Dynamic Copula Quantile Regressions and Tail Area Dynamic Dependence in Forex Markets

We introduce a general approach to nonlinear quantile regression modelling that is based on the specification of the copula function that defines the dependency structure between the variables of interest. Hence we extend Koenker and Bassett’s [1978] original statement of the quantile regression problem by determining a distribution for the dependent variable Y conditional on the regressors X and hence the specification of all the quantile regression functions. We use the fact that this multivariate distribution can be split into two parts: the marginals and the dependence function (or copula). We then deduce the form of the non linear conditional quantile relationship implied by the copula. Notice that this can be done with arbitrary distributions assumed for the marginals. Some properties of the copula based quantiles or c-quantiles are then derived. Finally, we develop an empirical application which examines conditional quantile dependency in the foreign exchange market and compare this approach with the standard tail area dependency measures.