2SLS - logit/probit in the second stage? I just have a quick question: what if I'm interested in estimating a logit/probit model in the second stage, can I follow this two-step procedure by running OLS in the first stage (endogenous variable = exogenous variables + instruments) and then replace the endogenous variable with the fitted value in the second stage when I run the logit/probit estimation?
I just saw this post (2SLS but second stage Probit) answers the above question, it seems the answer is positive, but does anyone have any references that I can cite regarding this issue? 
 A: The reference for this should be Newey (1987) "Efficient estimation of limited dependent variable models with endogenous explanatory variables", Journal of Econometrics, Vol. 36(3), pp. 231–250 (link). This is the estimator that is implemented with the probitiv command in Stata, for instance, where you can have an OLS first stage and probit second stage.
A: When googling this problem myself, I found the highly-cited article 

Terza, J.V., Basu, A. and Rathouz, P.J., 2008. Two-stage residual inclusion estimation: addressing endogeneity in health econometric modeling. Journal of health economics, 27(3), pp.531-543.

which proposes to use a method called 2-stage residual inclusion (2SRI) for the general linear model case. The method is very simple: Fit the first-stage model to get the residual and include both the residuals and the endogenous variable in the second-stage model.
Or more formally, let $_2$ be the endogenous variable, $_1$ till $_8$ the other exogenous control variables and $_1$ and $_2$ two instruments for $_2$. In the first stage, $_2$ is explained using linear regression 
$_2=_0+_1 _1+_2 _2+…+_8 _8+_9 _1+_10 _2+_2$,
with $$ as coefficients and $_2$ as error term. The equation splits $_2$ in an exogenous component $_0+_1 _1+_2 _2+…+_8 _8+_9 _1+_10 _2$ and omitted-variable component $_2$. The 2SRI method includes both the endogenous variable $_2$ and the residual $_2$ as estimator for the omitted variable in the model; i.e. $_1=logit(_0+_1 _1+_2 _2+…+_8 _8+_9 _2+_10 _2 )+_1$
with $_1$ being the dichotomous variable. The implementation with a statics software is straight forward. (However, getting the standard errors for the estimators is not.)
It has been shown by 

Burgess, S., & Thompson, S. G. (2012). Improving bias and coverage in instrumental variable analysis with weak instruments for continuous and binary outcomes. Statistics in medicine, 31(15), 1582-1600.

through a simulation that the 2SRI is better than 2SLS to provide another reference.
