Functional-Form Identification Without Excluded Variables

Why Functional Form Can Identify a Selection Correction

In a sample-selection problem, an outcome is observed only for units whose selection indicator equals one. If selection depends on unobserved factors related to the outcome, an outcome regression among observed units can have a conditional mean that differs from the population outcome equation. The Gaussian Heckman model represents this difference with an inverse Mills ratio.

An exclusion restriction can help separate the selection correction from the outcome regressors by changing selection while being absent from the outcome equation. But an excluded variable is not logically necessary for every form of identification. Even when the selection and outcome equations use the same observed covariates, the inverse Mills ratio is generally a nonlinear function of those covariates. If that nonlinear component varies in a way that cannot be represented as a linear combination of the outcome regressors, it can supply identification. The practical concern is that the curvature may be too slight to distinguish reliably in a finite sample.

Model and assumptions: Let \(D_i=1\) indicate that unit \(i\)'s outcome is observed. The selection rule is \(D_i=1[Z_i'\gamma+u_i>0]\), and the latent outcome is \(Y_i^*=X_i'\beta+\varepsilon_i\), observed when \(D_i=1\). Assume \((u_i,\varepsilon_i)\) is jointly normal conditional on the observed covariates, with \(E[u_i\mid Z_i,X_i]=E[\varepsilon_i\mid Z_i,X_i]=0\), \(\operatorname{Var}(u_i)=1\), \(\operatorname{Var}(\varepsilon_i)=\sigma^2\), and \(\operatorname{Cov}(u_i,\varepsilon_i)=\rho\sigma\). The unit variance for \(u_i\) is the probit scale normalization. The selection and outcome equations are correctly specified, and the covariates and selected observations provide the variation required by the rank condition below.

Here \(X_i\) and \(Z_i\) are vectors of observed regressors, while \(\beta\) and \(\gamma\) are their coefficient vectors. Functional-form identification without excluded variables is the case in which the same covariates appear in both equations, so that \(Z_i=X_i\) (up to the same specified transformations). Define the standard normal density and cumulative distribution function as \(\phi\) and \(\Phi\). The inverse Mills ratio for selection is \(\lambda(t)=\phi(t)/\Phi(t)\).

E[Y_i^*\mid D_i=1,X_i,Z_i] = X_i'\beta+\delta\lambda(Z_i'\gamma), \qquad \delta=\rho\sigma.

This conditional-mean expression follows from the assumed joint normality: among units satisfying \(u_i>-Z_i'\gamma\), the conditional expectation of \(\varepsilon_i\) is \(\rho\sigma\lambda(Z_i'\gamma)\). The coefficient \(\delta\) is the outcome-error covariance with the selection error, scaled by the selection-error standard deviation, which has been normalized to one. The correction vanishes if \(\delta=0\); otherwise, the selected-sample mean includes a term whose size depends on the selection index.

The identification question is whether that correction can be distinguished from \(X_i'\beta\). To isolate the issue, suppose first that \(\gamma\) has been identified from the selection equation and is known. The probit selection equation can identify \(\gamma\), subject to its usual specification and variation conditions, using selection indicators for selected and nonselected units. The question here is whether the selected observations then contain enough variation to distinguish \(\beta\) from \(\delta\).

Rank condition for the conditional mean: Let \(W_i\) be the vector formed by stacking \(X_i\) and \(\lambda(Z_i'\gamma)\). The parameters \(\beta\) and \(\delta\) in the selected-sample conditional mean are identified by its linear representation if the selected-population second-moment matrix \(E[W_iW_i'\mid D_i=1]\) has full rank. Equivalently, the inverse Mills ratio must not be an exact linear combination of the outcome regressors over the support of selected observations.

This condition is about exact population dependence, not merely whether the correction is visibly curved on a graph. If the correction column is exactly in the span of \(X_i\), different values of \(\beta\) and \(\delta\) produce the same conditional mean, so the decomposition is not unique. If it is not in that span, the decomposition is unique under the stated linear conditional-mean model. In a sample, the corresponding design matrix must also have full column rank to calculate the usual regression coefficients; near dependence can make those coefficients very imprecise.

A Short Derivation of the Rank Requirement

With \(\gamma\) fixed, write the selected-sample conditional mean as \(W_i'\theta\), where \(\theta\) stacks \(\beta\) and \(\delta\). Suppose two parameter vectors \(\theta_1\) and \(\theta_2\) generate the same conditional mean for all selected covariate values. Then \(W_i'(\theta_1-\theta_2)=0\) on that support. Multiplying by \(W_i\) and taking the selected-population expectation gives

E[W_iW_i'\mid D_i=1](\theta_1-\theta_2)=0.

If the matrix is full rank, its only null vector is zero, so \(\theta_1-\theta_2=0\), and the two parameter vectors are identical. Conversely, if the matrix is rank deficient, some nonzero vector \(a\) satisfies \(W_i'a=0\) on the selected support. Adding a multiple of \(a\) to \(\theta\) leaves the conditional mean unchanged there, so the parameters are not separately identified from that conditional mean. The proof also shows why the relevant variation is among the selected observations: only they contribute observed outcomes to this conditional-mean relationship.

When \(Z_i=X_i\), the nonlinear function is \(\lambda(X_i'\gamma)\). It will usually not be an exact linear combination of the components of \(X_i\) over a sufficiently rich support. This is the source of functional-form identification. It is not a guarantee that the rank condition holds for every regressor design: a very restricted support, redundant regressors, or a correction that is constant or linear over the support can defeat the condition.

Worked Example: Curvature Across a Broad Index Range

Consider a hypothetical selected-sample conditional mean with an intercept and one index regressor, \(X_i=(1,t_i)'\), where the selection index equals \(t_i\). Treat the selection parameter as known for this rank illustration, and use three index values, \(-1,0,1\). The approximate inverse Mills ratios are \(\lambda(-1)=1.5251\), \(\lambda(0)=0.7979\), and \(\lambda(1)=0.2876\). The design matrix has rows \((1,t_i,\lambda(t_i))\):

M= \begin{pmatrix} 1&-1&1.5251\\ 1&0&0.7979\\ 1&1&0.2876 \end{pmatrix}.

Subtract the middle row from the first and third rows. This row operation does not change the determinant, giving rows \((0,-1,0.7272)\), \((1,0,0.7979)\), and \((0,1,-0.5103)\). Expanding along the first column, including its cofactor sign, gives

\det(M) = -\left[(-1)(-0.5103)-(0.7272)(1)\right] = - [0.5103-0.7272] \approx 0.2169.

The determinant is nonzero, so these three columns are linearly independent for this illustrative design. With this support, the intercept, index slope, and inverse Mills ratio are algebraically distinguishable. This establishes neither precise estimation nor the validity of the Gaussian model; it only shows that the conditional-mean regressors are not exactly collinear at these index values.

Why Narrow Index Support Can Make Identification Weak

Nonzero curvature can be small. Over a narrow range of index values, a smooth function is often well approximated by a straight line. Since the outcome equation already includes a constant and an index regressor in the example below, a correction that is nearly linear adds little independent variation. The distinction is important: exact nonlinearity can satisfy a population identification condition while the finite-sample data provide very little information about the separate coefficients.

Worked Example: A Small Determinant Over a Narrow Range

Use the same intercept-and-index design, now at \(t=-0.1,0,0.1\). The approximate correction values are \(\lambda(-0.1)=0.8626\), \(\lambda(0)=0.7979\), and \(\lambda(0.1)=0.7353\). Thus

M= \begin{pmatrix} 1&-0.1&0.8626\\ 1&0&0.7979\\ 1&0.1&0.7353 \end{pmatrix}.

Subtract the middle row from the first and third. Expanding along the first column, again applying the cofactor sign, gives

\det(M) = -\left[(-0.1)(-0.0626)-(0.0647)(0.1)\right] = -(0.00626-0.00647) \approx 0.00021.

The same value follows from the equal-spacing second difference: \(0.1[0.8626+0.7353-2(0.7979)]=0.1(0.0021)=0.00021\). The determinant is not zero using the displayed rounded values, but it is much smaller than in the broad-range example. Its scale depends on how the regressors are measured, so determinant size alone is not a general measure of uncertainty. Here it illustrates that the correction column changes very little beyond what an intercept and index slope can already represent. In a real sample, noise and estimation of \(\gamma\) can then make the separate outcome coefficients unstable.

Worked Example: Recovering the Correction in a Toy Conditional Mean

Suppose a hypothetical outcome is measured in thousands of dollars, and its selected-sample conditional mean is \(m(t)=\beta_0+\beta_1t+\delta\lambda(t)\). For illustration, let the true values be \(\beta_0=10\), \(\beta_1=2\), and \(\delta=3\), with the broad-range index values and correction values from the first example. The three conditional means are

\begin{aligned} m(-1)&=10+2(-1)+3(1.5251)=12.5753,\\ m(0)&=10+2(0)+3(0.7979)=12.3937,\\ m(1)&=10+2(1)+3(0.2876)=12.8628. \end{aligned}

Their second difference eliminates both the intercept and the linear index term:

$$ m(-1)+m(1)-2m(0) = 12.5753+12.8628-2(12.3937) = 0.6507. $$

The corresponding second difference of the correction is \(1.5251+0.2876-2(0.7979)=0.2169\). Therefore \(\delta=0.6507/0.2169=3.000\) to the displayed precision. Then \(\beta_0=m(0)-\delta\lambda(0)=12.3937-3(0.7979)=10.000\). Finally,

\beta_1 = \frac{m(1)-m(-1)-\delta[\lambda(1)-\lambda(-1)]}{2} = \frac{0.2875-3(-1.2375)}{2} = 2.000.

This is an algebraic illustration, not an empirical estimate: three conditional means recover three coefficients because the design matrix is full rank. With a narrow index range, the correction's second difference would be small, and even modest errors in the conditional means could produce a much larger change in the recovered \(\delta\).

What Is and Is Not Identified

The conditional-mean argument identifies the outcome coefficients \(\beta\) and the correction coefficient \(\delta=\rho\sigma\), provided the selection parameters are identified and the rank condition holds. It does not, by itself, separately identify \(\rho\) and \(\sigma\): their product is the coefficient multiplying the inverse Mills ratio in this conditional mean. Additional distributional information in the selected outcomes may be relevant to other estimation approaches, but it is not supplied by the conditional-mean rank argument alone.

Functional-form identification also depends on the Gaussian specification. The particular inverse Mills ratio arises from the assumed joint normality and the probit selection rule. If the error distributions or dependence structure differ, the conditional expectation correction may have a different form. Thus a coefficient attributed to curvature is not automatically a robust, distribution-free correction for selection.

Finite-Sample Fragility and a Common Mistake

In practice, the inverse Mills ratio is usually estimated using the selection-equation coefficients, so uncertainty in \(\gamma\) also affects the generated correction. Even when the population regressors are linearly independent, a narrow range of fitted selection indices, limited numbers of selected observations, or a correction nearly collinear with the outcome regressors can leave little usable variation. The resulting estimates may have large standard errors, be sensitive to small specification changes, or show strong correlation between the estimated correction coefficient and outcome coefficients. This is weak practical identification, not necessarily a failure of exact population identification.

Common mistake: “There are no excluded variables, so the Heckman model cannot be identified.” That conclusion is too strong. Under the stated Gaussian model, the nonlinear inverse Mills ratio can distinguish the correction from the linear outcome regressors if the rank condition holds. The opposite claim is also mistaken: “The inverse Mills ratio is nonlinear, so identification must be strong.” Nonlinearity alone does not ensure useful curvature over the observed support, nor does it guarantee accurate finite-sample estimates or a correctly specified model.

The practical assessment should therefore ask whether the selection index spans a meaningful range, whether the correction has variation beyond the outcome regressors among selected observations, and whether estimates are sensitive to reasonable changes in specification. These checks describe the strength and credibility of functional-form identification; they do not turn a fragile source of identification into an exclusion restriction.

Key takeaway: Without excluded variables, Gaussian Heckman identification of the outcome correction can come from the inverse Mills ratio's nonlinearity. Exact identification requires that this correction not lie in the span of the outcome regressors on the selected support. If it is nearly in that span, estimates can be weak and unstable even when the population rank condition technically holds.

Check Your Understanding

Answer each question using the Gaussian selection model and the selected-sample conditional mean.

State the rank condition that allows \(\beta\) and \(\delta\) to be distinguished when the selection parameter \(\gamma\) is treated as known.
Why can the same covariates in the selection and outcome equations still provide identification in the population?
For index values \(-h,0,h\), explain why the determinant of the intercept, index, and correction columns depends on the correction's second difference and on \(h\).
Why does a nonzero determinant establish algebraic identification in an illustrative design but not guarantee precise estimates in a real sample?
What combination of the error parameters is the coefficient on the inverse Mills ratio in the conditional mean, and what does that imply about what the conditional-mean argument identifies?
Give one reason functional-form identification may be sensitive to the Gaussian specification.