Population Means and Predictions After Selection Correction

Why the Target Population Matters

A selection-corrected regression can produce more than one sensible prediction for the same covariate values. The prediction for an outcome among units known to be selected is not generally the same as the mean outcome for all units in the source population. Selection changes which units appear in the outcome sample, and, when selection and outcome errors are related, it also changes the average outcome error among selected units.

The correction therefore depends on the question being asked. A conditional prediction for a selected unit uses that unit’s selection information. A mean for a population averages predictions over that population’s covariate distribution. A mean outcome for nonselected units is different again: those outcomes are unobserved and their predictions rely on the selection model’s assumptions.

Model and Conditional Predictions

Let \(D_i=1\) indicate that unit \(i\) is selected and its outcome is observed. The latent outcome is \(y_i^*\); it is observed as \(y_i=y_i^*\) when \(D_i=1\). Write the outcome and selection equations as follows:

Here, \(x_i\) and \(z_i\) are observed covariate vectors, and \(\beta\) and \(\gamma\) are parameter vectors. The selection index is \(\eta_i=z_i'\gamma\). Use the probit normalization \(\operatorname{Var}(u_i)=1\). The Gaussian sample-selection model assumes that, conditional on the covariates, \((\varepsilon_i,u_i)\) has a bivariate normal distribution with means zero, variances \(\sigma_\varepsilon^2\) and 1, and covariance \(\rho\sigma_\varepsilon\). Covariates are treated as exogenous for these conditional-mean statements. The model also requires adequate identification and support for estimating its parameters.

Let \(\phi\) and \(\Phi\) denote the standard normal density and distribution functions. The probability of selection at a given \(z_i\) is \(\Phi(\eta_i)\). Define the inverse Mills ratio for selected units as \(\lambda(\eta_i)=\phi(\eta_i)/\Phi(\eta_i)\).

This conditional expectation is a fitted mean, not a guarantee about an individual’s realized outcome. Even with known parameters, a realized outcome can differ because \(\varepsilon_i\) varies among units with the same covariates.

Why the Selection Correction Appears

The correction follows from truncating the selection error. Given \(D_i=1\), the event \(u_i>-\eta_i\) has occurred. In a bivariate normal distribution, the conditional expectation of the outcome error given the selection error is \(E[\varepsilon_i\mid u_i]=\rho\sigma_\varepsilon u_i\). The mean of a standard normal variable truncated below at \(-\eta_i\) is \(\phi(\eta_i)/\Phi(\eta_i)\). Therefore:

Adding this conditional error mean to \(x_i'\beta\) gives the selected-unit prediction. When \(\rho=0\), selection does not shift the conditional mean of the outcome error, and the correction is zero. When \(\rho\ne 0\), the direction and size of the shift depend on both the error dependence and the selection index.

Means for Different Populations

A population mean requires a target population and its covariate distribution. For the source population, before conditioning on selection, the conditional mean at given covariates is \(E[y_i^*\mid x_i,z_i]=x_i'\beta\), since \(E[\varepsilon_i\mid x_i,z_i]=0\). If the source population has covariate distribution \(F_{XZ}\), its mean outcome is:

For a finite target set of \(N\) source-population units with covariates \((x_i,z_i)\), the corresponding model-based average is \(N^{-1}\sum_{i=1}^N x_i'\hat\beta\). Averaging over the source covariates is essential: the average \(x_i'\hat\beta\) among selected observations need not represent the source population.

For the selected population, the relevant target is \(E[y_i^*\mid D_i=1]\). Averaging the selected conditional mean over the selected covariate distribution gives:

The weighting follows because units with covariates \(x_i,z_i\) enter the selected population with probability \(\Phi(\eta_i)\). Equivalently, if a target source-population sample supplies covariates for every unit, weight each unit’s probability of selection and divide by the sum of those probabilities. If the target is instead a sample already drawn from the selected population, average its selection-corrected predictions over that sample’s covariates.

Predictions for Outcomes Missing Because of Nonselection

The latent outcomes of nonselected units are not observed, but the same Gaussian model implies a conditional mean for them. Conditional on \(D_i=0\), the selection event is \(u_i\leq-\eta_i\). The mean of a standard normal variable truncated above at \(-\eta_i\) is \(-\phi(\eta_i)/\Phi(-\eta_i)\). Thus:

This is a model-based prediction for an unobserved outcome, not an observed response and not a direct estimate from outcome data on that unit. Its credibility depends on the selection-model assumptions, including the specified dependence and error distribution. To estimate the average latent outcome among all nonselected units, average these predictions over the nonselected covariate distribution. In a source-population sample with known covariates, that distribution is represented by weights proportional to \(1-\Phi(\eta_i)=\Phi(-\eta_i)\).

Estimation and a Common Pitfall

In practice, replace unknown parameters by estimates from the selection-correction procedure, and calculate the index and correction for the covariates in the chosen target population. The target data must contain the covariates needed for the prediction; it need not contain outcomes for every unit. If the target covariate distribution is represented by a sample, averaging produces a plug-in estimate of the corresponding population mean. Inference for such an estimate must account for estimation of the selection and outcome parameters as well as the averaging step.

• Wrong: “The selected-sample average outcome is the estimated source-population mean because the regression is selection-corrected.”
• Correction: Selection correction provides conditional means under the model. The source-population mean must be averaged over source-population covariates as \(x_i'\hat\beta\); the selected-population mean instead averages corrected conditional means over the selected covariate distribution.
• Wrong: “The predicted outcome for a nonselected unit is its observed outcome, recovered by the correction.”
• Correction: That outcome remains unobserved. The model supplies a conditional expectation under its assumptions, not a recovered realization.