I understand that copulas can be used as a tool to add any conceivable dependence to a pair of random variables. However, I would like to add some dependence between two random vectors.
Let us suppose that we have random vectors $(X_1, Y_1)$ and $(X_2, Y_2)$. How do we construct their joint CDF, such that they are not independent? Can we use copulae for this purpose?
Let's say $F_1$ is the CDF of $(X_1, Y_1)$ and $F_2$ is the CDF of $(X_2, Y_2)$. Then $$ \Phi_1(x_1, y_1) := F_1(F^{-1}_{X_1}(x_1), F^{-1}_{Y_1}(y_1)) $$ is the copula that captures the relationship between $X_1$ and $Y_1$. Similarly,
$$ \Phi_2(x_2, y_2) := F_2(F^{-1}_{X_2}(x_2), F^{-1}_{Y_2}(y_2)) $$ is the copula of $X_2$ and $Y_2$.
In order to add a dependency between $(X_1, Y_1)$ and $(X_2, Y_2)$ we would need a special copula $\Phi(x_1, y_1, x_2, y_2)$, whose marginal distributions with respect to the two pairs of variables are the two partial copulae: $$ \lim_{x_1 \to \infty \\ y_1 \to \infty} \Phi(x_1, y_1, x_2, y_2) = \Phi_2(x_2, x_2) $$ and $$ \lim_{x_2 \to \infty \\ y_2 \to \infty} \Phi(x_1, y_1, x_2, y_2) = \Phi_1(x_1, x_1) $$
How can I construct such a copula? Are there some off-the-shelf copulae especially designed for the purpose of modelling inter-vector dependencies?
