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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!

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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.

And, it's important to remember to include the instruments for all the endogenous variables in each first-stage regression, which is something that was confusing me as well. So in the example I provided, We'd run the first-stage regression and the "1.5" stage regression (which, again, is really just multiple regressions, compacted in a nice format), and simply insert all the endogenous terms with their corresponding instrument. That makes it pretty easy, since the RHS formula is the same for all of them. I'm pretty confident this is right, but if not, let me know in the comments.

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