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Consider a case of $Corr(X,Z)$, often found to be high; where later, it was found that it holds exactly $Z = X + Y$. In effect, the previously found correlations were equal to $Corr(X, X+Y)$.

How can we interpret this correlation? Is it some type of error?

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If the given information is all you have, there is not much interpretation possible. There is always a random variable $Y:=Z-X$ such that $Z=X+Y$.

You said that $cor(X, Z)$ is high, which translates to $Y$ being small. Whether $Z$ can be interpreted as $X$ plus some small error $Y$ depends on the circumstances.

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  • $\begingroup$ Thanks. Could we say the same if Z=X*Y, or that would be a different situation ? $\endgroup$
    – amc____
    Commented Jun 10, 2022 at 5:16
  • $\begingroup$ @amc____ It would be similar. You would have to look out for division by zero when using $Y=Z/X$. And for the interpretation of "small error", $Y$ would have to be near one, not near zero. $\endgroup$
    – frank
    Commented Jun 10, 2022 at 5:52

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