Applications of state space models in finance

5.2 Stochastic volatility 69
likelihood (QML) approach as proposed by Harvey et al. (1994). It is based on the
transformation of the nonlinear Gaussian SV model into a linear non-Gaussian state
space model. In comparison to GMM procedures, QML has the attractive feature that
it can be easily extended to non-stationary and multivariate settings. Estimates of the
parameters as well as filtered and smoothed conditional volatility series can be readily
obtained via the Kalman filter.
While the estimation procedures related to GMM and QML are based on finding
solutions for analytical parameter functions, in many situations it is impossible to find
such analytical expressions. This is where computationally more demanding simulation
techniques come into play. They include the simulated methods of moments approach
by Duffie and Singleton (1993), the indirect inference estimator by Gouriéroux et al.
(1993) and the moment matching procedure by Gallant and Tauchen (1996).
5.2.2.2 Markov chain Monte Carlo
All estimation methods mentioned so far have in common that they are either based
on moments or on an approximation of the exact likelihood. Alternatively, inference
can be performed by computing the likelihood directly using computationally intensive
methods. The most prominent approach to date is the application of Markov chain
Monte Carlo (MCMC) techniques. Their origins trace back to the statistical physics
literature. The first fully efficient parametric inference procedures for SV models based
on MCMC have been developed by Shephard (1993) and Jacquier et al. (1994). Kim
et al. (1998) discussed alternative simulation-based strategies for actually implementing
MCMC to estimate SV models.
Following the outline of Andersen et al. (2005) MCMC can be used to deal with the
high-dimensional inference problem inherent to the likelihood of the SV model from
a Bayesian perspective. In the Bayesian MCMC approach, the model parameters are
considered to be random variables. The entire latent volatility process is treated as an
additional parameter. The focus is on the possibly high-dimensional joint distribution
of this parameter vector, conditioned by the data, which is referred to as the posterior
distribution. Âlthough this density is analytically intractable, it can be characterized
through a set of related conditional distributions. These allow a single parameter or a
whole group of parameters to be expressed conditional on the remaining parameters.
This feature is exploited by the MCMC procedure, which can be summarized as follows:
after initialization of the vector of parameters conditioned on the observed data,
the coefficients are drawn from the assumed prior distribution. The current draw for
the parameters is combined with the dynamics of the SV model and the observations.
A complete cycle through all conditional densities is called a sweep of the sampler.
Depending on the form of these distributions, different samplers such as the Metropolis-
Hastings algorithm or the Gibbs sampler or a combination of both can be used. Once
the sample generated from the joint posterior distribution is long enough, inference on
the parameters and the latent volatility process can be made. For a general introduction
to MCMC procedures, see Geman and Geman (1984), West and Harrison (1997, ¢ 15) or
Chib and Greenberg (1996). The idea of estimating SV models via MCMC procedures
is comprehensively illustrated by Jacquier et al. (1994).