Instrumental Variables For A Count Outcome

Why Use an Instrument for a Count Outcome?

A count outcome records how many times an event occurs during a defined period: for example, the number of hospital visits in a year. A count regression can describe how the mean count differs by treatment, but an observed treatment group may differ from an untreated group in causes of the outcome that were not measured. An instrumental variable (IV) analysis instead uses variation in treatment associated with an instrument, while relying on assumptions that make this variation informative about a causal effect.

Here the outcome is a nonnegative count \(Y\), the treatment is binary \(A\in\{0,1\}\), and the instrument is binary \(Z\in\{0,1\}\). Baseline covariates are denoted by \(X\). The goal is not to find a model merely because \(Y\) is a count. It is to define a causal contrast and specify a structural mean model that connects that contrast to an IV estimating equation.

An expected count can be fractional even though each observed outcome is an integer. For example, a mean ratio of 1.5 means a 50% higher expected count under the model; it does not mean that every person experiences half an additional event. If the untreated expected count is 2 events per person-year, a 1.5 mean ratio corresponds to 3 expected events per person-year, a difference of 1 event per person-year.

Model, Assumptions, and Identifying Moment

Let \(Y(a)\) denote the potential count outcome under treatment level \(a\), and let \(A(z)\) denote the treatment that would be received under instrument level \(z\). A causal IV analysis requires consistency (the observed outcome is \(Y(A)\)), no interference between units, and exclusion: changing \(Z\) does not change \(Y\) except through its effect on \(A\). It also requires conditional instrument independence: given \(X\), \(Z\) is independent of the potential outcomes and potential treatment assignments relevant to the analysis. Instrument assignment must have positive probability at each covariate pattern used for inference.

The structural mean restriction is an additional modeling assumption. For the multiplicative model used here, define the blipped-down outcome as \(Y\exp(-\psi A)\). The model asserts that, at the correct \(\psi\), its conditional mean does not depend on the instrument after conditioning on \(X\):

Here \(\mu(X)\) is an unrestricted or suitably modeled baseline mean function. This restriction encodes the causal treatment contrast on the multiplicative scale. It is not obtained from instrument independence alone: the structural mean model must also be a reasonable representation of the treatment effect. In particular, interpreting one \(\psi\) as a common structural log mean ratio requires the model's homogeneity restriction to be appropriate. With important effect heterogeneity, a single fitted value may instead be a model-dependent summary.

Define the instrument's conditional mean \(e(X)=P(Z=1\mid X)\). For binary \(Z\), the residual \(Z-e(X)\) has conditional mean zero given \(X\). Multiplying the structural mean restriction by this residual and taking expectations removes the unknown \(\mu(X)\), yielding the estimating equation:

This is the population moment used to estimate \(\psi\). Relevance is also necessary: treatment must vary with the instrument, conditional on \(X\), in the study population. For a binary instrument, inspect \(P(A=1\mid Z=1,X)\) versus \(P(A=1\mid Z=0,X)\), or a suitable conditional first-stage summary. If these probabilities are nearly equal, the moment contains little information about the treatment effect, and estimates can be unstable.

Deriving and Solving the Estimating Equation

The conditional mean restriction says that the blipped-down outcome has the same conditional mean for \(Z=1\) and \(Z=0\), at each \(X\). Since \(E[Z-e(X)\mid X]=0\), a quantity depending only on \(X\), including \(\mu(X)\), contributes zero to the residual-weighted expectation. Thus:

In a sample, replace the population expectation with an average and estimate \(e(X)\), for example with a correctly specified model for \(P(Z=1\mid X)\). Solve

The equation is generally nonlinear in \(\psi\), so it can be solved numerically. The count outcome itself is not required to follow a Poisson distribution for this moment equation to be valid; the key requirements are the IV assumptions, the structural mean restriction, and appropriate inference for the estimating procedure. Standard errors should account for estimation of the instrument model and the moment equation, commonly through a suitable sandwich estimator or bootstrap. If observations are clustered, resampling or variance estimation must respect that clustering.

For a binary instrument with no covariates and equal numbers observed in the two instrument groups, the sample moment is proportional to the difference between the two groups' mean blipped-down outcomes. The equation can then be solved from group summaries. With covariates, the appropriate residual is \(Z-\hat e(X)\), not automatically the residual from a linear regression of \(Z\) on a few covariates.

Using Covariates Correctly

When \(X\) predicts instrument assignment, the conditional instrument mean matters. The moment uses \(Z-e(X)\), where \(e(X)=P(Z=1\mid X)\). If \(e(X)\) is estimated incorrectly, the residual may retain a conditional mean that varies with \(X\), and the resulting moment need not remove the baseline term \(\mu(X)\). Flexible or well-justified modeling of \(e(X)\), adequate covariate overlap, and sensitivity to reasonable specifications are therefore important.

Ordinary covariate-adjusted two-stage least squares (2SLS) is not automatically the same calculation. Its linear projection residual for \(Z\) is orthogonal to the included regressors, but it need not equal \(Z-E[Z\mid X]\). The two coincide when the covariate specification correctly represents the conditional instrument mean, or under other conditions that make the residual conditionally mean-zero for the relevant baseline outcome. Without such conditions, a linear residual may remain associated with a nonlinear function of \(X\), so it may not implement the displayed moment.

Why a Naive Count Regression Can Differ

A naive count regression relates \(Y\) directly to observed treatment \(A\), without using \(Z\) as an instrument. For example, a Poisson regression with a log link may report an adjusted association as a rate ratio. That coefficient is not automatically causal: if treatment receipt is related to unmeasured causes of the count, treatment groups differ for reasons other than treatment. The IV structural mean estimate instead uses the instrument-induced treatment variation and depends on the IV and structural model assumptions.

Diagnostics, Alternatives, and Limits

Check the instrument's first stage, the distribution of \(X\) across instrument groups, and whether both instrument levels occur across the covariate patterns used. A weak first stage leads to imprecise estimates and can make conventional confidence intervals unreliable; use weak-instrument-robust inference when warranted rather than relying only on a routine standard error. Examine whether the fitted moment has a plausible root and whether estimates are sensitive to the instrument model, influential observations, and the structural mean specification. A moment that has no plausible solution is a warning about the model, data, or chosen estimating procedure, not permission to report an arbitrary root.

Instrument relevance can be assessed in the observed data, but exclusion and conditional independence cannot generally be verified from the outcome data alone. They require subject-matter justification and design knowledge. The analysis is also vulnerable to poor overlap in instrument assignment, measurement error, selection, and misspecification of the structural mean contrast. A single \(\psi\) may be inadequate when effects vary substantially across people or covariate strata.

The structural mean approach is useful because it defines an interpretable count-scale causal contrast without requiring the count distribution itself to be Poisson. It differs from a naive Poisson regression, which models an outcome mean conditional on observed predictors, and from 2SLS, which targets an additive linear contrast on its outcome scale. Those methods answer different model-based questions. The IV structural mean parameter should not be called a complier average effect without additional assumptions and a derivation connecting the target parameters.