Dear Statalist,

Hi. This is the first time I post questions on the forum. I apologize if my post is not sufficient informative or not in the best format.

I would like to estimate a model with both sample selection bias and endogenous treatment variable issue. I am concerned with three variable: Y, the outcome variable; I, the indicator of being selected into the treatment group; and Z, a continuous treatment variable. My goal is to evaluate the effect of Z on Y, but I encountered two problems
(1) Not everyone in my sample received the treatment Z. Some respondents self-select to the treatment group (I == 1) but the others do not (I == 0). In other words, I have something like this:

Code:
Z = . if I == 0
Z ∈ [1,10] if I == 1
(2) Meanwhile, I am also aware that the treatment and outcome may mutually determine each other. It seems that 2SLS is suitable for this problem.

I am a bit of puzzled by what methods should I use to account for my problems. In particular, would like to deal with the problem (1) because I would like to know "what are the effects of the Z for those who are not treated at all (I == 0)". I do not want to use I as the treatment variable, neither do I want to exclude all of those who have I == 0, because I believe that I is associated with Y in some ways. But I find that problem (1) is not a typical sample selection issue that Heckman selection model is used for, because the outcome variable Y is observed for the group with I == 0 as well. Could I use Heckman selection model in this case? If so, could I just include Inverse-Mills Ratio in 2SLS for adjustment?

For a real-life example, I could stand for childlessness, Z could stand for the relationship with children, and Y could stand for mental health. People self-select to be childless, but I am wondering what the effects of relationship with children on the childless people are had they have children.

I would appreciate if anyone could give me some advice on what models I should use or what theoretical frameworks should I use to understand my research questions.

Sincerely
Boyan