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?
To answer the questions below:
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?