*Bounty: 50*

*Bounty: 50*

I have seen papers at the US level where they include county fixed effects, and state-by-year fixed effects, i.e.:

$y_{c,t}$ = $beta$$x_{c,t}$ + $lambda_c$ + $mu_{s,t}$ + $eta_{c,t}$

where c indexes county, t time, and s states, opposed to a more typical county and just year fixed effect. $lambda_c$ are county fixed effects, $mu_{s,t}$ are state-year fixed effects, and $eta_{c,t}$ is the error term. They referred to this as ‘comparing counties within states’. How does this accomplish that goal? so is this estimator of $beta$ than estimating $dy/dx$ within a state, and then averaging the state effects over each state?