Cluster-robust SEs & random effects -- seeking some clarification

Hi JD,

Below are a couple of further thoughts on the questions you posed.

James

One further post perhaps framing my question slightly differently (or
altogether more generally):

What, specifically, do cluster-robust/robust SEs allow one to do with more
accuracy/precision *if* they are already using both random effects and
slopes to model relevant cluster-specific effects.

Just to be clear, using cluster-robust SEs does not change anything about
the accuracy or precision of the model's coefficient estimates. Using them
or not using them is purely a matter of how to estimate standard errors
(and thus build test statistics or confidence intervals) for those
coefficient estimates.

The advantage of using clustered SEs in a random effects model is that
doing so captures unmodeled sources of dependence or heteroskedasticity in
the errors. Thus, if you trust the specification of your random effects
structure, then there is no need to use clustered SEs. On the other hand,
if you (or your audience) are skeptical that you've got the right
specification, then clustered SEs are helpful. Think of them as an
insurance policy for your SEs/t-statistics/CIs, so that they remain valid
even in the event that your model might be incorrectly specified in some
respects.

Is it the case that
there may be any number of sources that could potentially account for
sources of heteroskedasticity (i.e., autoregressive structure in the case
of repeated measurements/time variables) that using the cluster robust SEs
would be of value for in making more precise inference assuming some
misspecification of the random effects structure of the model?

Yes.

Relatedly, is there a 'seminal' or 'key' paper that provides a deep dive on
the concept of heteroskedasticity? I have a few on hand, but wanted to see
if there was something I might not be aware of .

Cameron and Miller (noted in your subsequent paper) is an excellent,
thorough survey from the econometric perspective. McNeish and Kelley (2019;
https://doi.org/10.1037/met0000182) is a great resource that addresses the
fixed effects vs mixed effects modeling contexts. To be a bit
self-promotional, I have a working paper with Young Ri Lee that looks at
these issues in the context of multi-way clustering:
https://psyarxiv.com/f9mr2
The simulations in the paper illustrate the consequences of several
different forms of model mis-specification (such as omission of random
slopes).