Treatment-Effect Heterogeneity In Nonlinear IV Models

Why Heterogeneity Is Harder in Nonlinear IV Models

A fitted treatment coefficient can look like a single answer to the question “What is the effect?” In a nonlinear instrumental-variable (IV) model, that interpretation is especially risky. The coefficient may describe a shift in a latent index, while the causal contrast of interest is a difference in outcome probabilities or means. Those are not generally the same quantity. In addition, IV identifies effects through variation induced by the instrument, and that variation need not represent every subgroup equally.

The key distinction is between a model parameter and a causal estimand. Heterogeneity may be represented by treatment interactions in a structural model, may arise from the nonlinear link even without such interactions, or may appear in subgroup IV effects because different subgroups have different distributions of compliers. A subgroup interpretation requires assumptions connecting all these pieces.

Index Interactions and Causal Contrasts

Consider a binary-outcome structural probit model with a binary treatment and covariates \(X\):

Here \(d\) is an intervention setting of treatment. The model specifies the potential outcome under that setting, not merely the association among people observed with \(D=d\). For the following risk formula, assume the structural outcome error has a standard normal marginal distribution conditional on \(X\), so that \(P(Y(d)=1\mid X=x)=\Phi(\eta_d(x))\). In an IV setting, treatment may still be associated with \(\varepsilon\); the instrument and the model’s identification assumptions are what support estimation of the structural parameters.

The treatment coefficient \(\beta_D\) is the treatment shift in the latent index at \(X=0\). The interaction \(\beta_{DX}\) changes that shift as \(X\) changes. Neither coefficient is, by itself, a risk difference. Also, even if \(\beta_{DX}=0\), the risk difference can vary with \(x\), because the same index shift passes through different parts of the nonlinear link.

Why Nonlinearity Creates Heterogeneity

The risk difference implied by the model can be written as an integral over the index shift. Let \(\delta(x)=\beta_D+\beta_{DX}x\) and \(\eta_0(x)=\beta_0+\beta_Xx\). Then:

This equality follows from the definition of the normal cumulative distribution function: \(\Phi(b)-\Phi(a)=\int_a^b\phi(t)\,dt\). The risk contrast therefore depends both on the index shift and on the baseline index, through the values of the density \(\phi(t)\) along that interval. For a small shift, a first-order approximation is \(\tau(x)\approx\phi(\eta_0(x))\delta(x)\). It is an approximation, not an exact causal effect formula. This derivation explains why a constant latent-index treatment coefficient can imply different risk differences across baseline-risk subgroups.

Consequently, evidence that \(\beta_{DX}\) differs from zero is evidence of interaction on the model’s index scale, conditional on the model being correct. It is not a complete test of whether risk differences vary. Conversely, a varying risk difference does not require a nonzero index interaction. To discuss heterogeneity, specify the outcome scale first, then calculate or estimate contrasts on that scale.

Control Functions: Condition, Then Standardize

A control-function model accounts for endogeneity by including a control variable from the treatment-assignment process in the outcome model. The basic logic is that, under a suitable control-function sufficiency assumption, conditioning on the control variable makes the remaining treatment variation independent of the relevant outcome error. A fitted outcome regression might, for example, specify the conditional mean as \(m(d,x,v)=g^{-1}\{\eta(d,x,v)\}\), where \(v\) is a valid control variable and \(g^{-1}\) is the inverse link.

The causal subgroup mean under treatment setting \(d\) is not automatically \(m(d,x,v)\) at the observed value of \(v\). To obtain the mean for a target subgroup \(X=x\), the model-based prediction generally needs to be averaged over that subgroup’s target distribution of the control variable:

This calculation requires more than a convenient regression form. The control variable must satisfy the model’s sufficiency conditions; the outcome mean must be correctly specified over the relevant values of \(d,x,v\); and the distribution used for standardization must describe the intended target subgroup. With a generated residual, its construction and the first-stage model also matter. A generalized residual calculated from observed treatment is not automatically an observed, pre-treatment latent rank that can be held fixed under intervention. The structural control-function interpretation must justify the averaging operation.

Subgroup IV Effects and Compliers

A different source of heterogeneity comes from the population represented by the IV comparison. With a binary instrument \(Z\) and binary treatment \(D\), the familiar subgroup Wald ratio is \((E[Y\mid Z=1,X=x]-E[Y\mid Z=0,X=x])/(E[D\mid Z=1,X=x]-E[D\mid Z=0,X=x])\). Under instrument independence within \(X\), exclusion, relevance, and monotonicity, this ratio identifies the average treatment effect among compliers in subgroup \(X=x\), not generally the average effect for everyone in that subgroup. Monotonicity rules out individuals whose treatment moves opposite to the instrument; exclusion says the instrument affects outcomes only through treatment.

If the instrument changes treatment for different proportions of people across subgroups, a pooled IV effect can weight subgroup complier effects differently from an average across subgroups. Parametric nonlinear models can impose additional structure and produce other estimands, but a fitted interaction does not remove the need to state what population and effect scale the result represents.

Diagnostics, Alternatives, and Limits

A plausible-looking interaction can be misleading when the model is weakly supported or too restrictive. Before interpreting subgroup contrasts, inspect whether the instrument predicts treatment within the subgroups being reported, whether there is meaningful overlap in covariates and control-function values, and whether predictions rely on extrapolation. Very small subgroup first stages can make subgroup IV estimates unstable. In a nonlinear model, inspect predicted probabilities for impossible or implausible patterns and compare reasonable specifications of the outcome link and control-function terms. These checks can reveal problems; they cannot prove the exclusion restriction or validate the structural distributional assumptions.

The main alternatives answer different questions. A reduced-form analysis describes the effect of the instrument on the outcome, not the treatment effect unless additional identification conditions translate it. A subgroup Wald analysis makes its complier target explicit under binary-IV assumptions, but may be imprecise within small subgroups. A structural nonlinear model can use stronger functional-form and distributional assumptions to estimate standardized potential-outcome contrasts, potentially for a broader target population. It gains interpretability only to the extent those assumptions are credible. If treatment interactions or control-function interactions are omitted, the resulting model may force away heterogeneity; if they are added without enough data or support, estimates may be unstable.

The practical conclusion is not that parametric nonlinear IV models cannot describe heterogeneity. They can, but the fitted coefficient alone rarely settles the causal question. Report the outcome scale, subgroup definition, standardization distribution, and identification assumptions; give subgroup contrasts with uncertainty; and distinguish model-based population effects from complier effects.