How to isolate the impact of one variable via regression?

Question

I know the dependent variable $Y$ is a function of $X$ and $Z$, while $X$ and $Z$ are orthogonal. I want to quantify the impact of $X$ on $Y$, but the problem is that $X$ is unobservable to me. Is it plausible to regress $Y$ on $Z$ and take the residuals or $1-R^2$ to estimate the impact of $X$? To be more specific, let's assume a simple linear relationship between $Y$, $X$, and $Z$: $$Y = X +aZ $$ Here $Y$ and $Z$ are observable, but $X$ and $a$ are not. So in this case how can I estimate the value of $X$? Can it be done via regressing $Y$ on $Z$?

Answer 1

If $X$ is truly orthogonal to $Z$, and the coefficient on $X$ is truly equal to 1, and there is no error (i.e., $X$ and $Z$ are truly the only causes of $Y$), then regress $Y$ on $Z$ (with no intercept) and use the residuals from the model as $X$. That is, fit the model $$ Y = a Z + \varepsilon $$ And set $X = Y - \hat{a} Z$. This can be done because $\hat{a}$ is identified from the fact that $X$ and $Z$ are orthogonal, so the bias due to omitting $X$ is 0.

This requires an insane and highly unlikely set of assumptions and so would have no practical utility. But under the assumptions you wrote in your post (unreasonable as they are), estimating $X$ is straightforward. This is possibly related to this somewhat similar post involving an impossible set of assumptions.