A Simplied, General Approach to Simulating from Multivariate Copula Functions

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” (PPS) 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. ∗This work was supported by the U.S. Forest Service and the North Carolina Agricultural Research Service. The helpful comments of Andrew Patton are gratefully acknowledged. Goodwin is William Neal Reynolds Distinguished Professor in the Departments of Economics and Agricultural and Resource Economics at North Carolina State University. Direct correspondence to Box 8109, North Carolina State University, Raleigh, NC, 27695, E-mail: barry goodwin@ncsu.edu. A Simplified, General Approach to Simulating from Multivariate Copula Functions Copulas are an important tool for modeling relationships among (possibly) dependent random variables. A copula relates marginal distributions for individual variables in a representation of the joint distribution function. Sklar (1959) showed that, for any group of continuous random variables, the joint distribution function for the variables can be uniquely represented in terms of the marginals and a copula function. Specifically, Sklar’s Theorem implies that, for any continuous p−variate cumulative probability function F , a unique copula function C(·) exists for which F (x1, x2, ..., xp) = C(F1(x1), ..., Fp(xp); ξ) (1) where Fi(·) are marginal distributions and ξ is a set of parameters that measures dependence. This implies that the joint density function can be written as f(x1, x2, . . . , xp) = c(f1(X1), . . . , fp(xp) · Πpi=1fi(xi) (2) where c(·) is given by ∂C(·)/∂x and the fi terms are marginal density functions. Copula models have recently realized widespread application in empirical models of joint probability distributions. The models essentially use a “copula” function to tie together two marginal probability functions that may (or may not) be related to one another. Much of the work on copulas has been motivated by their applicability to the issues in risk management, insurance and financial economics (see, among others, Rodriguez (2003), Cherubini et al. (2004), Hu (2006), Patton (2006), and Jondeau and Rockinger (2006)). In the empirical literature, copula models have been used extensively in the design and rating of crop revenue insurance contracts, where the inverse correlation of prices and yields plays an important role in pricing revenue risk. A p–dimensional copula, C(u1, u2, . . . , up) , is a multivariate distribution function in the unit hypercube [0, 1] with uniform U(0, 1) marginal distributions. As long as the marginal For details on construction and properties of copulas, see among others, Joe (1997) and Nelsen (2006). 1 distributions are continuous, a unique copula is associated with the joint distribution, F , that can be obtained as: C(u1, u2, . . . up) = F (F −1 1 (u1), . . . , F −1 p (up)). (3) In a similar fashion, given a p–dimensional copula, C(u1, . . . up), and p univariate distributions, F1(x1), . . . , Fp(xp), the equation 1 is a p–variate distribution function with marginals F1, . . . Fp whose corresponding density function can be written as: f(x1, x2, . . . xp) = c(F1(x1), . . . , Fp(xp)) p ∏ i=1 fi(xi) (4) Provided that it exists, the density function of the copula, c, can be derived using equation 3 and marginal density functions, fi: c(u1, u2, . . . up) = f(F−1 1 (u1), . . . , F −1 p (up)) ∏p i=1 fi(F −1 i (ui)) . (5) There is a large number of parametric families of copulas applied in the literature. Two of the most commonly used copula families are elliptical copulas and Archimedean copulas. Gaussian and t–copulas are examples of elliptical copulas while the Clayton and Gumbel are among Archimedean copulas. Copulas are generally used to represent joint probability distributions. This joint distribution may be based upon different parametric families for the marginals. The individual marginals may also be nonparametric and thus represented using empirical distribution functions. Upon estimation of the relevant copula function, one is typically interested in simulating or predicting dependent random variables from a joint probability distribution. To this end, the marginal distribution functions and the copula representation of the joint likelihood function must be inverted. These methods make use of the fact that the inverse of the cumulative distribution function (cdf) for any random variable is uniformly distributed. Nelsen (2006) refers to methods that use this to simulate from a distribution as the “inverse distribution method.” Making use of the fact that F−1(x) is a uniform random variable u allows one to generate