# 2SLS with endogenous variable $X_1$ and interaction terms $X_1 \times X_2$, $X_2$ exogenous

I would like clarification on the correct way to set up the first-stage regression(s) for a hypothetical 2SLS regression. Say we have the following formula: \begin{align} y = x_1 + x_2 + x_1 \times x_2 + \text{controls} + \epsilon \end{align} Where $$x_1$$ is numeric and assumed endogenous, $$x_2$$ is categorical and assumed exogenous. Assume, also, that a variable $$z$$ is a valid instrument for $$x_1$$. Now, my first intuition was to use a single first-stage regression to find $$\hat{x_1}$$ like so: \begin{align} \text{First-Stage:} && \hat{x_1} = z + x_2 + z \times x_2 + \text{controls} + \epsilon \end{align} And then simply replace both the stand-alone $$x_1$$ and the instance of $$x_1$$ in $$x_1 \times x_2$$ in the second-stage regression like so: \begin{align} \text{Second-Stage:} && y = \hat{x_1} + x_2 + \hat{x_1} \times x_2 + \text{controls} + \epsilon \end{align} However, it came to my attention that it might actually be necessary to run a separate first-stage regression for $$x_1 \times x_2$$, and find fitted values for that, instead of simply using $$\hat{x_2}$$ hat from the previous first-stage regression. That is, we would need another first-stage regression, which I will call the 1.5 stage regression: \begin{align} \text{1.5-Stage:} && \widehat{x_1 \times x_2} = z + x_2 + z \times x_2 + \text{controls} + \epsilon \end{align} And our final second-stage regression would then be: \begin{align} y = \hat{x_1} + x_2 + \widehat{x_1 \times x_2} + \text{controls} + \epsilon \end{align} Which uses $$\hat{x_1}$$ from the first-stage equation, and $$\widehat{x_1 \times x_2}$$ from the 1.5-stage equation. I honestly thought the two methods would boil down to identical results, but I seem to get different results when doing some tests in $$\texttt{R}$$. The first method seems the most correct to me, however, I understand that in the scenario where you have two separate endogenous variables (and two instruments), you do indeed create two separate first-stage regressions, and get fitted values for each, like in method two. However, I was unsure if that holds up when, in this case, the second endgenous variable is an interaction term, and the endogenous half is already being instrumented.

I feel like I'm overlooking something obvoius, and overthinking this, but the fact that I get different results for the two methods is what confuses me. If I am making a mistake, please let me know. Thanks!

So it appears I was making a simple econometrics blunder. I believe the correct way to go about this is to indeed use multiple first-stage regressions, one for each of your endogenous variables. In the case of instrumenting a categorical-numeric interaction, that really just gives you $$n-1$$ additional endogenous variables, $$n$$ being the number of categories, and minus 1 because of the reference category.