I have a set of data composed of two variables $X$ and $Z$, where $Z = X + Y$. I want to make a statement about the relationship between $X$ and $Y$. For instance, I'd like to claim that $X$ and $Y$ vary together or correlate. Is there a way to do this with only measurements from $X$ and $Z$?
As an additional wrinkle, consider if X and Z are corrupted by some noise $X$-noise and $Z$-noise. $X$ is actually $X + X$-noise and $Z$ is $X + Y + Z$-noise. In addition, $X$ is scaled by some coefficient $a_X$ and $Z$ is scaled by some other coefficient $a_Z$. Therefore, the data I collected, which are paired measurements, are $[a_X*(X + X-\text{noise}), a_Z*(X + Y + Z-\text{noise})]$. Can I still estimate $Y$ from my measurements? If I do not know $a_X$ or $a_Z$ is this problem unsolvable?
