Sample Selection Model

Definition

The sample selection model (Heckman 1979) addresses bias arising when the outcome of interest is observed only for a non-random subsample of the population. Selection bias in ordinary least squares (OLS) occurs when unobserved determinants of the outcome are correlated with unobserved determinants of the selection process. The Heckman two-step estimator corrects for this by adding the inverse Mills ratio — estimated from a first-stage probit — as a regressor in the second-stage outcome equation.

The Problem

Consider a latent outcome $Y_i^* = \beta_0 + \beta_1 X_i + U_i$ and a selection indicator $I_i^* = \gamma_0 + \gamma_1 Z_i + V_i$ , with $I_i = 1$ if $I_i^* > 0$ (outcome is observed) and $I_i = 0$ otherwise. Within the selected subsample:

$E[Y_i \mid I_i = 1, X_i, Z_i] = \beta_0 + \beta_1 X_i + E[U_i \mid V_i > -\gamma_0 - \gamma_1 Z_i]$

If $U_i$ and $V_i$ are correlated ( $\rho \neq 0$ ), OLS on the selected subsample is inconsistent because the conditioning on $I_i = 1$ induces a non-zero expected value of $U_i$ . Standard examples: wages are only observed for employed workers (labor supply); test scores only for test-takers; disability insurance (DI) award outcomes only for applicants.

The Two-Step Estimator

Step 1: Estimate the selection equation via probit to obtain $\hat{\gamma}_0, \hat{\gamma}_1$ .

Step 2: Under bivariate normality of $(U_i, V_i)$ with correlation $\rho$ , the conditional expectation of $U_i$ given selection is $\rho\sigma \cdot \lambda(\hat{\gamma}_0 + \hat{\gamma}_1 Z_i)$ , where $\lambda(\cdot) = \phi(\cdot)/\Phi(\cdot)$ is the inverse Mills ratio (IMR). Add the estimated IMR as a regressor:

$Y_i = \beta_0 + \beta_1 X_i + \rho\sigma \cdot \lambda(\hat{\gamma}_0 + \hat{\gamma}_1 Z_i) + U_i^*$

OLS on this augmented equation gives consistent estimates for $\beta_1$ . The coefficient on $\lambda$ estimates $\rho\sigma$ and tests for selection ( $\rho \neq 0$ ).

The Exclusion Restriction Requirement

If the covariates $Z_i$ in the selection equation are identical to $X_i$ in the outcome equation, identification rests solely on the nonlinearity of the IMR. Because the IMR is approximately linear over most of its range, local identification becomes very weak without variation in $Z_i$ beyond $X_i$ . An exclusion restriction — a variable that affects selection but not the outcome — is therefore essential for robust identification. The exclusion restriction plays the same conceptual role as an instrument in Instrumental Variables estimation: it provides variation in the probability of selection that is independent of the outcome.

Key Assumptions and Critiques

Bivariate normality: The two-step form is derived under bivariate normality of $(U_i, V_i)$ . Results are sensitive to distributional misspecification; semiparametric alternatives (Powell 1987 censored least absolute deviations (LAD); Ahn and Powell 1993) are more robust but require larger samples.
Correct functional form in selection equation: Probit misspecification propagates into the IMR and biases the correction.
Correlation $\rho$ is hard to estimate precisely: The IMR collinearity with regressors when $Z_i = X_i$ inflates standard errors substantially; with a genuine exclusion restriction, precision improves but remains modest in typical applied samples.

Applications in This Wiki

Labor supply: The canonical application — wages observed only for employed workers; selection driven by wage–preference interactions. Blundell, Duncan & Meghir (1998) use tax reforms as the exclusion restriction.
DI application and outcomes: Outcomes (labor force participation, earnings post-DI) observed only for applicants and awardees — selection into the applicant pool is non-random relative to the full working-age population.
Disability and wage measurement: DI-induced selection out of the labor force means observed wages overrepresent healthier, more productive workers; cross-sectional wage regressions face selection bias.

Connection to IV and Potential Outcomes

The selection model can be recast in the Rubin Causal Model / Instrumental Variables framework: the exclusion restriction $Z_i$ is an instrument for selection $D_i$ . The Heckman two-step imposes a parametric functional form (bivariate normality) to recover $\beta_1$ — in contrast to the nonparametric IV/local average treatment effect (LATE) approach, which requires monotonicity but not distributional assumptions. The parametric assumptions enable identification without an exclusion restriction (via IMR nonlinearity) but create specification-sensitivity; the IV approach avoids distributional assumptions but requires a genuine exclusion restriction. Both are estimating something related to the average outcome conditional on selection.

Sources

Van der Klaauw 2014 — From Micro Data to Causality Forty Years of Empirical Labor Economics