Greetings!
Suppose I want to estimate a treatment effect model,
Y = b*X+a*D + e
D is the treatment and endogenous, where D = 1 if g*Z>0, and 0 otherwise.
If I understand correctly, treatreg in Stata does the following:
1. in the first stage using a probit model (regress D on probit(g*Z))
to estimate g.
2. In the second stage, add the inverse mills ratio to the equation Y
= Xb+a*D + e and estimate using OLS.
However, I have seen another approach to estimate the treatment effect:
1. in the first stage using a probit model (regress D on
probit(g*Z)) to estimate g.
2. replacing D with the estimated probabilities from the first stage
and then run the OLS.
I am not clear how this second approach is derived. I read through Lee
and Trost (1978 journal of econometrics) but there is not much
details.
Most important, which approach is the preferred one?
Thanks for your attention.
Gordon
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