Heckman Selection Model

The Heckman selection model corrects Sample Selection Bias with a two-equation system. The selection equation is a probit for whether an observation enters the sample: $\Pr(s_i = 1) = \Phi(\mathbf{z}_i'\boldsymbol\gamma)$. The outcome equation is OLS on the selected sample plus a correction term: $y_i = \mathbf{x}_i'\boldsymbol\beta + \rho\lambda_i + u_i$, where $\lambda_i$ is the Inverse Mills Ratio computed from the selection probit. If $\hat\rho$ is significant, selection bias is confirmed; the corrected $\hat{\boldsymbol\beta}$ is the unbiased outcome relationship.

When to use

Use Heckman whenever you suspect a non-random selection mechanism is shaping who appears in your data. The standard cases: Mroz wage equation (selection = labour-force participation), PS_2's fertility regression (selection = having any children), studies of firm exit / survival, returns to migration. Identification requires an exclusion restriction: at least one variable in the selection equation that is NOT in the outcome equation (PS_2 uses year.born). Without it, the model is identified only off the non-linearity of $\Phi$, which is fragile.