Archimedean copula parameters? Using R, I am attempting to fit data for 3 stock indices using 3 Archimedean copulas, Frank, Gumbel or Clayton. What are their parameters?
In class, we were taught to fit a t copula. Its parameters come from getting the correlation matrix (see last two lines of codes below).
This, this and this don't seem to help, I think. In the examples, there are some numbers plugged in to the copulas, but I don't really see how to determine the numbers to use? Help please?

library(copula)

dat=read.csv("index.csv")

hk=dat$Hangseng

jp=dat$Nikkei

cn=dat$SSE

new.dat=data.frame(hk,jp,cn)

new.ts=ts(new.dat)
plot(new.ts)
ret=diff(log(new.ts))
plot(ret)

uni=pobs(ret)

uni

uni.dat=data.frame(uni)

uni.dat

plot(uni.dat)

cor(uni.dat)
t.cop=tCopula(c(0.12,0.08,0.005), dim=3,dispstr="un",df.fixed=F)
 A: If you are asking for the parameters, you might not have seen the table in the section "Archimedean copulas" in the wikipedia entry on copulas (taken from Nelsen's introductory book). There, you will find the parameter spaces for a few Archimedean copulas including Frank, Gumbel and Clayton. You will realize that they are quite different and are not easy interpretable values such as the correlation matrix used for the t or Gaussian copula families. In the first place, they are just some arbitrary parameters - nothing more. In some cases, 1-1 relationships between a copula's parameter and Kendall's tau or Spearman's rho exist allowing for a re-parameterisation into an easier interpretable parameter space.
A limitation of higher (d>2) dimensional Archimedean copulas is that they still only have a single parameter that determines all dependencies in the d-dimensional data set. Hence they lack flexibility compared to t and Gaussian copulas that would e.g. take 3 correlation values for d=3. Nevertheless, the Archimedean copulas might in some cases still provide a better fit.
When it comes to estimation, as pointed out in other replies to your question, maximum likelihood estimation is the most frequently used approach. You will find an implementation of different estimation techniques bundled in the fitCopula function of the copula package in R. Take a look at the help page (?fitCopula) and try a few of the presented examples, this should enable you to use it for your own data set.
If you managed to get single family fits for your data set but are not yet satisfied (or just curious), I recommend to take a look at vine copulas (introduced as pair-copula construction and implemented in the VineCopula R package). Vine copulas allow to mix arbitrary bivariate copulas following  a regular vine to build multivariate copulas.
A: In general, you can fit any copula by maximizing the likelihood of its density:


*

*Compute empirical marginals:
$$\hat{F}_i(x_{it})=\dfrac{\#x_i\leq x_{it}}{T},\qquad t = 1...T,\,\,i = 1...n$$

*Copula ML: $$\max_{\theta_c\in\Theta_c} \,\, l(\theta_c)=\sum_{t = 1}^{T}\ln \, c(\theta_c|\hat{F}_1(x_{1t}),\dots,\hat{F}_n(x_{nt}))$$


$\theta_c$ is the Copula parameter set you want to estimate.
