# Effect of two time series over a different one

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.