Control-Function Estimation For A Binary Treatment Outcome

Why Include a First-Stage Residual?

When treatment is related to unmeasured causes of the outcome, a regression of a binary outcome on treatment and measured covariates need not estimate a causal treatment effect. An instrumental variable \(Z\) can help address this problem, but a binary outcome also makes the choice of outcome model consequential. Control-function estimation is one approach: model treatment using the instrument and covariates, then include a residual-like term from that first stage in a model for the outcome.

The procedure is not automatically valid just because a residual appears in the second-stage model. Its causal interpretation depends on instrument assumptions and on a correctly specified control-function relationship. In particular, a fitted logistic contrast is not, in general, the same estimand as a two-stage least squares coefficient or a Wald estimate.

Model and Assumptions

A convenient first-stage specification is a probit latent-index model. For individual \(i\), let

where \(U_i\sim N(0,1)\) conditional on \(Z_i,X_i\). The fitted treatment probability is \(\hat p_i=\Phi(\hat\eta_i)\), where \(\Phi\) is the standard normal cumulative distribution function. The instrument must contribute information to treatment assignment after conditioning on \(X_i\); in this model that requires a nonzero \(\gamma_Z\), with adequate strength in the observed sample.

The generalized residual is

where \(\phi\) is the standard normal density. The two expressions are algebraically equal. Unlike the raw residual \(D_i-\hat p_i\), this generalized residual uses the probit model's conditional expectation of its latent error.

A second-stage logistic control-function model can be written as

This outcome model is an assumption, not a consequence of fitting the first-stage probit. For causal interpretation, the instrument must be relevant, independent of potential outcomes and unmeasured treatment-outcome causes conditional on \(X\), and excluded from the outcome except through treatment. Treatment and outcome definitions must be consistent, there must be no interference between individuals, and there must be adequate treatment and instrument support in the covariate regions being analyzed. In addition, the chosen control function must sufficiently capture the treatment-related dependence in the outcome equation, and the outcome functional form must be suitable.

The logistic model specifies a conditional mean, not a risk difference directly. Its treatment coefficient \(\beta_D\) is a conditional log-odds contrast at fixed \(X\) and \(r\). To report a probability-scale contrast, calculate both fitted probabilities using the same covariates and control-function value, then subtract. If the control-function assumptions identify the relevant potential-outcome means, this contrast can be interpreted causally under that model. Otherwise it is a model-based adjusted contrast, not established evidence of a causal effect.

Why the Generalized Residual Has This Form

The probit first stage gives a direct derivation. Given \(\eta_i\), treatment occurs when \(U_i>-\eta_i\), so \(P(D_i=1\mid Z_i,X_i)=\Phi(\eta_i)\). For a standard normal variable, the mean above a threshold \(a\) is \(\phi(a)/[1-\Phi(a)]\), and the mean below \(a\) is \(-\phi(a)/\Phi(a)\). Applying these identities at \(a=-\eta_i\) yields

Combining the two cases gives

Thus the generalized residual estimates the conditional mean of the latent first-stage error. If that error is related to unobserved determinants of \(Y\), including an appropriate function of it can adjust the outcome model for this dependence. The derivation does not prove that a single linear residual term is adequate for every binary-outcome model. That sufficiency is an additional structural and functional-form assumption.

Fitted Contrasts on the Risk Scale

For fitted coefficients, define the logistic function \(\operatorname{expit}(a)=1/[1+\exp(-a)]\). At specified \(X=x\) and control value \(r\), calculate

The contrast is measured in probability units; multiplying by 100 expresses it in percentage points. For a target group, one may compute the two fitted risks for each person using the same \(X_i,r_i\) in both treatment settings and average the individual contrasts. The target group and the choice of control values must be stated. This averaging is a model-based standardization and does not repair invalid instrument assumptions or a misspecified control-function model.

Diagnostics, Inference, and Alternatives

Before interpreting results, examine whether the first stage is informative and whether the model is numerically stable. A weak conditional association between \(Z\) and \(D\) can make the control-function correction imprecise and highly sensitive to specification. Inspect fitted treatment probabilities for values near zero or one, sparse combinations of instrument and covariates, convergence warnings, and extreme generalized residuals. Extreme values can arise when the observed treatment is unlikely under the fitted first stage; they may signal limited support or model misspecification rather than a few observations that can safely be ignored.

Also assess whether the first-stage index is plausibly specified and whether the outcome model's functional form is adequate. The residual-inclusion coefficient \(\lambda\) is not a test that proves endogeneity has been removed: a nonsignificant estimate does not establish instrument validity or the correctness of the control-function restriction. Since the residual is estimated, standard errors from an outcome regression that treats it as fixed may be too small. A bootstrap that refits both stages in each resample is one practical way to reflect first-stage estimation uncertainty; inference should also respect the study's sampling and clustering structure.

The main alternatives answer somewhat different questions. Two-stage least squares with a linear probability outcome yields a linear-probability IV model and a coefficient on the probability scale, but fitted values may fall outside zero and one. A joint likelihood model for treatment and outcome can encode their dependence directly, but relies on its joint distributional assumptions. The Wald estimate for a binary instrument and binary treatment/outcome has its own assumptions and estimand; it is not interchangeable with this logistic control-function contrast. Choosing among approaches requires stating the target estimand and the assumptions that make it interpretable.

Common Mistake and Practical Limits

• Mistake: “The treatment coefficient in the second-stage logistic regression is the causal risk difference.” Correction: it is a conditional log-odds coefficient. Calculate fitted risks under both treatment values at the same covariate and control profile, and subtract them to obtain a probability-scale contrast.
• Mistake: “Any first-stage residual removes unmeasured confounding.” Correction: residual inclusion works only under a suitable first-stage model and a control-function restriction that adequately represents dependence in the outcome equation. A raw fitted-probability residual is not automatically equivalent to the probit generalized residual derived above.
• Mistake: “A strong first stage proves the instrument is valid.” Correction: relevance is only one requirement. Independence and exclusion are substantive assumptions and cannot be established from first-stage strength alone.

Finally, the logistic link is non-collapsible: conditional odds ratios can differ from marginal odds ratios even without confounding. That is one reason to report the actual fitted probability contrast, identify its target population, and explain the model assumptions behind its causal interpretation. Control-function estimates can also be sensitive to link choice and distributional assumptions, so they should not be presented as assumption-free consequences of having an instrument.