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I have the covariance matrix between log-returns of n variables. I suppose the distribution of the log-returns is normal for all the variables with average=0 but standard deviation in general $\neq$ 1. With the realized log return of n-1 variables I want to get an estimation of the last variable, let's call it K.

I thought about using Cholesky decomposition to get the triangular matrix L.

Then to get the best estimation I used:

$$\sum_{i=1}^{n-1} L_{n,i} * R_{i}$$ (where R is a vector of log return for n-1 variables)

And for the standard deviation I used:

$L{n,n} * std(K)$

I'm not sure this is right. In particular I suppose the general procedure is correct but there may be errors in "scaling". Maybe I have to rescale $R_{i}$ dividing them by the relative standard deviation to evaluate the best estimation and use $L{n,n}$ as the standard deviation of the best estimation.

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