When you type -margins, dydx(interaction)-, that will not hold t and treated at the correct levels. Assuming you used the i.interaction syntax, Stata will calculate the average of
F(a+b*post_i + c*treated_i + d*1)-F(a+b*post_i + c*treated_i + d*0),
where d is the coefficient on the interaction and F(.) is the standard normal cdf or the exponential in the Poisson case. Without the at() option, Stata will use whatever values for post and treated that it sees in the data, but that is not what you want for the DID. You want to fix post to 1 and treated to 1 for everyone. Hence you will get the wrong answer if you do this.
F(a+b*post_i + c*treated_i + d*1)-F(a+b*post_i + c*treated_i + d*0),
where d is the coefficient on the interaction and F(.) is the standard normal cdf or the exponential in the Poisson case. Without the at() option, Stata will use whatever values for post and treated that it sees in the data, but that is not what you want for the DID. You want to fix post to 1 and treated to 1 for everyone. Hence you will get the wrong answer if you do this.
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