Coarse fixed effects - which model?

[Ariel Karlinsky]

I have annual panel data on firms. Data is structured as firm-year. Firms are nested within states. I wish to investigate both within and between variation of different firm properties (some of which are time invariant) on some outcome variable. Thus I cannot use a firm FE model.
I do wish to account for different time-invariant properties of the states the firms operate in though - which can be done via state FE.
The question is, which model is more appropriate? Random Effects at the firm level with state FE or rather "regular" OLS with state FE, what are the different assumptions underlying either model?

"Speaking stata", the models are:

xtset firm year
*firm random effects with state FE*
xtreg y x1-x5 i.state, re

*OLS with state FE*
reg y x1-x5 i.state
areg y x1-x5, absorb(state)
reghdfe y x1-x5, absorb(state)

[Clyde Schechter]

I suspect you will get different opinions about this from different people. Basically, there is no 100% satisfactory solution, so it becomes a judgment about which of the several conflicting desiderata to sacrifice.

For my part, I would prefer your first approach because it represents each level of the data structure in the model. You might want to add -vce(cluster state)- to that one as well (assuming you have enough states to warrant that.)

By contrast, your -reg-, -areg-, and -reghdfe- models (which are essentially all the same except for some degrees of freedom issues) omit the firm level altogether. Firm-level effects are not represented in the model, with potential omitted variable bias, and it incorrectly calculates standard errors because the clustering of observations within firms is nowhere accounted for.

In truth, I would probably even go farther and do a full-blown three level random-effects model -mixed y x1-x5 || state: || firm:-. The usual objection to using random effects models is that they do not guarantee consistent estimation of parameters absent strong assumptions about the independence of the random effects from everything else. But if you use random effects at one level, you will likely have already bought that problem anyway. (Well, it is possible that the independence will hold at the state level but not the firm level, I suppose.) For my part, if I have to choose between consistent estimation of a mis-specified model and inconsistent estimation of a well-specified model, I will go for the latter in most circumstances.

[Ariel Karlinsky]

I have all US states and about a decade per firm (annually). I also thought that that having some firm level effect is preferable.
I wonder though whether to cluster on state or firm? noting of course that firms are completely nested within states (no firms cross state lines).

I am not too familiar with mixed/multi-level random models, will read more about it