Fitting a t-Copula to Empirical Marginals and Simulating Joint Samples

# how to apply a simple copula model


# how to apply a simple copula model












I'm playing around with copulas and wanted to generate some sample based on copula techniques in R. For this purpose I applied the following algorithm:

- Generate three sample vectors coming from different distributions (normal, t, uniform), i.e. $X=(X_1,\dots,X_{100})$,$Y=(Y_1,\dots,Y_{100})$,$Z=(Z_1,\dots,Z_{100})$

- Apply empirical distribution function to each of the three assets. I take the empirical distribution function, since in reality you may not know the actual marginal distribution. In math terms we estimate $F_X,F_Y,F_Z$ based on $X,Y,Z$ and apply it: $$U_1=F_X(X),U_2=F_Y(Y),U_3=F_Z(Z)$$

- Fit a t-copula to the resulting pseudo-unfiromly distributed samples, i.e. we fit a copula $c$ to $c(U_1,U_2,U_3)$

- generate $100$ sample matrices $A_i$, each of size $200$ per marginal sample, i.e. we generate $100$ matrices $A_i=[S_1^i,S_2^i,S_3^i]$ with $200$ rows and $3$ column vector $S_1^i,S^i_2,S^i_3$ containing generated samples coming from $c$.

- retransform back to obtain actual marginal distribution sample coming from a t-copula model, i.e. for each of the $A_i$ we look at $$B_i=[F_X^{-1}(S^i_1),F_Y^{-1}(S^i_2),F_Z^{-1}(S^i_3)] $$

here is a sample R code

```
a <- rnorm(1000,1,0.3)
b <- rt(1000,4)
c <- runif(1000)

data <- cbind(a,b,c)

#calculate emprical distribution

for (i in 1:ncol(data)){
  emp <- apply(data,2,ecdf)
}

hist(emp[[1]](data[,1]))
hist(emp[[2]](data[,2]))
hist(emp[[3]](data[,3]))

data.unif <- cbind(emp[[1]](data[,1]),emp[[2]](data[,2]),emp[[3]](data[,3]))

#fit copula

fitCopulat <- fit.tcopula(Udata = data.unif,method = "Kendall")

l1 <- list()

for(i in 1:100){
  l1[[i]] <- rcopula.t(200,df = fitCopulat$nu,Sigma = fitCopulat$P)
}

#transform back to get right marginal distributions

l2 <- l1
for(i in 1:100){
  for(j in 1:ncol(data)){
    l2[[i]][,j] <- quantile(data[,j],probs = l1[[i]][,j])
  }
}
```

I have the following two questions:

- Does the algorithm make sense?

- Is step 5. correctly implemented? I'm unsure if I used the function quantile correctly.