Marginal Effects And Counterfactual Risks From Nonlinear IV Models

From a Nonlinear IV Fit to Counterfactual Risks

A nonlinear IV model may report an outcome-equation coefficient on a probit or logit scale. That coefficient describes a change in a model’s linear predictor; it is not itself a change in the probability of the outcome. To report a treatment effect as a risk difference or risk ratio, first predict the outcome risk under treatment and under no treatment, then average those risks over a specified target population.

The predictions must come from the fitted structural or control-function model, not from an ordinary outcome regression that treats the endogenous treatment as if it were unconfounded. In particular, the IV assumptions and the model’s distributional or control-function assumptions are what support interpreting the predictions as counterfactual risks.

Model, Assumptions, and the Prediction Rule

For a bivariate probit IV model, one common specification has a latent outcome equation \(Y^*=X^\top\beta+\tau D+\varepsilon\), with \(Y=1\) when \(Y^*>0\), and a latent treatment equation \(D^*=Z^\top\gamma+X^\top\delta+u\), with \(D=1\) when \(D^*>0\). The errors \(\varepsilon\) and \(u\) are standard normal with correlation \(\rho\). The instrument \(Z\) is excluded from the outcome equation.

A causal reading requires more than fitting these equations. The instrument must be relevant for treatment, independent of the relevant potential outcomes conditional on included covariates, and excluded from the outcome equation except through treatment. The equations and joint error distribution must also be correctly specified; the bivariate probit interpretation relies on its distributional assumptions. There must be adequate instrument and covariate support for the population being studied. These assumptions are substantive and cannot be confirmed merely by a good numerical fit.

Under an intervention setting \(D=d\), the outcome latent index becomes \(X^\top\beta+\tau d\). The outcome error has its standard normal marginal distribution, so integrating over that error gives the counterfactual risk conditional on \(X=x\). The treatment-equation error correlation matters for modelling treatment selection and estimating the parameters; it does not mean that the intervention-specific risk is obtained by conditioning on the naturally observed treatment.

Here \(\Phi\) is the standard normal cumulative distribution function. With fitted coefficients, calculate a prediction for each member of the target population, or for each covariate pattern with its target-population weight. Average those predictions separately for \(d=1\) and \(d=0\), then contrast the two averages. If the target population is represented by a sample of \(n\) units, the plug-in standardization is

The average is over the covariate distribution of the stated target population, which need not be the estimation sample. If the target population is represented by strata with weights \(w_j\), use \(\widehat R(d)=\sum_j w_j\widehat P\{Y(d)=1\mid X=x_j\}\), where the weights sum to one.

Why a Link-Scale Coefficient Is Not a Risk Difference

In a probit model, \(\tau\) shifts the latent outcome index for a one-unit treatment change. Because \(\Phi\) is nonlinear, the resulting probability change depends on the starting index. For a binary treatment, the relevant conditional risk difference at covariate value \(x\) is the finite contrast \(\Phi(x^\top\beta+\tau)-\Phi(x^\top\beta)\), not \(\tau\) itself. A derivative such as \(\phi(\eta)\tau\) is a local approximation for a continuous change in an index; it is not the exact binary-treatment contrast.

The same distinction applies to logistic models: a treatment coefficient is a log-odds contrast conditional on the model’s other terms, whereas the marginal risk difference and risk ratio require predicted probabilities and averaging. Even when a coefficient is causal under a correctly specified IV model, its scale and conditioning differ from those of a population-average risk effect.

Averaging Predictions, Not Covariates

For nonlinear links, the risk at the mean covariate value need not equal the mean risk. Standardization therefore predicts under each treatment value at each target covariate pattern and then averages. This also makes the estimand explicit: changing the target population’s covariate distribution can change its average risks and risk difference.

Control-Function Models and Practical Diagnostics

The same risk-first principle applies to other nonlinear IV approaches, but the prediction rule must match the fitted model. A control-function outcome model includes a first-stage residual or generalized residual to account for endogeneity. Do not automatically replace \(D\) by 0 or 1 in an observed-data conditional prediction while holding every residual term fixed: some residual controls depend on the observed treatment and instrument. A causal counterfactual prediction requires a model-specific account of the latent control variable and its distribution under the intervention. If the model does not supply a justified counterfactual prediction rule, its fitted conditional probabilities alone do not establish counterfactual risks.

For a bivariate probit model, a direct diagnostic is to inspect whether predicted risks are within \([0,1]\), compare fitted and observed outcome patterns across covariate and instrument strata, and check for sparse or unsupported combinations that make standardization extrapolative. Also examine first-stage relevance and sensitivity to plausible changes in model specification. These checks can reveal instability or poor fit, but none proves instrument validity, exclusion, or the assumed joint error distribution.

Uncertainty in standardized risks and their contrasts must account for estimation of the model parameters. A delta-method calculation can use the gradient of the standardized estimand and the estimated parameter covariance matrix; a bootstrap can refit the full IV procedure and recompute both risks in each resample. For a bootstrap, resample at the independent sampling unit and preserve any design features that determine the target population. Report an interval for the estimand in context; do not interpret a confidence interval as a probability statement about a fixed parameter.

Choosing and Reporting the Effect

The risk difference is an absolute change in outcome probability and is often useful for expressing the number of additional or prevented outcomes per population size. The risk ratio compares relative risks, but it can be unstable or undefined when the no-treatment risk is zero or close to zero. Both are marginal summaries after standardization. They need not be constant across populations, even when the fitted structural treatment coefficient is common, because baseline risks and covariate distributions differ.

These model-based population contrasts should not be confused with the local effects identified by some IV designs or with an unadjusted association between observed treatment and outcome. Their interpretation depends on the model’s identifying assumptions and on the target population used for averaging. Report the model, target population, treatment-specific risks, contrast scale, units, and uncertainty. If a treatment effect is reported only as a link-scale coefficient, readers cannot directly infer the size of the average risk change.