When Should You Adjust Standard Errors for Clustering?
When Should You Adjust Standard Errors for Clustering? (arXiv: https://arxiv.org/abs/1710.02926) was written by Alberto Abadie, Susan Athey, Guido W. Imbens, and Jeffrey M. Wooldridge. The first draft was published online in 2017. It was then published in the Quarterly Journal of Economics, (vol. 138, no. 1) in 2023.
The authors formulate variance for OLS and fixed effects estimators. Specifically, they conceptualize a sequence of k populations with differential sampling and design parameters, and formulate the variance of estimators across these populations assuming a large k. For example, q is the cluster sampling probability; p is the individual sampling probability; A is the treatment assignment probability.
The authors then use this framework to demonstrate that components of variance approach zero under certain conditions. This explains which components dominate the calculation.
For OLS estimators, they show:
- conventional standard error calculations for OLS estimators are flawed.
robust standard errors are correct only when treatment effects are constant and when treatment assignment is not clustered (i.e., A is constant across clusters)
- they can otherwise overestimate or underestimate
cluster robust standard errors are accurate for small cluster sample rates (i.e., q), or when treatment effects are similar across clusters
- they are otherwise too conservative
novel method (causal cluster variance, CCV) is more appropriate if there are a substantial number of both treated and control cases in each cluster.
- an algorithmic resampling method (two-stage-cluster-bootstrap, TSCB) is more appropriate in the same conditions
The authors repeat this for fixed effect estimators.
In summary, cluster robust standard errors are not appropriate if sampling and treatment assignment are random, even if outcomes are correlated. If treatment assignment is clustered, cluster robust standard errors are required. Similarly, if clustered sampling is employed and either q or p are small, cluster robust standard errors are required and asymptotically correct. If clustering is employed for either reason, if clusters are large and there is variation of treatment assignment within clusters, CCV and TSCB can provide less conservative estimates.