# Heteroskedasticity-consistent Standard Errors for Difference Between Two Populations?

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?

• Not sure if the lack of answers/comments is because I have asked a very dumb question, or because it is not easy. As I keep trying to read more on this, it is looking more and more like the standard thing to do is just run a regression on the indicator variables. The the coefficient on the indicator describes the difference in population averages. – ely Apr 12 '12 at 19:23

Basically, both of your approaches are OK for the problem at hand.

On (1), you should look at the appropriate matrix algebra expressions and convince yourself that the coefficient of the dummy variable is indeed the difference between the two group means.

On (2), it has as much validity as taking the mean, in the first place. If the mean difference is in any way interpretable, so will be the permutation distribution. Typically, you get the $p$-value directly from the permutation distribution, rather than the standard error: the former is a consistent non-parametric procedure, while the latter would inquire some sort of asymptotic normality argument.

Angrist and Pischke are your two best friends here. They treat (1) at great lengths, although I don't think they appreciate permutation that much, so you would also want to make friends with Good.

You would also want to consider (3) $t$-test with Satterthwaite approximation for the degrees of freedom (default in Stata, BTW).

• Thank you for the answer. I should mention that I'm pretty familiar with the methods you mention, especially the Satterthwaite degrees of freedom calculation. We actually had to implement all of this ourselves during the first week of the course and all outside libraries were disallowed, to force us to understand the different things going on for different standard error calculation. It really was just trying to understand why the indicator variable coefficient gives you the mean difference that stumped me here. – ely Apr 12 '12 at 20:25
• I do have one follow-up question. In approach (2), will the standard error of the difference-in-population-averages represent the heteroskedasticity-consistent estimate (because you're not making any assumptions about the noise terms, you're just simulating), or not? In general, I find it difficult to understand the implications of assumptions like that (heteroskedasticity) as soon as I move away from the comfortable basic regression models. I generally favor Bayesian methods, and so it has been driving me nuts to figure out how you can see these same assumptions in simulational approaches. – ely Apr 12 '12 at 20:27
• I see. If you are a Bayesian, the only way for you to deal with heteroskedasticity is by modeling it. Permutation methods are at the heart of frequentist inference, so you may never be able to get them, as well as most other heteroskedasticity-robust techniques. In permutation, like in finite population sampling, the random variables are not the values of the response variables, but of the indicators, so this is a finite probability space from which you sample your particular allocation of 0s and 1s (with uniform measure on that space), and sampling distributions are defined over this space. – StasK Apr 12 '12 at 20:40