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So, I have the following linear model: $$y = \alpha + \beta x + u$$ and $x \in \{0,1\}$, i.e. the variable $x$ is boolean. Moreover $x$ may be endogenous, and I have a set of instrumental variables $\boldsymbol{z}$ which are exogenous. In this situation usually one uses a simple 2SLS regression and that's it. But I was wondering whether one could first regress $x$ on $\boldsymbol{z}$ thorough probit, and then take the fitted values $\hat{x}$ as instrumental variables in the second step of the regression, where we use $\hat{x}$ as instrumental variable for $x$ and use IV.
So I have replaced the OLS regression of the first step with a probit regression.

Is the result of this kind of two step regression consistent? Does it make sense to do so?

Thanks!

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    $\begingroup$ These answers suggest that the answer is not generally yes: stats.stackexchange.com/questions/94063/… stats.stackexchange.com/questions/125830/… $\endgroup$
    – Christoph Hanck
    Nov 24 '20 at 12:46
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    $\begingroup$ Angrist and Pischke section 4.6.1 is helpfull. Under the correct assumptions it can be shown in a GMM framework to be consistent and efficient to use the method you suggest. I ran a few numerical experiments which does not show a large efficiency gain so in any case I would follow the advice of Angrist and Pischke and stick with standard 2SLS. $\endgroup$
    – Jesper for President
    Nov 30 '20 at 16:59
  • $\begingroup$ have you ever found an answer? Cause I have a similar question? $\endgroup$
    – canIchangethis
    Feb 17 '21 at 21:01

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