Compare regression coefficients across groups in panel data

#1
Dear TalkStats users,

I am wondering how to test for differences in regression coefficients across groups in panel data (after a fixed-effects regression).

Particularly, I can't think of a solution of how to construct interaction terms if the groups you are interested in are not the same than the groups that you set your fixed-effects at.

Example:

I have two types of survey respondents (2 groups): adults and children. They answer questions about different sorts of ice-cream (fixed effects at ice-cream level).

Dependent variable: Rating (how much do you like the ice-cream?)
Independent variables: Cost, Size

I could run the regressions separately on two datasets (one for adults, one for children), but how could I test for significant differences then?

If I aggregate the data of the two groups to one dataset (panel structure: one sort of ice-cream per day), I'm loosing the information about the groups (e.g. for vanilla on day 1, I aggregate the responses of two adults and three children)..

I'm really puzzled by this, and appreciate you help very much. Also, I'm happy to provide more details or clarity on the problem.


Thank you!