Structural And Reduced-Form Models For Nonlinear Outcomes

Two Models, Two Questions

With a nonlinear outcome, it is especially important to distinguish a model for the outcome under treatment from a model for the outcome across instrument values. They can both predict probabilities, but those probabilities answer different questions. A structural model asks what the outcome would be under a specified treatment intervention; a reduced-form model asks how the outcome varies with the instrument. Confusing the two can turn a valid description of instrument groups into an unsupported claim about treatment effects.

Let \(Y\) be an outcome, \(D\) a treatment, \(Z\) an instrument, and \(X\) observed covariates. For a binary outcome, \(Y\) takes values 0 and 1, so its conditional mean is a probability. Potential outcomes \(Y(d)\) denote the outcome that would be observed under treatment level \(d\). The structural and reduced-form perspectives connect these quantities to different predictors.

A common structural specification for a binary outcome is a latent-index model:

Here \(U\) collects unobserved determinants of the outcome, and \(\mathbf{1}\{\cdot\}\) equals 1 when its condition holds and 0 otherwise. If \(U\mid X\sim N(0,1)\), the structural probability under treatment \(d\), at covariate value \(x\), is

where \(\Phi\) is the standard normal cumulative distribution function. The coefficient \(\beta\) shifts the latent index. It is not, in general, a change in outcome probability or a risk difference. The probability change depends on the starting index and hence can differ across covariate values.

A reduced-form regression instead models \(E[Y\mid Z,X]\). For binary \(Z\), its conditional means are the outcome probabilities in the two instrument groups. A linear probability model might describe their difference directly; a probit or logit reduced form uses a nonlinear link. In either case, the predictor is \(Z\), not \(D\). The model does not by itself decompose an instrument–outcome pattern into a treatment effect.

Assumptions Behind Interpretation

A structural equation can be given a causal interpretation only when its treatment intervention is well defined and the model’s assumptions support its counterfactual predictions. For the probit expression above, the assumed distribution of \(U\mid X\) is part of the model, not a consequence of observing binary outcomes. If treatment is endogenous, \(D\) may be related to \(U\); fitting a probit of \(Y\) on observed \(D\) does not automatically recover the structural parameters. Instrumental-variable assumptions and additional modeling choices are needed for identification and estimation.

A reduced-form conditional mean \(E[Y\mid Z=z,X=x]\) is always a descriptive quantity for the observed data, subject to adequate data at those values. To interpret differences across \(z\) as causal effects of the instrument, one needs, for example, conditional independence of \(Z\) from relevant potential outcomes given \(X\). To use \(Z\) as an instrument for \(D\), one additionally needs an exclusion restriction: \(Z\) affects \(Y\) only through \(D\). Relevance is also needed for the instrument to generate treatment variation. These assumptions have distinct roles: instrument randomization can support an effect of assignment on \(Y\), while exclusion is needed to attribute that effect to treatment.

The reason the two models cannot generally be swapped is visible by averaging over treatment and unobserved outcome factors. The reduced-form probability is a mixture of structural outcomes across the treatment and latent-error distribution in each instrument group:

The distribution being averaged can change with \(z\), and it can reflect dependence between \(D\) and \(U\). Because the outcome response is nonlinear, evaluating a structural response at a predicted or average treatment does not generally equal this mixture. This is the central distinction: structural predictions concern specified treatment interventions; reduced-form predictions average what happens under the instrument’s observed treatment and latent-factor distributions.

Reduced-Form Predictions and Their Meaning

For a binary instrument, a reduced-form risk difference compares the outcome probabilities at two instrument values. If \(Z\) is randomly assigned, the difference can be interpreted as the causal effect of instrument assignment on the outcome, even when \(Y\) is binary. It is not automatically the effect of treatment. The distinction matters because assignment may change treatment for only some people, and because treatment can be related to unobserved outcome determinants.

A reduced-form probit has its own coefficient scale. If it specifies \(P(Y=1\mid Z,X)=\Phi(a+bZ+cX)\), then \(b\) shifts the reduced-form latent index for \(Z\). It is not a structural treatment coefficient, nor is it a risk difference. To report a probability-scale contrast, calculate predicted probabilities at the two instrument values and subtract them, keeping \(X\) fixed or averaging over a stated covariate distribution.

Structural Predictions Need Appropriate Averaging

Even when a structural model is correctly specified, a probability contrast at one covariate value need not equal an average contrast for a target population. For a binary treatment, a population-average structural risk difference can be calculated by predicting both intervention-specific risks for each covariate pattern and averaging those risks over the target distribution of \(X\). This is often called standardization. It preserves the nonlinearity rather than applying the link function to an average covariate value.

Model Checks, Alternatives, and Limits

Structural and reduced-form models are complementary rather than competing descriptions of the same coefficient. A structural model can answer intervention questions and support predictions under alternative treatment levels, but its conclusions depend on structural form, distributional assumptions, and identification. A reduced-form model is closer to the observed instrument–outcome relationship and can provide a useful assignment-effect estimate when instrument assignment is exogenous. It generally does not provide the full treatment-response function.

Useful checks include comparing predicted and observed outcome probabilities across instrument and covariate groups, checking whether predictions are systematically too high or low, and examining whether conclusions change under reasonable alternative specifications. For structural predictions, also check whether the treatment and covariate combinations being predicted are supported by the data; extrapolation can make an apparently precise model-based prediction unreliable. A good fit to observed probabilities does not prove instrument validity or validate counterfactual predictions. Conversely, a reduced-form model can be useful even when it cannot answer a structural treatment question.

A frequent mistake is to take a reduced-form coefficient on \(Z\), rename it the effect of \(D\), and interpret it as a structural treatment effect. That reasoning skips the role of the first stage, exclusion, and the assumptions connecting instrument-induced treatment changes to the target effect. The correction is to state which predictor generated the coefficient, identify its scale, and name the assumptions needed for the causal interpretation. For nonlinear models, report predicted probabilities or probability contrasts when those are the quantities of substantive interest.