Q: Copula Behaviour : Gaussian vs Student T -Numerical Stability I would like to get your opinion on the following topic:
I am comparing the behaviour of Gaussian and Student-t Copulas. 
I employ the follwing procedure:


*

*Simulate N=100,000 samples from a Student Copula with 3 DoF, Student Copula with 100 DoF and a Gaussian Copula.The dimension of the copulas is 3.

*I then transform these samples via a Student-t Distribution with 3 DoF. 

*Finally I compute the empirical quantile function of the sum of the marginals  and look at the tail of the distribution between 99% and 99.9%. For people familiar with the domain of finance, this is very close to be computing the VaR.


The following graphs are obtained with a Correlation of 0.1. 

The following graphs are obtained with a Correlation of 0.9.
 
I reach the following conclusions:


*

*When the correlation is high, the different copulas behave similarly. This seems logical because with high correlation, the "distributional " properties of the different copulas will become "almost" void

*I notice that the sum of correlated  marginals of a Student copula with 100 DoF is very close and often smaller than the sum of correlated marginals from the Gaussian copula. This seems to be a numerical problem as the two results should be very close ?

*In the case of high correlation (i.e 0.9), depending on the chosen seed for the generation the results from the  Student copula with 3 DoF are below the Gaussian copula. How can this be explained ?(numerical instability?).


I would be greatful if someone could confirm/correct my conclusions, and shed some additional light on these results with a more mathematical proof/approach.
Below the code I used (Copula package)
seed1 <- runif(1,0,100000)

n<- 100000
cor <- c(0.9, 0.9,0.9)
t.cop <- tCopula(cor, dim = 3, dispstr = "un",df = 3)
t.cop100 <- tCopula(cor, dim = 3, dispstr = "un",df = 100)
n.cop <- normalCopula(cor, dim = 3, dispstr = "un")
set.seed(seed1)
tCop <- rCopula(n, t.cop)
set.seed(seed1)
tCop100 <- rCopula(n, t.cop100)
set.seed(seed1)
nCop <- rCopula(n, n.cop)

StudentN <- qt(nCop,3)
StudentT <- qt(tCop,3)
StudentT100 <- qt(tCop100,3)

# StudentN <- qnorm(nCop)
# StudentT <- qnorm(tCop)
# StudentT100 <- qnorm(tCop100)

Seq_L <- seq(0.99,0.999,0.00001)
plot(Seq_L,quantile(rowSums(StudentT),Seq_L), type="l", col="red")
lines(Seq_L,quantile(rowSums(StudentN),Seq_L), type="l", col="blue")
lines(Seq_L,quantile(rowSums(StudentT100),Seq_L), type="l", col="green")

nam <- c("Student Copula with 3 DoF", "Gaussian Copula", "Studentr Copula with 100 DoF")
legend('topleft',  nam, 
       lty=1, col=c('red', 'blue', 'green'), bty='n', cex=.75)
title("Comparison of 3 Copulas with correlation: 0.9")

