Multiple Endogenous Treatments In Control-Function Models

Why Two Endogenous Treatments Need a Joint Control

When two treatments are endogenous, each may be associated with unobserved factors that also affect the outcome. Adjusting for one treatment’s first-stage residual alone does not generally account for the unobserved component associated with the other. Nor is it generally enough to include two separate residual terms while assuming their joint relationship is irrelevant. A control-function approach instead models both treatments and includes controls for their joint first-stage disturbances in the outcome model.

The key idea is an extension of residual inclusion: if the first-stage disturbances capture the sources of treatment-outcome dependence, conditioning on them can make the treatment terms interpretable within a specified structural outcome model. This requires substantive assumptions, adequate instrument variation, and a sufficiently appropriate model for how the residuals enter the outcome equation.

Identification and Model Assumptions

For clarity, consider linear first stages for continuous treatments:

Here \(m_1\) and \(m_2\) are conditional mean functions, and \(V_1,V_2\) are the treatment disturbances. In a linear first stage, they are the population residuals after projecting each treatment on the included covariates and instruments. In practice, the residuals are estimated. For nonlinear or discrete first stages, ordinary linear residuals are generally not the appropriate controls; generalized residuals or controls derived from a correctly specified joint treatment model are needed.

A causal interpretation requires more than estimating these equations. The instruments must be relevant for the treatments, independent of the structural outcome disturbance conditional on \(X\), and excluded from the outcome except through the treatments. The control-function sufficiency condition must also hold: conditional on \(X,V_1,V_2\), the remaining outcome disturbance must not retain dependence on the treatment choices. This condition is the bridge that permits residual inclusion to address endogeneity; it is not guaranteed merely because residuals have been computed.

In a linear first-stage system, the standard relevance condition is a rank condition. After accounting for \(X\), the excluded instruments must shift the two treatments in linearly independent ways. If there are exactly two excluded instruments, let \(\Pi\) be the \(2\times2\) matrix of their coefficients in the two first stages. The condition is that \(\Pi\) have rank two. With more instruments, the corresponding coefficient matrix must still have rank two. Having two instrument names is not sufficient if they generate only one effective direction of treatment variation.

These conditions do not establish that the instruments are valid. Relevance can be examined empirically, but exclusion and independence require substantive justification. The included covariates \(X\) must also be handled consistently in both stages, and the structural outcome model must be correctly specified for the interpretation claimed.

Joint Residual Inclusion in a Nonlinear Outcome Model

For a binary outcome, one possible structural model is a probit latent-index model. Suppose \(Y=1\) when the latent outcome \(Y^*\) is positive, and specify:

If \(\varepsilon\) is standard normal and independent of \(D_1,D_2\) conditional on \(X,V_1,V_2\), then the conditional probability is the standard normal CDF of the index. One illustrative control function is \(h(V_1,V_2)=\gamma_1V_1+\gamma_2V_2+\gamma_{12}V_1V_2\). The interaction permits the outcome association with one disturbance to vary with the other. It is a modeling choice, not a universal requirement; a different application may justify another joint function.

The parameters \(\beta_1,\beta_2\) describe shifts in the specified latent index conditional on the other treatment, covariates, and controls. They are not, by themselves, changes in outcome probability. Probability-scale effects require comparing predicted probabilities under treatment values and, for population summaries, averaging those contrasts over a defined covariate distribution.

Why the Joint Control Works

The justification is conditional, not automatic. Suppose the first stages give \(D_j=m_j(X,Z)+V_j\), and suppose the structural outcome disturbance is independent of the treatment choices once \(X,V_1,V_2\) are held fixed. Then the conditional outcome distribution can be written as a function of the treatments, observed covariates, and the two control variables. Including an adequate function \(h(V_1,V_2)\) represents that conditional relationship in the outcome model.

The two residuals must be considered together because their dependence may contain information about the unobserved factors that jointly influence both treatments and the outcome. If the outcome relationship depends on a combination of the disturbances, two additive residual terms can fail to represent it. Conversely, adding a residual interaction without justification does not repair invalid instruments or an incorrect control function.

This argument identifies the role of the controls, but it does not uniquely determine \(h\). The required function depends on the treatment and outcome models and on the way unobserved causes enter them. A probit index with linear residual terms and an interaction is one parametric specification. It should be defended as an approximation to the relevant conditional structure, not treated as a general theorem that a particular polynomial always suffices.

Discrete Treatments and Generalized Residuals

For a binary treatment, the fitted probability is not an observed treatment value, and subtracting it from the treatment is not generally equivalent to a control based on the treatment’s latent disturbance. For example, if a binary treatment follows a probit selection equation with index \(\eta\) and a standard normal latent disturbance \(U\), its generalized residual is the conditional mean of \(U\) given the observed treatment and predictors. For \(D=1\), this is \(\phi(\eta)/\Phi(\eta)\); for \(D=0\), it is \(-\phi(\eta)/[1-\Phi(\eta)]\), where \(\phi\) and \(\Phi\) are the standard normal density and CDF.

Diagnostics, Alternatives, and Limitations

Diagnostics can reveal problems but cannot certify causal validity. Inspect first-stage relevance for each treatment, including whether estimated instrument effects are too weak or nearly collinear to distinguish treatment variation. Examine residual distributions and their joint pattern, and check whether predictions and residuals show serious departures from the assumed first-stage form. In the outcome stage, compare substantively plausible control-function forms and assess whether conclusions depend strongly on including a joint term. Such comparisons are specification checks, not proof that the selected function is correct.

A major practical issue is uncertainty from estimating the first stages. Treating estimated residuals as if they were observed can produce incorrect standard errors. Inference should account for both stages, for example through an appropriate stacked estimating-equation variance calculation or a bootstrap that repeats the first-stage and outcome estimation together, with resampling that respects the sampling design.

The approach is distinct from ordinary two-stage least squares. In a linear outcome model, replacing endogenous treatments with their fitted values is associated with familiar linear IV procedures. In a nonlinear outcome model, substituting fitted treatment values into a probit or logit generally does not preserve the required conditional outcome model and does not, by itself, solve endogeneity. Residual inclusion instead models outcome dependence conditional on the first-stage controls. A full-information joint likelihood is another option: it specifies and estimates the treatment and outcome distributions together, but can be more sensitive to joint distributional assumptions. Structural mean models target different structural parameters through their own assumptions and estimation strategies.

Control-function estimates can be unreliable when instruments are weak, when excluded instruments are invalid, when the first-stage distributions are misspecified, or when the chosen joint residual function does not adequately capture dependence. Strong functional-form assumptions can appear to identify effects even when the instruments provide little independent variation; this should not be mistaken for robust empirical identification. Results should be reported with the instrument rationale, first-stage specification, outcome model, control terms, and uncertainty procedure made explicit.

Common Mistake

• Wrong reasoning: “There are two endogenous treatments, so I can include one residual for each and need not consider how the residuals relate.”
• Correction: Both controls are required, but their joint dependence may matter as well. Specify and justify a function of \(V_1,V_2\) appropriate to the model; assess whether conclusions are sensitive to plausible joint terms. No choice of residual function compensates for invalid or uninformative instruments.
• Wrong reasoning: “The probit coefficient on \(D_1\) is the percentage-point effect of treatment 1.”
• Correction: It is a conditional latent-index coefficient. Compute predicted probabilities or defined probability contrasts, and interpret them under the identification and model assumptions.