I have a sample of 3000 observations. I want to study the impact of covariates on a binary dependent variable (i.e. two categories: "yes" or "no") using either a logistic regression or a linear probability model. The sample is imbalanced (95% "yes", 5% "no"). The two models have pretty bad predictions (i.e. they do not predict the "no's" at all) and very low R2 (or pseudo R2 for the logistic regression).
I understand that wrong predictions can be due to the imbalanced sample, but what about the low R2/pseudo R2, can it also be due to the imbalanced sample?
What are the advantages/disavantages of using one of these two models when dealing with rare events? What could I do to have better predictions (or maybe measure predictions of these models more accurately?