Sensitivity To Control-Function Specification

Why Control-Function Specification Deserves Attention

A control-function analysis can produce a precise-looking estimate while depending strongly on how a residual enters the nonlinear outcome model. A linear residual term, a squared residual, and a flexible residual function impose different restrictions on how unobserved selection relates to the outcome. A different first-stage distribution can also produce a different control function. The resulting estimates therefore warrant sensitivity analysis, not automatic confidence in whichever model fits best.

The purpose of this tutorial is to assess that dependence. A sensitivity analysis can show that conclusions are robust, or reveal that they depend on a particular functional form. It cannot establish that the instrument is valid: no comparison of outcome-model specifications proves the instrument’s independence or exclusion restriction.

Conditions for Interpreting the Comparisons

A common binary-outcome control-function model writes the conditional outcome probability as

where \(G\) is a link function such as the logistic or normal cumulative distribution function, and \(h\) describes the control-function terms. For example, \(h\) might be \(\lambda R\), or \(\lambda_1R+\lambda_2(R^2-\overline{R^2})\). Centering the squared term can make its coefficient easier to interpret and can reduce unnecessary collinearity with the intercept; it does not change the model’s fitted probabilities.

This equation alone does not identify a causal effect. A control-function interpretation requires a relevant instrument, instrument independence from unobserved outcome determinants conditional on \(X\), and exclusion of \(Z\) from the outcome equation except through \(D\). It also requires a correctly specified first stage and an adequate control function: conditional on the chosen \(R\) and \(X\), the remaining dependence between \(D\) and the outcome’s unobserved determinants must be handled well enough for the outcome model to be valid. The link and the other outcome-model terms must also be appropriate. These conditions are substantive and cannot be verified just by comparing fits.

For a continuous treatment with a linear first stage, \(R\) may be the first-stage residual. For a nonlinear first stage, a generalized residual is used; its definition depends on the first-stage model. In either case, \(R\) is estimated, not observed as a known covariate. Uncertainty calculations should account for this generated regressor, for example by re-estimating both stages in a bootstrap that resamples independent observational units.

Why Alternative Residual Terms Can Matter

The basic rationale is that if the conditional distribution of the outcome’s unobserved component given \(D,Z,X\) depends on selection into \(D\) through a latent first-stage component, a control function can summarize that dependence. If the summary is adequate, conditioning on it makes the remaining outcome uncertainty compatible with the specified outcome model. But a single linear term assumes a particular way that selection affects the outcome’s conditional index. A quadratic or flexible function relaxes that restriction.

For example, suppose a latent outcome index contains \(aR+bR^2\). A model containing only \(R\) omits \(bR^2\) when \(b\ne0\). If \(R\) is associated with treatment, the omitted curvature can be attributed partly to the treatment term, shifting the treatment estimate. Conversely, adding many residual terms in a small sample can inflate uncertainty, make estimates unstable, and fit noise. Greater flexibility is not automatically better.

The same concern arises when treatment effects vary with selection. An interaction \(D R\) lets the treatment association on the model’s index scale vary with \(R\). If such variation is plausible, omitting the interaction imposes a restriction. Including it changes the interpretation: a single treatment coefficient is no longer the treatment contrast at every value of \(R\). For nonlinear links, even a constant index coefficient does not equal a constant risk difference.

Distributional Assumptions and Generalized Residuals

A generalized residual is tied to the assumed first-stage distribution. For example, suppose a binary treatment \(D\) is modeled by a probit first stage with index \(\eta=Z^\mathsf{T}\pi+X^\mathsf{T}\delta\). Under the standard normal latent-error assumption, a generalized residual is

where \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution functions. This is a conditional mean of the latent first-stage error, not an ordinary observed-minus-predicted residual. Changing the first-stage link or distribution changes the residual values and therefore can change the control-function outcome estimates.

Compare Effects on a Common Scale

Outcome-model coefficients from different links are not directly comparable: a logit coefficient and a probit coefficient use different index scales. Even within a single link, changes in other model terms can alter coefficient interpretations. A useful comparison reports a common estimand, such as an average predicted risk difference, computed by the same procedure in each model. For each person, predict the outcome under the treatment values being compared, average each set of predictions over the target sample, and subtract.

When the treatment is binary, this comparison should say which population is averaged over and how the control-function value is handled for each counterfactual prediction. In many applications, simply changing \(D\) while holding an estimated residual fixed is a model-based contrast, not automatically the desired population causal effect. The assumptions connecting the control function to a causal estimand must justify the chosen prediction procedure. Report the estimand and procedure rather than presenting a model coefficient as though it were universally interpretable.

Practical Diagnostics and Alternatives

A defensible comparison begins by holding the sample, covariates, first-stage information, and target estimand fixed where possible. Then vary a small set of substantively defensible choices: linear versus nonlinear residual terms, an interaction when justified, and plausible first-stage specifications. Plot the outcome against residual values or compare observed outcomes with model-predicted risks across residual groups. Such plots can reveal curvature or systematic lack of fit, but do not establish identification.

• Check residual support. Sparse tails make squared terms and splines unstable. Report how many observations support the extreme values and check whether treatment comparisons rely on extrapolation.
• Check convergence and precision. Large changes in estimates, very large standard errors, nearly redundant terms, or convergence warnings can indicate weak information or overfitting.
• Use inference that reflects estimation of \(R\). A bootstrap should repeat the first stage and outcome stage together. Treating the estimated residual as fixed can understate uncertainty.
• Compare predictions and the target effect. Report average risks or risk differences on a common scale, not only link-specific coefficients.
• Separate fit from validity. A likelihood-ratio test or information criterion evaluates aspects of model fit under its assumptions. Neither tests instrument exclusion or independence.

Alternatives answer different questions and rely on different restrictions. A control-function model specifies how selection is represented in the conditional outcome model. Structural mean models and G-estimation instead formulate treatment effects through structural restrictions; they are not merely alternative residual transformations. A conventional regression without a control function may be useful descriptively, but it does not solve endogeneity just because it includes observed covariates. The appropriate comparison depends on the estimand and assumptions, rather than on choosing the specification with the smallest information criterion.

A central limitation is that functional-form sensitivity analysis cannot diagnose every failure. Similar estimates across several specifications do not prove instrument validity, and different estimates do not identify which specification is correct. Weak instruments, incorrect exclusion restrictions, unmeasured effect heterogeneity, and first-stage misspecification can remain consequential across all the models compared. Report the range of substantively defensible estimates and explain the assumptions that distinguish them.