Heteroskedasticity and Clustered Covariances from a Bayesian Perspective

We show that root-n-consistent heteroskedasticity-robust and cluster-robust regression estimators and confidence intervals can be derived from fully Bayesian models of population sampling. In our model, the vexed question of how and when to “cluster” is answered by the sampling design encoded in the model: simple random sampling implies a heteroskedasticity-robust Bayesian estimator, and clustered sampling implies a cluster-robust Bayesian estimator, providing a Bayesian parallel to the work of Abadie et al. (2017). Our model is based on the Finite Dirichlet Process (FDP), a well-studied population sampling process that apparently originates with R.A. Fisher, and our findings may not be surprising to readers familiar with the frequentist properties of the closely related Bayesian Bootstrap, Dirichlet Process, and Efron “pairs” or “block” bootstraps. However, our application of FDP to robust regression is novel, and it fills a gap concerning Bayesian cluster-robust regression. Our approach has several advantages over related methods: we present a full probability model with clear assumptions about a sampling design, one that does not assume that all possible data-values have been observed (unlike many bootstrap procedures); and our posterior estimates and credible intervals can be regularized toward reasonable prior values in small samples, while achieving the desirable frequency properties of a bootstrap in moderate and large samples. However, our approach also illustrates some limitations of “robust” procedures.