Questions regarding Modelling both Sample Selection Bias and Endogenous Treatment Bias

[Boyan Zheng]

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:

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

[FernandoRios]

Dear Boyan
i think that what you have is pretty much a case of endougenous continuous treatment. Which can be considered as a Tobit type 3.
there are 2 options that use community contributed commands. One is oheckman. This is like Hickman, but the selection term comes from an ordered probit, and a regression is run for every point of the ordered variable. Which in this case is your treatment.
The second option is a command named ctreatreg which I think is only available from the stata journal archives. I’m less familiar on how that commands deals with the problem.

both commands have articles in stata journal, so that can help you understand what is going on behind each one and see which one is more appropriate.

i think there is a third option. In stata 15 one has access to the commands eregress, which can handle tobit selection models. I would think that etregress also has an option for tobit selection models.
hth
fernando

[FernandoRios]

Hi Boyan
An update to my answer. I just checked the helpfile on my computer, and etregress does not have a tobit selection model as an option. I think, however, you can still estimate that model by hand, using a bootstraped procedure to correct for the two-step estimator.
Fernando