Nonparametric Forecasting of Multivariate Probability Density Functions

The study of dependence between random variables is the core of theoretical and applied statistics. Static and dynamic copula models are useful for describing the dependence structure, which is fully encrypted in the copula probability density function. However, these models are not always able to describe the temporal change of the dependence patterns, which is a key characteristic of financial data. We propose a novel nonparametric framework for modelling a time series of copula probability density functions, which allows to forecast the entire function without the need of post-processing procedures to grant positiveness and unit integral. We exploit a suitable isometry that allows to transfer the analysis in a subset of the space of square integrable functions, where we build on nonparametric functional data analysis techniques to perform the analysis. The framework does not assume the densities to belong to any parametric family and it can be successfully applied also to general multivariate probability density functions with bounded or unbounded support. Finally, a noteworthy field of application pertains the study of time varying networks represented through vine copula models. We apply the proposed methodology for estimating and forecasting the time varying dependence structure between the S&P500 and NASDAQ indices.