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Let us say we have data grouped in $m$ different classes, each of size $n_j$ for $j = 1,...,m$. We denote as $X_k^{(j)}$ the $k$-th member of group $j$. We want to simulate unit-variance random variables with a structure such that the "inner" correlation in group $j$ is $\rho_j$, and the "outer" correlation between groups $j$ and $i$ is $\rho_{ij}$, with the $\rho$'s as parameters. This is, $$ \begin{aligned} Var(X_k^{(j)}) &= 1 \quad \forall k,j \\ Cov(X_k^{(j)}, X_r^{(j)}) &= \rho_j \quad \forall k \neq r, j = 1,...,m\\ Cov(X_k^{(j)}, X_r^{(i)}) &=\rho_{ij} \quad \forall i\neq j, k\neq r \end{aligned} $$ Basically, I have grouped data and want to create a relationship structure both inside the groups and between the groups. I have tried approaching the problem by using the Gaussian copula and establishing $$ X_k^{(j)} = \sqrt{\beta_j}\, Z + \sqrt{\rho_j - \beta_j} \, Z_j + \sqrt{1-\rho_j} \, e_k^{(j)} $$ assuming both inner and outer correlations are positive, with $Z, Z_j, e_k^{(j)}$ i.i.d. $N(0,1)$. However, one finds that $$ \beta_i \beta_j = \rho_{ij}^2 $$ which I believe is not generally solvable, given there are more equations than variables. I even used non-linear optimization techniques to try and find an approximate solution, without luck (cannot approximate each $\rho_{ij}$ appropriately).

I would like to know if there is any literature regarding this kind of problems or models, or if it is even possible. Even though $\rho_j$ seem to be always positive in my data, I would like the model to be as flexible as possible.

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    $\begingroup$ Yes this will most likely not work and you will have to deal with positive definiteness as well. You should search for "hierarchical copulas" there is quite a literature on this topic. $\endgroup$
    – g g
    Mar 16, 2021 at 19:44
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    $\begingroup$ This would be easier to answer with more description of the variables. Maybe the groups are teams and you’re looking at the correlation of offensive and defensive contributions? Right now I just see a jumble of subscripts and superscripts. $\endgroup$
    – Matt F.
    Apr 25, 2021 at 13:37
  • $\begingroup$ I am trying to simulate loss distributions for credit portfolios of different size. Each portfolio $j$ has $n_j$ credits, and each credit has an inner correlation (inside the portfolio) and outer correlation (with the rest of the portfolios). The final goal is to simulate the loss distributions with given inner and outer correlations. Hence, generate r.v.'s with such structure. $\endgroup$ Apr 29, 2021 at 21:55

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I found an answer, which was easier than expected, using factor models for credit risk. Given $m$ groups each with $n_j$ members, for $j=1,...,m$, we define $$ X_k^{(j)} = \sqrt{\rho_j}Z_j + \sqrt{1-\rho_j}e_k^{(j)} $$ for $k=1,...,n_j$, where $e_k^{(j)}$ are i.i.d. for each and all $k$ and $j$, and independent of $Z_j$ (idiosyncratic noise). Moreover, $\pmb{Z}= (Z_1,...,Z_m)\sim N(0, \Sigma)$ where the entries of $\Sigma$ are given by $$ (\Sigma)_{ij} = \frac{\rho_{ij}}{\sqrt{\rho_i \rho_j}}. $$ Thus, note that $E[X_k^{(j)}] = 0$ and $Var(X_k^{(j)}) = 1$, given the independence of $Z_j$ and $e_k^{(j)}$. Moreover, given the linearity of the normal distribution, $X_k^{(j)} \sim N(0,1)$.

For the correlations, note that (we obviate subindexes for they are not necesarry) $$ \begin{aligned} Corr(X^{(i)}, X^{(j)}) &= Cov(X^{(i)}, X^{(j)}) \quad \text{(Unitary variance)} \\ &= Cov(\sqrt{\rho_i}Z_i + \sqrt{1-\rho_i}e^{(i)},\sqrt{\rho_j}Z_j + \sqrt{1-\rho_j}e^{(j)}) \\ &= \sqrt{\rho_i}\sqrt{\rho_j}Cov(Z_i, Z_j) \quad \text{(Linearity of $Cov$ and ind.)} \\ &= \sqrt{\rho_i}\sqrt{\rho_j}(\Sigma)_{ij} \\ &= \sqrt{\rho_i}\sqrt{\rho_j}\frac{\rho_{ij}}{\sqrt{\rho_i \rho_j}} \\ &= \rho_{ij} \end{aligned} $$ as required. When both the variables are from the same group, we just get $Corr(X^{(j)}, X^{(j)}) = \rho_{jj}$ so we can establish $\rho_{jj} = \rho_j$ w.l.o.g.. In fact, one can prove that $\pmb{X} = (X_1^{(1)},X_2^{(1)},...,X_{n_1}^{(1)},X_1^{(2)},..,X_{n_m}^{(m)})$ the whole vector of $n = \sum_j n_j$ variables is multivariate normal.

Therefore, to simulate such a $\pmb{X}$ one can just construct the whole covariance matrix of size $n$ and simulate a multivariate normal from that (which is cumbersome given that $n$ is large in our application), or you could use the approach above and simulate an $m$ sized vector for $\pmb{Z}$ with covariance matrix $\Omega$, and $n$ i.i.d. $N(0,1)$. We have found that, for our application, the later approach is much faster given that $n$ is in the millions and $m$ around 10.

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