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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.

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Rubin explains in his great free book, Basic Concepts of Statistical Inference for Causal Effects in Experiments and Observational Studies (pdf), what's called "The fundamental problem of causal inference" (see, e.g., p. 4). I'd suggest starting form there.

In short: the only way to truly assess a causal effect is to compare things that don't really exist - potential outcomes. The observed and the unobserved (usually, the mean is used as a representative statistic).

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