The Control-Function Logic For Endogenous Treatments

Why a Control Function Is Needed

In a nonlinear outcome model, treatment endogeneity cannot generally be handled by replacing treatment with its first-stage fitted value. The fitted value does not, by itself, account for the dependence between treatment and unobserved determinants of the outcome. A control function takes a different route: it uses information from the first stage to construct a variable that, under explicit conditions, captures the relevant part of that dependence. The outcome model then conditions on this control as well as on treatment.

The key point is conditional, not automatic. A first-stage residual is useful only if conditioning on it adequately controls the association between treatment and the unobserved outcome determinants. The link function, the way residuals enter the outcome model, and the first-stage specification must also be appropriate. Without those conditions, residual inclusion is just another potentially misspecified regression.

Conditions Behind the Logic

Let \(Y(d)\) denote the outcome that would be observed under treatment level \(d\), \(D\) the received treatment, \(Z\) an instrument, and \(X\) observed covariates. A useful control-function argument has several parts.

• Instrument validity. Conditional on \(X\), \(Z\) is independent of the potential outcomes and the unobserved variables governing treatment assignment. The instrument also has no direct effect on the outcome except through treatment.
• Relevance and support. The distribution of treatment changes with \(Z\), and the data provide adequate overlap to estimate the outcome model at the treatment and control values of interest.
• A correctly specified first stage. The first stage describes treatment as a function of \(Z\), \(X\), and unobserved treatment variation. For example, \(D=m(Z,X)+V\), with \(E[V\mid Z,X]=0\), defines a residual \(V=D-m(Z,X)\). The conditional mean \(m\) and residual construction must fit the treatment process.
• Control sufficiency. Conditional on \(V\) and \(X\), treatment assignment contains no further information about the potential outcome: \(Y(d)\perp D\mid V,X\). This is the substantive condition that makes residual conditioning useful.
• A suitable outcome model. The conditional distribution or mean of \(Y\) given \(D,X,V\) must have the chosen functional form. For a probit model, for example, the outcome probability is modeled through the standard normal cumulative distribution function with a specified index in \(D,X,V\).

The last two conditions are not consequences of using an instrument. Instrument validity does not prove that a particular residual is sufficient, or that a linear residual term is the correct way to model its effect. Those are additional modeling assumptions. In particular, a residual from a linear first stage is not automatically appropriate when the treatment process is nonlinear or discrete.

Deriving the Conditional Outcome Model

The derivation makes clear what the residual must accomplish. Suppose the structural conditional response at treatment \(d\), covariates \(x\), and unobserved outcome determinant \(U\) is described by a function \(q(d,x,U)\). Assume the instrument is excluded from this outcome response, and that the residual \(V\) is a sufficient control: the conditional distribution of \(U\) given \(D,Z,X\) depends on those variables only through \(V,X\). Then

The first equality averages the structural response over the distribution of unobserved outcome determinants among observations with the stated treatment and covariates. The second uses the control-sufficiency condition. The final expression shows that, after conditioning on \(V\) and \(X\), the conditional outcome response can be modeled as a function of treatment and the control, rather than as a function of the instrument itself.

For a binary outcome, one possible specification is a probit conditional mean:

where \(\Phi\) is the standard normal cumulative distribution function. This equation is a functional-form assumption about the conditional probability. It does not follow merely from the existence of a first-stage residual. Also, \(\beta_D\) describes the treatment term in the conditional probit index; the probability-scale effect depends on the values of \(D,X,V\), so the coefficient is not itself a constant probability change.

Worked Calculations

Diagnostics, Alternatives, and Limits

A control-function analysis should be scrutinized at both stages. At the first stage, inspect whether the instrument predicts treatment in the relevant covariate strata, whether the model captures the treatment pattern, and whether residuals show systematic structure against fitted values or covariates. These checks can reveal a poor first-stage functional form, but they cannot establish that the residual is a sufficient control for unobserved outcome determinants.

At the outcome stage, assess whether the conditional model is plausible, whether predicted probabilities stay within \([0,1]\), and whether the data support the treatment and control values used for interpretation. Compare reasonable specifications of the control’s contribution when theory permits, such as allowing a nonlinear residual function rather than imposing a single linear term. Such comparisons are sensitivity checks, not a guarantee that one specification has removed all endogeneity. An instrument that has a direct outcome effect, a residual that fails to capture relevant dependence, or substantial lack of overlap can invalidate the interpretation even if the regression converges.

The main alternative is to model the joint treatment and outcome process directly, with assumptions linking their unobserved components. Such a model can make the dependence structure explicit, but it brings its own distributional and functional-form assumptions. A control-function model can be more modular: estimate a treatment model, construct a control, and include it in the conditional outcome model. Neither approach is assumption-free, and neither becomes credible merely because an instrument is available.

Finally, distinguish conditional predictions from population effects. A fitted probability at fixed \(D,X,V\) is conditional on those values. To report an average contrast, predict outcomes under two treatment levels while holding each observation’s covariates and control value fixed, form the contrasts, and average over the target population. This averaging step is especially important with nonlinear links.

Common Mistakes

• Assuming any residual removes endogeneity. A residual is a control only if the first-stage structure and sufficiency condition justify it. A poorly specified residual can leave confounding uncontrolled.
• Ignoring exclusion. If the instrument affects the outcome directly, its influence cannot be represented solely through treatment and the proposed control under this setup.
• Treating the probit coefficient as a risk difference. The coefficient changes the latent index; the probability-scale change depends on the index and the values at which predictions are made.
• Replacing treatment by its fitted value and calling that residual inclusion. These procedures are different. Fitted-value substitution does not condition on the observed treatment residual and, with a nonlinear link, need not reproduce averaged conditional predictions.
• Confusing model fit with identification. A good-looking fit or stable estimate does not establish instrument validity, residual sufficiency, or adequate support.