This is a homework question, so I am looking for help in getting the right idea so that I can execute the rest of the work on my own.
I have some data regarding military veteran status and log-earnings. One of the questions says the following:
Estimate the returns to veteran status (an indicator variable, 0-not veteran, 1-veteran) by taking the difference in average log-earnings for veterans and non-veterans. Calculate the standard error for this estimate, and describe how you calculated it. Calculate both the homoskedasticity based standard errors and the robust (that is, the heteroskedasticity-consistent) standard errors.
In our course, we only discussed the distinction of homoskedasticity/heteroskedasticity-consistent standard errors in the context of basic regression, about 10 weeks ago. The relevant material for this question is all about instrumental variables models and methods to test causal effectiveness of treatments.
So here are some very naive things I would normally try to compute a standard error for the effect of veteran status on log-wages:
(1) Perform a basic regression with a constant term and a coefficient on the indicator of veteran status. Then won't it be true that the coefficient on veteran status represents the "veteran status premium"? And so the basic formulas for the different standard errors for this estimated coefficient should apply? I'm not confident that this is correct. It's not clear to me how the coefficient would represent the difference in population averages for the veteran population and the non-veteran population.
(2) Pretend it is a randomized experiment. Draw permutations of the veteran-status vector over all of the individuals and pretend like their observed wages always represent their veteran status from the drawn permutation vector. Compute the average wage over the veteran population, minus the average wage over the non-veterans. Do this for many iterations to get a Monte Carlo average for this veteran-vs-non-veteran difference, and a standard error by virtue of the simulation. I'm extremely skeptical of this since there would be a lot of causal effects related to veteran status that one ignores by assuming randomization, and further, it's not clear at all how one could use this approach to get the heteroskedasticity-consistent estimate.
I don't have any way to check whether what I am doing is correct, so it's a bit like taking a stab in the dark. I could write a lot of code that performs the above operations, but how do I really know if it's addressing the question? Are there any good, readable references for the way to do regression on indicator variables, especially when the statistic of interest is the difference in averages over the populations where the indicator differs?