Regression Discontinuity Group Level Program Evaluation

Question

I'd like to use a regression discontinuity design to evaluate a program where the discontinuity/assignment to treatment occurs at the group level. However, I'd like to measure the outcome at the individual level (as opposed to measuring the outcome as an average at the group level).

I have three questions (feel free to answer any):

1. What are the criteria/assumptions for doing an RD with assignment at the group level?
2. Would using HLM be appropriate?
3. Do you know of any studies that do this?

For example, consider a very large firm where workers make widgets independently, but are divided into teams where they can consult with each other or ask for advice a few times a week. The firm calculates how long it takes each worker to make a widget on average, and then calculates an average widgets per hour for each team. They provide additional training for groups where the group average is below a certain threshold and no additional training for groups above the threshold, hence the discontinuity. The firm then wants to evaluate the effect of additional training on an individual.

Answer 1

I don't think it is particularly useful to think about this as an RD problem. In RD, you have treatment assigned as function of a running variable $Z$ and you compare post-training outcome $Y$ for those near the cutoff on the left (C) versus the right (T). The prototypical example of this is a unionization drive. Firms have elections. Some firms have 49% pro vote, so they don't get a union. Others have 51% vote, and so they do. You compare wages in the two groups after the election to get the union wage premium, "throwing away" the data away from the magic cutoff.

Your treatment for observation $i$ is based in part on the pre-training outcome of observation $i$. Averages are notoriously non-robust to outliers. Also, if these shocks are serially correlated, this violates the assumption that observations on one side or the other are exchangeable. Another way exchangeability fails is that agents can manipulate their treatment assignment: they can alter output because training is either fun or annoying. They may also convince other group members to do likewise. Or everyone may slack off, thinking the rest of the team will pull through. Or they can work extra hard because for them the training would be very useful.

My first intuition was that since group assignment is random, you can use the initial composition of the group as an instrumental variable. But if skills can diffuse (and perhaps bad habits can also be contagious) within groups, this method would fail as well depending on how skill transfer works. In short, I think this is a pretty hard problem than any partial equilibrium estimator will have trouble with.