A basic question about ATT and statistical notation I'm trying to find an answer to what I think should be very basic. It's about the formal equation of the average treatment effect on the treated (ATT):
$$E[Y_{1i}|D_i = 1] - E[Y_{0i}|D_i = 1] $$
$Y_{1i}$ is the potential outcome if person $i$ is treated ($D_i = 1$) and
$Y_{0i}$ is the potential outcome if the person $i$ is not treated ($D_i = 0$).
I understand that we are dealing with potential outcomes, $D_i$ can only be 0 or 1 in this example. Now my question is specifically about one thing, in this seemingly simple formula. For me, the first part makes sense $E[Y_{1i}|D_i = 1]$, however, this part does not: $E[Y_{0i}|D_i = 1] $.
In plain words, I read the second part as what is the expected outcome of the untreated group($Y_{0i}$), give that the group is treated ($D_i = 1]$). Now that just doesn't make any sense - how  Does anyone care to explain?

ATT is the average causal effect of treatment in those that received treatment. This is defined as the difference between two things:

*

*the expected outcome of those that were given the treatment, if they had received the treatment (which they actually did, so ignoring issues like missing data, we directly observe this) and

*the expected outcome of those that were given the treatment, if they had not received the treatment (we do not directly observe this).

You paraphrased the second bullet slightly incorrectly, which I think is the source of the confusion.
And, yes, the second thing is a hypothetical thing that we somehow need to figure out (e.g. via some appropriate modeling) and cannot normally observe directly. There's of course experimental designs that make this a lot easier, e.g. if treatment assignment is via randomization, then you know that the expectation in the second bullet has the same expectation as the observed outcomes of those randomly assigned to be untreated.
