# How can I find out if effects vary by race if I am including family-level fixed effects?

I don't have a working example, because this question is more conceptual. Let's say I'm running a linear regression using the plm package in R on the relationship between graduating from college and getting lunch-subsidies as a child.

I have panel data that includes observations from individuals that may be in the same family, so I add family-level fixed effects to allow for arbitrary correlations within the family.

If I were to add race covariates, however, R would omit them because of multicollinearity in the family level.

How would I test the hypothesis, then, that blacks have differential effects than whites, but still resolving the issues solved by fixed effects?

Let's say I have the following regression, attempting to test for the hypothesis that different races and sexes are helped differentially from a free lunch program insofar as it relates to graduating college:

plm(graduate_college ~ free_lunch + black + male + hispanic + data=data, index=c("mother_id"), model="within")


Mother_ID just tracks siblings from the same mother. Now, if I want to test the hypothesis that blacks have different effects than whites as it relates to the effect of free lunch on graduating college, how would I test this? My guess is to remove fixed effects, add clustered standard errors at the mother_id level and add an interaction term for black*free_lunch?

• Can you show your model in standard mixed effects notation? The family level fixed effect seems odd to me. Let's say you have 100,000 observations. If you have 3 people in family in average, then it's 30,000 fixed effects. This sounds a little bit crazy, unless I;m missing something – Aksakal Apr 4 '16 at 17:41
• Just to be clear, you're nesting individuals within families, correct? Also, you've tried fitting race into your model as a main effect? R returns it as a linear combination (collinear) with the "family" level? That doesn't make sense to me if it's a simple main effect. It would make more sense to me if you were to take the interaction of race and lunch subsidies. – Mike Hunter Apr 4 '16 at 17:45
• Also, what's your hypothesis? That somehow some races are not being helped by free lunches? I'm just curious – Aksakal Apr 4 '16 at 17:47
• I tried to answer these questions, @Aksakal To be clear: I'm just learning this stuff. – Parseltongue Apr 4 '16 at 18:26
• Let's say you make this regression work, what would this mean though? Are you trying to see whether subsidized lunch help higher education? Would it say that it's not income that's preventing some races getting higher education? It's not clear what's the research question here, – Aksakal Apr 4 '16 at 18:29