  Parametric bivariate copula families have been known to flexibly capture
enough various dependence patterns, e.g., either positive or negative
dependence in either the lower or upper tails of bivariate distributions.
However, to the best of our knowledge, there is not a single parametric model
adaptable enough to capture several of these features simultaneously. To
address this, we propose a mixture of 4-way rotations of a parametric copula
that is able to capture all these features. We illustrate the construction
using the Clayton family but the concept is general and can be applied to other
families. In order to include dynamic dependence regimes, the approach is
extended to a time-dependent sequence of mixture copulas in which the mixture
probabilities are allowed to evolve in time via a moving average type of
relationship. The properties of the proposed model and its performance are
examined using simulated and real data sets.
