I'm interested in the way migrants vote compared to the rest of the population. To do so, I'd like to use a matching method (PSM, CEM..., with the treatment corresponding to the fact of being migrant or not). However, the fact that not all migrants have the right to vote may lead to a selection bias. Therefore I thought about using Heckman correction to take this selection bias into account. To do Heckman correction, I would estimate the probability that someone has the right to vote, and then match people so that their only difference is whether they are migrant or not, among all of those who can vote.
Is it possible to use Heckman correction before applying a matching method? Is it correct to include the Inverse Mills Ratio in the matching procedure like other covariates?
