivariate specifications purely reflect
the respective volatility processes.
5.3.2 Multivariate stochastic volatility
In recent years, the basic univariate SV model has been extended to cope with multivariate
N × 1 time series vectors y t = [y1,t · · · yN,t] ′ . Important works on the issue
include, among others, Harvey et al. (1994), Pitt and Shephard (1999), Aguilar and
West (2000), Chan et al. (2005), Jungbacker and Koopman (2005a) and Chib et al.
(2006). Similar to the case of multivariate GARCH models, the literature on multivariate
SV (MSV) focusses on (i) conditions that guarantee positive definiteness of the
conditional covariance matrix, and (ii) restrictions that reduce the number of unknown
parameters. According to Asai et al. (2006) the different approaches to address these
two issues can be categorized as follows:
A basic model with constant correlations.
Asymmetric models.