Bounds Under Imperfect Exclusion

Why Bound an Imperfect Instrument?

An instrumental-variable analysis often relies on exclusion: changing the instrument affects the outcome only by changing treatment. In some applications, a small direct effect is plausible even when an exactly zero direct effect is not. Rather than silently treating exclusion as true, we can state how large a direct effect we are willing to allow and determine what that permits for the treatment effect.

The result is generally a bound, not a point estimate. The bound below applies to the average treatment effect for compliers under monotonicity. It is a valid outer bound under the stated assumptions; it need not be the exact set of treatment effects compatible with all the observed data and restrictions.

Setup and Assumptions

Let $Z\in\{0,1\}$ be the instrument, $D\in\{0,1\}$ treatment, and $Y$ the outcome. Write $D(z)$ for the treatment a unit would receive if the instrument were set to $z$, and $Y(d,z)$ for the outcome if treatment and instrument were set to $d$ and $z$. The observed values satisfy $D=D(Z)$ and $Y=Y(D,Z)$. Expectations refer to the population under study.

Assumptions for the bound: (1) Consistency and no interference, so observed outcomes agree with the relevant potential outcomes and one unit’s assignment does not change another unit’s outcome. (2) Instrument independence: $Z$ is independent of the collection of potential treatments and outcomes. (3) Monotonicity: $D(1)\ge D(0)$ for every unit. (4) Relevance: $E[D\mid Z=1]-E[D\mid Z=0]>0$. (5) Bounded direct effect: for each treatment status $d$, $\lvert Y(d,1)-Y(d,0)\rvert\le\delta$, where $\delta\ge0$ is specified in outcome units.

The bounded direct-effect condition is a restriction on each unit’s potential outcomes, not merely on an overall average. It permits the instrument to affect the outcome at fixed treatment, but limits the size of that effect. In the special case $\delta=0$, it becomes the exclusion restriction.

Monotonicity partitions units into always-takers, who have $D(0)=D(1)=1$; never-takers, who have $D(0)=D(1)=0$; and compliers, who have $D(0)=0,D(1)=1$. Defiers, with $D(0)=1,D(1)=0$, are ruled out. Let $\pi_C=P(D(1)>D(0))$ be the population share of compliers. Under instrument independence and monotonicity, $ \pi_C=E[D\mid Z=1]-E[D\mid Z=0] $. The target is the complier average treatment effect at instrument level zero: $ \tau_C(0)=E[Y(1,0)-Y(0,0)\mid D(1)>D(0)] $.

Deriving the Bound

Define the instrument’s outcome contrast and treatment contrast as $ \Delta_Y=E[Y\mid Z=1]-E[Y\mid Z=0] $ and $ \Delta_D=E[D\mid Z=1]-E[D\mid Z=0] $. Under the assumptions above, $\Delta_D=\pi_C>0$. The outcome contrast combines treatment changes among compliers with direct effects of changing $Z$ at fixed treatment among all three principal strata.

\begin{aligned} \Delta_Y &=E\!\left[Y(D(1),1)-Y(D(0),0)\right]\\ &=\pi_C\tau_C(0)+B,\\ B &=E\!\left[Y(D(1),1)-Y(D(1),0)\right]. \end{aligned}

To see the decomposition, a complier contributes $Y(1,1)-Y(0,0)$. Add and subtract $Y(1,0)$: this equals its treatment effect at $Z=0$, $Y(1,0)-Y(0,0)$, plus its direct effect at treatment $1$, $Y(1,1)-Y(1,0)$. An always-taker contributes only $Y(1,1)-Y(1,0)$, and a never-taker contributes only $Y(0,1)-Y(0,0)$. Thus $B$ collects the relevant direct effects, weighted over the full population.

For every unit, the direct-effect restriction gives $\lvert Y(D(1),1)-Y(D(1),0)\rvert\le\delta$. Taking expectations and applying the triangle inequality yields $\lvert B\rvert\le\delta$. Since $\pi_C=\Delta_D>0$, rearranging the decomposition gives the bound.

Result: Under the stated assumptions, $$ \frac{\Delta_Y-\delta}{\Delta_D} \le \tau_C(0)\le \frac{\Delta_Y+\delta}{\Delta_D}. $$ This is a valid, potentially non-sharp outer bound. Every compatible value must lie in this interval, but not every value in the interval must be compatible with the full restrictions and observed data.

The bound’s width is $2\delta/\Delta_D$. A larger allowed direct effect widens the interval, while a stronger first stage (larger $\Delta_D$) limits how much a given direct-effect allowance shifts the implied treatment effect. This widening describes the formula’s sensitivity; it does not prove that every point in the widened interval can occur.

Worked Example: A Small Direct-Effect Allowance

Consider a hypothetical population with $E[D\mid Z=0]=0.30$, $E[D\mid Z=1]=0.50$, $E[Y\mid Z=0]=10.00$, and $E[Y\mid Z=1]=10.12$. Suppose the assumptions above are maintained and the outcome is measured in points. Set the unit-level direct-effect limit at $\delta=0.03$ points.

First stage. The treatment contrast is $ \Delta_D=0.50-0.30=0.20 $. Under monotonicity and instrument independence, the complier share is therefore $0.20$.

Outcome contrast. $ \Delta_Y=10.12-10.00=0.12 $ points.

Bound. Substitution gives $ (\Delta_Y-\delta)/\Delta_D=(0.12-0.03)/0.20=0.09/0.20=0.45 $ and $ (\Delta_Y+\delta)/\Delta_D=(0.12+0.03)/0.20=0.15/0.20=0.75 $. The valid outer bound is therefore $[0.45,0.75]$ outcome points for the complier average treatment effect at $Z=0$. These are compatible-value limits implied by this inequality, not a confidence interval and not a claim that every value inside is feasible.

How the Allowed Direct Effect Changes the Bound

Keeping the contrasts in the example fixed, the lower endpoint decreases and the upper endpoint increases as $\delta$ grows. The table makes this mechanical widening explicit. It does not establish sharpness at any row.

Direct-effect limit $\delta$ (points)	Lower endpoint	Upper endpoint	Outer bound for $\tau_C(0)$ (points)
0.00	0.60	0.60	[0.60, 0.60]
0.01	0.55	0.65	[0.55, 0.65]
0.03	0.45	0.75	[0.45, 0.75]
0.05	0.35	0.85	[0.35, 0.85]

At $\delta=0$, the stated restriction is exact exclusion and the bound collapses to the usual complier ratio, $0.12/0.20=0.60$ points. When $\delta>0$, the ratio alone no longer identifies the treatment effect: some of the outcome contrast may reflect direct effects. The formula allows for that ambiguity, but can be conservative because it uses only a bound on the aggregate direct-effect contribution.

Worked Example: Intersecting With Outcome Support

Suppose a hypothetical outcome is known to lie in $[0,1]$, and the maintained assumptions hold with $\Delta_D=0.40$, $\Delta_Y=0.32$, and $\delta=0.10$. All quantities are in the outcome’s units.

Apply the direct-effect bound. $ (0.32-0.10)/0.40=0.22/0.40=0.55 $ and $ (0.32+0.10)/0.40=0.42/0.40=1.05 $. So the direct-effect argument alone gives the outer bound $[0.55,1.05]$.

Apply outcome support. Because both $Y(1,0)$ and $Y(0,0)$ lie in $[0,1]$, their difference lies in $[-1,1]$; its conditional average $\tau_C(0)$ must also lie in $[-1,1]$. Intersecting the two restrictions gives $[0.55,1.00]$. This intersection remains an outer bound, not necessarily the sharp identified set.

If the same contrasts instead had $\delta=0.20$, the formula would give $[0.30,1.30]$, and support would reduce it to $[0.30,1.00]$. The larger direct-effect allowance cannot tighten the result; support merely prevents treatment effects outside their logically possible range.

Discrete Outcomes and a Crucial Qualification

The numerical size of $\delta$ must be interpreted in the scale and support of the outcome. For a binary outcome, $Y(d,z)\in\{0,1\}$, so the unit-level direct difference $Y(d,1)-Y(d,0)$ can only be $-1$, $0$, or $1$. If $\delta<1$, the restriction $\lvert Y(d,1)-Y(d,0)\rvert\le\delta$ forces that difference to equal zero for every unit and each $d$. Thus this particular pointwise restriction is exact exclusion, not permission for small nonzero direct effects. A small allowance can be substantively different for a continuous outcome.

Worked Example: Binary Outcomes With a Limit Below One

Suppose a hypothetical binary-outcome setting has $E[D\mid Z=0]=0.41$, $E[D\mid Z=1]=0.59$, $E[Y\mid Z=0]=0.42$, and $E[Y\mid Z=1]=0.54$. Assume instrument independence, monotonicity, relevance, and the unit-level restriction with $\delta=0.03$.

The treatment contrast is $\Delta_D=0.59-0.41=0.18$, and the outcome contrast is $\Delta_Y=0.54-0.42=0.12$. Since the outcome is binary, each fixed-treatment direct difference is in $\{-1,0,1\}$. Its absolute value being at most $0.03<1$ forces it to be zero. The restriction therefore implies exact exclusion, and $ \tau_C(0)=\Delta_Y/\Delta_D=0.12/0.18=0.6667 $ outcome units.

Blindly substituting into the general inequality would give the outer bound $ [(0.12-0.03)/0.18,\,(0.12+0.03)/0.18]=[0.500,0.8333] $. That interval is valid as an outer bound, but its interior values are not compatible with the binary-outcome restriction here: the restriction forces exact exclusion and the target is $0.6667$. The example shows why a valid inequality should not automatically be described as the sharp set of compatible values.

Interpretation and Common Pitfall

A bound answers a different question from a point estimate. It says that, given the maintained assumptions and the chosen direct-effect limit, the target must lie within stated limits. It does not assign probabilities to values in that range. Nor does widening the bound show that the instrument’s direct effect actually equals the chosen $\delta$; $\delta$ is an assumption used to describe sensitivity.

Pitfall: treating the outer bound as sharp. The mistaken reasoning is that every value between the calculated endpoints must fit the data. The inequality proves only that compatible values lie within those endpoints. Additional restrictions, including outcome discreteness, can rule out values inside.
Pitfall: dividing the outcome contrast by the first stage and calling it the treatment effect. That ratio is justified as the complier effect when exclusion holds. With imperfect exclusion, the outcome contrast also contains a direct-effect component; the bound accounts for that component rather than assuming it away.
Pitfall: choosing $\delta$ without units or outcome support in mind. A limit of 0.03 points may be small for one continuous outcome, while for a binary outcome the same pointwise limit forces zero direct effect. State the outcome scale and the substantive meaning of the restriction.

The analysis is conditional on the maintained assumptions, including instrument independence and monotonicity. A numerical bound cannot establish those assumptions. Its value is that it makes the exclusion restriction’s role explicit and shows how conclusions change under a stated allowance for direct effects.

Check Your Understanding

Use the assumptions and definitions in this tutorial to answer the following.

Why does monotonicity make the first-stage contrast equal to the complier share?
If $\Delta_Y=0.10$, $\Delta_D=0.25$, and $\delta=0.02$, calculate the outer bound for $\tau_C(0)$. State its outcome units if the outcome is measured in dollars.
What happens to the width of the bound when $\delta$ doubles while $\Delta_D$ stays fixed?
For a binary outcome, what does a pointwise direct-effect limit $\delta=0.4$ imply about $Y(d,1)-Y(d,0)$?
Explain why the interval from the general inequality should not automatically be called the sharp identified set.