PhD Thesis

Auxiliary Variable Sampling for some Time-Varying Volatility Models
 

Supervisor: Prof. Petros Dellaportas

Examiners:

• Prof. Stephen G. Walker (University of Bath),

• Prof. Peter Muller (U.T. M.D. Anderson Cancer Center),

• Prof. Paul Damien (University of Texas at Austin),

• Prof. Ε. Xekalaki (AUEB),

• Prof. Ν. Fragos (AUEB),

• Prof. Ε. Panas (AUEB)

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ABSTRACT

The phenomenon of changing variance and covariance is often encountered in financial time series. As a result, during the last years researchers focused on the time-varying volatility models. These models are able to describe the main characteristics of the financial data such as the volatility clustering.

 In addition, the development of the Markov Chain Monte Carlo Techniques (MCMC) provides a powerful tool for the estimation of the parameters of the time-varying volatility models, in the context of Bayesian analysis.

In this thesis, we adopt the Bayesian inference and we propose easy-to-apply MCMC algorithms for a variety of time-varying volatility models.

We use a recent development in the context of the MCMC techniques, the Auxiliary variable sampler. This technique enables us to construct MCMC algorithms, which only consist of Gibbs steps. We propose new MCMC algorithms for the following models:

Univariate models

• Stochastic volatility model,

• Unobserved ARCH model,

• ARCH model,

• GARCH model,

Multivariate models

• Stochastic volatility model,

• Unobserved ARCH,

• Latent factor ARCH model,

• Latent factor GARCH model.

By using the auxiliary variable techniques and non-linear transformations of the parameters space, we propose new MCMC algorithms - for the aforementioned models - which are easily applied and allow estimation of the posterior distribution of model parameters.

Furthermore, we apply the proposed MCMC algorithms to real data and compare the above models based on their predictive distribution

Finally, we propose a new MCMC convergence diagnostic that is based on the subsampling theory.

Keywords: Time-Varying models, Stochastic Volatility, Gibbs, Auxiliary Sampler (Slice Sampling), Unobserved components, ARCH, GARCH