Tightening Instrumental-Variable Bounds with Monotonicity

Why Monotonicity Can Tighten Bounds Without Exclusion

A binary instrument can change treatment uptake without satisfying the exclusion restriction: it may also affect the outcome directly. In that setting, the familiar instrumental-variable point identification argument is unavailable. But assumptions about uptake can still restrict which causal response patterns are compatible with observed data. This tutorial uses monotonicity to impose one such restriction and shows how to obtain sharp bounds while allowing the outcome to depend directly on the instrument.

We focus on binary instrument \(Z\), binary treatment \(D\), and binary outcome \(Y\). The target is the average treatment effect among compliers, averaged over the two instrument settings. Because exclusion is not assumed, the outcome potential under treatment \(d\) can differ between \(Z=0\) and \(Z=1\). The bounds therefore retain those direct-effect possibilities rather than treating them as zero.

The uptake types, also called principal strata, are defined by the pair \((D(0),D(1))\). Under monotonicity, the possible types are always-takers \((1,1)\), compliers \((0,1)\), and never-takers \((0,0)\). A defier would have type \((1,0)\), which monotonicity excludes. We assume the instrument is randomized, or otherwise independent of all the potential quantities just defined, and use consistency: the observed treatment is \(D=D(Z)\) and the observed outcome is \(Y=Y(Z,D)\).

What Uptake Monotonicity Implies

Let \(p_z=P(D=1\mid Z=z)\). Under instrument independence and consistency, the instrument-specific treatment rates equal the probabilities of taking treatment under each instrument setting. Monotonicity therefore gives a simple observable implication:

where \(\pi_A,\pi_C,\pi_N\) are the population proportions of always-takers, compliers, and never-takers. These proportions sum to one. If \(p_1<p_0\), the data contradict monotonicity under the stated independence and consistency conditions. If \(p_1=p_0\), monotonicity implies that the complier proportion is zero.

Allowing Direct Effects in the Outcome Model

For a binary outcome, each person has four binary outcome potentials: \(Y(0,0),Y(0,1),Y(1,0),Y(1,1)\). Exclusion would impose \(Y(0,d)=Y(1,d)\) for both treatment values \(d\). We do not impose those equalities. In particular, an observed outcome difference across instrument arms may reflect a direct effect of the instrument, a change in treatment uptake, or both.

The complier estimand used here is \(\tau_C=\frac{1}{2}E[Y(0,1)-Y(0,0)+Y(1,1)-Y(1,0)\mid C]\), where \(C\) denotes the complier stratum. It averages the treatment effect at \(Z=0\) and at \(Z=1\), giving each instrument setting equal weight. The definition is explicit about what is held fixed: each contrast changes treatment while holding the instrument setting fixed. Other weightings could be used if a different target is scientifically appropriate.

Here is the linear-program representation. Let \(x_{s,\mathbf y}\) be the population probability of uptake stratum \(s\in\{A,C,N\}\) and outcome vector \(\mathbf y=(y_{00},y_{01},y_{10},y_{11})\in\{0,1\}^4\), where \(y_{zd}\) represents \(Y(z,d)\). Each \(x_{s,\mathbf y}\) is nonnegative. The variables must sum to one, and their stratum totals must equal \((p_0,p_1-p_0,1-p_1)\).

For each observed instrument value \(z\), treatment value \(d\), and outcome value \(y\), add a constraint that the latent probability of that observed cell equals its empirical population probability. In the constraint, retain a latent type only if its \(D(z)\) equals \(d\), and retain an outcome vector only if its \(y_{zd}\) equals \(y\). For instance, the constraint for \(Z=0,D=1,Y=1\) sums \(x_{s,\mathbf y}\) over types with \(D(0)=1\) and outcome vectors with \(y_{01}=1\). The two observed conditional distributions supply all these constraints.

When \(\pi_C>0\), the denominator is fixed by the uptake rates under monotonicity. The objective is therefore linear in the latent probabilities. Minimizing and maximizing it subject to the observed-cell constraints gives the lower and upper bounds. Since there are three uptake strata and 16 outcome vectors, this is a finite linear program with at most 48 latent probability variables.

Why the Linear-Program Bounds Are Sharp

The word sharp means that every value inside the reported interval is compatible with the assumptions and observed distribution, and that no value outside it is compatible. The justification follows from the latent representation, not from an approximation to the data.

This also explains how to compare with bounds that do not impose monotonicity. The unrestricted latent model includes the defier stratum \((1,0)\) and does not fix the other stratum proportions from \(p_0,p_1\). Every distribution feasible under monotonicity is still feasible in that larger model. Therefore, for a common linear target, adding monotonicity can only weakly narrow the interval: its lower endpoint cannot decrease, and its upper endpoint cannot increase. A strict improvement is not guaranteed; it depends on whether the excluded defier models would otherwise attain an endpoint.

A Fully Compliant Example With No Exclusion

A transparent special case shows how to calculate outcome bounds without equating outcomes across instrument settings. Suppose \(p_0=0\) and \(p_1=1\). Under monotonicity, every person is a complier: \(\pi_C=1\). The observed data reveal \(Y(0,0)\) for everyone when \(Z=0\), and \(Y(1,1)\) for everyone when \(Z=1\). They do not reveal \(Y(0,1)\) or \(Y(1,0)\).

Practical Interpretation and Common Pitfalls

In a general data set, the observed outcome distributions are mixtures of principal strata. For example, among \(Z=0,D=0\) participants, both compliers and never-takers may appear. Their outcome distribution cannot be separated using that cell alone. Likewise, \(Z=1,D=1\) combines compliers and always-takers. The linear program handles these mixtures and their shared latent stratum proportions simultaneously; assigning each mixture cell wholly to one type would generally produce incorrect bounds.

Monotonicity is a statement about treatment uptake, not about outcomes. It does not say that treatment benefits everyone, that the instrument has no direct effect, or that outcomes are monotone in treatment. The exclusion restriction is a separate assumption, \(Y(0,d)=Y(1,d)\), and adding it would alter the feasible set and potentially the estimand’s bounds. Do not infer exclusion from the fact that monotonicity has been imposed.

The main benefit of monotonicity for bounds is that it removes latent defier models and, when compatible with the data, fixes the shares of the remaining uptake strata. This can tighten bounds for targets that depend on those strata, as the complier effect does. The improvement may be small or absent for particular observed distributions, and the bounds remain an expression of partial identification rather than a point estimate.