Question about fixed effects, and state-by -time fixed effects

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

$$y_{c,t}$$ = $$\betax_{c,t}$$ + $$\lambda_c$$ + $$\mu_{s,t}$$ + $$\eta_{c,t}$$

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?

• You also have a county x time FE here, but don't mention this in the text. Or is that meant to be the error term? Jul 13 '20 at 15:59
• Sorry yes $\eta_c,t$ is the error term. I will edit that into the question Jul 13 '20 at 16:08

Consider first the simpler regression: $$y_c = \beta x_c + \mu_s + \eta_c$$ In this regression, the idea is to use the variability of $$x$$ across counties within states to identify $$\beta$$. Why would you want that? Maybe because other policies are implemented at the state level that might influence both $$x$$ and $$y$$. In this case, omitting the state fixed effects $$\mu$$ would allow one to use both the cross-county and the cross-state variability of $$x$$, which might cause a bias.

Running this fixed-effect regression is exactly equivalent to centering your $$y$$ and your $$x$$ by removing the state average to each county-level measure and run the simple regression of the centered $$y$$ on the centered $$x$$. Or, as you put it, running one regression per state, and then average all state-specific estimates.

Now, let's go back to your specification: $$y_{ct} = \beta x_{ct} + \lambda_{c} + \mu_{st} + \eta_{ct}$$

We are not only using the within-state variability in this case. For instance, if $$x$$ was varying across counties but constant over time, we could not identify $$\beta$$. The presence of $$\lambda_c$$ means that we are identifying $$\beta$$ from the variability of $$x$$ over time and across counties within states.

This is a very flexible specification because it allows the different states to have arbitrarily different time evolutions in $$y$$, and counties to have arbitrarily different levels of $$y$$. For this reason, it is also quite demanding on the data, and you end up throwing away a lot of the variability in $$x$$, leading to less precise estimators.

Note that differentiating this equation over time leads to: $$\Delta y_{ct} = \beta \Delta x_{ct} + \Delta \mu_{st} + \Delta \eta_{ct}$$ with $$\Delta X_t$$ defined as $$X_t - X_{t-1}$$. In differences, this model is therefore close to the first one: we rely on the within-state cross-county variability of the time difference of $$x$$.

• Thank you! I think this is making a lot of sense to me, particularly thinking of the logic in the differenced specification Jul 24 '20 at 19:57