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Tail dependence in the pseudo-observations of bivariate copula imply that extreme upper or lower samples move together in some way suggesting correlation between the two marginals' (variables') extreme samples.

Kurtosis on the other hand describes the fat-tails found in the extreme samples of only one non-Gaussian univariate (marginal) distribution, therefore lacking the dependence feature.

Since both concepts address extreme samples, but with tail dependence being a concept applied to bivariate data, and kurtosis applied to univariate data (therefore correlation of extreme samples does not apply to kurtosis), is there some sort of connection between copula tail dependence and univariate (or maybe multivariate) kurtosis? i.e. does high tail dependence indicate kurtosis somehow, or vice versa?

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There is no reason for tail dependence to vary with kurtosis or the other way around.

Consider the copula model of the joint distribution. The copula function specifies tail dependence for probability integral transforms (PITs) of the marginal distributions, not for the marginal distributions themselves. Meanwhile, kurtosis is a feature of a marginal distribution alone. The possible PIT mappings are in no way bound by the copula function or its features such as tail dependence; they can correspond to any kind of univariate cumulative distribution function (CDF).

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  • $\begingroup$ if there is no connection between tail dependence and kurtosis, then it can be said that copula cannot capture fat-tails (leptokurtosis)? $\endgroup$
    – develarist
    Sep 6, 2020 at 22:53
  • $\begingroup$ @develarist, yes, that is correct. Copula only captures interdependence between the variables, not features of any single variable taken separately. $\endgroup$ Sep 7, 2020 at 6:00
  • $\begingroup$ if the copula function is equal to joint CDF then both completely fail to account for fat tails? $\endgroup$
    – develarist
    Sep 7, 2020 at 6:02
  • $\begingroup$ @develarist, no, because joint CDF incorporates information about fat tails from each individual variable via its marginal distribution. $\endgroup$ Sep 7, 2020 at 6:06
  • $\begingroup$ so joint CDF uses the actual marginals, whereas the copula function uses transformed (uniform) marginals, allowing it, but not the copula, to capture heavy tails? $\endgroup$
    – develarist
    Sep 7, 2020 at 6:09

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