Bayesian Inference for Stochastic Volatility Models
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Date
2012-09-26T17:32:44Z
Authors
Men, Zhongxian
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
Stochastic volatility (SV) models provide a natural framework for a
representation of time series for financial asset returns. As a
result, they have become increasingly popular in the finance
literature, although they have also been applied in other fields
such as signal processing, telecommunications, engineering, biology,
and other areas.
In working with the SV models, an important issue arises as how to
estimate their parameters efficiently and to assess how well they
fit real data. In the literature, commonly used estimation methods
for the SV models include general methods of moments, simulated
maximum likelihood methods, quasi Maximum likelihood method, and
Markov Chain Monte Carlo (MCMC) methods. Among these approaches,
MCMC methods are most flexible in dealing with complicated structure
of the models. However, due to the difficulty in the selection of
the proposal distribution for Metropolis-Hastings methods, in
general they are not easy to implement and in some cases we may also
encounter convergence problems in the implementation stage. In the
light of these concerns, we propose in this thesis new estimation
methods for univariate and multivariate SV models. In the simulation
of latent states of the heavy-tailed SV models, we recommend the
slice sampler algorithm as the main tool to sample the proposal
distribution when the Metropolis-Hastings method is applied. For the
SV models without heavy tails, a simple Metropolis-Hastings method
is developed for simulating the latent states. Since the slice
sampler can adapt to the analytical structure of the underlying
density, it is more efficient. A sample point can be obtained from
the target distribution with a few iterations of the sampler,
whereas in the original Metropolis-Hastings method many sampled
values often need to be discarded.
In the analysis of multivariate time series, multivariate SV models
with more general specifications have been proposed to capture the
correlations between the innovations of the asset returns and those
of the latent volatility processes. Due to some restrictions on the
variance-covariance matrix of the innovation vectors, the estimation
of the multivariate SV (MSV) model is challenging. To tackle this
issue, for a very general setting of a MSV model we propose a
straightforward MCMC method in which a Metropolis-Hastings method is
employed to sample the constrained variance-covariance matrix, where
the proposal distribution is an inverse Wishart distribution. Again,
the log volatilities of the asset returns can then be simulated via
a single-move slice sampler.
Recently, factor SV models have been proposed to extract hidden
market changes. Geweke and Zhou (1996) propose a factor SV model
based on factor analysis to measure pricing errors in the context of
the arbitrage pricing theory by letting the factors follow the
univariate standard normal distribution. Some modification of this
model have been proposed, among others, by Pitt and Shephard (1999a)
and Jacquier et al. (1999). The main feature of the factor SV
models is that the factors follow a univariate SV process, where the
loading matrix is a lower triangular matrix with unit entries on the
main diagonal. Although the factor SV models have been successful in
practice, it has been recognized that the order of the component may
affect the sample likelihood and the selection of the factors.
Therefore, in applications, the component order has to be considered
carefully. For instance, the factor SV model should be fitted to
several permutated data to check whether the ordering affects the
estimation results. In the thesis, a new factor SV model is
proposed. Instead of setting the loading matrix to be lower
triangular, we set it to be column-orthogonal and assume that each
column has unit length. Our method removes the permutation problem,
since when the order is changed then the model does not need to be
refitted. Since a strong assumption is imposed on the loading
matrix, the estimation seems even harder than the previous factor
models. For example, we have to sample columns of the loading matrix
while keeping them to be orthonormal. To tackle this issue, we use
the Metropolis-Hastings method to sample the loading matrix one
column at a time, while the orthonormality between the columns is
maintained using the technique proposed by Hoff (2007). A von
Mises-Fisher distribution is sampled and the generated vector is
accepted through the Metropolis-Hastings algorithm.
Simulation studies and applications to real data are conducted to
examine our inference methods and test the fit of our model.
Empirical evidence illustrates that our slice sampler within MCMC
methods works well in terms of parameter estimation and volatility
forecast. Examples using financial asset return data are provided to
demonstrate that the proposed factor SV model is able to
characterize the hidden market factors that mainly govern the
financial time series. The Kolmogorov-Smirnov tests conducted on
the estimated models indicate that the models do a reasonable job in
terms of describing real data.
Description
Keywords
Bayesian inference, Stochastic volatility, Markov Chain Monte Carlo