When performing a treatment study (e.g. a difference-in-differences design) in multi-dimensional panel data (e.g., firm-individual-time, bank-firm-time, etc.) is it ok if the "smaller" cross-sectional unit is in both the treatment and control group?
That is, if treatment is defined based on some firm-individual characteristic (or bank-firm, area-individual, etc.) then it is possible for the individual to satisfy the rule assigning treatment on some firm relationships but not on others.
It seems like a bad idea for the same individual dealing with a different firm to act as a control for the actions of that individual with a different firm. But what if they only served as a control for the actions of other individuals?
- Say I have data on purchases and sales credit between upstream firms and downstream firms over 4 years. There are 100 upstream firms but 100,000s of downstream firms. Each downstream firm can buy from multiple upstream firms (say 10 downstream-upstream relationships is the limit).
- A reform happens at some time that reduces the legal protection given to downstream firms for any net sales (sales-credit) over some limit X.
- The treated group are the upstream-downstream pairs with sales-credit>X. I have constructed a control group based on pairs who have sales>X and the downstream firm in this pair has "similar" total credit to the treatment group across all relationships.
- The problem here is that an individual downstream firm can have (sales-credit)>X in one relationship (and is treated) and have (sales-credit)$<$X but sales>X in another relationship (and is untreated).
- The final regression model looks at how the downstream firm changes their purchasing from upstream firms based on the upstream firm characteristics, after the reform. I want to do this regression model at the upstream-downstream level. Consequently, it seems worrying to have the same downstream firm in a control relationship and a treatment relationship.