Combining Multiple Instruments for Partial Identification

Why Multiple Instruments Can Sharpen Bounds

With several binary instruments, each instrument supplies information about how treatment and outcome change when that instrument is varied. If the assumptions and target are aligned, all those pieces of information must hold at once. Their joint implications can be more restrictive than the bounds from any one instrument alone.

The crucial qualification is that restrictions can be intersected only when they concern the same causal target under a coherent set of assumptions. Different instruments may shift treatment for different people, so their Wald ratios need not estimate one common effect when effects are heterogeneous. This tutorial develops two ways to combine information: a transparent common-effect sensitivity model for bounded exclusion violations, and a latent-type linear program that allows limited violations of monotonicity.

Bounded Exclusion Violations for a Common Effect

Begin with a deliberately explicit sensitivity model. Let \(D\) be binary treatment and \(Y\) an outcome. For instrument \(j\), compare two assignment regimes that differ only in \(Z_j\), holding the other instruments at a specified reference setting. Define the population reduced-form contrasts \(\Delta D_j=E[D\mid Z_j=1]-E[D\mid Z_j=0]\) and \(\Delta Y_j=E[Y\mid Z_j=1]-E[Y\mid Z_j=0]\), where the expectations refer to those regimes.

Assume a common structural treatment effect \(\tau\) for the changes in treatment induced by these comparisons, and write \(\Delta Y_j=\tau\Delta D_j+\gamma_j\). Here \(\gamma_j\) is the instrument-specific direct-effect contribution to the outcome contrast. The sensitivity restriction is \(|\gamma_j|\leq\delta_j\), for a specified nonnegative bound \(\delta_j\). Instrument assignment must also be independent of the relevant potential outcomes and treatment responses, or the contrasts must be adjusted under a stated design that justifies the same relation.

The bounded direct effect gives a pair of inequalities for each instrument. The bounds from multiple instruments are imposed together, not chosen one at a time.

When \(\Delta D_j>0\), rearranging gives \((\Delta Y_j-\delta_j)/\Delta D_j\leq\tau\leq (\Delta Y_j+\delta_j)/\Delta D_j\). When \(\Delta D_j<0\), dividing reverses the inequality directions; a safe method is to solve the two-sided inequality first and then order its endpoints. If \(\Delta D_j=0\), this instrument imposes no finite bound on \(\tau\) when \(|\Delta Y_j|\leq\delta_j\), and the restrictions are incompatible when \(|\Delta Y_j|>\delta_j\).

Simultaneous Restrictions With Latent Response Types

A common-effect interval is useful when that model is substantively defensible, but it is not a substitute for modeling heterogeneous responses. A more general formulation treats the treatment response to all instrument settings as a latent type. For two binary instruments, write an assignment setting as \(z=(z_1,z_2)\in\{0,1\}^2\). A treatment-response pattern is \(r=(r_{00},r_{10},r_{01},r_{11})\), where \(r_z\in\{0,1\}\) is the treatment the unit would receive under setting \(z\).

Let \(Y(0)\) and \(Y(1)\) be binary potential outcomes under exclusion, and let \(\pi_{r,y_0,y_1}\) be the population share with response pattern \(r\), \(Y(0)=y_0\), and \(Y(1)=y_1\). These shares are nonnegative and sum to one. Under consistency, exclusion, and independence of the instrument vector from these latent potential outcomes and treatment responses, the observed conditional distribution in every assignment cell is a sum over the same latent shares.

Here \(\mathbf{1}\{\cdot\}\) equals one when its condition holds and zero otherwise. Each observed-data equation is linear in the latent shares. The average treatment effect is also linear: \(E[Y(1)-Y(0)]=\sum_{r,y_0,y_1}\pi_{r,y_0,y_1}(y_1-y_0)\). Consequently, minimizing and maximizing this expression over the shares consistent with the observed cell probabilities gives lower and upper bounds on the average treatment effect.

Relaxing Monotonicity Inside the Joint Model

Monotonicity for instrument \(j\) means that changing \(Z_j\) from zero to one cannot reduce anyone’s treatment, with the other instrument setting held fixed. In the latent-type representation, this requires \(r_{(1,z_{-j})}\geq r_{(0,z_{-j})}\) for every setting \(z_{-j}\) of the other instrument. Rather than require this for every unit, allow a limited share \(\eta_j\) of types to violate it. Define \(V_j(r)=1\) if the response pattern violates this condition at any setting of the other instrument, and zero otherwise. The relaxation is the linear constraint \(\sum_{r,y_0,y_1}\pi_{r,y_0,y_1}V_j(r)\leq\eta_j\).

Both instruments’ restrictions are imposed on the same \(\pi\). Thus the combined analysis requires the observed-data equations for every instrument setting, nonnegative shares summing to one, and each specified limit on the share violating monotonicity. Exact monotonicity is the special case \(\eta_j=0\). Increasing \(\eta_j\) weakens that restriction and can widen the treatment-effect bounds; it cannot remove the observed-data equations.

This is a linear program: its objective and constraints are linear in the latent shares. The bounds are sharp relative to the stated model when every feasible share vector corresponds to a potential-outcome population satisfying the assumptions and observed probabilities. This approach avoids the invalid shortcut of calculating separate instrument-specific bounds and intersecting them when those bounds refer to different estimands or incompatible latent populations.

What the Intersection Does—and Does Not—Say

Intersecting restrictions can sharpen partial identification without selecting one instrument as the uniquely valid one. It also provides a model check: an empty feasible set indicates that at least one maintained restriction, the data inputs, or their compatibility must be reconsidered. It does not identify which assumption failed, and a nonempty set does not prove instrument validity.

Population restrictions and sampling uncertainty are distinct. In applications, contrasts and conditional cell probabilities are estimated from data. An interval formed by plugging in sample estimates is not automatically a confidence region for the partially identified target. Inference must account for uncertainty in the restrictions and for the fact that bounds can be sensitive to estimated endpoints.