Reverse DiD: or using always treated as control

Question

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?

Answer 1

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.

In the case you mention in the question there is just one extra twist that you also reverse order of the time periods, but that itself should not affect the properties of the estimator. Only interpretation is different.

Also, you do not have just analogue of parallel trend assumption, you still need to satisfy the parallel trend assumption, except since you reversed the order of time periods the parallel trend assumption needs to hold not before but after.

So for example, if you would run this DiD with 2 time period 2020 and 2021, you would need data on outcomes for 2022, 2023, 2024, 2025... and so on and see if there is common/parallel trend assumption.

Answer 2

For those interested, there is work on this: Kim, K., & Lee, M. J. (2019). Difference in differences in reverse. Empirical Economics, 57(3), 705-725.