Jeffrey Wooldridge, a famous econometrician, posed the following question to Bayesians on twitter:

> I think frequentists and Bayesians are not yet on the same page, and
> it has little to do with philosophy. It seems some Bayesians think a
> proper response to clustering standard errors is to specify an HLM.
> But in the linear case, HLM leads to GLS, not OLS.
> 
> Moreover, a Bayesian would take the HLM structure seriously in all
> respects: variance and correlation structure and distribution. I'm
> happy to use an HLM to improve efficiency over pooled estimation, but
> I would cluster my standard errors, anyway. A Bayesian would not.
> 
> There still seems to be a general confusion that fully specifying
> everything and using a GLS or joint MLE is a costless alternative to
> pooled methods that use few assumptions. And the Bayesian approach is
> particular unfair to pooled methods.
> 
> One only needs to think of something like a simple time series
> regression with serial correlation. I think there are four common
> things one might do.
> 
> 1. OLS with usual (nonrobust) SEs
> 2. OLS with Newey-West SEs
> 3. Prais-Winston with usual SEs
> 4. P-W with N-W SEs
> 
> In my view, choice (3) is almost as bad as (1). Choices (2) and (4)
> make sense, with (4) requiring strict exogeneity. But at least we're
> then comparing applies with applies.
> 
> Again, what is the Bayesian version of (4) after priors and
> distributional assumptions are imposed?


What is the answer to Wooldridge's question?

Glossary:

HLM = Hierarchical Linear Model, GLS = Generalized Least Squares, OLS = Ordinary Least Squares, SE = Standard Error, MLE = Maximum Likelihood Estimator.