Generalized Elliptical Distributions: Theory and Applications

The thesis recalls the traditional theory of elliptically symmetric distributions. Their basic properties are derived in detail and some important additional properties are mentioned. Further, the thesis concentrates on the dependence structures of elliptical or even meta-elliptical distributions using extreme value theory and copulas. Some recent results concerning regular variation and bivariate asymptotic dependence of elliptical distributions are presented. Further, the traditional class of elliptically symmetric distributions is extended to a new class of `generalized elliptical distributions' to allow for asymmetry. This is motivated by observations of financial data. All the ordinary components of elliptical distributions, i.e. the generating variate, the location vector and the dispersion matrix remain. Particularly, it is proved that skew-elliptical distributions belong to the class of generalized elliptical distributions. The basic properties of generalized elliptical distributions are derived and compared with those of elliptically symmetric distributions. It is shown that the essential properties of elliptical distributions hold also within the broader class of generalized elliptical distributions and some models are presented. Motivated by heavy tails and asymmetries observed in financial data the thesis aims at the construction of a robust dispersion matrix estimator in the context of generalized elliptical distributions. A `spectral density approach' is used for eliminating the generating variate. It is shown that the `spectral estimator' is an ML-estimator provided the location vector is known. Nevertheless, it is robust within the class of generalized elliptical distributions. The spectral estimator corresponds to an M-estimator developed 1983 by Tyler. But in contrast to the more general M-approach used by Tyler the spectral estimator is derived on the basis of classical maximum-likelihood theory. Hence, desired properties like, e.g., consistency, asymptotic efficiency and normality follow in a straightforward manner. Not only caused by the empirical evidence of extremes but also due to the inferential problems occuring for high-dimensional data the performance of the spectral estimator is investigated in the context of modern portfolio theory and principal components analysis. Further, methods of random matrix theory are discussed. They are suitable for analyzing high-dimensional covariance matrix estimates, i.e. given a small sample size relative to the number of dimensions. It is shown that the Marchenko-Pastur law fails if the sample covariance matrix is used in the context of elliptically of even generalized elliptically distributed and heavy tailed data. But substituting the sample covariance matrix by the spectral estimator resolves the problem and the classical arguments of random matrix theory remain valid

Bayesian Multivariate Regression Analysis with a New Class of Skewed Distributions

In this paper, we introduce a novel class of skewed multivariate distributions and, more generally, a method of building such a class on the basis of univariate skewed distributions. The method is based on a general linear transformation of a multidimensional random variable with independent components, each with a skewed distribution. Our proposed class of multivariate skewed distributions has a simple, intuitive form for the pdf, moment existence only depends on the existence of the moments of the underlying symmetric univariate distributions, and we avoid any conditioning on unobserved variables. In addition, we can freely allow for any mean and covariance structure in combination with any magnitude and direction of skewness. In order to deal with both skewness and fat tails, we introduce multivariate skewed regression models with fat tails, based on Student distributions. We present two main classes of such distributions, one of which is novel even under symmetry. Under standard non-informative priors on both regression and scale parameters, we derive conditions for propriety of the posterior and for existence of posterior moments. We describe MCMC samplers for conducting Bayesian inference and analyse two applications, one concerning the distribution of various measures of firm size and another on a set of biomedical data.Asymmetric distributions; Heavy tails; Linear regression model; Mardia's measure of skewness; Orthogonal matrices; Posterior propriety.

Dynamic Specification Tests for Static Factor Models

We derive computationally simple score tests of serial correlation in the levels and squares of common and idiosyncratic factors in static factor models. The implicit orthogonality conditions resemble the orthogonality conditions of models with observed factors but the weighting matrices refl ect their unobservability. We derive more powerful tests for elliptically symmetric distributions, which can be either parametrically or semipametrically specified, and robustify the Gaussian tests against general non-normality. Our Monte Carlo exercises assess the finite sample reliability and power of our proposed tests, and compare them to other existing procedures. Finally, we apply our methods to monthly US stock returns.ARCH, Financial returns, Kalman filter, LM tests, Predictability

On the efficiency and consistency of likelihood estimation in multivariate conditionally heteroskedastic dynamic regression models

We rank the efficiency of several likelihood-based parametric and semiparametric estimators of conditional mean and variance parameters in multivariate dynamic models with i.i.d. spherical innovations, and show that Gaussian pseudo maximum likelihood estimators are inefficient except under normality. We also provide conditions for partial adaptivity of semiparametric procedures, and relate them to the consistency of distributionally misspecified maximum likelihood estimators. We propose Hausman tests that compare Gaussian pseudo maximum likelihood estimators with more efficient but less robust competitors. We also study the efficiency of sequential estimators of the shape parameters. Finally, we provide finite sample results through Monte Carlo simulations.Adaptivity, ARCH, Elliptical Distributions, Financial Returns, Hausman tests, Semiparametric Estimators, Sequential Estimators.

Numerical Reconstruction of the Covariance Matrix of a Spherically Truncated Multinormal Distribution

We relate the matrix SB of the second moments of a spherically truncated normal multivariate to its full covariance matrix Σ and present an algorithm to invert the relation and reconstruct Σ from SB. While the eigenvectors of Σ are left invariant by the truncation, its eigenvalues are nonuniformly damped. We show that the eigenvalues of Σ can be reconstructed from their truncated counterparts via a fixed point iteration, whose convergence we prove analytically. The procedure requires the computation of multidimensional Gaussian integrals over an Euclidean ball, for which we extend a numerical technique, originally proposed by Ruben in 1962, based on a series expansion in chi-square distributions. In order to study the feasibility of our approach, we examine the convergence rate of some iterative schemes on suitably chosen ensembles of Wishart matrices. We finally discuss the practical difficulties arising in sample space and outline a regularization of the problem based on perturbation theory