I am interested in the impact of Unemployment (U) on Gross Domestic Product (GDP). I also want to check if the effect is different for developed countries and developing countries. I am using 10 developed countries and 10 developing countries data between 2000 and 2015 as the sample. I use the following fixed effects model.

GDPit=αi+β1GDPit-1+β2Uit+β3Dit+β4(Uit^Dit)+β3Xit+ϵi

where Dit is a dummy variable with value 1 for developed countries and 0 for developing countries and Xit is a set of controls.

Given that Dit is a time-invariant characteristic, my fixed effects variable omits this variable. Hence my new model is:

GDPit=αi+β1GDPit-1+β2Uit+β4(Uit^Dit)+β3Xit+ϵi

I am interpreting this model as follows. is the effect of unemployment on GDP. is the by how much the effect of unemployment on GDP increases when the country is developed. Is this correct?

Also since Dit is omitted due to being time-invariant my model omits developed country main effect. Is this a problem? Is including the interaction without the main effect an issue?