If fixed effects and robust standard errors both necessary, do they have to be at the same level, and why?

Question

I am working on an empirical paper using repeated cross-sectional data, and a reviewer has asked that we cluster our standard errors at the same level as our geographic fixed effects. Given the structure of our data, it makes the most sense to cluster at the level of the enumeration area (based on Abadie et al, 2017 and Angrist and Pischke, 2009). But since treatment occurs at the level of the enumeration area, I believe an enumeration area fixed effect would result in multicollinearity. Originally, we used sub-district fixed effects, but this reviewer seems to believe that consequently, we need to match our standard error clustering so that it is also at the sub-district level.

I have never in my studies come across any theoretical work supporting the claim that if fixed effects and robust standard errors are both necessary, then they must be at the same level. Could anyone please point me in the direction of any relevant citations, or could anyone briefly offer an explanation as to theoretically why we want the level of the fixed effects to match the level at which we cluster the standard errors?

Answer 1

I believe I understand your concern. Fixed effects estimation at your lowest level wouldn't be feasible. Suppose you estimated dummies for all units below the sub-district level. In the case of repeated sampling of units within sub-districts, some of these would be singletons. As you already noted, you do not observe the same enumeration areas within sub-districts over time. You obtain completely independent samples of enumeration areas at each sampling period.

It appears there is no well-defined level of aggregation at a higher level. In other words, there is no "treatment assignment" at the district level. However, treatment is assigned at the enumeration level (i.e., areal census unit). But, as you already indicated in the comments, you are dealing with repeated cross-sections, which means you acquire different samples of enumeration areas across time. In each sampling round, some tracts are treated and some tracts are untreated. It is difficult to advise you how to proceed in this setting. Some suggest matching the level of clustering performed in the analysis to the level at which treatment was assigned (see here for more). Others suggest clustering at a coarser level (see here for more).

In my opinion, it is not mandatory to cluster at the same level as your fixed effects. What I want to know is why not observe the same enumeration areas over time? Typically, treatment is administered at a higher level (e.g., district, county, state, etc.), and all you need to do is sample from within these higher level units at the various time periods.

I will update this answer if anything is unclear. I also encourage you to review section 4 of this working paper. It is a good resource.