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Suppose we have that $\left(Y(1), Y(0)\right)$ are potential outcomes with $X$ being the covariate and $Z$ the treatment assignment. Typically in causal inference, one will assume strong ignorability or unconfoundedness to equate these two:

$$ E[ Y(1) \mid Z=1, X ] = E[Y(1) \mid X] $$

What I am wondering is why we cannot just use iterated expectations to get:

\begin{align*} E\bigg( E\left( Y(1) \mid Z=1, X \right) \mid X\bigg) &= E[Y(1) \mid X] \end{align*}

since it appears to me that $X$ is a subcollection of $(Z=1, X)$?

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    $\begingroup$ Two things: if you are taking the expectation over $Z$, it can't be fixed to 1. Also, your goal is not to get rid of $Z$ (you observe $Z$) but to replace counterfactual quantities (like Y(1) which is unobserved) with observed quantities (like Y). $\endgroup$ – Carlos Cinelli Nov 13 '18 at 3:59
  • $\begingroup$ What is the reason that we cannot fix $Z=1$ when taking the expectation over $Z$? $\endgroup$ – user321627 Oct 14 '19 at 13:05
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Ignorability assumption, a.k.a. unconfoundedness (Rosenbaum and Rubin, 1983) is based on the idea that we have measured ENOUGH features so that Y(1),Y(0) are independent of the treatment conditional to the covariates X. If you numerically test it with a toy example, using random variables in the mix, you will notice that you need a LOT of Y(1),Y(0) to get the assumption correct.

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