2SLS with a boolean regressor

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!

• These answers suggest that the answer is not generally yes: stats.stackexchange.com/questions/94063/… stats.stackexchange.com/questions/125830/… Nov 24 '20 at 12:46
• 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. Nov 30 '20 at 16:59
• have you ever found an answer? Cause I have a similar question? Feb 17 '21 at 21:01