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let me explain you the process, I have random variables in a matrix $X_1$: $260\times3$.

I have my correlation matrix $\rho_1$: $3\times3$ from my matrix $X_1$.

Now I use a Cholesky decomposition and I create $X_2$ from the covariance in $X_1$.

As expected correlation $\rho_2$ of $X_2$ and $\rho_1$ of $X_1$ are very close.

When I create $X_3$ as the $X_1$ in which the first column has been replaced by the first one in $X_2$, the correlation matrix is very far from $\rho_1$ and $\rho_2$.

1) I don’t really get why as my correlation matrix are very close between $X_1$ and $X_2$ so I was expected $X_3$ to have a correlation close to the two other ?

2) Is there a way to perform this simulation / using Cholesky to in the end keep the two last column of $X_1$ and recreate the first one while keeping the correlation (or very close) of $X_1$ ?

Thank you for your help

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  • $\begingroup$ If your sample is rank dependent, (you said you set X3=X1), then there's no way to estimate the covariance matrix without having some form of quasi-inverse of the precision matrix. Depending on what linear algebra package you use, setting the correlation of two identical columns to 0 can be one valid form of a quasi inverse matrix. $\endgroup$ – AdamO Apr 8 at 19:29
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If I understand correctly you want to create only a new first column whose population correlation with the 2 later columns given those later column values is equal to the sample correlations with the original first column. I will assume you want to continue to use Cholesky rather than any other choice of decomposition.

  1. Reorder your original variables so the first column is last in the covariance matrix.

  2. Find the Cholesky decomposition of this covariance matrix.

  3. Now simulate only the last variable given the other two

That's your new first variable in the original framework.

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