# 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 $$X$$, say, $$\hat{X}$$. 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 :)

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).

• Hello! Thank you for your answer! I guess you have explained why we need to include the controls in the first stage, but I suppose it still does not really explain why we do also need to include them in the second stage? Hopefully others can chip in too! Aug 27, 2022 at 5:02
• @EthanMark It's for the same reason. If you read the quote carefully, ALL variables are interrelated in a system of equations. The control variables help explain both X and Y, so they have to be included in both. X and Y are different variables. In the first stage, they explain variations in X and their indirect effect on Y through X. In the second stage, they explain the variations in Y (direct). Aug 27, 2022 at 12:13