Wishart Processes in Statistics and Econometrics: Theory and Applications

The objective of this project is to make a series of theoretical and empirical contributions. We will concentrate on the following three subprojects. First, we plan to study the distribution of the quadratic forms under non-normality. Particularly we will concentrate on truncated Gaussian distribution and the general family of elliptical distributions. Second, a goodness-of-fit test for Wishart models will be derived and applied to the existent Wishart processes. The econometric part will be partially modified in the light of obtained theoretical and empirical results. Direct modelling of Wishart matrices is problematic due to potentially non-positive definite forecasts. Taking Cholesky decomposition of the matrices guarantees the positive definiteness of the predictions, depends, however, on the ordering of the time series. We will analyse the impact of the autocorrelation structure in the volatility processes, i.e. arising from Autoregressive Wishart processes, on the optimal asset ordering for the Cholesky decomposition. This appears to the first step while extending the current models to specific features of the Wishart processes. These problems can be analyzed empirically and analytically. Moreover, we will introduce the inverse Wishart process in a similar autoregressive fashion as the WAR(p). This model is important in practice, since the inverse covariance matrix, frequently named precision matrix, appears in numerous applications, i.e. in the expressions for portfolio weights in finance.