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Heckman selection correction

Heckman selection correction models outcome observation and a continuous outcome jointly. Learn the inverse Mills ratio, sign conventions, identification requirements and two-step inference through an exact reporting example.

Heckman selection correction is a model-based method for estimating an outcome regression when the outcome is observed only for a selected subset and unobserved influences on selection are correlated with the outcome error. It models both observation and the outcome, then corrects the selected sample's conditional mean. In quantitative research, this matters when a database contains voluntarily reported performance: a regression among reporters may describe a different relationship from the one across all eligible managers.
The method addresses a particular form of selection bias. Its credibility comes from assumptions about the observation mechanism, rather than from adding a correction variable to an arbitrary incomplete dataset.

Two equations, two different unobserved quantities

Let Yi∗Y_i^* be a continuous outcome that exists for every unit in the target population. Write

Yi∗=xi′β+εi,Si=1{zi′γ+ui>0}.Y_i^*=\mathbf{x}_i'\boldsymbol\beta+\varepsilon_i, \qquad S_i=\mathbf{1}_{\{\mathbf{z}_i'\boldsymbol\gamma+u_i>0\}}.

We observe Yi=Yi∗Y_i=Y_i^* only when Si=1S_i=1. A missing outcome for Si=0S_i=0 is not a zero. Selection indicators and selection covariates zi\mathbf z_i must be available for both groups; outcome regressors xi\mathbf x_i are needed for the selected observations used in estimation.

In the classical model, (εi,ui)(\varepsilon_i,u_i) has a centered bivariate normal distribution, independent of the covariates, with
Var(εi)=σ2,Var(ui)=1,Cov(εi,ui)=ρσ.\text{Var}(\varepsilon_i)=\sigma^2,\qquad \text{Var}(u_i)=1,\qquad \text{Cov}(\varepsilon_i,u_i)=\rho\sigma.
The unit selection-error variance fixes the scale of a latent index whose sign alone is observed. Constant σ\sigma and ρ\rho, the linear outcome equation and the normal joint error law are substantive restrictions. Separate normal marginal distributions would not establish joint normality. The basic cross-sectional estimator also assumes independent sampling. These are the model assumptions, not properties demonstrated by a normal-looking histogram of reported outcomes. Stata's official explanation gives the joint-normal selection setup.

Set ai=zi′γa_i=\mathbf z_i'\boldsymbol\gamma, and let ϕ\phi and Φ\Phi denote the standard normal density and cumulative distribution function. Then

P(Si=1∣xi,zi)=Φ(ai),λ(ai)=ϕ(ai)Φ(ai).\mathbb P(S_i=1\mid\mathbf x_i,\mathbf z_i)=\Phi(a_i), \qquad \lambda(a_i)=\frac{\phi(a_i)}{\Phi(a_i)}.
Joint normality gives E[εi∣ui]=ρσui\mathbb E[\varepsilon_i\mid u_i]=\rho\sigma u_i. Averaging over the retained tail ui>−aiu_i>-a_i therefore yields the conditional expectation
E[Yi∗∣xi,zi,Si=1]=xi′β+ρσλ(ai).\mathbb E[Y_i^*\mid\mathbf x_i,\mathbf z_i,S_i=1] =\mathbf x_i'\boldsymbol\beta+\rho\sigma\lambda(a_i).
The inverse Mills ratio λ\lambda captures the selection-related mean shift. Omitting it can distort an ordinary least squares regression; which slopes change depends on how this omitted term varies with the included regressors.
The sign follows the selection rule. Our rule retains ui>−aiu_i>-a_i, so λ(ai)>0\lambda(a_i)>0 and the shift has the sign of ρ\rho. Writing the threshold as c=−aic=-a_i gives the equivalent expression ϕ(c)/(1−Φ(c))\phi(c)/(1-\Phi(c)). For the other group, the conditional error mean is −ρσϕ(ai)/(1−Φ(ai))-\rho\sigma\phi(a_i)/(1-\Phi(a_i)). Reversing the selection indicator changes the index and associated correlation convention. A memorized ratio without its inequality can give the wrong correction. The Stata FAQ explains these conventions.

A reported mean of 10.798 can coexist with a population mean of 10

Consider a fictional manager score at one fixed covariate profile. Stipulate an all-candidate mean of 1010, outcome-error standard deviation σ=2\sigma=2, selection index a=0a=0, and error correlation ρ=0.5\rho=0.5. These are chosen model inputs, not fitted manager data.

Half the candidates report because Φ(0)=0.5\Phi(0)=0.5. Among reporters,

λ(0)=1/2π1/2=2π≈0.79788456,\lambda(0)=\frac{1/\sqrt{2\pi}}{1/2} =\sqrt{\frac2\pi}\approx0.79788456,

so

E[ε∣S=1]=(0.5)(2)2π≈0.79788456,E[Y∗∣S=1]≈10.79788456.\mathbb E[\varepsilon\mid S=1] =(0.5)(2)\sqrt{\frac2\pi}\approx0.79788456, \qquad \mathbb E[Y^*\mid S=1]\approx10.79788456.

The mean of 1010 is unconditional on reporting, while holding the covariates fixed. Reporters have a higher expected error because the reporting error and score error are positively correlated. This does not establish manager skill or a causal benefit from reporting. Nor does it demonstrate slope bias: this single-profile example isolates a mean shift.

Exact fictional outcome densities at fixed covariates. All candidates have mean score 10 and outcome-error standard deviation 2. With latent selection index 0 and error correlation 0.5, half report and the reported score mean is 10.798. Both outcome densities extend below and above score 10.
Exact fictional outcome densities at fixed covariates. All candidates have mean score 10 and outcome-error standard deviation 2. With latent selection index 0 and error correlation 0.5, half report and the reported score mean is 10.798. Both outcome densities extend below and above score 10.
Original exact normal-model calculation, not a simulation or observed performance. Selection acts on a separate latent error, so it does not create a hard cutoff in the outcome score.

Estimation needs information about non-reporters

The familiar two-step procedure is:

  1. Fit a probit selection model for SiS_i using both observed and unobserved-outcome units. Compute λ^i=ϕ(zi′γ^)/Φ(zi′γ^)\widehat\lambda_i=\phi(\mathbf z_i'\widehat{\boldsymbol\gamma})/\Phi(\mathbf z_i'\widehat{\boldsymbol\gamma}).
  2. Among Si=1S_i=1 units, regress YiY_i on xi\mathbf x_i and λ^i\widehat\lambda_i. Its additional coefficient estimates δ=ρσ\delta=\rho\sigma, not ρ\rho by itself.
If the archive lists only reporters, its contents alone cannot supply the first-stage reporting denominator. A defined eligible population and selection data are needed. The two-step construction and its covariance formula appear in Stata's methods documentation.
The second-stage disturbance has selection-dependent variance, and λ^i\widehat\lambda_i is an estimated regressor. Ordinary second-stage standard errors ignore first-stage uncertainty. Use inference derived for the two-step estimator, or a justified bootstrap that refits both stages and respects the sampling structure. Applying heteroscedasticity-robust errors only to the final regression does not generally account for the generated regressor. Full maximum likelihood is another estimation route under the joint model; it does not remove its assumptions.

Identification is more than fitting a probit

A defensible exclusion restriction provides a variable in z\mathbf z that changes observation probability but has no direct effect on the outcome conditional on x\mathbf x, and is exogenous to the outcome error. For a reporting study, a randomly assigned administrative reminder might be a candidate. It would still require evidence that it changes reporting and cannot alter the measured performance. A variable is not valid merely because leaving it out makes the calculation work.
Under the normal model, nonlinearity of λ\lambda can supply identification without an excluded variable when the covariate support provides sufficient variation. This is functional-form identification, not a general guarantee: constant or insufficiently varied selection indices can leave the augmented regression unidentified. Over a restricted index range, λ\lambda can be nearly linear in the outcome regressors, creating multicollinearity, weak separation of selection from the outcome relationship, and unstable estimates. A weak excluded predictor can leave the same practical difficulty. Honoré and Hu's author manuscript explains the dependence of identification on distributional and exclusion assumptions.
Correlated errors allow missingness not at random: after conditioning on observed covariates, whether the outcome is seen can still depend on its unobserved component. The normal selection model specifies one such mechanism; the observed data do not validate every extrapolation to missing outcomes. With ρ=0\rho=0, this model's conditional mean shift disappears, but an insignificant estimate does not prove every selection problem absent.
A Tobit model instead censors an outcome at a known boundary under its own structure. Reporting selection, omitted confounders, failed strategies omitted from a search log and unavailable delisting returns need their own data and assumptions. Heckman correction is not a universal repair for these problems, nor does it confer a causal interpretation on β\boldsymbol\beta.
For the regression foundation, Regression Assumptions and Diagnostics explains why a failure of the conditional mean requires a different response from a failure of the error covariance. The public page is a preview; the full lesson requires membership.

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