  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 new representations which allow to
build new parametric factor copulas with a varying conditional dependence
structure. We discuss estimation and properties of these representations. In
order to distinguish between conditionally independent and conditionally
dependent factor copulas, we provide a novel statistical test which does not
assume any parametric form for the conditional dependence structure.
Illustrations of our framework are provided through examples, numerical
experiments, as well as a real data analysis.
