Building conditionally dependent parametric one-factor copulas

So far, one-factor copulas induce conditional independence with respect to a latent factor. In this paper, we extend one-factor copulas to conditionally dependent models. This is achieved through two representations which allow to build new parametric one-factor copulas with a varying conditional dependence structure. Moreover, the latent factor's distribution can be estimated despite it being unobserved. In order to dis- tinguish between conditionally independent and conditionally dependent one-factor copulas, we provide with a novel statistical test which does not assume any parametric form for the conditional dependence structure. Illustrations of the approach are provided through examples, numerical experiments as well as a real data analysis where we capture the intrinsic state of a financial market and the dependence structure of its individual assets.