On two-stage-least-squares regression My professor has just covered the concept of two-stage-least-squares (TSLS) regression today.
In particular, we explored the over-identified case where we have a single endogenous regressor, $X$, with $m$ instruments and $r$ controls. It was explained that, in the first stage, we regress $X$ on the $m$ instruments and $r$ controls to get the ordinary-least-squares estimators of $\hat{X}$. Subsequently, in the second stage, we regress $Y$ on $\hat{X}$ and the $r$ controls to get the TSLS estimators overall.
Now, I noticed that we were regressing on the $r$ controls in both stages of the TSLS regression and I wondered why this was so, since my thoughts are that the effects of the $r$ controls are already captured in $\hat{X}$ by the first stage, so regressing on these $r$ controls again in the second stage would either be redundant or "double-counting the effects of the $r$ controls".
I discussed this with my professor and he seemed stumped by this too; he said that this had not crossed his mind before and he could not come up with a good answer for me. Thus, I am hoping that some of the great minds here can shed some light on this intuitively :)
 A: I have not thought about this, but let me have a crack at it.
The first stage of 2-SLS is an attempt to remove the bias caused by endogeneity. For $ \hat X $ to be a good instrument for $ X $, it must be correlated with $ X $ but not with the error/residuals of the main equation.
As with any other regression, we need to include all the independent variables that explain variations in $ X $ in the first stage. Then, we can obtain a good instrument $ \hat X $ and use it in the second stage. Therefore, we will need to include some instruments $ m $ in the first stage.
These instruments are the variables that affect $ X $ and also have some association with $ Y $ through $ X $.
Along with these instruments, we also include all the control variables (all exogenous variables) in the first stage. Why? Because they belong to a system of equations and are interrelated. All these variables affect $ Y  $ and $ X $ either directly or indirectly through other variables in the system. This means that they also affect $ X $ either directly or indirectly and we need them in the first stage to get a good instrumental variable $ \hat X $.

As put by A. Koutsoyiannis in "Theory of Econometrics":
Every exogenous variable, regardless of the equation in which it
appears, affects all the endogenous variables of the system either
directly (if it is present in the equation) or indirectly (by
influencing other endogenous variables in the equation in which it
does not appear explicitly).

