Gaussian Sample-Selection Model and Its Assumptions

Why Model Selection and Outcomes Together?

When an outcome is recorded only for units that satisfy a selection rule, a regression using the observed outcomes describes a selected group. If the unobserved factors that influence selection are related to the unobserved factors that influence the outcome, the selected group can have systematically different outcome errors from the population. The Gaussian sample-selection model makes this dependence explicit by specifying a selection equation and an outcome equation together.

The model is not simply a probit regression followed by an unrelated linear regression. Its central distributional assumption concerns the joint behavior of the two equation errors. That assumption implies a particular form for the selection probability and explains when conditioning on being observed shifts the outcome error.

The Two Equations

Definition: For unit \(i\), let the latent selection index be \(S_i^*=z_i'\gamma+v_i\), and define the observed selection indicator by \(S_i=1\) if \(S_i^*>0\), and \(S_i=0\) otherwise. Let the latent outcome be \(Y_i^*=x_i'\beta+u_i\). The outcome is observed when \(S_i=1\); it is not observed when \(S_i=0\). Here \(z_i\) and \(x_i\) are observed covariate vectors, and \(\gamma\) and \(\beta\) are coefficient vectors.

The outcome equation describes the outcome that would apply to unit \(i\) whether or not that unit is selected. For a unit with \(S_i=0\), \(Y_i^*\) is therefore a latent or counterfactual outcome, not a recorded value. This distinction matters: the model aims to describe the outcome process in the population, even though the outcome data are available only for selected units.

The selection rule is a threshold rule. A positive latent index means selection; a nonpositive index means no selection. The zero threshold is a normalization: a different fixed threshold could be absorbed into the intercept of the selection equation.

Joint Normality and the Probit Selection Probability

The Gaussian specification assumes that, conditional on the observed covariates, the errors have a joint normal distribution:

\begin{pmatrix}u_i\\v_i\end{pmatrix}\,\bigg|\,x_i,z_i \sim N\left[ \begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix} \sigma^2 & \rho\sigma\\ \rho\sigma & 1 \end{pmatrix} \right], \qquad \sigma>0,\quad -1<\rho<1.

The variance of \(v_i\) is fixed at one. Only the sign of the latent selection index determines \(S_i\), so the scale of the selection equation cannot be separately identified from its coefficients and error standard deviation without a normalization. Setting that standard deviation to one gives the conventional probit scale. The outcome error has standard deviation \(\sigma\), and \(\rho\) is the correlation between the outcome and selection errors. In particular, their covariance is \(\rho\sigma\), not \(\rho\).

Because \(v_i\) is standard normal under this specification, the selection probability follows directly from the threshold rule:

P(S_i=1\mid x_i,z_i) =P(v_i>-z_i'\gamma\mid x_i,z_i) =\Phi(z_i'\gamma),

where \(\Phi\) is the standard normal cumulative distribution function. This probability depends on \(z_i\) and \(\gamma\), not directly on the outcome equation. The joint normality assumption connects the two equations through \(\rho\); it does not change the marginal probit form of the selection probability.

Worked Example: Calculating a Selection Probability

Suppose a hypothetical employment study uses the selection equation \(S_i^*=-0.4+0.08\,\text{experience}_i+v_i\), with \(v_i\sim N(0,1)\). A worker has 10 years of experience. Find the model-implied probability that this worker is selected into the observed-employment sample.

The selection index is \(z_i'\gamma=-0.4+0.08(10)=0.4\). The model therefore gives \(P(S_i=1\mid z_i)=\Phi(0.4)\approx0.6554\). Thus the probability is about 65.54%. This calculation uses the unit-variance normalization for \(v_i\) and the specified rule \(S_i=1\) when \(S_i^*>0\). It is a model-based probability, not a guarantee that a particular worker will be selected.

How Error Correlation Creates Selection Bias

The outcome error has mean zero in the population conditional on the covariates: \(E[u_i\mid x_i,z_i]=0\). But an outcome regression among selected units concerns \(E[u_i\mid S_i=1,x_i,z_i]\), which need not be zero. Selection restricts the distribution of \(v_i\) to values above the threshold \(-z_i'\gamma\). When \(u_i\) and \(v_i\) are correlated, that restriction also changes the distribution of \(u_i\).

The direction is intuitive under joint normality. Conditional on \(v_i\), the mean of \(u_i\) is \(\rho\sigma v_i\). Selection keeps units whose \(v_i\) is above a covariate-dependent cutoff. The selected units consequently tend to have outcome errors shifted upward when \(\rho>0\), and shifted downward when \(\rho<0\). The size of the shift depends on the selection threshold as well as on the error scale and correlation. It is not generally a constant intercept difference across covariate values.

This is selection bias in the outcome equation: among selected units, the residual may have a nonzero conditional mean and may vary systematically with the covariates that predict selection. A linear regression fit only to the observed outcomes can then combine the structural outcome relationship with this selection-related component. It should not automatically be interpreted as estimating \(\beta\).

Worked Example: Reading the Sign and Scale of Error Dependence

Suppose a hypothetical outcome is measured in thousands of dollars, with \(\sigma=10\), and the standardized selection error has unit variance. Let \(\rho=0.30\). Calculate the covariance between the outcome and selection errors and interpret its sign.

The covariance is \(\operatorname{Cov}(u_i,v_i\mid x_i,z_i)=\rho\sigma=0.30(10)=3.00\). Its units are thousands of dollars times the standardized selection-error unit. The positive covariance means that higher-than-usual values of the selection error tend to accompany higher-than-usual outcome errors. Since selection requires \(v_i>-z_i'\gamma\), selected units are drawn from the upper part of the selection-error distribution. The model therefore predicts an upward selection-related shift in their outcome errors, conditional on their observed covariates.

This calculation establishes the sign and covariance scale; it does not by itself give the conditional mean shift for a particular selection threshold. That shift depends on the threshold as well as on \(\sigma\) and \(\rho\).

A Useful Independence Result

The special case \(\rho=0\) clarifies the role of the joint-error assumption. Because a jointly normal pair with zero correlation is independent, \(u_i\) is independent of \(v_i\) conditional on \(x_i,z_i\). The selection indicator is determined by \(v_i\) and the observed covariates. Conditioning on \(S_i=1\) therefore does not change the conditional mean of \(u_i\):

\rho=0 \quad\Longrightarrow\quad E[u_i\mid S_i=1,x_i,z_i] =E[u_i\mid x_i,z_i] =0.

Consequently, in this special case the selected-sample conditional mean of the outcome remains \(x_i'\beta\), provided the outcome equation and other assumptions are correct. This is a statement about selection through the specified errors, conditional on the observed covariates; it does not imply that every possible source of sample or model bias has been ruled out.

Justification: Under joint normality and \(\rho=0\), the two errors are conditionally independent. For fixed \(x_i,z_i\), \(S_i\) is a function of \(v_i\) alone. Thus conditioning on \(S_i=1\) supplies information about \(v_i\), but not about the independent error \(u_i\). Its conditional expectation remains zero.

Assumptions Required for the Model

The equations are meaningful only with a clear observation rule and a set of distributional and specification assumptions. The standard Gaussian sample-selection model requires the following:

A well-defined latent outcome. \(Y_i^*\) is a meaningful outcome for selected and nonselected units, even though it is observed only when \(S_i=1\).
A correctly specified selection index and threshold. Selection occurs exactly when \(z_i'\gamma+v_i>0\), with the relevant observed predictors included in \(z_i\).
A correctly specified linear outcome conditional mean. The population outcome equation is \(Y_i^*=x_i'\beta+u_i\), with the intended covariates and functional form represented by \(x_i\).
Conditional zero means and exogeneity. Given the observed covariates, \(E[u_i\mid x_i,z_i]=E[v_i\mid x_i,z_i]=0\). The covariates are not carrying unmodeled conditional means in the errors.
Joint conditional normality with constant parameters. Given \(x_i,z_i\), the error pair has the stated bivariate normal distribution, with the selection variance normalized to one and \(\sigma\) and \(\rho\) common across units. This excludes unmodeled heteroskedasticity or a different dependence structure.
A valid observation pattern. The selection indicator and the covariates used in the equations are observed for the relevant sample, while the outcome is observed according to the stated selection rule. The missing outcomes are not being treated as observed values.
An appropriate sampling basis. The sampling process must be compatible with the model’s population interpretation; in the usual independent-observation version, units are sampled independently from the target population.

These assumptions do not all have the same status. The threshold and observation rule describe how outcomes become available. The linear and probit indices specify functional forms. Joint normality and constant \(\rho\) impose substantive restrictions on unobserved heterogeneity. A good fit to observed outcomes alone cannot establish that the unobserved errors are jointly normal.

The model specification also does not by itself guarantee that every coefficient can be credibly distinguished from the selection mechanism in a particular application. The covariates and the information in the data matter for that question. Here the model’s role is to state the joint selection and outcome structure precisely, not to claim that its assumptions are automatically satisfied.

Worked Example: Checking a Proposed Error Specification

A hypothetical analyst proposes \(\sigma=8\) outcome units and \(\rho=-0.25\), with selection-error variance normalized to one. Check the implied covariance matrix for validity and interpret the dependence.

The covariance is \(\rho\sigma=(-0.25)(8)=-2.00\). The proposed conditional covariance matrix is therefore \(\begin{pmatrix}64&-2\\-2&1\end{pmatrix}\), because the outcome-error variance is \(\sigma^2=8^2=64\).

Its determinant is \(64(1)-(-2)^2=60>0\), and its diagonal entries are positive, so it is positive definite and corresponds to an admissible nondegenerate bivariate normal distribution. The negative correlation means that higher selection errors tend to accompany lower outcome errors. Since selection retains units with selection errors above a cutoff, this dependence implies a downward selection-related shift in outcome errors among selected units, conditional on the covariates.

The matrix check only verifies that the chosen variances and covariance can form a valid joint distribution. It does not verify that the data-generating errors are actually normal, that the covariates are exogenous, or that the proposed equations match the application.

Common Mistake: Treating the Observed Sample as the Population

A common mistake is to reason, “The outcome equation has a zero-mean error, so its error must average to zero among the units whose outcomes are observed.” The first statement is a population conditional-mean assumption; the second conditions on the selection event. When \(\rho\ne0\), that event contains information about the outcome error, so the second statement does not follow. The correction is to distinguish \(E[u_i\mid x_i,z_i]=0\) from \(E[u_i\mid S_i=1,x_i,z_i]\), and to recognize that the latter can be nonzero.

A related mistake is to interpret \(\rho\) as the correlation between the observed outcome and the selection indicator. It is instead the correlation between the two unobserved equation errors. The observed indicator is a thresholded function of the selection index, so its relationship with the outcome is not summarized by simply reading \(\rho\) as an observed-variable correlation.

Key takeaway: The Gaussian sample-selection model combines a threshold-based probit selection equation with a linear latent outcome equation. Joint normality makes the error correlation precise: when it is nonzero, conditioning on selection generally changes the outcome-error distribution; when it is zero, the two errors are independent under the model.

Check Your Understanding

Use the equations and assumptions above to answer these questions.

Write the selection probability when the selection index is \(z_i'\gamma\), and state which variance normalization makes it a probit probability.
If \(\sigma=5\) and \(\rho=-0.4\), what is the covariance between \(u_i\) and \(v_i\)? Include its sign.
Why does zero correlation imply no selection-induced shift in the conditional mean of \(u_i\) under joint normality?
Explain why \(Y_i^*\) must be defined for units with \(S_i=0\), even though it is unobserved for them.
Does a positive \(\rho\) mean that the observed outcome and selection indicator have correlation \(\rho\)? Explain.