A while ago I read (probably on stata forums) that FE can be on a lower level than SE clusters, but not vice versa. For example, firm/household FE with industry/state SE are good, but doing state FE with household SE would be wrong. Intuitively it seems to make sense. If treatment assignment is on a household level, shouldn't household FE be the first choice? But it seems that there are papers in different fields that use higher level FE with lower level SE. What am I missing?

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