In what instances do we need to worry about Heckman correction for selection bias? I understand that Heckman selection models attempt to address selection bias by running two-part models. In the oft cited example of determinants of wage offers: 1) we have a group of people who work and do not work due to various reasons, 2) we only have wage offers for those who are working, and 3) there may be unobserved forces that drive people to work versus not work that may also impact wage offers.
My question is, what if we are only interested in the subgroup of people who are working? Does the potential for selection bias still apply since we are not interested in determinants of wage offers for both working and not working people? Why can't we just restrict to the population of working people, so long as this is the only group we are interested in?
 A: There are tons of math proof you can check out from the internet. I'm offering the intuition here to help out, as this question has bogged me down for quite a while when people first questioned my research (quite a while ago). 
The simple answer is, sadly, it DOES matter even if you are not interested in the factors that drive the subsample (that you are interested in) being selected into your analysis. For example, if you just restrict to working people, the estimate you get from the subsample will be biased due to the selection effect you fail to account for. 
Often, we are NOT interested in those factors driving your sample selection (they are omitted variables), but we still need to be interested in the conclusion being correct. Right? In other words, you don't want the pattern you observe from a sub-sample to be driven by the selection process. So, get used to this type of questions, and try to control the selection effect. And you can convince that your finding is a robust and convincing one.
