COPULA Procedure
Archimedean Copulas
Overview of Archimedean Copulas
Let function 
be a strict Archimedean copula generator function and suppose its inverse 
is completely monotonic on 
. A strict generator is a decreasing function 
that satisfies 
and 
. A decreasing function 
is completely monotonic if it satisfies
An Archimedean copula is defined as follows:
The Archimedean copulas available in the COPULA procedure are the Clayton copula, the Frank copula, and the Gumbel copula.
Simulation
Suppose the generator of the Archimedean copula is 
. Then the simulation method using the Laplace-Stieltjes transformation of the distribution function is given by Marshall and Olkin (1988) where 
:
Generate a random variable V with the distribution function F such that
. Draw samples from independent uniform random variables
. Return
.
The Laplace-Stieltjes transformations are as follows:
For the Clayton copula,
, and the distribution function F is associated with a Gamma random variable with shape parameter 
and scale parameter one. For the Gumbel copula,
, and F is the distribution function of the stable variable 
with 
. For the Frank copula with
, 
, and 
is a discrete probability function 
. This probability function is related to a logarithmic random variable with parameter value 
.
For more information about simulating a random variable from a stable distribution, see Theorem 1.19 in Nolan (2010). For more information about simulating a random variable from a logarithmic series, see Chapter 10.5 in Devroye (1986).
For a Frank copula with 
and 
, the simulation can be done through conditional distributions as follows:
Draw independent
from a uniform distribution. Let
. Let
.
Fitting
One method to estimate the parameters is to calibrate with Kendall’s tau. The relation between the parameter 
and Kendall’s tau is summarized in Calibration Estimation for the three Archimedean copulas.
Table 2: Calibration Using Kendall’s Tau
| Copula Type |  |
Formula for
|
|---|---|---|
| Clayton |  |
 |
| Gumbel |  |
|
| Frank | 
|
No closed form |
In Table 2, 
for , and 
for . In addition, for the Frank copula, the formula for has no closed form. The numerical algorithm for root finding can be used to invert the function 
to obtain as a function of .
Alternatively, you can use the MLE or the CMLE method to estimate the parameter given the data 
and 
. The log-likelihood function for each type of Archimedean copula is provided in the following sections.
Fitting the Clayton Copula
For the Clayton copula, the log-likelihood function is as follows (Cherubini, Luciano, and Vecchiato 2004, Chapter 7):
Let 
be the derivative of 
. Then the first-order derivative is
The second-order derivative is
Fitting the Gumbel Copula
A different parameterization 
is used for the following part, which is related to the fitting of the Gumbel copula. For the Gumbel copula, you need to compute 
. It turns out that for 
,
where 
is a function that is described later. The copula density is given by
where 
, 
, 
,
,
, and 
.
The log density is
Now the first-order derivative of the log density has the decomposition
Some of the terms are given by
where
The last term in the derivative of the 
is
Now the only remaining term is 
, which is related to 
. Wu, Valdez, and Sherris (2007) show that 
satisfies a recursive equation
with 
.
The preceding equation implies that 
is a polynomial of x and therefore can be represented as
In addition, its coefficient, denoted by 
, is a polynomial of 
. For simplicity, use the notation 
. Therefore,
Fitting the Frank Copula
For the Frank copula,
When , a Frank copula has a probability density function
where 
.
The log likelihood is
Denote
Then the derivative of the log likelihood is
The term in the last summation is
The function satisfies a recursive relation
with 
. Note that is a polynomial whose coefficients do not depend on ; therefore,
where
For the case of and , the bivariate density is
