Applications of state space models in finance

74 5 Conditional heteroskedasticity models
with ˆσ 2 ∗ denoting the ML estimate of σ 2 ∗. The estimator of hT +1 given all T observations
and its MSE, denoted as ˆ h T +1|T and p T +1|T , are given by
ˆh T +1|T =
p T +1|T =
� M
� M
i=1 wih T +1|T θ (i)
� M
i=1 wi
, (5.42)
i=1 wipT +1|T θ (i)
�M i=1 wi
. (5.43)
Based on the draw θ (i) , the Kalman filter is applied to the approximating model g(θ|y)
to obtain h T +1|T θ (i) and p T +1|T θ (i) .
5.2.4 Extensions
The SV model can be extended in various directions. As in the case of the GARCH
framework, the most important extensions cope with alternative error distributions for
the conditional mean innovations and leverage effects. An account of recently published
papers related to extensions of the basic SV model, applying both MCMC and efficient
MCL methods, is given in ¢ 5.2.2.
5.2.4.1 Heavy-tailed distributed errors
Empirical evidence of heavy-tailed ɛt in the context of SV models has been provided,
for example, by Gallant et al. (1997). The efficient MCL method using importance sampling
techniques presented in ¢ 5.2.3.2, can be adapted to consider Student-t distributed
errors by replacing the Gaussian density p(y|θ) by a Student-t density with ν degrees of
freedom. This leads to an altered pair of equations, which can be solved to obtain an approximating
model through at and bt. The importance density itself remains Gaussian;
see Lee and Koopman (2004, Appendix B) for details.
Even though the basic SV model can be generalized by allowing for Student-t distributed
mean errors, throughout this thesis only SV models with Gaussian errors will
be considered. According to Ghysels et al. (1996) the SV model in its basic form is able
to capture the excess kurtosis usually found in financial time series by considering yt
as a mixture of distributions, where the degree of mixing is governed by the parameter
σ2 η. As outlined in ¢ 5.2.1.2, the kurtosis of yt is equal to 3 exp � σ2 �
h which can take any
nonnegative value. Besides, the empirical analyses in this thesis are strictly based on
weekly series, for which the evidence in favor of fat-tails is not as strong as for daily
data (cf. Jacquier et al. 2004).
5.2.4.2 Asymmetric effects
As outlined in ¢ 5.1.2, asymmetric effects represent a well documented empirical stylized
fact for many financial time series. It has been demonstrated in the previous section how
to employ nonlinear GJR-GARCH and EGARCH extensions to specify the conditional
volatility as a function of the sign and/ or size of past returns. For discrete time SV
models, leverage effects can be implemented by allowing for contemporaneously negatively
correlated observation and state disturbances. One of the earliest studies on