Design-Based Multi-Way Clustering

This paper extends the design-based framework to settings with multi-way cluster dependence, and shows how multi-way clustering can be justified when clustered assignment and clustered sampling occurs on different dimensions, or when either sampling or assignment is multi-way clustered. Unlike one-way clustering, the plug-in variance estimator in multi-way clustering is no longer conservative, so valid inference either requires an assumption on the correlation of treatment effects or a more conservative variance estimator. Simulations suggest that the plug-in variance estimator is usually robust, and the conservative variance estimator is often too conservative.