Finding mover–stayer quantile difference due to unobservables using quantile selection corrections

For movers ((Formula presented.)) from a state and stayers ((Formula presented.)), it is of interest to know how they differ in their distributions of a response variable (Formula presented.) given some observed regressors (Formula presented.) (e.g., wage (Formula presented.) given education (Formula presented.)), which is equivalent to their quantile differences in (Formula presented.) due to unobservables (e.g., ability) given (Formula presented.). In many mover/stayer cases, however, (Formula presented.) is observed only for the movers, enabling the identification of only quantiles of the movers; for example, moving is migration and (Formula presented.) is the wage in the host country. For this, we propose a practical parametric way to find quantile differences between movers and stayers despite unobserved (Formula presented.) for the stayers. Our approach has the advantage comparing the movers to the stayers, whereas the existing semiparametric alternatives compare the movers only to the population to dilute the mover–stayer differences. Also, our approach easily allows heteroskedasticity and heterocorrelation in the (Formula presented.) and (Formula presented.) equations. An empirical example demonstrates these points.