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I am referring to this paper on school quality and returns to schooling. The econometric specification has been described by eqn (1) on page 4.

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The authors have taken "cohort specific fixed effect for each state of birth" and "cohort specific fixed effect for each state of residence". Am I correct in understanding this is is a form of interaction between cohort and state, which I can obtain by creating group identifiers based on cohort and state such that different combinations of values of cohort-state are generated?

If so, what situations would justify using such fixed effects instead of just using cohort fixed effects and state fixed effects separately?

Moreover, there is an interaction between years of education and returns to education, where returns to education consists of two components: cohort and state of birth effects, and cohort and region of residence effect. While I have gone through their rationale in that section, I'm still having a hard time understanding what exact purpose this form of specification serves.

Would appreciate it if someone takes some time to help resolve these doubts. Thanks!

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  • $\begingroup$ (1) Don't have enough time for a formal answer, but a few thoughts that might help (sorry if they don't!): Generally state-year FEs are used if you're concerned about idiosyncratic "noise." Just taking state and year FEs individually imposes more structure on the regression: you're assuming the effects of each state are relatively constant and the effects of each year are relatively uniform. Take Covid as an example--say you were trying to estimate some other effect and were worried about Covid as a lurking variable. If you added state and year fixed effects, you'd be assuming the impact of... $\endgroup$
    – AndrewC
    Aug 31 at 13:48
  • $\begingroup$ (2) being in any state was relatively constant across years, and the year effects were also relatively uniform across states. However, we observe some states that were hit really hard in 2020, and some that were hit much harder in 2021. State-year FEs would match that observed pattern (since they'd show things like being in NY in 2020 was really bad, being in FL was kind of bad in both 2020 and 2021, while being in Vermont wasn't too bad in either 2020 or 2021), where state and year FEs wouldn't (since those might show being in NY was "kind of bad" generally, averaging over 2020 and 2021)... $\endgroup$
    – AndrewC
    Aug 31 at 13:53
  • $\begingroup$ (3) I can't really defend the exact specification they use (again, not really enough time to look in more detail, sorry!) including these two terms separately was intended to account for the fact that returns to schooling might look "really good" if either: the schools were excellent (and therefore substantially increased skills), or the local labor market had very high returns for worker. If these two weren't included separately, we might see cases where there appears to be high returns to schooling purely because of macroeconomic effects, or low returns driven just by a poor labor market. $\endgroup$
    – AndrewC
    Aug 31 at 13:57
  • $\begingroup$ Thank you so much for explaining in such detail. Much appreciated $\endgroup$
    – Jan3
    Sep 8 at 13:14
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From a quick scan, I think what the authors really want to do is see if data they've collected on school quality by state and cohort (table 1) explains returns to education. The point of the interacted cohort/state dummies in equation (1) above is to generate average differences in returns to education by state/cohort (table 2), which they can then use as the dependent variable in their returns to education quality regressions (table 3). Equation (1) is estimated using individual-level census data, but only the state/cohort dummies are really used.

As I say,that's from a quick scan. I'm sure there's a good reason why they don't just put the quality data straight into eqn (1),but I can't think of it offhand.

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