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In causal inference, the unconfoundedness assumption is usually stated as:

$$ Y(1), Y(0) \perp Z \mid X $$ or $$ P(Z \mid Y(1), Y(0), X) = P(Z \mid X) $$

where $Z$ is the treatment assignment, $(Y(1), Y(0))$ the potential outcomes, and $X$ the covariates.

My own interpretation of the above is that: "The covariates $X$ contain all necessary information for determining which of the two potential outcomes were received."

Another interpretation I have seen is that: "Conditional on the covariates $X$, the treatment assignment doesn't depend on the potential outcomes."

This is a bit strange for me to understand as it appears to me that our focus should be on the potential outcomes, instead of the treatment assignment. Am I missing something deeper from the italicized statement above? Thanks.

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One way of thinking about this statement is to consider that the potential outcomes occur earlier in time than the treatment assignment does. Before treatment is assigned, each individual has two potential outcomes. The assignment to treatment essentially "opens the door" of one of those potential outcomes, and that is the one that then becomes the observed outcome. The only causes of the potential outcomes are covariates $X$. Treatment can't cause potential outcomes because potential outcomes occur prior to treatment; rather, treatment reveals potential outcomes (only one per individual).

If those same covariates $X$ that cause the potential outcomes cause selection into treatment, there is an association, or dependence, between treatment assignment and the potential outcomes. This is what we call confounding. The assumption of unconfoundedness means we have observed all $X$ that are sufficient to eliminate confounding (i.e., all variables that yield an association between the potential outcomes and treatment).

I think it would be far less confusing to write $Z \perp \{Y(0),Y(1)\}|X$. That prioritizes the treatment as being causally separate (i.e., independent) from the potential outcomes. This is not a statement about predicting counterfactuals; it's a statement about the treatment assignment mechanism. In particular, it says the treatment assignment mechanism is independent from (i.e., not associated with) the potential outcomes given $X$. It is this statement about the treatment assignment mechanism that allows us to estimate the treatment effect using only the observed outcomes, the treatment, and the covariates, even though the causal claim we want to make involves only the potential outcomes.

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  • $\begingroup$ Thanks very much for this example. It is quite clear! One thing I am always confused about is why the outcomes are not written explicitly as a function of the covariates $X$, such as: $\left(Y(Z=1,X), Y(Z=0,X)\right)$ instead of the standard $(Y(1),Y(0))$? Is there a very specific point to this? $\endgroup$ – user321627 Apr 25 '19 at 17:21
  • $\begingroup$ Well, that's not very general, because the potential outcomes don't have to depend on any measured covariates. They could be purely stochastic given treatment, or depend on variables otherwise irrelevant to the analysis (i.e., unrelated to treatment). It's also extra unnecessary notation. The focus in using potential outcomes notation is to identify the potential outcomes corresponding to intervened levels of the treatment. The covariates are not being intervened on. $\endgroup$ – Noah Apr 26 '19 at 3:04
  • $\begingroup$ I have a related question I was wondering if you had insight on. My understanding of unconfoundedness exists primarily through the equation: $E(Y(1)\mid Z=1,X) = E(Y(1)\mid X)$. I am trying to intuitively understand what it means for $Z$ to be independent of the outcomes in this equation. I see the equation as saying that by computing at the outcome mean at levels of $X$, the observed mean of those who were treated represents the "true" outcome mean of treatment. Is there a way to integrate the flipped statement $Z \perp \{Y(0),Y(1)\}|X$ above into this equation? Thank you! $\endgroup$ – user321627 Apr 28 '19 at 4:43

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