Why Nonlinear IV Results Are Easy to Misread
Instrumental-variable analysis becomes especially delicate when the outcome model is nonlinear. A coefficient in a logit, probit, or count model is expressed on the model's link scale; it is not automatically a risk difference, a risk ratio, or a population-average causal effect. Likewise, predicting an endogenous treatment from an instrument and inserting that prediction into a nonlinear outcome equation is not generally equivalent to a valid IV analysis. Finally, a control-function term can model residual dependence, but its presence or statistical significance does not establish that the instrument satisfies the causal assumptions.
Let \(Y\) denote an outcome, \(D\) an endogenous treatment, \(Z\) an instrument, and \(X\) observed covariates. Write \(Y(d)\) for the outcome that would be observed under treatment level \(d\). A causal interpretation requires more than fitting a nonlinear regression. At minimum, the proposed instrument must be relevant to treatment, independent of potential outcomes and treatment-related unmeasured causes conditional on \(X\), and excluded from the outcome except through treatment. The treatment and outcome models must also be appropriate for the particular estimator. In settings with heterogeneous effects, additional assumptions determine which population or local effect is identified.
Error 1: Reading a Link Coefficient as a Causal Effect
In a logistic regression, a coefficient is a change in log odds for a one-unit increase in a predictor, conditional on the other included predictors. Exponentiating that coefficient gives a conditional odds ratio. Neither quantity is a risk difference. In an IV analysis, even this link-scale interpretation becomes causal only if the structural model and identification assumptions justify it. A coefficient from an outcome regression that simply includes \(D\) is not made causal merely by calling \(Z\) an instrument.
Worked Example: Logistic Coefficient Versus Risk Difference
Suppose a hypothetical structural logistic model for a binary outcome has linear predictor \(-1.3863 + 0.7000D\), with other covariates held fixed. Assume, for this illustration, that the model is appropriate for the causal contrast being considered. The baseline risk at \(D=0\) is approximately \(0.2000\), because \(\operatorname{logit}(0.2000)=\log(0.2/0.8)\approx-1.3863\).
The treatment coefficient is \(0.7000\) on the log-odds scale. The corresponding odds ratio is
At \(D=1\), the linear predictor is \(-1.3863+0.7000=-0.6863\), so the predicted risk is
Thus, at this covariate value, the risk difference is approximately \(0.3349-0.2000=0.1349\), or 13.49 percentage points. The coefficient \(0.7000\) is not a 0.7000 increase in risk; it corresponds to a conditional odds ratio of about 2.014. The 13.49-percentage-point contrast is specific to the stated baseline risk. If the model and IV assumptions do not support a causal interpretation, these are model-based associations rather than causal effects.
For nonlinear links, the marginal effect can vary with the covariate values and baseline risk. A population-average causal risk difference generally requires predicting the outcome under each treatment level for the target population and averaging those predictions, or using an estimator explicitly targeting that contrast. It cannot be read off the coefficient alone. Odds ratios are also non-collapsible: a conditional odds ratio can differ from a population-average odds ratio even without confounding. Do not interpret this difference, by itself, as evidence of bias.
Error 2: Substituting Fitted Treatment Values Without Justification
In two-stage predictor substitution (2SPS), an analyst predicts \(D\) from \(Z\) and \(X\), then substitutes the fitted treatment \(\hat D\) into an outcome model. In linear models, substitution may be justified under particular conditions. In a nonlinear outcome model, however, the conditional mean of the outcome is not generally obtained by applying the nonlinear outcome function to the conditional mean of treatment. Nonlinearity, unobserved heterogeneity, and the estimator's target all matter.
The basic mathematical issue is that, for a nonlinear function \(g\), \(E[g(W)]\) is generally not equal to \(g(E[W])\). For example, let \(g(t)=1/(1+\exp(-t))\), the logistic function. Then
The discrepancy shows why replacing a varying quantity by its average inside a nonlinear function can change the answer. For an endogenous treatment, the relevant discrepancy is not resolved simply by estimating \(E[D\mid Z,X]\). The outcome model needs a justification that connects its fitted values to the desired causal contrast.
Worked Example: A Fractional Fitted Treatment Is Not an Endpoint Contrast
Consider a hypothetical first stage with \(E[D\mid Z=0]=0.2000\) and \(E[D\mid Z=1]=0.8000\). Suppose an analyst specifies the outcome prediction function \(g(t)=1/(1+\exp[-(-2+t)])\) and plugs the first-stage means into it.
For \(Z=0\), the fitted treatment is \(0.2000\), and the plug-in prediction is
For \(Z=1\), the fitted treatment is \(0.8000\), giving
The plug-in difference is about \(0.2315-0.1419=0.0896\). But if the same function is interpreted as a causal risk model for binary treatment, its endpoint predictions are \(g(0)=1/(1+\exp(2))\approx0.1192\) and \(g(1)=1/(1+\exp(1))\approx0.2689\), a difference of approximately \(0.1497\). The plug-in calculation evaluates the model at fractional treatment values; it does not calculate the \(D=0\) versus \(D=1\) contrast. This numerical comparison does not prove that every 2SPS estimate is biased. It shows why a fitted-value substitution needs a model-based argument matching the treatment scale, outcome model, and target estimand.
A useful check is to ask what values the fitted treatment can take and what the outcome model says those values mean. For binary \(D\), \(\hat D\) is often a probability between zero and one, whereas the observed treatment takes only zero or one. Treating that probability as an observed treatment level may impose an unmotivated extension of the structural model. In other settings, even when fitted values lie in the observed range, substitution may still fail because conditional averaging and nonlinear transformation do not commute.
Error 3: Treating a Control-Function Term as Proof of Instrument Validity
A control-function approach adds a function of first-stage residual information to the outcome model to account for dependence between treatment and unobserved outcome causes. With a binary first stage, a generalized residual is one possible residual measure. Under a probit first stage with \(P(D=1\mid Z,X)=\Phi(\eta)\), define the generalized residual as
Here \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution function. A nonlinear outcome model might then include \(R\), for example as an additional predictor in a logit or count regression. The causal argument requires the first-stage model and the control-function relationship to be correctly specified, as well as valid instrument assumptions. A control function is an adjustment device under assumptions; it is not a test or certificate of those assumptions.
Worked Example: A Generalized Residual Does Not Validate the Instrument
Suppose a hypothetical binary-treatment probit first stage gives \(\eta=0\) for an observation. Then \(\phi(0)=0.3989423\) and \(\Phi(0)=0.5\). If \(D=1\), the generalized residual is
If \(D=0\), it is
Suppose an outcome regression that includes this residual estimates a nonzero residual coefficient, with a hypothetical \(p\)-value of \(0.02\). That result may indicate residual dependence of the type represented by the fitted control-function specification. It does not establish that \(Z\) is independent of potential outcomes, that \(Z\) has no direct effect on \(Y\), or that the first-stage model is correctly specified. For example, an instrument that directly affects the outcome could still be associated with a nonzero control-function term. The numerical residual is a model-derived quantity, not a diagnostic that verifies exclusion or independence.
A nonsignificant residual coefficient does not prove exogeneity either. It can reflect low power, a misspecified residual function, or an inadequate model for the dependence. In addition, significance tests for endogeneity address a narrower question than instrument validity. Instrument relevance, exclusion, and independence need separate substantive justification and, where possible, appropriate empirical checks. No single statistical test can verify the untestable parts of the IV assumptions.
A Practical Diagnostic Sequence
When a nonlinear IV result seems surprising, work through the following checks before interpreting its coefficient.
Specify whether the target is a risk difference, risk ratio, odds-scale parameter, mean count contrast, or another quantity, and identify the population to which it refers.
Assess relevance, conditional independence, exclusion, and any assumptions required for the intended treatment-effect interpretation. Explain the substantive basis for each.
Check whether the first-stage and outcome specifications, residual construction, and estimation procedure justify the reported parameter. Do not assume that an approach valid in a linear model transfers to a nonlinear one.
Calculate predictions or contrasts on the scientific outcome scale. Examine sensitivity to plausible control-function specifications and look for weak first-stage prediction, poor fit, extrapolation, or influential observations.
These checks can reveal warning signs, but they do not turn an invalid instrument into a valid one. Weak relevance can make estimates unstable and inference unreliable. Poor fit or sensitivity to residual specifications can weaken confidence in the model-based result. A strong first stage and stable estimates do not establish exclusion or independence.
How the Main Approaches Differ
| Approach | What it models | Common interpretive risk |
|---|---|---|
| Two-stage predictor substitution | Uses fitted treatment values in an outcome model | Assuming nonlinear substitution identifies the causal parameter without a supporting model |
| Control-function or residual inclusion | Includes first-stage residual information in an outcome model | Treating the residual term as proof of instrument validity or reading the treatment coefficient as a marginal effect |
| Structural mean models with g-estimation | Targets a specified structural treatment-effect contrast under its identifying assumptions | Ignoring the chosen effect scale or assuming the model identifies effects beyond its stated target |
| Joint likelihood approaches | Specify a joint model for treatment and outcome, such as a bivariate probit model | Overlooking the dependence on distributional and functional-form assumptions |
These methods are not interchangeable recipes. The introductory material on structural mean models and control functions establishes their distinct modeling strategies; the essential practice here is to keep each estimator's assumptions and target explicit. A nonlinear link alone does not select an IV method, and a method's name does not guarantee causal interpretation.
Common Pitfalls and Their Corrections
- Wrong reasoning: “The logistic treatment coefficient is the causal risk increase.” Correction: It is a link-scale coefficient; derive the desired causal contrast from justified counterfactual predictions or use a method targeting that scale.
- Wrong reasoning: “The predicted treatment is an adequate replacement because it is the first-stage expectation.” Correction: In nonlinear models, a function of an expectation need not equal the expectation of a function; justify substitution for the specified model and estimand.
- Wrong reasoning: “The residual coefficient is significant, so the instrument is valid.” Correction: It concerns dependence represented by a particular control-function model. It cannot establish exclusion or independence.
- Wrong reasoning: “A strong first stage and good outcome-model fit confirm the IV assumptions.” Correction: Relevance and fit do not verify the instrument's independence or exclusion restrictions.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A logistic model has a treatment coefficient of \(0.7000\). What quantity does its exponential represent, and why is it not automatically a risk difference?
- For the probit generalized residual at \(\eta=0\), calculate \(R\) for \(D=1\) and \(D=0\), using \(\phi(0)=0.3989423\) and \(\Phi(0)=0.5\).
- An analyst reports a significant control-function residual coefficient. Name one conclusion it may support under the fitted model and two instrument assumptions it does not establish.
- Why can substituting \(E[D\mid Z,X]\) into a nonlinear outcome function give a different result from averaging predictions over treatment-related heterogeneity?
Solutions. (1) \(\exp(0.7000)=2.014\), a conditional odds ratio under the logistic model. It is not a risk difference because the risk difference depends on baseline risk and must be calculated on the probability scale. (2) For \(D=1\), \(R=0.3989423/0.5=0.7979\); for \(D=0\), \(R=-0.3989423/0.5=-0.7979\), to four significant figures. (3) It may support residual dependence represented by that control-function specification, conditional on the model. It does not establish instrument exclusion or independence from potential outcomes. (4) A nonlinear function does not generally commute with averaging: in general, \(E[g(W)]\ne g(E[W])\). Therefore a conditional treatment mean alone need not preserve the outcome prediction or causal contrast.