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In Dependence Modeling with Copulas (Harry Joe) I'm struggling to interpret the meaning of a statement. In Chaper 5.1, it is stated:

Parametric inference for copulas

For dependence modeling with copulas, most common is the use of parametric copula families. Advantages of parametric inference methods for copula models are the following.

  • They are easier to numerically implement than non-parametric approaches.
  • They can be used in high dimensions.
  • With the use of the likelihood, it is the same theory for continuous, discrete or mixed response variables, and censored or missing data can be accommodated.
  • They can be adapted to univariate margins that are regressions or time series models, or to time-varying dependence. Covariates can be included in univariate or dependence parameters.
  • They can easily be used to compare competing models.

My question relates to the fourth point, specifically to They can be adapted to univariate margins that are regression. It seems to imply that the copula specification requires some additional parameterisation due to the fact that the marginals are now regressions, dependent on covariates.

Intuitively, I can appreciate this as the covariates would determine the actual distribution of the marginals and therefore the copula would then also need to respond to this. However, I can also intuit that the copula has already been estimated using data from the marginal regressions and so already captures this information from the covariates.

That is, the $(u,v)$ used to estimate the copula already reflect the covariates from those marginal distributions. Is it ok for the copula specification to then be agnostic of those covariates?

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  • $\begingroup$ In both "regression" and "time series model" settings, a collection of errors (or "innovations") is modeled as a random vector. That's really just a special example of the "used in high dimensions" application in the second bullet. $\endgroup$
    – whuber ♦
    Commented Jan 16 at 17:20
  • $\begingroup$ Thanks for the comment @whuber. My question relates to the fourth point. Was your comment related to that? $\endgroup$
    – statsplease
    Commented Jan 17 at 0:28
  • $\begingroup$ Yes, it was, as I had hoped the phrases in quotes indicated. $\endgroup$
    – whuber ♦
    Commented Jan 17 at 1:03
  • $\begingroup$ Maybe my question isn't worded clearly. My question revolves around the implication (by Joe) that something about changing from a static marginal (fixed distribution, not dependent on covariates) to a regression marginal changes the specification of the copula and requires adaptation. $\endgroup$
    – statsplease
    Commented Jan 17 at 11:09

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