Frank copula

[Eleanor Cawston]

Hi,
I am confused about the Frank copula, or perhaps what the tail dependency vs association means.

In the notes in section 5.1 and again in section 7 (from Core Reading), we have that the Frank copula has zero dependence in both tails.

However, in the question at the end of Section 7, the answer to part (b) is that the Frank copula with a high positive parameter has a strong positive association through out and to part (e) is that with a high negative parameter the Frank copula has a strong negative association throughout.

How are these both true? Is there a difference between "dependence" and "association"?
Thanks,
Eleanor

 

[Andrew Martin]

Hi Eleanor

There are quite a few terms flying around this chapter! Association means that there is some form of statistical relationship between the variables whereas the coefficients of tail dependence are specifically looking at the relationship of the variables in the tails (extremes) of the distribution.

It's possible to have different levels of association at different points in the distribution. For example, consider the scatter plot relating to the Clayton copula on page 29 of the CMP, there is positive association throughout but it is stronger in the lower tail compared to the rest of the distribution.

Now consider the scatter plot relating to the Frank copula on page 30. There again appears to be some broad level of positive association throughout. However, for example, the association in the lower tail is much weaker than that shown in the example graph for the Clayton copula. Even when u is small, v still takes a somewhat wide range of values. It isn't strong enough association to have a positive coefficient of lower tail dependence.

As another example, try running the following R code which looks at the Frank copula with alpha = 100:

#install.packages(copula)

library(copula)

# create copula object for frank copula with parameter 100:

frank.100 = frankCopula(100)

# generate sample:

set.seed(100)
sample.frank100 = rCopula(10000, frank.100)

# plot sample:

plot(sample.frank100, pch = 20)

# looks like much stronger positive association than when alpha = 5! However is there any tail dependence? Let's look closer at the tails, remember the coefficient of tail dependence is the limit as we go right to the extreme:

plot(sample.frank100, pch = 20, ylim = c(0, 0.1), xlim = c(0, 0.1))

# this looks similar (ish) to the full plot when the parameter was 5. There doesn't appear to be any tail dependence.

# Let's try the same thing with the Clayton copula even with just parameter 5:

# creating copula object:

clay.5 = claytonCopula(5)

# simulating a sample:

set.seed(100)
sample.clay5 = rCopula(10000, clay.5)

# full plot:

plot(sample.clay5, pch = 20)

# plot that is more zoomed in on lower tail:

plot(sample.clay5, pch = 20, ylim = c(0, 0.1), xlim = c(0, 0.1))

Hopefully this demonstrates the idea of tail dependence a bit more.

One final note, just because variables are associated or there is a positive coefficient of tail dependence doesn't mean the variables depend on each other directly (ie there may not be a direct causal relationship between them).

Hope this helps

Andy