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I'm reading this paper by A. J. Patton
http://www.christoffersen.com/CHRISTOP/2007/Patton_IER_2006.pdf
on dynamic copulas whose parameter is driven, indirectly via Kendall's tau, by some ARMA-type evolution path. However he only proposes the evolution path and logistic transformation for the symmetrized Joe-Clayton (SJC) copula. Does anyone know if the same evolution path can be applied to other copulas e.g. Joe, Frank, Gumbel, etc.? Thank you.

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The paper in your post discusses this issue already:

  • It is quite clear that knowing the marginal distributions and linear correlation is not sufficient to describe a joint distribution: Clayton’s copula, for example, has contours that are quite peaked in the negative quadrant, implying greater dependence for joint negative events than for joint positive events. Gumbel’s copula implies the opposite.

  • We will specify and estimate two alternative copulas, the “symmetrized Joe–Clayton” copula and the normal (or Gaussian) copula, both with and without time variation. The normal copula may be considered the benchmark copula in economics, though Chen et al. (2004) find evidence against the bivariate normal copula for many exchange rates. The reason for our interest in the symmetrized Joe–Clayton specification is that although it nests symmetry as a special case, it does not impose symmetric dependence on the variables like the normal copula.

Just have a closer look at the paper, it even proposes new copulas.

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