# Copulas and tail dependence, part 2

An alternative to describe tail dependence can be found in the Ledford & Tawn (1996) for instance. The intuition behind can be found in Fischer & Klein (2007)). Assume that  and   have the same distribution. Now, if we assume that those variables are (strictly) independent,

But if we assume that those variables are (strictly) comonotonic (i.e. equal here since they have the same distribution), then

So assume that there is a  such that
Then =2 can be interpreted as independence while =1 means strong (perfect) positive dependence. Thus, consider the following transformation to get a parameter in [0,1], with a strength of dependence increasing with the index, e.g.

In order to derive a tail dependence index, assume that there exists a limit to

which will be interpreted as a (weaktail dependence index. Thus define concentration functions

for the lower tail (on the left) and

for the upper tail (on the right). The R code to compute those functions is quite simple,
```> library(evd);
> data(lossalae)
> X=lossalae
> U=rank(X[,1])/(nrow(X)+1)
> V=rank(X[,2])/(nrow(X)+1
> fL2emp=function(z) 2*log(mean(U<=z))/
+ log(mean((U<=z)&(V<=z)))-1
> fR2emp=function(z) 2*log(mean(U>=1-z))/
+ log(mean((U>=1-z)&(V>=1-z)))-1
> u=seq(.001,.5,by=.001)
> L=Vectorize(fL2emp)(u)
> R=Vectorize(fR2emp)(rev(u))
> plot(c(u,u+.5-u[1]),c(L,R),type="l",ylim=0:1,
+ xlab="LOWER TAIL      UPPER TAIL")
> abline(v=.5,col="grey")```

and again, it is possible to plot those empirical functions against some parametric ones, e.g. the one obtained from a Gaussian copula (with the same Kendall’s tau)

```> tau=cor(lossalae,method="kendall")[1,2]
> library(copula)
> paramgauss=sin(tau*pi/2)
> copgauss=normalCopula(paramgauss)
> Lgaussian=function(z) 2*log(z)/log(pCopula(c(z,z),
+ copgauss))-1
> Rgaussian=function(z) 2*log(1-z)/log(1-2*z+
+ pCopula(c(z,z),copgauss))-1
> u=seq(.001,.5,by=.001)
> Lgs=Vectorize(Lgaussian)(u)
> Rgs=Vectorize(Rgaussian)(1-rev(u))
> lines(c(u,u+.5-u[1]),c(Lgs,Rgs),col="red")```

or Gumbel copula,

```> paramgumbel=1/(1-tau)
> copgumbel=gumbelCopula(paramgumbel, dim = 2)
> Lgumbel=function(z) 2*log(z)/log(pCopula(c(z,z),
+ copgumbel))-1
> Rgumbel=function(z) 2*log(1-z)/log(1-2*z+
+ pCopula(c(z,z),copgumbel))-1
> Lgl=Vectorize(Lgumbel)(u)
> Rgl=Vectorize(Rgumbel)(1-rev(u))
> lines(c(u,u+.5-u[1]),c(Lgl,Rgl),col="blue")```

Again, one should look more carefully at confidence bands, but is looks like Gumbel copula provides a good fit here.

# Copulas and tail dependence, part 1

As mentioned in the course last week Venter (2003) suggested nice functions to illustrate tail dependence (see also some slides used in Berlin a few years ago).

• Joe (1990)’s lambda

Joe (1990) suggested a (strong) tail dependence index. For lower tails, for instance, consider

i.e

• Upper and lower strong tail (empirical) dependence functions

The idea is to plot the function above, in order to visualize limiting behavior. Define

for the lower tail, and

for the upper tail, where is the survival copula associated with , in the sense that

while

Now, one can easily derive empirical conterparts of those function, i.e.

and

Thus, for upper tail, on the right, we have the following graph

and for the lower tail, on the left, we have

For the code, consider some real data, like the loss-ALAE dataset.

```> library(evd)
> X=lossalae```

The idea is to plot, on the left, the lower tail concentration function, and on the right, the upper tail function.

```> U=rank(X[,1])/(nrow(X)+1)
> V=rank(X[,2])/(nrow(X)+1)
> Lemp=function(z) sum((U<=z)&(V<=z))/sum(U<=z)
> Remp=function(z) sum((U>=1-z)&(V>=1-z))/sum(U>=1-z)
> u=seq(.001,.5,by=.001)
> L=Vectorize(Lemp)(u)
> R=Vectorize(Remp)(rev(u))
> plot(c(u,u+.5-u[1]),c(L,R),type="l",ylim=0:1,
+ xlab="LOWER TAIL          UPPER TAIL")
> abline(v=.5,col="grey")```

Now, we can compare this graph, with what should be obtained for some parametric copulas that have the same Kendall’s tau (e.g.). For instance, if we consider a Gaussian copula,

```> tau=cor(lossalae,method="kendall")[1,2]
> library(copula)
> paramgauss=sin(tau*pi/2)
> copgauss=normalCopula(paramgauss)
> Lgaussian=function(z) pCopula(c(z,z),copgauss)/z
> Rgaussian=function(z) (1-2*z+pCopula(c(z,z),copgauss))/(1-z)
> u=seq(.001,.5,by=.001)
> Lgs=Vectorize(Lgaussian)(u)
> Rgs=Vectorize(Rgaussian)(1-rev(u))
> lines(c(u,u+.5-u[1]),c(Lgs,Rgs),col="red")```

or Gumbel’s copula,

```> paramgumbel=1/(1-tau)
> copgumbel=gumbelCopula(paramgumbel, dim = 2)
> Lgumbel=function(z) pCopula(c(z,z),copgumbel)/z
> Rgumbel=function(z) (1-2*z+pCopula(c(z,z),copgumbel))/(1-z)
> u=seq(.001,.5,by=.001)
> Lgl=Vectorize(Lgumbel)(u)
> Rgl=Vectorize(Rgumbel)(1-rev(u))
> lines(c(u,u+.5-u[1]),c(Lgl,Rgl),col="blue")```

That’s nice (isn’t it?), but since we do not have any confidence interval, it is still hard to conclude (even if it looks like Gumbel copula has a much better fit than the Gaussian one). A strategy can be to generate samples from those copulas, and to visualize what we had. With a Gaussian copula, the graph looks like

```> u=seq(.0025,.5,by=.0025); nu=length(u)
> nsimul=500
> MGS=matrix(NA,nsimul,2*nu)
> for(s in 1:nsimul){
+ Xs=rCopula(nrow(X),copgauss)
+ Us=rank(Xs[,1])/(nrow(Xs)+1)
+ Vs=rank(Xs[,2])/(nrow(Xs)+1)
+ Lemp=function(z) sum((Us<=z)&(Vs<=z))/sum(Us<=z)
+ Remp=function(z) sum((Us>=1-z)&(Vs>=1-z))/sum(Us>=1-z)
+ MGS[s,1:nu]=Vectorize(Lemp)(u)
+ MGS[s,(nu+1):(2*nu)]=Vectorize(Remp)(rev(u))
+ lines(c(u,u+.5-u[1]),MGS[s,],col="red")
+ }```

(including – pointwise – 90% confidence bands)

```> Q95=function(x) quantile(x,.95)
> V95=apply(MGS,2,Q95)
> lines(c(u,u+.5-u[1]),V95,col="red",lwd=2)
> Q05=function(x) quantile(x,.05)
> V05=apply(MGS,2,Q05)
> lines(c(u,u+.5-u[1]),V05,col="red",lwd=2)```

while it is

with Gumbel copula. Isn’t it a nice (graphical) tool ?

But as mentioned in the course, the statistical convergence can be slow. Extremely slow. So assessing if the underlying copula has tail dependence, or not, it now that simple. Especially if the copula exhibits tail independence. Like the Gaussian copula. Consider a sample of size 1,000. This is what we obtain if we generate random scenarios,

or we look at the left tail (with a log-scale)

Now, consider a 10,000 sample,

or with a log-scale

We can even consider a 100,000 sample,

or with a log-scale

On those graphs, it is rather difficult to conclude if the limit is 0, or some strictly positive value (again, it is a classical statistical problem when the value of interest is at the border of the support of the parameter). So, a natural idea is to consider a weaker tail dependence index. Unless you have something like 100,000 observations…

# a short word on profile likelihood

Profile likelihood is an interesting theory to visualize and compute confidence interval for estimators (see e.g. Venzon & Moolgavkar (1988)). As we will use is, we will plot

But more generally, it is possible to consider

where . Then (under standard suitable conditions)

which can be used to derive confidence intervals.

```> base1=read.table(
+ "http://freakonometrics.free.fr/danish-univariate.txt",
> library(evir)
> X=base1\$Loss.in.DKM
> u=5```

The function to draw the profile likelihood for the tail index parameter is then

```> Y=X[X>u]-u
> loglikelihood=function(xi,beta){
+ sum(log(dgpd(Y,xi,mu=0,beta))) }
> XIV=(1:300)/100;L=rep(NA,300)
> for(i in 1:300){
+ XI=XIV[i]
+ profilelikelihood=function(beta){
+ -loglikelihood(XI,beta) }
+ L[i]=-optim(par=1,fn=profilelikelihood)\$value }
> plot(XIV,L,type="l")```

It is possible to use it that profile likelihood function to derive a confidenceinterval,

```> PL=function(XI){
+ profilelikelihood=function(beta){
+ -loglikelihood(XI,beta) }
+ return(optim(par=1,fn=profilelikelihood)\$value)}
> (OPT=optimize(f=PL,interval=c(0,3)))
\$minimum
[1] 0.6315989

\$objective
[1] 754.1115
> up=OPT\$objective
> abline(h=-up)
> abline(h=-up-qchisq(p=.95,df=1)/2,col="red")
> I=which(L>=-up-qchisq(p=.95,df=1)/2)
> lines(XIV[I],rep(-up-qchisq(p=.95,df=1)/2,length(I)),
+ lwd=5,col="red")
> abline(v=range(XIV[I]),lty=2,col="red")```

This is done with the following code

```> library(ismev)
> gpd.profxi(gpd.fit(X,5),xlow=0,xup=3)```