I have two random series, $X$ and $Y$, and I've defined a third series $Z$ as:
$Z = X - Y$
Now, I want to compare two other series, $Y1$ and $Y2$, to understand why $Y1$ is generally closer to $Y$ than $Y2$. $Y1$ and $Y2$ are defined as:
$Y1 = X - 2 \cdot \text{mean}(Z)$
$Y2 = X - \text{mean}(Z)$
I'm looking to explain why, under certain conditions or assumptions, $Y1$ tends to be closer to $Y$ than $Y2$ in terms of values. I need a mathematical explanation for this observation.
Thank you.