I have been under the impression that for a 2SLS FE model you got to have the fixed effects in both stages. So if you have a time invariant instrument you can forget about your FE, as they already capture all effects of your instrument. In fact, if I remember correctly, some people here made fun of someone who considered going for such an approach. Now I found this paper: https://onlinelibrary.wiley.com/doi/full/10.1111/j.1468-0297.2011.02420.x?casa_token=ov2jna-_8ZUAAAAA:THfv2kmtVtai6bP4r653TPTJQU2MFtmA7_95xmPVvJ8VugRgjixZO8va9QCtsEtMJSpbhakFjLrgCg
Admittedly, I have not read it from front to back, but skimmed it for their methodology. They use a time invariant instrument (whether a country has fixed telephone lines) to predict broadband introduction. So far so good, but then they go on stating that one of their second stage models includes Fixed Effects. So have they use FE in a second stage but apparently not in their first stage.
Why can they get away with that? Could someone kindly explain the rationale behind it?