There is something conceptual I don't understand with average treatment effect on the treated. Say you have two binary outcomes $Y_i(W_i)$ in which $W_i$ is either 0 or 1 (treatment). So I either observe $Y_i(1)$ or $Y_i(0)$.
What I am not sure I understand is what is $E(Y_i(0)|W_i=1)$. Since $Y_i(0)$ is a random draw from the population who did not receive a treatment, I am not sure what the expectation of this given treatment means. In other words conditioning on treatment while drawing outcomes from the population who didn't receive treatment doesn't make sense to me.
I guess I am missing something important here.