Endogenous Selection

Endogenous selection is the technical condition behind Sample Selection Bias: who enters the estimation sample is determined by a process whose unobservables are correlated with the outcome equation's unobservables. Formally, the selection indicator $s_i$ and the outcome error $u_i$ have $\text{cov}(\text{unobservables in selection eq}, u_i) \neq 0$.

When to use

Diagnose endogenous selection whenever the sample-inclusion decision is itself an economic choice (working vs not working, having children vs not, exiting the market vs surviving). Heckman tests for it via the significance of the Inverse Mills Ratio coefficient $\rho$; a significant $\rho$ confirms endogenous selection and the corrected estimate is preferred to naïve OLS.