Dear Statalisters,
Using panel data, I estimated treatment effect with different timings of introduction of the treatment by etregress and other methods as follows.
Q1: Why are the results of treatment effect, active_t, and lambda or IMR, different between Models 1,2 and 3?
Q2: Which is correct, and why?

Notation:
treat: a binary dummy variable that varies between individuals and does not vary over time within an individual
active_t: a binary dummy variable that varies between individuals and varies over time within an individual
IMR: inverse Mills ratio calculated from a probit estimation (probit active_t z x) which is identical with lambda in etregress
id: individual
year: year

Model 1: OLS
regress y x i.id i.year i.active_t i.treat IMR
Source | SS df MS Number of obs = 456
-------------+---------------------------------- F(106, 349) = 28.39
1.treat | 0.000 (omitted)
1.active_t | -0.131 0.490 -0.27 0.790 -1.095 0.834
IMR | 0.242 0.292 0.83 0.409 -0.333 0.817

Model 2: FE
xtreg y x i.year i.active_t i.treat IMR,fe
Fixed-effects (within) regression Number of obs = 456
Group variable: id Number of groups = 78
1.treat | 0.000 (omitted)
1.active_t | -0.131 0.490 -0.27 0.790 -1.095 0.834
IMR | 0.242 0.292 0.83 0.409 -0.333 0.817

Model 3: etregress
etregress y x i.id i.year i.treat ,treat(active_t= z x ) twostep
Linear regression with endogenous treatment Number of obs = 456
Estimator: two-step Wald chi2(127) = 3930.86
1.treat | 0.000 (omitted)
active_t | 0.069 0.438 0.16 0.876 -0.790 0.927
hazard |
lambda | 0.102 0.275 0.37 0.709 -0.436 0.640

Nobuya Fukugawa