Direction of a copula

See also: Copulas in ModelRisk, the Vose Bivariate Copula window, the Vose Multivariate Copula window

The three Archimedean copulas in ModelRisk exhibit positive correlation (pane 1 below).  In the multivariate versions of these copulas the positive correlation pattern must be maintained.

However, for a bivariate copula we have the flexibility to transform the relationships by taking one or both of the u,v variables and transform to 1-u or 1-v. This gives us three more patterns, shown in panes 2-4 below.

The Clayton and Gumbel copulas can be set to directions 1, 2, 3 and 4 corresponding to the directions illustrated below:

 

As the Frank copula is symmetric under rotations of 180° it can only take 1 (default) or 2 as direction parameter.

The T copula remains identical under any number of 90° rotations of the plane so it does not take a direction parameter.

Changing the direction of the Normal copula corresponds to changing the sign of its covariance parameter.

 

 

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