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?

  • $\begingroup$ 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? $\endgroup$
    – dimitriy
    Jul 13 '20 at 15:59
  • $\begingroup$ Sorry yes $\eta_c,t$ is the error term. I will edit that into the question $\endgroup$
    – Steve
    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 $x$.

  • 1
    $\begingroup$ Thank you! I think this is making a lot of sense to me, particularly thinking of the logic in the differenced specification $\endgroup$
    – Steve
    Jul 24 '20 at 19:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.