# Why can you not bypass the strong ignorability/unconfoundness assumption via iterated expectations?

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 $$(Z=1, X)$$?

• 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). – Carlos Cinelli Nov 13 '18 at 3:59
• What is the reason that we cannot fix $Z=1$ when taking the expectation over $Z$? – user321627 Oct 14 '19 at 13:05