Endogenous Treatments In Nonlinear Outcome Models

Why Treatment Endogeneity Needs Special Care in Nonlinear Models

Suppose a treatment indicator \(D\) records whether a person receives a program, a clinical intervention, or a subsidy, and \(Y\) is an outcome such as spending, employment, or a health measure. If unobserved characteristics that affect \(Y\) also influence treatment selection, then \(D\) is endogenous. A nonlinear outcome model does not remove this problem: fitting a logit or a log-mean equation directly to \(Y\) and \(D\) can still confound treatment differences with selection differences.

A control function addresses this dependence by modeling treatment assignment and including a function of the first-stage information in the outcome equation. Its role is not merely to improve prediction. The assumptions behind the control function determine whether a fitted contrast is a conditional outcome association or a causal treatment effect.

A Binary Selection Model and Its Control

Let \(X\) denote observed covariates included in both equations, and let \(Z\) denote excluded instruments: variables that help predict treatment but, conditional on \(X\), have no direct effect on the outcome. A common selection model is a latent-index probit:

Here \(D=1\) indicates treatment, and \(V\) is an unobserved treatment-selection disturbance. Assume \(V\mid X,Z\sim N(0,1)\). The first-stage treatment probability is \(P(D=1\mid X,Z)=\Phi(\eta(X,Z))\), where \(\Phi\) is the standard normal cumulative distribution function. Instrument relevance requires \(Z\) to contribute useful variation in this probability after conditioning on \(X\).

To see how selection creates endogeneity, suppose an unobserved outcome disturbance \(U\) is correlated with \(V\). Under a joint normal model, write \(U=\rho\sigma_U V+\epsilon\), where \(\epsilon\) is mean-independent of \(V\), \(\rho\) is their correlation, and \(\sigma_U\) is the standard deviation of \(U\). Conditional on treatment status, the expected value of \(U\) is generally not zero. A selection correction is therefore a function of treatment status and the first-stage index:

The function \(\phi\) is the standard normal density. The displayed conditional expectation follows from the truncated-normal means of \(V\): treated observations have \(V\geq-\eta\), while untreated observations have \(V<-\eta\). In an applied model, the selection correction may be included directly or multiplied by an estimated coefficient. The normal selection model is one particular justification, not a universal recipe for every endogenous binary treatment.

From the Control Function to a Nonlinear Outcome Equation

For a positive outcome, one possible specification is a log conditional-mean model. For a binary outcome, a logit conditional-probability model is another. In either case, let \(C\) denote the control term actually used in the outcome equation. For example, under the selection model above, \(C=h(D,\eta)\) is an observed, generated function of the first-stage treatment status and index. The outcome model might take the form

or, for a binary \(Y\),

These equations specify conditional means or probabilities, not the entire distribution of \(Y\). Their functional forms and the control adjustment must be justified for the application. In particular, including a first-stage residual or correction does not by itself guarantee that all dependence between treatment and outcome has been removed.

What a Nonlinear Treatment Contrast Means

In a nonlinear model, the treatment coefficient alone is usually not a difference in the outcome's original units. For the log conditional-mean model, \(\tau\) is a difference in log conditional means holding \(X\) and \(C\) fixed. Exponentiating it gives the conditional mean ratio \(\exp(\tau)\); the corresponding difference in outcome units depends on the covariate and control values. In a logit model, \(\tau\) is a conditional log-odds difference, and \(\exp(\tau)\) is a conditional odds ratio. A probability difference must be calculated from the fitted probabilities.

When Can the Contrast Be a Treatment Effect?

Define \(Y(1)\) and \(Y(0)\) as the potential outcomes under treatment and no treatment. Consistency says the observed outcome is \(Y=DY(1)+(1-D)Y(0)\); no interference means one individual's treatment does not change another individual's potential outcome. To identify causal effects, the control must do more than improve the fit of the observed conditional mean. In a control-sufficiency formulation, conditioning on observed covariates \(X\) and a valid control \(C\) must remove treatment selection for the potential outcomes:

This conditional independence is an assumption, not a consequence of adding a generated variable to a regression. For a generic control-sufficiency approach, there must also be overlap: for covariate and control values used in the target population, both treatment states must be possible. This condition does not hold for the treatment-specific \(C=h(D,\eta)\) defined above, since \(h\) is positive when \(D=1\) and negative when \(D=0\). Thus, conditioning on this \(C\) cannot identify treatment contrasts over a common \((X,C)\) distribution; causal effects using this correction require structural control-function assumptions and counterfactual adjustment or integration. The outcome and first-stage specifications must be adequate, the excluded instruments must be relevant, and instrument exogeneity and exclusion must hold. Exclusion means \(Z\) has no direct effect on potential outcomes once the stated conditioning variables are accounted for. These requirements are demanding, especially when the control is obtained from a parametric selection model.

When the assumptions hold, define \(m_d(x,c)=E[Y(d)\mid X=x,C=c]\). The conditional average treatment effect at \((x,c)\) is \(m_1(x,c)-m_0(x,c)\). The average treatment effect for a specified target population is the average of that contrast over the target distribution of \(X,C\), not the coefficient \(\tau\) in general:

Computing this average requires evaluating both treatment states for the same target covariate-and-control distribution and then averaging. It is not generally correct to compare the mean among observed treated people with the mean among observed untreated people: those groups may have different distributions of \(X\) and the control. In selection-model implementations, the constructed correction \(h(D,\eta)\) depends on observed \(D\). A causal calculation therefore needs a structural interpretation that provides the relevant counterfactual adjustment or integration; simply holding each observed treatment-specific correction fixed while switching \(D\) need not produce an ATE.

Why the Selection Correction Has Its Form

The inverse Mills terms above are conditional means of the latent selection disturbance. For \(D=1\), the selection rule implies \(V\geq-\eta\). A standard normal variable truncated below at \(a\) has conditional mean \(\phi(a)/(1-\Phi(a))\). Taking \(a=-\eta\), and using \(\phi(-\eta)=\phi(\eta)\) and \(1-\Phi(-\eta)=\Phi(\eta)\), gives

For \(D=0\), the truncation is \(V<-\eta\). A standard normal variable truncated above at \(a\) has conditional mean \(-\phi(a)/\Phi(a)\), so

Finally, \(U=\rho\sigma_U V+\epsilon\), with \(E[\epsilon\mid V,D,X,Z]=0\), implies \(E[U\mid D,X,Z]=\rho\sigma_U E[V\mid D,X,Z]\). This yields the stated selection correction. The derivation depends on the latent-index and joint-normal assumptions; different treatment and outcome dependence may require a different control function.

Common Mistakes and Practical Checks

• Calling a fitted conditional contrast a treatment effect. The control-adjusted outcome equation describes conditional outcomes under its assumptions. A causal interpretation additionally requires that the control handles selection for potential outcomes and that overlap and identification conditions hold.
• Reading a nonlinear coefficient on the outcome scale. In a log-mean model, \(\tau\) is a log-mean contrast; in a logit model, it is a log-odds contrast. Compute mean or probability contrasts at relevant covariate values.
• Using a raw first-stage residual without justification. For binary treatment, the selection correction depends on treatment status and the first-stage probability. The ordinary residual \(D-\hat p\) is not generally the same control.
• Ignoring generated-control uncertainty. The control is estimated in a first stage. Inference for the outcome model must account for that estimation rather than treating the generated values as fixed data.
• Assuming an instrument is valid because it predicts treatment. Relevance does not establish exogeneity or exclusion. A variable can strongly predict treatment and still directly affect the outcome.

Control-function methods are useful because they let researchers formulate nonlinear conditional outcome models while addressing a specific source of endogeneity. Their value depends on keeping three questions separate: whether the first-stage model is credible, whether the outcome equation adequately describes conditional outcomes, and whether the resulting contrast is identified as causal for the target population.