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I understand that the coefficient of the interaction term in a standard diff-in-diff model is the average treatment effect on the treated (ATT). But I was wondering how/when we can derive the average treatment effect (ATE) from a diff-in-diff regression.

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I'm not sure but ATE doesn't seem to be identified under the parallel trends assumption for the potential untreated outcomes because ATU isn't. Here is why.

First, let us show that ATU should be identified in order for ATE to be identified when ATT is. This part is straightforward because \begin{align} ATE &= E(y^1 - y^0)\\ &= E(y^1-y^0|d=1)P(d=1) + E(y^1-y^0|d=0) P(d=0)\\ &= ATT\cdot P(d=1) + ATU \cdot P(d=0), \end{align} where ATT is identified by DID (as OP mentions, under the parallel trends assumption for $y^0$) and $P(d=1)$ and $P(d=0)$ are obviously identified. So ATE is identified if and only if ATU is identified unless $P(d=0)\ne 0$, which is true. Thus, the question is whether ATU is identified by a DID regression.

For this second question, we have $$ATU = E(y^1 - y^0|d=0) = E(y^1|d=0) - E(y|d=0),$$ where $E(y|d=0)$ is obviously identified. Without assumptions on $E(y^1|g,t)$, it is impossible to identify $E(y^1|d=0)$. Therefore, with parallel trends only for $y^0$, ATU remains unidentified, and correspondingly ATE is unidentified.

What, then, happens if parallel trends are assumed for the potential treated outcomes $y^1$? Well, in that case, ATU should be identified but ATT and ATE aren't.

The bottom line is that identifying ATE requires some assumptions on both $y^0$ and $y^1$. This is my reasoning. What would you think?

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