Tools for intuition about sample selection bias and its correction

We provide mathematical tools to assist intuition about selection bias in concrete empirical analyses. These new tools do not offer a general solution to the selection bins problem; no method now does that. Rather, the techniques we present offer a new decomposition of selection bias. This decomposition permits an analyst to develop intuition and make reasoned judgments about the sources, severity, and direction of sample selection bins in a particular analysis. When combined with simulation results, also presented in this paper our decomposition of bias also permits a reasoned empirically-informed judgment of when the well-known two-step estimator of Heckman (1976, 1979) is likely to increase or decrease the accuracy of regression coefficient estimates. We also use simulations to confirm mathematical derivations.