Standard error clustering under treatment assignment in groups of varying size.

[Simon Heß]

Basic setup:
Unit of observation is the individual. Treatment (binary) is assigned on city level. Every state contains 4 cities, 2 get randomly chosen for treatment, 2 control. There are only 5 states. The outcome of interest is likely to be regionally clustered.

Question:
How to cluster standard errors for treatment evaluation?

Cameron and Miller (2014) state that

[If] either the regressors or the errors are likely to be uncorrelated within a potential group, then there is no need to cluster within that group [...] If a key regressor is randomly assigned within clusters [...] then the within-cluster correlation of the regressor is likely to be zero. Thus there is no need to cluster standard errors, even if the model’s errors are clustered.

Following this logic, it would not be necessary to cluster at the state level, as city-treatment is random within state. However, varying city sizes introduce within state correlation of treatment. Yet, the small number of states keeps me from being able to cluster at the state level.
Additionally, I think, because the exact character of the within cluster correlation of treatment is known (city size), there must be a more efficient way to correct for this.

Reference:
A. Colin Cameron and Douglas L. Miller (2014), A Practitioner’s Guide to Cluster-Robust Inference, Journal of Human Resources: http://www.econ.ucdavis.edu/faculty/...14_July_09.pdf

[Simon Heß]

I might have found a solution.
Basically this simply allows for state level random effects, while remaining the normal clustering at the city level. State level random effects might be justified here because states are the randomization strata. In particular, intra-state correlation of individual treatment arises only through city-size differences, to which treatment is orthogonal.
In Stata I estimate this by:

xtset state bootstrap treatment=_b[treatment], cluster(city): xtreg outcome treatment, mle Which gives the right rejection rates on simulated data at least. I am not sure it's the simplest/most efficient way though. Comments and ideas are very welcome!



(Here: http://stats.stackexchange.com/quest...f-varying-size, I pose this very question in more detail)