Does there exist a principled way to choose the optimal length of the pre-period in a difference-in-differences model?
I imagine there's the usual variance-bias tradeoff from going back too far, but I have never come across discussion of this in the literature.
To clarify, I am thinking of the multi-period version of the model, rather than the simpler two-period/pre-post diff-in-diff.
This paper by Chabé-Ferret (2010) may be interesting in this context. He provides different scenarios under which a DID estimator using pre-post-treatment pairs with equal time distance to the treatment is consistent while using just the most recent pre-treatment period is not consistent. His framework is somewhat restrictive though and also he tries to answer a different question. In terms of the literature this will be the closest related to your question which, as it stands, has not yet been looked at in particular (afaik).
Other papers like Slaughter (2001) play around with the pre-treatment window as a robustness check. In his case the results don't change when he lengthens or shortens the pre-treatment period used in the analysis. Unfortunately few people provide such evidence in their studies.
To my knowledge there is no paper yet that considers the optimal pre-treatment window in DID analysis. I would assume that this hasn't been looked at so far because many papers use micro data which usually do not come with large numbers of time periods to begin with. The other reason might be that as long as treatment and control groups exhibit parallel trends before the treatment the only effect from changing the pre-treatment window on the estimated treatment parameter should come from negligible sampling variation. If it is not negligible one might as well question whether a DID makes sense to begin with. Nonetheless this is an important question and one that is fairly underexplored (at least in the econometrics literature).
