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!
