comparing regression across models with whole dataset and divided dataset I have a dataset of census tracts (say 4000 tracts). There are 2500 tract with high density. Now I can run regression on the whole dataset. Or I can also run regression on dense part or low dense part SEPARATELY. Now how can I figure out if this splitting (only high density or only low density) is yielding better models, or the previous model (with the whole dataset) is better? 
can I compare this with the likelihood ratio test? 
(same target variable, the same set of independent variable; nothing with train test things.)
 A: If i get you correct, you fit a regression and get a coefficient (which you refer to as parameter). You have done it for all observations and separately for two groups, and your question is whether you can compare whether fitting separately or together as one group, is a better model so as to say.
If you have a linear model already, let's say your dependent variable is y, independent is x, and the group is a categorical variable called G, the formula will be (below is what's normally used in R or python):
y ~ x + G + x:G

You can do an anova or t.test for the interaction term x:G, this will tell you whether the coefficient significantly differs between the two groups. I think this is more straight-forward.
You cannot really comment on whether one is overfitting or underfitting, because there is no prediction etc involved. You fit everything to the model. If the interaction is significant, you can say there is evidence to suggest the parameter x, is different between the two groups in G.
The "comparing likelihood" you are referring to is possible, so if would be a LRT test, something like:
m1: 1 paramter
m2: 2 paramter

LR = -2*(logLik(m2) - logLik(m1))

And you test it with a chi square, 1 degree of freedom. 
