Why Exclusions Matter
In a sample-selection model, the outcome is observed only for units whose selection indicator equals one. If selection is related to the unobserved determinants of the outcome, the mean outcome among observed units need not equal the population outcome equation. The Heckman two-step procedure represents this selection-related difference with an inverse Mills ratio. For that correction to be distinguished from the ordinary outcome regressors, it is useful to have a variable that shifts selection while leaving the outcome equation unchanged.
Such a variable is an exclusion restriction. Its value is not established just because it predicts selection, or because it was omitted from an outcome regression. The proposed variable must be relevant for selection, vary in a useful way after conditioning on the included covariates, and have a defensible substantive reason for not entering the outcome equation.
The Model and the Exclusion Restriction
Let \(Y_i^*\) be the latent outcome for unit \(i\), and let \(S_i\) indicate whether that outcome is observed. Write \(X_i\) for the included outcome covariates and \(W_i\) for variables proposed for exclusion. The selection covariate vector is \(Z_i=(X_i,W_i)\). A standard Gaussian sample-selection model is
where \(Y_i=Y_i^*\) is observed when \(S_i=1\). Conditional on the covariates, the errors have mean zero and are jointly normal, with \(\operatorname{Var}(u_i)=1\), \(\operatorname{Var}(\varepsilon_i)=\sigma^2\), and \(\operatorname{Cov}(\varepsilon_i,u_i)=\rho\sigma\). The covariates are exogenous to these errors, the selection index is correctly specified, and the outcome equation has the stated linear form. The scale of \(u_i\) is normalized to one because only the sign of the selection index determines selection.
The exclusion restriction is that \(W_i\) enters the selection index but not the latent outcome equation. This requires more than a choice of notation: conditional on \(X_i\), changes in \(W_i\) must not directly change \(Y_i^*\), and \(W_i\) must satisfy the model’s exogeneity assumptions. The exclusion does not require \(W_i\) to be unrelated to selection. On the contrary, it must help predict selection for the correction to gain useful variation.
Under the model, the probability of selection given \(Z_i\) is \(\Phi(Z_i'\gamma)\), where \(\Phi\) is the standard normal cumulative distribution function. Define the inverse Mills ratio for selection as \(\lambda(a)=\phi(a)/\Phi(a)\), where \(\phi\) is the standard normal density. The conditional mean of the observed outcome is then
The coefficient \(\rho\sigma\) is the selection correction’s scale. If \(\rho=0\), this conditional-mean correction is zero; if \(\rho\ne0\), it changes with the selection index. The two-step estimator first estimates the selection index, constructs \(\hat{\lambda}_i\), and then includes it alongside \(X_i\) in an outcome regression for selected observations. In that second step, the correction is distinguishable from the outcome coefficients only if it contributes variation not already spanned by the outcome regressors.
How an Exclusion Helps Identify the Correction
The key idea can be seen by comparing two units with the same \(X_i\) but different \(W_i\). Their outcome-regressor component \(X_i'\beta\) is the same. If the exclusion shifts the selection index, it can change \(\lambda(Z_i'\gamma)\); the difference in their conditional outcome means then contains information about \(\rho\sigma\).
This comparison is informative when the bracketed difference is nonzero and the exclusion is valid. In a sample, a corresponding design condition is that the correction variable is not perfectly explained by the outcome regressors among selected observations. More generally, the selected-sample regressor matrix with columns \(X_i\) and \(\hat{\lambda}_i\) must have full column rank. If the correction is nearly a linear combination of \(X_i\), the model may be identified in principle but its estimates can be imprecise.
Worked Example: A Selection Shift at Fixed Outcome Covariates
Suppose a hypothetical fitted selection index is \(Z_i'\gamma=-0.5+W_i\), with \(X_i\) held fixed. Compare \(W_i=0\) and \(W_i=1\). For illustration, take the selection-correction coefficient \(\rho\sigma\) to be 2 outcome units and the common outcome component \(X_i'\beta\) to be 20 outcome units.
At \(W_i=0\), the index is \(-0.5\), so the selection probability is \(\Phi(-0.5)=0.3085\). Since \(\phi(-0.5)=0.3521\), the inverse Mills ratio is
At \(W_i=1\), the index is \(0.5\), and \(\Phi(0.5)=0.6915\). Thus
The model-implied conditional means are \(20+2(1.141)=22.282\) outcome units when \(W_i=0\), and \(20+2(0.5092)=21.0184\) outcome units when \(W_i=1\). Their difference is \(22.282-21.0184=1.2636\) outcome units. This difference comes from the correction, not from a change in \(X_i'\beta\). The example shows the identifying comparison; it does not establish that any real variable is a valid exclusion.
Relevance, Conditional Variation, and Support
Three related questions help assess whether an exclusion can do useful identification work. They concern different properties and should not be collapsed into a single test of whether a coefficient is statistically significant.
Does \(W_i\) help shift the selection index, conditional on \(X_i\) and the other selection covariates? If its selection coefficient is zero, it cannot generate index changes through the proposed exclusion.
Does \(W_i\) vary among units with comparable \(X_i\)? Variation that is entirely predictable from \(X_i\) does not supply a clean within-\(X_i\) comparison. In the selected sample, does the resulting correction vary beyond the span of the outcome regressors?
Are there enough observations at the relevant covariate values, including selected observations, to estimate the comparison? Separately, is there a convincing reason why \(W_i\) affects selection but has no direct effect on the latent outcome?
Support is about where the data actually provide comparisons, not just whether a variable has different values somewhere in the full sample. A variable may vary in the full population but have little variation among selected units, or its different values may occur in covariate regions with few observations. Selection probabilities close to zero also mean that relatively few outcomes are observed in the corresponding region. Such patterns can make the correction hard to estimate precisely. Exact identification is a rank property; practical informativeness also depends on how much data support the relevant comparisons.
Worked Example: Checking Conditional Variation and the Correction Design
Consider a hypothetical selection index \(Z_i'\gamma=-1+0.5X_i+W_i\). The following table gives four covariate combinations and their index values. The last column reports the corresponding inverse Mills ratio, rounded to four significant figures.
| \(X_i\) | \(W_i\) | Index \(Z_i'\gamma\) | \(\lambda(Z_i'\gamma)\) |
|---|---|---|---|
| 0 | 0 | -1.0 | 1.525 |
| 0 | 1 | 0.0 | 0.7979 |
| 1 | 0 | -0.5 | 1.141 |
| 1 | 1 | 0.5 | 0.5092 |
For example, at \(X_i=0,W_i=0\), the selection probability is \(\Phi(-1)=0.1587\) and the correction is \(\lambda(-1)=\phi(-1)/\Phi(-1)=0.2420/0.1587=1.525\). At \(X_i=0,W_i=1\), the index is zero, so \(\lambda(0)=0.3989/0.5=0.7979\). Thus, at the same \(X_i=0\), changing \(W_i\) changes the correction by \(0.7979-1.525=-0.7271\). At \(X_i=1\), the corresponding change is \(0.5092-1.141=-0.6318\). This is conditional variation in the selection correction.
With the outcome regressors \(1\) and \(X_i\), the correction values are not a linear function of those two columns: they differ between \(W_i=0\) and \(W_i=1\) within each \(X_i\) value. Therefore, if the selected sample contains observations from these combinations, the three columns \(1,X_i,\lambda(Z_i'\gamma)\) can have full rank. But the cell with index \(-1\) has a selection probability of only 0.1587, so selected outcomes from that cell may be comparatively scarce. The table demonstrates a favorable design pattern; actual support and sample counts still need examination.
Substantive Credibility Is a Separate Requirement
An exclusion restriction is an economic or scientific claim about the outcome equation, not a conclusion delivered by a first-stage regression. A variable may be a strong predictor of selection and still affect the outcome directly. For example, for an employment-selection model with wages as the latent outcome, local childcare availability may predict employment. But it could also affect which jobs a person can take, their hours, or their bargaining position, each of which could alter wages. Whether the exclusion is defensible depends on the outcome definition, the covariates held fixed, and the institutional setting.
A careful justification asks what pathways connect the proposed variable to the latent outcome. It also considers whether omitted factors jointly affect that variable and the outcome disturbance, whether the variable is measured before selection, and whether the claimed omission remains plausible after conditioning on \(X_i\). No single statistical significance test can establish all these points. The strongest case combines a clear design argument with evidence that the variable shifts selection and has useful conditional support.
Worked Example: Relevance Evidence Is Not an Exclusion Test
Suppose a hypothetical probit selection equation for employment includes a standardized measure of local childcare-slot availability \(W_i\). Conditional on the included covariates, the estimated coefficient is \(0.30\), with standard error \(0.10\). The test statistic for a zero selection coefficient is
Using a standard normal reference distribution, the two-sided p-value is approximately \(2\{1-\Phi(3)\}=0.00270\). This is evidence against a zero coefficient in the specified selection equation. At an index of zero for otherwise fixed covariates, a one-unit increase in \(W_i\) changes the predicted selection probability from \(\Phi(0)=0.5000\) to \(\Phi(0.30)=0.6179\), an increase of \(0.1179\), or 11.79 percentage points.
These calculations support selection relevance on the probit scale and illustrate a potentially meaningful probability shift at that index. They do not show that childcare availability has no direct effect on wages. If, for instance, it changes access to higher-paying jobs, excluding it from the wage equation may be substantively implausible. The relevance calculation and the exclusion argument answer distinct questions; both must be considered before treating \(W_i\) as a valid identifying variable.
Common Mistake: Treating a Significant First Stage as Proof
A common mistake is to reason, “The proposed exclusion has a significant coefficient in the selection equation, so it is a valid instrument for the selection correction.” This establishes neither the outcome exclusion nor exogeneity. It also says little by itself about whether there is enough conditional variation and selected-sample support for precise estimation.
The correction is that relevance is necessary for the proposed design but not sufficient. State the substantive reason for excluding the variable from the outcome equation, assess its conditional variation and support, and check that the estimated correction is not redundant with the outcome regressors. A large coefficient or small p-value cannot substitute for those steps. Conversely, a plausible substantive argument without a meaningful selection shift does not provide a useful exclusion.
Check Your Understanding
Answer each question by distinguishing what the data can show from what must be justified substantively.
- In the Gaussian selection model, what does it mean for \(W_i\) to be excluded from the outcome equation?
- Why can a variable that predicts selection fail to provide useful conditional variation for identifying the correction?
- At fixed \(X_i\), how does a change in \(W_i\) help distinguish the selection correction from \(X_i'\beta\)?
- Why does a small p-value for a selection-equation coefficient not establish that an exclusion restriction is valid?
- What does it imply for the two-step outcome regression if the estimated inverse Mills ratio is perfectly explained by the outcome regressors among selected observations?