When to use cluster robust standard errors in random intercept multilevel modeling?

[Felix Kaysers]

I’ve recently revisited multilevel modeling for my research project. My data consist of a cross-sectional survey of individuals (Level 1) nested within regions (Level 2). I am estimating a random-intercept logistic model (melogit) with a binary outcome. Multilevel models are often presented as an alternative to OLS regression when the independence assumption is violated. However, some argue that one can instead use standard regression with clustered standard errors (Link 1).
Since these two approaches are usually positioned as alternatives, I was surprised that Stata’s multilevel models also allow the use of clustered standard errors. What added to my confusion is the idea of correlated random effects, which are used when Level 2 effects may be correlated with Level 1 regressors. I am uncertain whether this approach is appropriate in purely cross-sectional data. According to Stata’s official website, cluster-robust standard errors are available to “relax distributional assumptions and allow for correlated data.” I have searched the help files and the applied literature in my field for guidance or examples but have found very little. This makes me wonder whether I am overcomplicating the issue or overlooking something obvious.
So, my question is when to use cluster robust standard errors in multilevel logistic regression and how this is different from correlated random effects?

[Erik Reinbergs]

This (awesome) paper from psychology goes into detail about difference modeling choices for clustered data and possibilities of combining those choices (such as combining a multilevel model with clustered standard errors). https://psycnet.apa.org/doiLanding?d...7%2Fmet0000620

There is also this paper from political science: https://www.cambridge.org/core/journ...53FC878F416980

If i understand right, correlated random effects helps correct for a bias in the estimates themselves where clustered standard errors are just helping to correct the standard errors (not the estimates). And that cluster robust standard errors in an MLM helps correct the standard errors for the assumption that the random intercepts are normally distributed.

But hopefully someone with more expertise can chime in.

[Jeff Wooldridge]

As I discuss in my 2003 AER P&P paper on clustering, there are good reasons to cluster even if nominally you think you’re doing GLS (“random effects” to econometricians) or MLE using a mixed model. The GLS estimator is often more efficient than OLS even if it’s not true GLS. But you should cluster your standard error with a sufficient number of clusters. Correlated random effects has different meanings, but usually it means allowing the group (second level) effects to be correlated with level one covariates — as Felix stated. In the case with just additive heterogeneity, the CRE approach reproduces the fixed effects estimates on the level on covariates. See my 2010 MIT Press book — the chapter on clustering— or my 2019 Journal of Econometrics paper. Advantages of CRE over FE is you can include level two covariates and it leads to a simple test of RE versus FE. But you should always cluster your standard errors unless you have few clusters!