Clinical Professor, Decision Sciences & MSISLeBow College of Business, Drexel University
Covariance Matrices and Skewness: Modeling and Applications in Finance
This Ph.D. dissertation is concerned with using model based and computation intensive statistical approaches to gain insight into substantive issues in finance related topics. It addresses the use of posterior (predictive) simulation for Bayesian inference in high dimensional and analytically intractable models. It consists of three studies. One study focuses on the issue of covariance estimation. I propose prior probability models for variance-covariance matrices. The proposed models address two important issues. First, the models allow a researcher to represent substantive prior information about the strength of correlations among a set of variables. Second, even in the absence of such substantive prior information, it provides increased flexibility. This is achieved by including a clustering mechanism in the prior probability model. Clustering is with respect to variables or with respect to pairs of variables. This leads to shrinkage towards a mixture structure implied by the clustering, instead of towards a diagonal structure as is commonly done. With departure from the standard MLE or inverse-Wishart prior there are many computational difficulties that arise in modeling covariances. A second study follows from the computational challenges that arise in the covariance estimation study. Because of the requirement that the correlation matrix be positive (semi-) definite, the high dimensional space of the correlation matrix is analytically intractable. Calculating the normalizing constant is then left to numerical techniques. I examine a method for estimating the constant, and also an argument for ignoring it in certain cases. An additional strategy called the ``shadow prior'' is explored by introducing an additional level of hierarchy into the model. A third study follows the foundational work of Markowitz's portfolio selection in 1952. Issues that he pointed out but didn't address include parameter uncertainty, utility optimization, and inclusion of higher moments. I build upon his foundation by incorporating a skew-normal model. Using Bayesian methods, higher moments and parameter uncertainty can be accommodated in a natural way in the portfolio selection process. Preference over portfolios is framed in terms of utility function maximization, which are optimized using Bayesian methods. Findings suggest that skewness is important for distinguishing between `good' and `bad' variance.