Working example: Assume I have standard panel data with observations on children with several time points, and I want to determine the effect of the social composition of the child's school on its educational success.
Children on 'worse' schools have also worse educational success. But obviously, I am stuck in the classic counterfactual problem. I have only observational data and I cannot tell, how much of the differences are determined by the families instead of the schools (since the family background pretty much determines the school and has also a strong impact on the educational sucess).
Since, I have panel data, I can apply fixed effect models to eliminate the efffect of families. However, this assumes, that the family background is time-invariant, which is a strong assumption. If I want to take this into account, I would need to apply the random effects approach. However, I cannot take into account all individual time-constant heterogeneity (i.e. individual ability and unobserved familiy characteristics).
My question is, can I use a random effect model with random slopes for this? Or in other words: does a random slope random effects model combine advantages of fixed and random effects? The idea is that since the random slopes allow unexplained heterogeneity for each individual, the model should take into account both time-invariant effects and individual heterogeneity.
I think, I am missing something, because otherwise many more studies probably would apply random slope models instead of fixed effects.
Any answers and references are highly appreciated. Many thanks!