A note on a simplified and general approach to simulating from multivariate copula functions

Abstract

Copulas have become an important analytic tool for characterizing multivariate distributions and dependence. One is often interested in simulating data from copula estimates. The process can be analytically and computationally complex and usually involves steps that are unique to a given parametric copula. We describe an alternative approach that uses ‘Probability-Proportional-to-Size’ random sampling with weights formed from the copula likelihood function. The method is flexible and can be applied to parametric and nonparametric marginal density estimates. The precision of the simulation can be calibrated by adjusting the density of the multidimensional grid used in the simulation process. The approach is fully transparent to any copula function with continuous random variables. An example evaluates a number of goodness-of-fit criteria and provides strong support for the validity and practicality of the method.

Keywords:

• copulas
• simulation
• nonparametric marginal densities

JEL Classification:

• C15
• C10

Acknowledgements

This work was supported by the US Forest Service and the North Carolina Agricultural Research Service. The helpful comments of Andrew Patton and Sujit Ghosh are gratefully acknowledged.

Notes

¹ For details on construction and properties of copulas, see among others, Joe (1997) and Nelsen (2006).

² Estimation and inferences were accomplished using the ‘Probability-Proportional-to-Size’ ‘copula’ and ‘CD Vine’ packages of the R language. Details are available in Gambino (2012), Brechman and Schepsmeier (2012) and Kojadinovic and Yan (2010). Excellent overviews of the R‐copula packages and implementation issues are presented by Yan (2007) and Czado (2011).

³ Values above the diagonal correspond to positive dependence. The outer envelope represents perfect correlation.