Sensitivity to Violations of Instrument Independence

Why Instrument Independence Needs a Sensitivity Analysis

Instrumental-variable conclusions can be sensitive to the assumption that instrument assignment is independent of potential outcomes. In an encouragement design, for example, people assigned different encouragements may differ in their potential outcomes even before any treatment effect is considered. If those differences are small enough to bound, it is possible to retain a useful partial-identification analysis rather than treating independence as either certainly true or entirely unusable.

This tutorial quantifies that idea for a binary instrument and binary treatment. The key result is a reduced-form decomposition: the observed outcome contrast equals a complier-effect component plus terms capturing outcome differences between instrument groups. Bounds on those terms produce bounds on the complier effect. The analysis also allows imperfect exclusion, so it keeps separate the direct effect of the instrument and dependence between the instrument and baseline potential outcomes.

Setup and Sensitivity Restrictions

Let \(Z\in\{0,1\}\) be the instrument, \(D\in\{0,1\}\) the treatment, and \(Y\) the outcome. Write \(D(z)\) for the treatment a person would take under instrument value \(z\), and \(Y(d,z)\) for the outcome under treatment \(d\) and instrument value \(z\). The pair \(S=(D(0),D(1))\) is the person's compliance type. Assume consistency, no interference, finite conditional means, and positive probability of each instrument value.

Equal type proportions across instrument groups are a restriction on the distribution of compliance types, not full instrument independence. In particular, people of the same type may have different potential-outcome means across the two instrument groups. To describe that difference, define

The superscript on \(\mu_{s,d}^{(z)}\) identifies the observed instrument group used for conditioning; the outcome itself is evaluated at instrument value zero. Thus \(b_{s,d}\) measures a difference in baseline potential-outcome means across instrument groups within a compliance type. It is a sensitivity parameter, not an observed contrast: type and counterfactual outcomes are generally not jointly observed.

The target is the complier effect among compliers in the \(Z=0\) group:

This target is deliberately group-specific. Equal type proportions do not imply equal potential-outcome means among compliers in the two instrument groups, and therefore do not imply that \(\tau_C^{(0)}\) is the effect among compliers with \(Z=1\).

Finally, allow exclusion to fail. Define the average direct-effect contribution among those assigned \(Z=1\) as

A sensitivity analysis can bound \(|g|\) by a chosen value \(\Gamma\). This is distinct from bounding the \(b_{s,d}\): \(\Gamma\) concerns changing the instrument value in an outcome potential, whereas \(b_{s,d}\) concerns differences in baseline potential outcomes across the groups actually assigned different instrument values.

Reduced-Form Decomposition

Let \(\Delta_Y=E[Y\mid Z=1]-E[Y\mid Z=0]\) be the reduced-form outcome contrast, and let \(\Delta_D=E[D\mid Z=1]-E[D\mid Z=0]\) be the first-stage contrast. Under consistency, monotonicity, and equal compliance-type proportions, \(\Delta_D=\pi_C\): always-takers and never-takers contribute no change in treatment rate, while each complier contributes one.

For the outcome contrast, first compare \(Y(D(1),0)\) in the \(Z=1\) group with \(Y(D(0),0)\) in the \(Z=0\) group. Then add the direct-effect contribution \(g\) to account for the fact that the first group actually has instrument value one. Within each type, the baseline contrast is:

• Always-takers: \(\mu_{A,1}^{(1)}-\mu_{A,1}^{(0)}=b_{A,1}\).
• Never-takers: \(\mu_{N,0}^{(1)}-\mu_{N,0}^{(0)}=b_{N,0}\).
• Compliers: \(\mu_{C,1}^{(1)}-\mu_{C,0}^{(0)}=\tau_C^{(0)}+b_{C,1}\).

Weighting these contrasts by the common type proportions gives the decomposition and the corresponding sensitivity bound.

Suppose the sensitivity restrictions are \(|b_{A,1}|\le\varepsilon_{A,1}\), \(|b_{N,0}|\le\varepsilon_{N,0}\), \(|b_{C,1}|\le\varepsilon_{C,1}\), and \(|g|\le\Gamma\), where all bounds are specified in outcome units. Define

The triangle inequality implies that the total non-complier-effect contribution has absolute value at most \(R_{\max}\). Since \(\pi_C=\Delta_D>0\), solving the decomposition for \(\tau_C^{(0)}\) yields:

Why the Bound Has This Form

The logic can be checked directly. Let \(B=\pi_A b_{A,1}+\pi_N b_{N,0}+\pi_C b_{C,1}+g\). The restrictions imply \(|B|\le R_{\max}\), and the decomposition gives \(\tau_C^{(0)}=(\Delta_Y-B)/\pi_C\). Therefore \(-R_{\max}\le B\le R_{\max}\). Subtracting \(B\) from \(\Delta_Y\), and dividing by positive \(\pi_C\), gives the stated interval. The interval is an identification region under the restrictions, not a sampling confidence interval.

The type-specific terms in \(R_{\max}\) are weighted by the prevalence of their type. A given possible mean difference matters less when the relevant type is less common. The direct-effect allowance \(\Gamma\), by contrast, bounds the average direct contribution itself and is not multiplied by a type proportion.

Calculating an Identification Interval

Interpretation, Limitations, and a Common Pitfall

This analysis relaxes independence only in a specific way. It permits bounded differences in potential-outcome means across instrument groups within compliance types, while retaining equal type proportions. It does not cover arbitrary dependence of the instrument on compliance type. If type proportions differ across \(Z\), then \(\Delta_D\) need not equal \(\pi_C\), and additional composition terms enter the reduced form. The displayed bound should not be applied unchanged in that setting.

Likewise, a bound on \(|g|\) is not automatically implied by a bound on the \(b_{s,d}\). The former concerns imperfect exclusion; the latter concerns instrument-group differences in baseline potential outcomes. When exclusion is credible, set \(\Gamma=0\). When it is not, state and justify a separate allowance.

Common pitfall: One might reason that equal compliance-type proportions mean the compliers in the two instrument groups are interchangeable, and therefore interpret \(\tau_C^{(0)}\) as the effect among \(Z=1\) compliers too. That conclusion is not justified. Equal proportions constrain how many people of each type are in each group, not the groups' potential-outcome means. The target here is specifically the effect among compliers with \(Z=0\); a claim about the \(Z=1\) complier effect requires additional assumptions or a separate sensitivity analysis.

A second practical caution is that these are population identification bounds. In an empirical analysis, \(\Delta_Y\), \(\Delta_D\), and type proportions are estimated, so sampling uncertainty must also be accounted for when drawing statistical conclusions. Identification uncertainty from the sensitivity restrictions and sampling uncertainty are different sources of uncertainty.