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

For movers (D=1$D=1$) from a state and stayers (D=0$D=0$), it is of interest to know how they differ in their distributions of a response variable Y*$Y^{\ast }$ given some observed regressors Z$Z$ (e.g., wage Y*$Y^{\ast }$ given education Z$Z$), which is equivalent to their quantile differences in Y*$Y^{\ast }$ due to unobservables (e.g., ability) given Z$Z$. In many mover/stayer cases, however, Y*$Y^{\ast }$ is observed only for the movers, enabling the identification of only quantiles of the movers; for example, moving is migration and Y*$Y^{\ast }$ 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 Y*$Y^{\ast }$ 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 D$D$ and Y*$Y^{\ast }$ equations. An empirical example demonstrates these points.