Continuous Instruments and Local IV

From Instrument Changes to Local Effects

A binary instrument can compare outcome and treatment changes between two instrument values. A continuously varying instrument allows a different question: what treatment effect is associated with a small change in the instrument around a particular value? Under suitable assumptions, the ratio of the resulting outcome slope to the treatment-probability slope identifies the effect for individuals at a specific margin of treatment take-up.

That interpretation is stronger than saying that the instrument predicts treatment, and more local than an average treatment effect. The key is a model in which instrument values move a common threshold for treatment take-up. The location of that threshold determines which latent-resistance group is marginal at each instrument value.

Assumptions and the Threshold Model

The argument uses potential outcomes and a scalar latent resistance \(U\), normalized to be uniformly distributed on \([0,1]\) conditional on \(X=x\). The treatment-selection model is \(D=1\{U\leq p(Z,x)\}\), where \(p(z,x)=P(D=1\mid Z=z,X=x)\). Thus \(p(z,x)\) is both the treatment probability at \(z,x\) and the threshold in the resistance scale. Individuals with lower \(U\) are more inclined to take treatment.

• Conditional independence: \(Z\) is independent of \((Y(0),Y(1),U)\) conditional on \(X\). This excludes confounding of instrument variation by the potential outcomes or latent resistance, given the covariates.
• Exclusion: the instrument affects the observed outcome only through treatment; the outcome under treatment status \(d\) is \(Y(d)\), not a separate function of \(Z\).
• Monotone treatment selection: treatment follows the threshold rule \(D=1\{U\leq p(Z,X)\}\). As the instrument changes, it shifts the threshold rather than causing some people to move into treatment while others move out in an unrestricted way.
• Regularity and relevance: the conditional means exist, \(p(z,x)\) is differentiable at the value considered, and its derivative with respect to \(z\) is nonzero. The treatment probability must respond locally to the instrument.

Here, monotonicity concerns how treatment take-up responds to the instrument. It is not the separate outcome restriction sometimes called monotone treatment response, \(Y(1)\geq Y(0)\). That outcome restriction is not required for the derivative-ratio result. If it is imposed as an additional assumption, it implies nonnegative individual treatment effects, but it is conceptually distinct from monotone treatment selection.

Define \(m_d(u,x)=E[Y(d)\mid U=u,X=x]\) and \(\Delta(u,x)=m_1(u,x)-m_0(u,x)\). The quantity \(\Delta(u,x)\) is the mean treatment effect at resistance level \(u\), conditional on \(X=x\). It need not be constant across \(u\); that heterogeneity is precisely why a local effect can differ from a population average.

Deriving the Local IV Ratio

For a fixed \(x\), write \(g(z,x)=E[Y\mid Z=z,X=x]\). Conditional independence and exclusion let us calculate this mean by integrating the potential-outcome means over the resistance values assigned to each treatment status. People with \(U\leq p(z,x)\) receive treatment; those with \(U>p(z,x)\) do not. Since \(U\mid X=x\) is uniform,

Rearranging this expression is important. It gives a term that does not depend on the instrument and a single integral with a moving upper endpoint:

Suppose \(\Delta(u,x)\) is continuous at \(u=p(z,x)\). The fundamental theorem of calculus and the chain rule then give the outcome-mean derivative. The treatment mean is \(E[D\mid Z=z,X=x]=p(z,x)\), so its derivative is the first-stage slope.

When the first-stage slope is nonzero, division cancels the instrument-induced change in the threshold:

This is the local IV estimand. It identifies the conditional mean treatment effect at the resistance threshold \(u=p(z,x)\), not necessarily the effect for every individual who changes treatment over a broad interval of instrument values.

Regularity: Differentiate the Combined Mean

The derivation does not require \(m_0\) and \(m_1\) to be individually continuous at the threshold. It requires continuity of their difference \(\Delta\) there, so the combined integral has a derivative. This distinction matters: separate moving-endpoint integrals can each have a corner even when their sum is differentiable.

Local Slopes and Finite Instrument Changes

A derivative ratio uses an infinitesimal instrument change at a particular \(z\). A finite comparison between \(z_0\) and \(z_1\) generally has a different interpretation. Under the same threshold model, if \(p(z_1,x)>p(z_0,x)\), the finite outcome change is the integral of \(\Delta(u,x)\) over the resistance interval between the two thresholds. Dividing by the treatment-probability change averages those effects over that interval. It equals the local effect at one endpoint only under additional conditions, such as a constant treatment effect over the interval.

Interpretation, Scope, and Common Pitfalls

The local IV result is useful when treatment effects vary with latent resistance: it explains what a continuous instrument's local outcome slope, relative to its local first-stage slope, can identify. It does not by itself identify the average treatment effect, and it does not establish that the effect is representative of people whose resistance levels lie elsewhere. With covariates, the result is conditional on \(X=x\); combining results across covariate values requires a separately specified aggregation.

• Calling the ratio an ATE. The ratio identifies \(\Delta(p(z,x),x)\), a local mean effect at a threshold, rather than an average over the full population.
• Using a finite Wald ratio as a derivative ratio. A finite instrument contrast generally averages effects across the threshold interval it traverses.
• Ignoring a weak local first stage. If \(\partial p(z,x)/\partial z=0\), the derivative ratio is undefined. If it is near zero, estimation can be unstable even when the population ratio is mathematically defined.
• Assuming an instrument is valid because it is continuous. A continuum of instrument values does not establish conditional independence or exclusion; these remain substantive assumptions.
• Differentiating the two moving-endpoint integrals separately. Their derivatives may fail to exist when the component means are discontinuous. Rewrite the outcome mean using the fixed baseline integral and the continuous difference \(\Delta\) before applying the chain rule.

The identification statement is about population conditional means and their derivatives. Estimating those derivatives from data requires additional statistical choices, such as how to smooth the conditional means and how to quantify uncertainty. Those choices do not replace the causal assumptions or change the target estimand.