Instrumental Variable Assumptions as Potential-Outcome Restrictions

Why State Instrument Assumptions in Potential-Outcome Form?

An instrumental variable is useful only when it changes treatment in a way that supports a causal interpretation of the resulting outcome variation. The assumptions behind that interpretation concern different things: how the instrument is assigned, whether it has a direct route to the outcome, whether it moves treatment in the population, and how individuals respond to it. Writing each assumption in potential-outcome notation makes these distinctions explicit.

Consider a binary instrument $Z\in\{0,1\}$, a binary treatment $D\in\{0,1\}$, and an outcome $Y$. The instrument might be an offer or encouragement, while treatment is uptake. The notation below describes each unit, even though only one instrument value and one treatment outcome are observed for that unit.

Definition: Let $D(z)$ be the treatment that would be received if the instrument were set to $z$. Let $Y(d,z)$ be the outcome that would occur if treatment were set to $d$ and the instrument to $z$. Consistency means that the observed values satisfy $D=D(Z)$ and $Y=Y(D,Z)$. In the absence of interference, one unit’s observed outcome is not changed by other units’ instrument or treatment assignments.

The notation $Y(d,z)$ allows the outcome to depend directly on the instrument as well as on treatment. That flexibility matters: exclusion is not built into the definition of a potential outcome. It is a separate assumption that removes the direct dependence on $z$. Throughout, assumptions refer to the population or assignment mechanism being studied, not merely to the realized sample.

Four Assumptions, Four Different Restrictions

Independence: In an unconditional design, the instrument is independent of the full vector of potential variables: $$ Z\ \perp\!\!\!\perp\ \{D(0),D(1),Y(d,z):d,z\in\{0,1\}\}. $$ This says that instrument assignment carries no information about units’ potential treatment responses or potential outcomes. With observed covariates $X$, a common alternative is conditional independence of this vector and $Z$ given $X$.

Independence is a statement about assignment relative to potential variables. Random assignment of $Z$ can justify it by design. In observational settings, it is instead an identifying assumption about how instrument values arose; controlling for $X$ is useful only if the conditional independence claim is credible. Conditional independence also requires overlap: within covariate values relevant to the target population, both instrument values must have positive probability.

Exclusion: For every unit and each treatment level, changing the instrument while holding treatment fixed does not change the outcome: $$ Y(d,1)=Y(d,0),\qquad d\in\{0,1\}. $$ When exclusion holds, write the common potential outcome as $Y(d)$.

Exclusion rules out every direct effect of the instrument on the outcome, including effects through channels other than the treatment being studied. For example, if an encouragement changes treatment uptake but also changes outcomes through reassurance or access to additional services, exclusion may fail unless those pathways are included in the definition of treatment.

Relevance: The instrument must change treatment in the target population. For a binary instrument, a population-level version is $$ E[D(1)-D(0)]\ne 0. $$ A positive first stage uses the stronger directional condition $E[D(1)-D(0)]>0$.

Relevance is a population condition on the average treatment response to the instrument. It does not say that every unit changes treatment, nor does it require every unit to respond in the same direction. In data, the corresponding observed treatment contrast is informative about this potential-outcome contrast only when the assignment assumptions needed to connect them are also credible.

Monotonicity: The instrument does not move treatment in opposite directions for different units. Under the chosen orientation $Z=1$ as the higher-encouragement value, this restriction is $$ D(1)\ge D(0)\quad\text{for every unit}. $$

Monotonicity is an individual-level restriction on the pair of potential treatments. It is not the same as saying that the average first stage is positive. A positive average can coexist with units whose treatment decreases when the instrument changes, as long as increases outweigh decreases. Conversely, monotonicity alone does not ensure relevance: if no one’s treatment changes, the inequality holds but the instrument has no first stage.

What Random Assignment Connects to the First Stage

The link between a potential-outcome restriction and an observed contrast can be shown directly. Suppose consistency holds and $Z$ is independent of $\{D(0),D(1)\}$. For either $z\in\{0,1\}$, the observed treatment among units with $Z=z$ equals $D(z)$. Therefore,

E[D\mid Z=z] =E[D(z)\mid Z=z] =E[D(z)].

The first equality uses consistency; the second uses independence. Subtracting the $z=0$ expression from the $z=1$ expression gives

E[D\mid Z=1]-E[D\mid Z=0] =E[D(1)-D(0)].

Thus, under these conditions, the observed population treatment contrast equals the average potential response to the instrument. This justifies assessing relevance through a population first stage in a randomized design. Without independence, the observed contrast can also reflect differences in the potential treatment responses of the groups assigned different instrument values.

Worked Example: Auditing the Restrictions in a Randomized Encouragement

Suppose a hypothetical population consists of four equally weighted units. Their potential treatments and outcomes are as follows. The outcome columns give $Y(0)$ and $Y(1)$, and suppose each unit’s outcome depends only on treatment.

Unit	$D(1)$	$Y(0)$	$Y(1)$
A	1	2	5
B	1	1	2
C	0	4	4
D	0	3	6

Suppose $Z$ is assigned by randomization, independently of all these potential variables. Then the independence condition holds by design in the assignment mechanism. Because the outcome values are specified as $Y(d)$, with no additional dependence on $Z$, exclusion holds in this hypothetical population.

For relevance, calculate the average treatment under each instrument value: $$ E[D(0)]=\frac{0+0+0+0}{4}=0,\qquad E[D(1)]=\frac{1+1+0+0}{4}=0.5. $$ So $E[D(1)-D(0)]=0.5-0=0.5$, and relevance holds with a positive first stage. For monotonicity, compare each row: $1\ge0$ for A and B, and $0\ge0$ for C and D. It holds for every unit.

The population observed treatment means under random assignment are therefore $E[D\mid Z=0]=0$ and $E[D\mid Z=1]=0.5$. Their difference, 0.5 treatment uptake, matches the potential-outcome calculation. In a finite randomized sample, realized group means need not match these population quantities exactly.

Conditional Independence and the Role of Overlap

An instrument need not be randomized unconditionally to be plausibly independent after conditioning on observed covariates. A conditional formulation is

Z\ \perp\!\!\!\perp\ \{D(0),D(1),Y(d,z):d,z\in\{0,1\}\}\mid X.

This condition says that within covariate values $X=x$, instrument assignment is unrelated to the potential variables. To learn from both instrument values in each relevant covariate group, one also needs positivity, $0<P(Z=1\mid X=x)<1$. If one instrument value never occurs for a covariate group, its within-group contrast is not identified by observed comparisons there.

Worked Example: A Conditional First Stage With Two Covariate Groups

Consider a hypothetical target population with two covariate groups: group A has population weight 0.75 and group B has weight 0.25. Suppose instrument assignment is randomized within each group, with both instrument values possible. The potential-treatment means are:

Group	Population weight	$E[D(0)\mid X]$	$E[D(1)\mid X]$	Conditional contrast
A	0.75	0.20	0.60	0.40
B	0.25	0.50	0.60	0.10

The conditional first-stage contrasts are $0.60-0.20=0.40$ in group A and $0.60-0.50=0.10$ in group B. Averaging over the target population’s covariate distribution gives $$ E[D(1)-D(0)] =0.75(0.40)+0.25(0.10) =0.30+0.025 =0.325. $$

Under conditional independence and consistency, each group’s observed treatment contrast identifies its conditional potential-treatment contrast. The population contrast, standardized to the stated group weights, is therefore 0.325. This is a positive average response to the instrument in this population. The calculation does not show that every unit responds positively, and it does not establish exclusion; those are distinct assumptions.

Why the Assumptions Are Not Interchangeable

A key discipline is to ask what kind of claim each assumption makes. Independence concerns assignment relative to potential variables; exclusion concerns potential outcomes under different instrument values while treatment is held fixed; relevance concerns a population average; monotonicity concerns each unit’s two potential treatments. Evidence for one does not establish the others.

Worked Example: A Positive Average Does Not Imply Monotonicity

Imagine a hypothetical population in which half the units have $D(0)=0,D(1)=1$, and the other half have $D(0)=1,D(1)=0$. The average treatment under $Z=0$ is $$ E[D(0)]=0.5(0)+0.5(1)=0.5, $$ and under $Z=1$ it is $$ E[D(1)]=0.5(1)+0.5(0)=0.5. $$ The average first stage is therefore $E[D(1)-D(0)]=0$, even though treatment changes for every unit.

Now change the proportions: let 60% have $(D(0),D(1))=(0,1)$ and 40% have $(D(0),D(1))=(1,0)$. Then $$ E[D(1)-D(0)]=0.60(1)+0.40(-1)=0.20. $$ Relevance holds with a positive average first stage, but monotonicity fails because the 40% whose treatment falls violate $D(1)\ge D(0)$. A positive population contrast is not proof of a common individual direction.

Exclusion has a different failure mode. Suppose an encouragement can improve an outcome directly, even for a unit whose treatment remains fixed. Then $Y(d,1)\ne Y(d,0)$ for at least some units and treatment levels. Randomizing the encouragement does not remove this direct pathway; it ensures independence of assignment, not exclusion. Likewise, an instrument can satisfy exclusion and independence but have no effect on treatment, violating relevance.

What Can Be Learned From Observed Data?

For each unit, only one value of $Z$, one realized treatment $D(Z)$, and one realized outcome $Y(D,Z)$ are observed. The alternative potential values are not jointly observed for that unit. Consequently, the assumptions involving those unobserved values generally cannot be confirmed just by inspecting the observed data.

A randomized design can provide a strong design-based rationale for independence, while conditional adjustment relies on the adequacy of measured covariates and the credibility of conditional independence. A nonzero observed first stage is evidence relevant to relevance, but its interpretation as a potential-treatment contrast relies on the assignment condition. Exclusion and monotonicity are substantive restrictions that require subject-matter justification; neither follows from randomization. Observed patterns may reveal problems or challenge a design, but a successful diagnostic does not prove these restrictions.

Common Mistake

Wrong reasoning: “The treatment rate is higher when $Z=1$, so the instrument is valid and monotonicity holds.”
Correction: A higher observed treatment rate is a first-stage pattern. Connecting it to the average potential response requires an appropriate independence condition and consistency. Even a positive average potential response does not imply monotonicity for every unit, and neither fact establishes exclusion.

Key takeaway: State the instrument assumptions separately. Independence is about assignment, exclusion removes a direct instrument-to-outcome path, relevance requires a nonzero population treatment response, and monotonicity constrains every unit’s treatment response. Their scopes and justifications differ.

Check Your Understanding

Use potential-outcome notation in your answers where appropriate.

Write the binary-instrument monotonicity condition and explain why it is stronger than a positive average first stage.
What does exclusion require of $Y(d,z)$ when treatment $d$ is held fixed?
Under consistency and independence, why does the observed population first stage equal $E[D(1)-D(0)]$?
Why is overlap needed when independence is assumed conditional on covariates $X$?
Can random assignment of $Z$ by itself establish exclusion? Explain the distinction between the two assumptions.

Unit	\(D(1)\)	\(Y(0)\)	\(Y(1)\)
A	1	2	5
B	1	1	2
C	0	4	4
D	0	3	6