Difference-in-differences in 2SLS regression

Question

Usually when we do a difference-in-differences estimation, we do it in a OLS reduced form as follows: $$ Y_{it}=\alpha After_t+\gamma Treatment_i+\delta After*Treatment_{i,t}+X_{it}\beta+\epsilon_{i,t} $$ However, I was wondering, if the $Treatment$ group is endogenous (e.g. self-selected), but we can define an "eligible" group for the treatment, whether it would be more precise to estimate a diff-in-diff in a OLS/2SLS form as: $$ Treatment_{i,t}=constant+\alpha After_t+\gamma Eligible_i+\delta After*Eligible_{i,t}+\epsilon_{i,t} $$and get $\hat{Treatment_{i,t}}$, then

$$ Y_{i,t}=X_{it}\beta+\delta\hat{Treatment_{i,t}}+\epsilon_{i,t} $$

How should we understand the diff-in-diff in a OLS/2SLS form? Are there any paper using this particular identification strategy that I could take a look?

Thank you very much in advance!

Answer 1

Well, if you believe that treatment is endogenous (which depends on the problem at hand here and is not an inherent feature of the model), then using eligibility as an instrumental variable will help you to get rid of the biases due to the safe selection in treatment. (Incidentally, DID is intended to do the same, but won't do as good a job as a well chosen instrument, so there is some doubts whether applying both of them is better then resorting to only one). However it is up to you to decide whether eligibility is exogenous, as it well may be, that those who are expecting higher return to treatment made sure to be eligible.

Taking that we believe that there are some biases that arenot eliminated by DID and that eligibility can help us, there is still considerations of efficiency. In many cases eligibility may happen to be a weak instrument and then the reduction is bias will come at a cost of significant efficiency loss.

And taking a look at the particular specification that you have sugested, it seems not very reasonable in general setting. You may choose when you believe that eligibility is changing quickly, or the interaction term in second equation will be generally unhelpful. Inclusion of time After in that equation can have even more drastic consequences, as it is likely to be endogenous and will weaken the bias reduction effect. If not endogenous, it is likeliy to be negligible as well as interaction, unless Treatment is rapidly changing on it's own.

So in this case I would recommend leaving only the eligibility as an instrument in the first equation and specifying the third one in a DID form.

With respect to interpretation, my specification does not allow for a nice interpretation of difference in changes in two subgroups and should be interpreted as a difference in changes in two hypothetical subgroup where each person is divided between them with some weights.

Your specification, however, loses all interpretation as DID, because you do not use the resulting interaction coefficient, but just employ more variables as instruments for treatment.

Unfortunately, probably due to the aformentioned reasons, I was unable to recall or find any appropriate paper, sorry about that.

Answer 2

The issue of selection into treatment based on some observable variable that does not enter the outcome equation is solved with a latent index approach or a Heckman 2-step method. A difficulty with Heckman 2-step is the requirement to find a valid instrument, but if you already have one, it will solve your endogenous treatment issue.