Wishart Processes in Statistics and Econometrics: Theory and Applications

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The purpose of this project is to make a series of theoretical and practical contributions in two major fields. The first field deals with statistical properties of the Wishart distribution and its several extensions. First, we analyze the Wishart distribution not isolated, but in a combination with a Gaussian vector. This fact is motivated by applications in discriminant analysis and portfolio theory, where the key quantities are given by a product of inverse Wishart matrix and a Gaussian vector. Second, in the light of the increasing importance of asymmetric distributions, we extend the concept of the Wishart matrices to matrices obtained from truncated and skewed Gaussian vectors. Third, we use the results to extend the paper of Bodnar et al. (2009) by developing statistical monitoring procedures for new distribution classes. The second field of contribution is econometrically oriented. Recent econometric developments show an increasing interest in time-series models for Wishart processes. Within the project we will extend the results of Gourieroux et al. (2009) and Chiriac and Voev (2010) to develop the theory of autoregressive (inverse, singular) Wishart processes. This type of the models is particularly important for multivariate modelling of the volatility dynamics and realized volatilities based on high-frequency data. Furthermore, a new multivariate class of stochastic volatility processes will be suggested which relies on the square root of the autoregressive Wishart process. This is a new and interesting alternative to current multivariate GARCH models.