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I have three time series data $(X_1, X_2, X_3)$ that are correlated with $Y$ as follows: $0.95, 0.75, 0.65$. I want to make a weighted mean of the three time series based on their correlation with $Y$ to make $X_4$. In effect, $X_4$ is the ensemble mean but weighted according to their correlation with $Y$. To make a simple mean is counter-intuitive to the fact that $X_1$ has the highest correlation with $Y$ and its effects will be diluted by $X_3$.

Stochastic
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user2543
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    Why do you want to do this? You will probably get a more helpful answer if you explained what you were trying to achieve. The short answer is that you should use X1, X2 and X3 to forecast Y and then use the fitted values, but how one does that (and whether it is a good idea) would depend on your purpose. – Peter Ellis Apr 23 '20 at 03:35
  • These are three forecast time series. Considering X1 has the highest correlation with Y1, I would choose X1. But I find that X2 and X3 still has some merit and I do not want to discount both in the ensemble mean so I am asking if there is a method to create a weighted mean using the correlation values.

    My first take is like this:

    mean = x10.5+x20.3+x3*0.2

    X1 is the time series with highest correlation, X2 has the second highest, X3 has the least correlation. This way, I am putting more weight to the data with highest correlation than simple mean where their weight is equal.

     – user2543 Apr 23 '20 at 04:54
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    There's just not enough information (yet) to provide good advice. This could be framed as a regression problem (how to find a linear combination of the $X_i$ with the greatest correlation with $Y$) and, if so, the solution depends on the covariances among the $X_i.$ It's not clear (yet) whether this is your objective, though. – whuber Apr 23 '20 at 13:37
  • Thank you @PeterEllis – user2543 Apr 25 '20 at 12:06

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