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 of acronyms:

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