The joint distribution of stock returns is not elliptical

Using a large set of daily US and Japanese stock returns, we test in detail the relevance of Student models, and of more general elliptical models, for describing the joint distribution of returns. We find that while Student copulas provide a good approximation for strongly correlated pairs of stocks, systematic discrepancies appear as the linear correlation between stocks decreases, that rule out all elliptical models. Intuitively, the failure of elliptical models can be traced to the inadequacy of the assumption of a single volatility mode for all stocks. We suggest several ideas of methodological interest to efficiently visualise and compare different copulas. We identify the rescaled difference with the Gaussian copula and the central value of the copula as strongly discriminating observables. We insist on the need to shun away from formal choices of copulas with no financial interpretation.Comment: 12 figure

Generalized Additive Modeling For Multivariate Distributions

In this thesis, we develop tools to study the influence of predictors on multivariate distributions. We tackle the issue of conditional dependence modeling using generalized additive models, a natural extension of linear and generalized linear models allowing for smooth functions of the covariates. Compared to existing methods, the framework that we develop has two main advantages. First, it is completely flexible, in the sense that the dependence structure can vary with an arbitrary set of covariates in a parametric, nonparametric or semiparametric way. Second, it is both quick and numerically stable, which means that it is suitable for exploratory data analysis and stepwise model building. Starting from the bivariate case, we extend our framework to pair-copula constructions, and open new possibilities for further applied and methodological work. Our regression-like theory of the dependence, being built on conditional copulas and generalized additive models, is at the same time theoretically sound and practically useful