I have three time series $X_1$, $X_2$ and $Y$. All of them starting from the same date and with the same length.

I know that $Y$ contains information of $X_1$, $X_2$ and other variables that I cannot observe. This is because I constructed $Y$ as an index that has all the external factors that affect the development of some payments. More information about this can be found on the Separation Method section of: https://www.casact.org/library/astin/vol9no1and2/219.pdf

I would like to know how much of $X_1$ (clean from the information of $X_2$) is in $Y$ (as a percentage $w_1$). And I would like to have the same for $X_2$.

Notice that is expected that $w_1+w_2\leq 1$ (not equal to 1 because $Y$ might be explained also by other variables that I cannot observe).

The idea is that $w_1\cdot X_1$ explains the change in $Y$ because of $X_1$ (without the information of $X_2$) and that $w_2\cdot X_2$ explains the change of $Y$ because of $X_2$ (without the information of $X_1$).

Im doing a program with R and I would like to iterate this process every time I change my time series $X_1$, $X_2$ and $Y$.

My objective is to understand better the index $Y$ by checking how my payments change only under the effect of $X_1$ an $X_2$. I tried to have a linear combination, but $X_1$ and $X_2$ might have a strong relation.


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