The way that "eteffects" attempts to identify the selection bias

Dear Statalist,


While going through the PDF manual entry of "eteffects", I found some skipped but I think critical steps in two places. I really appreciate it if someone can help me understand the reasoning by completing the missed steps.



# 1 It is not that obvious to me that under (3) and (5):
E[\epsilon_ij | E(t|z_i)+v_i]=E[\epsilon_ij | v_i]

Although \epsilon_ij and v_i is mean independent of z_i respectively, it may not be the case in the expectation specified as above where it is conditional on v_i and z_i simultaneously.

Also E[\epsilon_ij | v_i] = v_i*beta_2j, I think such linearity is something assumed rather than that can be derived from the existing assumptions.

# 2 in equation (7), given E[\epsilon_ij | v_i] = v_i*beta_2j, I am curious how to derive E[\epsilon_ij | x_i, v_i,t_i=j] = v_i*beta_2j as this is additionally conditional on x_i and t_i.


Kind regards,
Yugen