Standard errors for fixed effects

[greimel]

Is there a way to get standard errors for the fixed effects?

I have an application where the estimates for the fixed effects are of interest.
(I have > 10 million observations and a few thousand fixed effects)

[felixholub]

It's a good question and I have no answer.

For a Wald test in which you force all your few thousand fixed effects to zero, you need an estimate of the covariance matrix, i.e., invert X'X.
I first thought that it might be possible to implement a likelihood ratio test. You can obtain the likelihood as -0.5N [1 + ln(2pi) + ln(SSR/N)]. This is actually implemented in the Stata package reghdfe. However, this assumes that errors are independent and will not even work out under heteroskedasticity.

In principle, one might also be able to bootstrap, but there are usually very very few observations per unit.

There is a paper by Kline, Saggio and Solvsten that discusses inference on fixed effects: https://eml.berkeley.edu/~pkline/papers/KSS2020.pdf It might be that their method has something to say on estimating standard errors, but I have not seen it discussed in the paper.

[felixholub]

I just read this tweet by Jeff Wooldridge and remembered this issue.
https://mobile.twitter.com/jmwooldridge/status/1551768919194669062
He suggests to test for the fixed effects by running a regression that includes the unit specific averages of x.