| Περίληψη : | The scope of this Thesis is to provide an original contribution in the area of Multivariate Volatility modeling. Multivariate Volatility modeling in the present context, involves developing models that can adequately describe the Covariance matrix process of Multivariate financial time series. Developmentof efficient algorithms for Bayesian model estimation using Markov Chain Monte Carlo (MCMC) and Nested Laplace approximations is our main objective in order to provide parsimonious and flexible volatility models. A detailed review of Univariate Volatility models for financial time series is first introduced in this Thesis. We illustrate the historical background of each model proposed and discuss its properties and advantages as well as comment on the several estimation methods that have emerged. We also provide a comparative analysis via a small simulation example for the dominant models in the literature. Continuing the review from the univariate models we move on to the multivariate case and extensively present competing models for Covariance matrices. The main argument presented is that currently no model is able to capture the dynamics of higher dimensional Covariance matrices fully, but their relative performance and applicability depends on the dataset and problem of interest. Problems are mainly due to the positive definiteness constraints required by most models as well as lack of interpretability of the model parameters in terms of the characteristics of the financial series. In addition, model development so far focus mostly in parameter estimation and in sample fit; it is our goal to examine the out-of-sample fit perspective of these models. We conclude the review section by proposing some small improvements for existing models that will lead towards more efficient parameter estimates, faster estimation methods and accurate forecasts. Subsequently, a new class of multivariate models for volatility is introduced. The new model is based on the Spectral decomposition of the time changing covariance matrix and the incorporation of autocorrelation modeling or the time changing elements. In these models we allow a priori for all the elements of the covariance matrix to be time changing as independent autoregressive processes and then for any given dataset we update our prior information and decide on the number of time changing elements. Theoretical properties of the new model are presented along with a series of estimation methods, bayesian and classical. We conclude that in order to estimate these models one may use an MCMC method for small dimension portfolios in terms of the size of the covariance matrix. For higher dimensions, due to the curse of dimensionality we propose to use a Nested Laplace approximation approach that provides results much faster with small loss in accuracy. Once the new model is proposed along with the estimation methods, we compare its performance against competing models in simulated and real datasets; we also examine its performance in small portfolios of less than 5 assets as well in the high dimensional case of up to 100 assets. Results indicate that the new model provides significantly better estimates and projections than current models in the majority of example datasets. We believe that small improvements in terms of forecasting is of significant importance in the finance industry. In addition, the new model allows for parameter interpretability and parsimony which is of huge importance due to the dimensionality curse. Simplifying inference and prediction of multivariate volatility models was our initial goal and inspiration. It is our hope that we have made a small step towards that direction, and a new path for studying multivariate financial data series has been revealed. We conclude by providing some proposals for future research that we hope may influence some people into furthering this class of models.
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