17.3 Sample selection
Wages are observed only for those who work, and those who work are not a random sample of the population: they are likely to be those with high wage offers. Heckman (1979) modelled this with two equations: \[ \text {selection: } s_i^* = z_i'\gamma + v_i, \quad s_i = 1 \text { if } s_i^* > 0; \qquad \text {outcome: } w_i = x_i'\beta + u_i, \text { observed if } s_i = 1, \] with \((v_i, u_i)\) bivariate normal, \(\Var (v_i) = 1\), \(\Var (u_i) = \sigma _u^2\) and \(\Corr (u_i, v_i) = \rho \).
Theorem 17.3.1 (Heckman). For the selected sample, \[ E(w_i \mid x_i, z_i, s_i = 1) = x_i'\beta + \rho \sigma _u\lambda (z_i'\gamma ) . \] Least squares of \(w_i\) on \(x_i\) in the selected sample is therefore inconsistent unless \(\rho = 0\). The two-step estimator first estimates \(\gamma \) by probit on the full sample, then regresses \(w_i\) on \(x_i\) and \(\hat \lambda _i = \lambda (z_i'\hat \gamma )\) in the selected sample; it is consistent for \(\beta \) and for \(\rho \sigma _u\).
Proof. For a bivariate normal pair, \(E(u_i \mid v_i) = \rho \sigma _uv_i\), since the regression coefficient of \(u\) on \(v\) is \(\Cov (u, v)/\Var (v) = \rho \sigma _u\). Hence \(E(u_i \mid s_i = 1) = E(u_i \mid v_i > -z_i'\gamma ) = \rho \sigma _uE(v_i \mid v_i > -z_i'\gamma ) = \rho \sigma _u\lambda (z_i'\gamma )\), by the calculation in the proof of Theorem 17.2.1(i) with \(\sigma = 1\). Adding \(x_i'\beta \) gives the conditional mean. In the second step, \(\hat \lambda _i\) converges to \(\lambda _i\), and the regression is a correctly specified linear regression with the extra regressor \(\lambda _i\). □
The second-step standard errors must be corrected, because \(\hat \lambda _i\) is estimated and the disturbance of the second step is heteroscedastic. Identification deserves care. If \(z_i = x_i\), the model is identified only by the non-linearity of \(\lambda \), which is nearly linear over much of its range, so \(\hat \lambda _i\) is highly collinear with \(x_i\) and the estimates are imprecise and fragile. A credible exclusion restriction, a variable that affects participation but not wages (such as the number of young children, or non-labour income), is needed in practice. The test of \(\rho \sigma _u = 0\), the \(t\) test on \(\hat \lambda _i\), is a test for selection bias.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.