If X=Y+Z, Is it ever useful to regress X on Y? If we have X and Y that are mathematically dependent: X = Y + Z, is it 'forbidden' to use Y as a predictor to X in linear regression?  I'm trying to find a concise explanation for why it is, or isn't.
One explanation I've found is that you shouldn't use simple linear regression in this case, but multiple linear regression, with Y and Z as predictors, but this doesn't seem right. The mathematical relationship is exact, and the multiple regression would return coefficients 0, 1, and 1  for X = 0 + 1Y + 1Z. So, there is something more fundamental I'm missing.
The purpose for linear regression is predicting the value of one variable from the other. But we already know that X can be calculated as the sum of Y and Z, so effectively, we are calculating the regression between Y and Y + Z. Is this 'forbidden'?
But, if we can only measure Y, does the regression give us a tool to predict X? Isn't it better to find the regression between Y and Z? I'm confused.
 A: Linear regression is a tool that is used to achieve a goal. So any answer will depend on the goal to be achieved. As said already in the answer of @Noah, if you already know $X$, $Y$, and $Z$, I can't see any goal worth achieving that is achieved by this regression. Why would you want imprecise predictions of $X$ if you can have precise ones?
If however you don't know $Z$, linear regression may work well for predicting $X$ from $Y$. There is nothing that formally forbids you from trying that out (in fact if $Z$ is independent of $Y$ and distributed according to standard regression model assumptions, the standard regression model is just fulfilled). But then, depending on the exact nature of the data (particularly the distribution of $Z$ and how it is related to $Y$), it may not work that well, and/or other techniques may work better. This can be explored for example using bootstrap or cross-validation.
A: Two people step on a scale. The scale outputs only the total weight of the couple.
If you know the weight of the first person ($Y$) can you guess what the scale will output ($X$) without knowing anything about the weight of the second person ($Z$)?
Intuitively, there has to be some connection. If $Y$ is very heavy, it is likely that the total weight $X$ will also be high.
Regressing X on Y here is just trying to put a number to that connection by looking at a bunch of real-life examples of total weight $X$ given first person weight $Y$.
It is not forbidden, as you say. It is simply trying to make the best guess given the information you have.
Of course if you know the weight of both people $Y$ and $Z$ and you know how scales work, you don't need to make observations and regression.
A: If you know $X = Y + Z$ and you have $Y$ and $Z$ measured, why would you need to run a regression of $X$ on $Y$ and $Z$? It provides no additional information and does not allow you to make "better" predictions about $X$ (since you know $X$ exactly from $Y$ and $Z$). But if you don't have $Z$, don't know the exact relationship between $Y$ and $X$, and want to predict $X$ from $Y$, then you absolutely can (and should!) regress $X$ on $Y$. The true marginal relationship between $X$ and $Y$ depends on the correlation between $Y$ and $Z$, which is not stated in the problem, so you won't automatically know what the marginal relationship between $X$ and $Y$ is  from the formula for $X$ alone. Indeed, this is the usual case that we do regression: we assume a (possibly) deterministic relationship between the outcome and some other variables, but many of those variables are unmeasured, so their influence is captured in the error term. In the absence of measured $Z$, the effect of $Z$ will be captured in the error term, and the marginal effect of $Y$ will be estimated by the model.
