I was told that it's possible to run a two-stage IV regression where the first stage is a probit and the second stage is an OLS. Is it possible to use 2SLS if the first stage is a probit but the second stage is a probit/poisson model?
What was proposed to you is sometimes referred to as a forbidden regression and in general you will not consistently estimate the relationship of interest. Forbidden regressions produce consistent estimates only under very restrictive assumptions which rarely hold in practice (see for instance Wooldridge (2010) "Econometric Analysis of Cross Section an Panel Data", p. 265-268).
The problem is that neither the conditional expectations operator nor the linear projection carry through nonlinear functions. For this reason only an OLS regression in the first stage is guaranteed to produce fitted values that are uncorrelated with the residuals. A proof for this can be found in Greene (2008) "Econometric Analysis" or, if you want a more detailed (but also more technical) proof, you can have a look at the notes by Jean-Louis Arcand on p. 47 to 52.
For the same reason as in the forbidden regression this seemingly obvious two-step procedure of mimicking 2SLS with probit will not produce consistent estimates. This is again because expectations and linear projections do not carry over through nonlinear functions. Wooldridge (2010) in section 15.7.3 on page 594 provides a detailed explanation for this. He also explains the proper procedure of estimating probit models with a binary endogenous variable. The correct approach is to use maximum likelihood but doing this by hand is not exactly trivial. Therefore it is preferable if you have access to some statistical software which has a ready-canned package for this. For example, the Stata command would be
ivprobit (see the Stata manual for this command which also explains the maximum likelihood approach).
If you require references for the theory behind probit with instrumental variables see for instance:
- Newey, W. (1987) "Efficient estimation of limited dependent variable models with endogenous explanatory variables", Journal of Econometrics, Vol. 36, pp. 231-250
- Rivers, D. and Vuong, Q.H. (1988) "Limited information estimators and exogeneity tests for simultaneous probit models", Journal of Econometrics, Vol. 39, pp. 347-366
Finally, combining different estimation methods in the first and second stages is difficult unless there exists a theoretical foundation which justifies their use. This is not to say that it is not feasible though. For instance, Adams et al. (2009) use a three-step procedure where they have a probit "first stage" and an OLS second stage without falling for the forbidden regression problem. Their general approach is:
- use probit to regress the endogenous variable on the instrument(s) and control variables
- use the predicted values from the previous step in an OLS first stage together with the control (but without the instrumental) variables
- do the second stage as usual
A similar procedure was employed by a user on the Statalist who wanted to use a Tobit first-stage and a Poisson second stage (see here). The same fix should be feasible for your estimation problem.
if you want a more detailed (but also more technical) proof, you can have a look at the notes by Jean-Louis Arcand on p. 47 to 52.
This does not seem to be the case. The Arcand discussion is not about the functional form; instead, it is about the inclusion of different covariate sets in the first-stage versus the second-stage models. "In words, the correct 2SLS procedure entails including all of the exogenous covariates that appear in the structural equation in the first-stage reduced form. The forbidden regression involves leaving some or all of them out."
Going back to the original question, I would recommend using an OLS for the first stage, and the probit for the second. While this may be technically biased, it is likely (assuming you have a good instrument) to be less biased than the non-IV approach.
I wanted to add an answer here because there seems to be A LOT of confusion about the forbidden regression. In my opinion the OP falls in this regression, only when he wished for the second stage to be a probit/poisson model. I base my answer on comments from Wooldridge himself on Statalist, and Wooldridge (2010) Econometric Analysis of Cross Section and Panel Data.
I went through some of Wooldridge's own comments on Statalist (and his books) and in contrast to what Andy comments (please don't shoot me), it seems that the forbidden regression, is using fitted values of a first stage into a non-linear second stage. I base this on two threads:
Here Wooldridge explains that another poster falls in the forbidden regression trap: https://www.statalist.org/forums/forum/general-stata-discussion/general/1308457-endogeneity-issue-negative-binomial. I quote him: "You cannot, in most cases, simply insert fitted values for the EEV into a nonlinear function."
In this post Wooldridge even suggest using an ordinal probit in the first stage (and use the fitted probabilities in the second stage), which therefore apparently does not pose any issue: https://www.statalist.org/forums/forum/general-stata-discussion/general/1381281-iv-estimation-for-ordinal-variable?_=1617356656297.
Please also note that in my opinion Wooldridge (2010) mentions that you can still use 2SLS in this case, but not mimic it by using fitted values! See chapter 9.5.2 titled "Estimation".