When it comes to decisions between, e.g., a "-mixed- or a -regress and cluster- approach," what is increasingly clear (to me, anyway--and reflected in a helpful StataList discussion here) is that no firm, hard rules exists. What is clear, however, is that model specification decisions need to pivot on a granular understanding of the data and their underlying structure.
One classic context that frequently gives rise to such a decision involves efforts to assess student-level educational outcomes given hypothesized varied influences at the classroom- and school-levels into which the individual students nest. While a regression and clustering specification may yield results similar to a multi-level specification under certain circumstances (e.g., "if intraclass correlations at the school and class levels are very close to zero"), as the comments note: "The -regress- approach, even with -vce(cluster ...)- does not adjust for potential confounding due to systematic differences among classes or schools. The -mixed- model does so."
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