# Diff-in-diff and ATE vs ATT

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.

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?