Applications of state space models in finance

5.2 Stochastic volatility 73
3. An estimate for the unobserved process θ can be computed from the state space
model given by (5.26) and (5.39) with (5.29) and (5.30) by making use of the
Kalman smoother as introduced in ¢ 3.3.2.
4. Steps #2 and #3 are repeated using the smoothed estimates of θ until convergence
of θ, at and bt is achieved.
5. Finally, with the importance density g � θ (i) |y � being evaluated using the Kalman
smoother, the likelihood given by (5.38) can be estimated with the samples θ (i)
that are drawn from g � θ (i) |y � via simulation smoothing.
In order to avoid numerical problems, this is usually done by working with logarithms of
the likelihood. Let w(θ) = p(y|θ)/g(y|θ) and wi = w � θ (i)� for i = 1, . . . , M. According
to Durbin and Koopman (1997) an approximately unbiased estimator of log L(ψ) is
given by
log ˆ L(ψ) = log LG(ψ) + log ¯w + s2w , (5.40)
2M ¯w 2
where ¯w = M −1 � M
i=1 wi and s 2 w = (M − 1)−1 � M
i=1 (wi − ¯w) 2 represent the sample
mean and variance of wi, respectively. The last term on the right hand side corrects
for the bias that can be shown to arise from taking logarithms of (5.38). According to
Durbin and Koopman (2000) the bias is usually so small that it can be neglected in practice.
Numerical maximization of the simulated loglikelihood (5.40), using an iterative
numerical optimization method as discussed in ¢ 3.4.2, yields the MCL estimate of ψ.
Starting values can be obtained from log LG(y|ψ). The accuracy of the approximation
of the likelihood function depends positively on M, the number of draws. Sandmann
and Koopman (1998) showed that N = 5 is sufficient in empirical applications.
In this thesis, the SV model is estimated using Ox 3.30 by Doornik (2001) together
with the packages SsfPack 2.3 and SsfNonG by Koopman et al. (1999). The relevant
code for implementing the estimation of SV models in Ox has been downloaded from
www.feweb.vu.nl/koopman/sv/.
5.2.3.3 Filtering, smoothing and forecasting
Once estimates for the unknown model parameters have been found as demonstrated
above, another interest focuses on obtaining filtered, smoothed and predicted estimates
of the conditional volatility process. In contrast to the GARCH model, where filtered
estimates of the volatility process can be calculated recursively once the parameters
are known, for an SV model the latent volatility has to be estimated. In addition to
the filtered volatility also smoothed estimates are available via the Kalman filter and
smoother (cf. ¢ 3.3); see Sandmann and Koopman (1998) for details.
According to Koopman et al. (2004) the one-step ahead forecast of conditional volatility
can be calculated as
ˆσ 2 T +1 = ˆσ2 ∗ exp
�
ˆh T +1|T + 1
2 p �
T +1|T , (5.41)