Semiparametric Control Functions for Selected Outcomes

Why Use a Semiparametric Control Function?

The familiar inverse Mills ratio correction is tied to a particular Gaussian selection model. A control-function approach keeps the central selection idea—selection changes the average outcome error among observed units—but allows that correction to be an unknown function rather than a normal-theory formula. This flexibility is useful when the outcome is observed only for selected units and a credible parametric error distribution is unavailable.

Flexibility does not remove the need for assumptions. In particular, a selection probability can summarize the relevant selection threshold only when the distribution of the selection error is stable across covariates. Excluded-variable variation is also important: it can change selection while leaving the outcome equation unchanged, helping distinguish a selection correction from covariate effects.

Model and Control-Function Representation

Let \(D_i=1\) indicate that the outcome is observed for unit \(i\). For clarity, begin with a latent-threshold selection rule and an outcome equation:

Here \(X_i\) contains outcome covariates, \(Z_i\) contains selection covariates, and \(U_i,V_i\) are unobserved outcome and selection components. The function \(\eta\) determines the selection threshold. The exclusion restriction is that \(Z_i\) affects selection but does not enter the outcome equation directly. The outcome \(Y_i\) is observed only when \(D_i=1\).

The independence assumption is a transparent sufficient condition, not the only possible set of conditions for a control-function representation. Its role here is crucial: it makes the selection probability depend on the threshold through a common distribution, rather than through an unmodelled covariate-specific distribution.

Write \(c(X,Z)=-\eta(X,Z)\) for the selection threshold. Then the conditional probability of selection is \(p(X,Z)=P(D=1\mid X,Z)=1-F_V(c(X,Z))\). For an interior probability \(0<p(X,Z)<1\), the selected conditional outcome mean is \(E[Y\mid X,Z,D=1]=m(X)+g(p(X,Z))\), where \(g\) is an unknown selection correction. This statement follows because the distribution of \(U,V\) is stable across \(X,Z\), and conditioning on selection truncates \(V\) at the threshold corresponding to \(p\).

Why the Correction Is a Function of the Selection Probability

A short derivation makes the distributional requirement explicit. Define \(W=F_V(V)\). Since \(F_V\) is continuous and strictly increasing, \(W\) is uniformly distributed on \((0,1)\). The event \(V>c\) is equivalent to \(W>F_V(c)\). Since \(p=1-F_V(c)\), it is also equivalent to \(W>1-p\). Let \(q(r)=E[U\mid W=r]\), understood as a conditional-mean function defined almost everywhere. Then:

The denominator is \(p\) because \(W\) is uniform. The numerator has the displayed integral because \(E[U\mid W=r]=q(r)\). Thus the correction is a function of \(p\) alone under the stated stable joint distribution of \((U,V)\). No Gaussian distribution is needed. At \(p=1\), selection is certain, so \(g(1)=E[U]=0\).

This result does not say that the correction is necessarily linear, smooth, or well approximated by a low-degree polynomial. Those are approximation choices to be assessed against the data and the available support. If the conditional distribution of \(V\) changes with \(X,Z\), the formula \(1-F_V(c)\) using a single marginal CDF need not give the conditional selection probability, and the reduction to one common \(g(p)\) need not hold.

Series Estimation in Two Stages

A practical semiparametric procedure estimates the selection probability first and then estimates the selected conditional mean using flexible basis functions. A series is a finite collection of known functions—such as powers, splines, or interactions—whose number can be increased as the sample grows. The series does not require the selection error to be normal.

For the additive model, the second stage approximates \(E[Y\mid X,D=1,Z]=m(X)+g(p(X,Z))\) with terms such as \(A_J(X)'\beta+B_L(p)'\gamma\). Here \(A_J\) and \(B_L\) are chosen bases, and \(J,L\) index their dimensions. The conditional mean does not depend on \(Z\) once \(X\) and the true \(p\) are held fixed; \(Z\) helps identify the correction by generating changes in \(p\).

The decomposition into \(m\) and \(g\) needs a normalization. In this model \(g(1)=0\), so \(m(X)\) is the mean outcome when selection is certain, under the maintained outcome equation and error-mean assumption. If probabilities near one are absent, estimating that normalization may require extrapolation. The selected conditional mean can still be estimable over observed support even when the structural component \(m(X)\) is not precisely recoverable there.

Identification, Support, and Inference

An exclusion restriction is useful when it produces meaningful variation in selection probability while holding outcome covariates fixed. In a flexible additive model, the data need enough such variation to distinguish changes in \(m(X)\) from changes in \(g(p)\). If \(p\) is effectively determined by \(X\), the two components may be difficult or impossible to separate without stronger functional-form restrictions. A nominal exclusion that barely moves selection probabilities may provide little practical information.

Support matters in two ways. First, selected observations must exist over the covariate and probability region where the conditional mean is to be estimated. Second, the normalization \(g(1)=0\) is informative only if the observed support reaches one or comes sufficiently close for a credible approximation. A flexible series cannot create information in regions with no data; estimates there are extrapolations.

The estimated probability is a generated regressor: it is calculated using parameters estimated from the same data. Consequently, ordinary second-stage standard errors that act as if \(\hat p_i\) were fixed can be incorrect. Inference should account for first-stage estimation and series approximation, using an appropriate joint or resampling procedure. The earlier material on Murphy-Topel inference discusses the general issue of generated corrections in two-step estimation.

Common Mistakes

• Using a marginal selection-error distribution as if it were conditional. The equality \(p(X,Z)=1-F_V(-\eta(X,Z))\) requires a common conditional distribution of \(V\mid X,Z\), as ensured by the independence assumption here. Without it, use the appropriate conditional distribution and establish that a stable control representation is valid.
• Assuming that “semiparametric” means assumption-free. The unknown correction function avoids a specified Gaussian formula, but the control-function representation, exclusion restriction, support, and approximation conditions still matter.
• Claiming that an exclusion alone guarantees identification. An excluded variable must actually shift selection at comparable outcome covariates, and the resulting variation must be sufficiently rich and supported.
• Treating the fitted correction as causal evidence. The estimated conditional mean describes outcomes under the maintained selection and outcome assumptions. It does not by itself establish a causal effect of \(Z\), which is excluded from the outcome equation.
• Ignoring the first-stage estimate in standard errors. The control is estimated, not observed. Inference must reflect that estimation and the chosen series approximation.