Bounding Effects Under Limited Exclusion Violations

Why Bound an Exclusion Violation?

The exclusion restriction says that an instrument affects the outcome only by changing treatment. In an encouragement study, this means that assignment to the encouragement has no effect on the outcome for someone whose treatment status is held fixed. The restriction can be difficult to defend: an encouragement might also change attention, expectations, or access to related services. Treating exclusion as either exactly true or entirely unusable can conceal a more informative possibility: the instrument’s direct contribution may be limited to a scientifically credible range.

This tutorial derives bounds for a complier treatment effect when that direct contribution is known only to lie in an interval. The result is a sensitivity bound, not automatically a sharp description of every effect compatible with all the observed data. That distinction matters because restrictions such as the range of a binary outcome can rule out values inside the algebraic interval.

Setup and Assumptions

Let $Z$ be a binary instrument, $D$ a binary treatment, and $Y$ an outcome. Write $D(z)$ for the treatment that would be received under instrument value $z$, and $Y(d,z)$ for the outcome under treatment $d$ and instrument value $z$. The discussion uses a randomized instrument, so assignment is independent of these potential outcomes. Consistency means that the observed treatment and outcome equal the potential values corresponding to the instrument and treatment actually received. Assume no interference between units and that the instrument is relevant.

Also assume monotonicity: $D(1)\geq D(0)$ for every unit. Thus, there are no defiers. Units with $D(1)>D(0)$ are compliers; units with $D(1)=D(0)=1$ are always-takers; and units with $D(1)=D(0)=0$ are never-takers. This tutorial’s target is the average treatment effect among compliers when $Z=1$, denoted $\tau_C^{(1)}$. With direct effects allowed, specifying the instrument context for the treatment effect is important.

Definition: The first stage is $FS=E[D\mid Z=1]-E[D\mid Z=0]$, and the reduced form is $RF=E[Y\mid Z=1]-E[Y\mid Z=0]$. Under random assignment and monotonicity, $FS=P(\text{complier})$. Define the aggregate direct-effect contribution by $\delta=E[Y(D(0),1)-Y(D(0),0)]$, the average change in outcome caused by changing the instrument from 0 to 1 while holding each unit’s treatment at $D(0)$. Define $\tau_C^{(1)}=E[Y(1,1)-Y(0,1)\mid \text{complier}]$.

The direct-effect quantity $\delta$ is in outcome units and is an average over the population. It is the direct contribution in the decomposition below, not necessarily the direct effect for every person or at every treatment level. A restriction on some other direct-effect parameter can be used only after showing how it implies a restriction on this $\delta$. For example, if the direct effect is constant across units and treatment levels, that constant equals $\delta$.

Decomposition and the Bound

Random assignment and consistency identify the population reduced form as the average contrast between $Y(D(1),1)$ and $Y(D(0),0)$. Add and subtract $Y(D(0),1)$. The first resulting contrast is a treatment change at $Z=1$; the second is the direct contribution $\delta$.

\begin{aligned} RF &=E[Y(D(1),1)-Y(D(0),0)]\\ &=E[Y(D(1),1)-Y(D(0),1)] +E[Y(D(0),1)-Y(D(0),0)]\\ &=FS\,\tau_C^{(1)}+\delta. \end{aligned}

For the last equality, an always-taker or never-taker has $D(1)=D(0)$, so contributes zero to the first expectation. A complier has $D(1)=1$ and $D(0)=0$, so contributes $Y(1,1)-Y(0,1)$. Averaging over the population therefore gives the proportion of compliers multiplied by their average treatment effect at $Z=1$. Under monotonicity, that proportion is $FS$. This establishes the decomposition without assuming a common treatment effect across units.

Suppose the direct contribution is constrained by $L\leq\delta\leq U$, with $L$ and $U$ expressed in the same outcome units as $RF$. If $FS>0$, rearranging the decomposition gives the following interval.

Theorem: Under the assumptions above, if $FS>0$ and $L\leq\delta\leq U$, then $$ \frac{RF-U}{FS}\leq\tau_C^{(1)}\leq\frac{RF-L}{FS}. $$ This is an outer bound: the assumptions imply that the effect lies in this interval, but additional restrictions from the observed data or the potential-outcome model may make some values in it incompatible.

The ordering of the endpoints follows because subtracting the larger possible direct contribution $U$ gives the smaller treatment-effect value. If $FS<0$, the instrument can be recoded so that the first stage is positive; without recoding, division reverses the endpoint order. If $FS=0$, this decomposition does not permit division to recover the complier effect. A first stage close to zero also makes the bound wide, since the direct-effect uncertainty is magnified by division by $FS$.

Worked Examples

Worked Example: A Positive First Stage and a Narrow Direct-Effect Range

Suppose a hypothetical randomized encouragement has $FS=0.30$ and $RF=0.12$, where $Y$ is a binary outcome and contrasts are expressed as proportions. Assume monotonicity and suppose substantive knowledge supports $-0.03\leq\delta\leq0.05$. Calculate the resulting bound and compare it with exact exclusion.

The first stage is positive, so the bound is $$ \left[\frac{RF-U}{FS},\frac{RF-L}{FS}\right] = \left[\frac{0.12-0.05}{0.30},\frac{0.12-(-0.03)}{0.30}\right] = [0.2333,0.5000]. $$

Thus the assumptions and the direct-effect restriction imply an outer bound of about 0.233 to 0.500 on the complier risk difference at $Z=1$, or 23.3 to 50.0 percentage points. If exact exclusion were imposed, $\delta=0$, and the Wald estimate would be $RF/FS=0.12/0.30=0.400$, or 40.0 percentage points. It lies inside the bound because zero is inside the allowed interval for $\delta$. The calculation does not establish that every value between the endpoints is compatible with all details of the data.

Worked Example: A Direct Effect Restricted to Be Nonnegative

In a second hypothetical study, let $RF=0.06$, $FS=0.20$, and $0\leq\delta\leq0.04$. Assume the instrument is randomized, treatment uptake is monotone, and the stated interval applies to the aggregate direct contribution defined above.

The lower endpoint uses the largest direct contribution, and the upper endpoint uses the smallest: $$ \left[\frac{0.06-0.04}{0.20},\frac{0.06-0}{0.20}\right] = [0.10,0.30]. $$

The treatment effect is therefore bounded between 0.10 and 0.30 outcome units per treated complier, on the scale of $Y$. Exact exclusion would instead give $0.06/0.20=0.30$. Here zero is an endpoint, not an interior value, of the allowed direct-effect interval, so the exact-exclusion estimate is the upper endpoint of the outer bound. The result depends on the stated restriction being about $\delta$; a bound on a different direct-effect contrast cannot be substituted without a valid link between the parameters.

Worked Example: Enforcing the Feasible Range for a Binary Outcome

Consider hypothetical population contrasts $RF=0.09$ and $FS=0.10$, with $-1\leq\delta\leq1$. For a binary outcome, a risk difference such as $\tau_C^{(1)}$ must lie between $-1$ and $1$, because it is the difference of two probabilities.

The algebraic outer bound is $$ \left[\frac{0.09-1}{0.10},\frac{0.09-(-1)}{0.10}\right] = [-9.10,10.90]. $$ Intersecting this with the binary-outcome feasible range gives $$ [-9.10,10.90]\cap[-1,1]=[-1,1]. $$

The wider algebraic interval cannot be treated as a set of values all compatible with the data: values below $-1$ or above $1$ are impossible risk differences. These contrasts are feasible, for example, with 10% compliers whose treatment effect is zero and 90% never-takers among whom the instrument directly changes the binary outcome for 10% of units from 0 to 1. Then $FS=0.10$, $\delta=0.90(0.10)=0.09$, and $RF=FS(0)+\delta=0.09$. This construction illustrates why the raw formula must be called an outer bound; other observed-data restrictions could narrow the compatible set further.

Interpreting and Using the Result

The bound describes sensitivity to a stated range of direct effects. It is not a confidence interval and does not account by itself for sampling variation in $RF$, $FS$, $L$, or $U$. In an empirical analysis, plugging sample contrasts into the formula gives a sample-based sensitivity calculation. Statistical uncertainty in those quantities requires separate treatment; a bound computed from point estimates should not be described as having a specified confidence level.

The width of the algebraic interval is $(U-L)/FS$ when $FS>0$. Consequently, a narrower credible range for the direct contribution tightens the bound, while a weaker first stage widens it. The exact-exclusion Wald estimate is $RF/FS$. It is inside the algebraic interval whenever $0\in[L,U]$; if zero is ruled out, the exact-exclusion estimate need not satisfy the sensitivity restriction. For outcomes with known limits, intersect the outer bound with the treatment-effect range implied by those limits. Such an intersection is necessary but may not be sufficient for sharpness.

State the target and assumptions.
Specify the instrument, treatment, outcome, randomized assignment, consistency, monotonicity, and the treatment-effect context being bounded.

Define the direct-effect restriction.
Check that its interval applies to the aggregate contribution $\delta$ in the reduced-form decomposition, with units matching the outcome contrast.

Calculate and qualify the bound.
Use the reduced form and first stage, then apply outcome-scale or other known feasibility restrictions. Label the result an outer bound unless sharpness has been established.

Common Mistake

A common mistake is to compute the two algebraic endpoints and claim that every value between them is compatible with the observed data. The decomposition proves only that the effect must lie within that interval given the direct-effect restriction. It does not prove that each candidate effect can be paired with a feasible distribution of potential outcomes. For example, with a binary outcome, a calculated risk-difference bound extending above 1 must be intersected with $[-1,1]$, and further restrictions may still apply. Another mistake is to use an interval for an individual direct effect or a different conditional contrast as if it were automatically a bound on the population quantity $\delta$; the relationship must be justified.

Key takeaway: A restriction $L\leq\delta\leq U$ replaces exact exclusion with a range of possible direct contributions and yields an outer bound on the complier effect. Its interpretation depends on a precise definition of $\delta$, a nonzero first stage, and any additional feasibility constraints imposed by the outcome and observed data.

Check Your Understanding

Work through these questions using the definitions and assumptions above.

For $RF=0.08$, $FS=0.25$, and $-0.02\leq\delta\leq0.03$, calculate the outer bound and the exact-exclusion Wald estimate.
Why does monotonicity make the first stage equal to the proportion of compliers?
Explain why the direct-effect interval must refer to the aggregate contribution in the decomposition, rather than an unspecified direct effect.
If an algebraic bound for a binary-outcome risk difference is $[-2,3]$, what additional restriction must be applied, and why might the resulting interval still fail to be sharp?
What happens to the width of the bound when the first stage becomes smaller while the direct-effect interval remains fixed?