# Reverse DiD: or using always treated as control

I would be interested also in the generalised case, but let's start with 2x2 to keep it simple.

Say you have two groups $$i \in \{1,2\}$$ and two time periods $$t$$ and $$t-1$$, as the classical DiD case. One group is untreated in $$t-1$$, i.e. $$D_{t-1}=0$$, and starts receiving treatment in $$t$$, i.e. $$D_{t}=1$$. However, the other group is treated in both periods $$D_t=1$$. We are interested in the Average Treatment Effect on the Treated (ATT) $$ATT = E[Y_t(1)-Y_t(0)|D=1]$$.

Can we do some kind of "reverse" DiD here? And if so, what would be the analogue of the parallel trends assumption?

In principle yes, but you would not use 'reverse treatment' but standard terminology. Here just withdrawal of some stimuli is the treatment. That is you would still code the withdrawal as $$D=1$$.
For example, suppose you have two groups of people where everyone drink. If you implement policy that forces one group to stop consuming alcohol, that will be the treatment (with $$D=0$$ for groups that drink and $$D=1$$ for abstinence). This would give you the effect of withdrawal of alcohol consumption.