Lots of analysis in the US collapse CPS or Census data and run regressions on the group means. I wonder why not run the regression on the individual data? Absent measurement error, should we expect the regression on individual and aggregated data to be the same?
Here is a Stata example with random data where the regressions at the individual level or on the means seem to deliver different estimates.
Following BB King's comments, I changed the example to include a "shock" at the group level. In this case, it seems like the individual regression 1. and the collapsing on transformed data regression 3. do deliver similar results absent covariates. Is this a general result?
** make some random data set seed 35135 clear set obs 1000 gen state = ceil(_n/100) gen shock = runiformint(0,1000) bysort state (shock): replace shock = shock[_N] gen wage = abs(int(rnormal() * 1000)) gen age = floor(abs(rlogistic()) * 15 + abs(rlogistic() * 5)) ** 1. individual data gen age2 = age^2 gen ln_wage = ln(wage) reg ln_wage age age2 shock reg ln_wage age age2 shock i.state ** 2. aggregated data preserve collapse (mean) wage age shock, by(state) gen age2 = age^2 gen ln_wage = ln(wage) reg ln_wage age age2 shock restore ** 3. transform before collapsing collapse (mean) ln_wage age age2 shock, by(state) reg ln_wage age age2 shock