Counterfactual Notation for Instrumental Variables

Why Counterfactual Notation Matters

An instrumental-variable analysis distinguishes the value of the instrument a unit receives from the treatment that unit actually takes. The instrument may change treatment receipt, while the outcome is connected to treatment through a separate set of counterfactuals. Potential-outcome notation makes those links explicit: it tells us which hypothetical value corresponds to each possible instrument and treatment setting.

This notation is bookkeeping for causal questions, not a method for establishing that an instrument is valid. In particular, defining potential outcomes does not imply that the instrument is independent of them, that it affects the outcome only through treatment, or that its effect on treatment has a particular direction. Those are additional restrictions. Here the task is narrower: define the potential variables and state how the values under the realized settings correspond to the observed data.

Instrument and Treatment-Receipt Potential Outcomes

For unit $i$, let $Z_i$ denote the observed instrument and $D_i$ the observed treatment receipt. Both are binary: $Z_i,D_i\in\{0,1\}$. The labels 0 and 1 identify the two instrument states and the two treatment states; their substantive meanings depend on the application. For example, $Z_i=1$ might denote eligibility for an offer, while $D_i=1$ denotes receipt of a service.

Definition: For each unit $i$, $D_i(z)$ is the potential treatment receipt that would occur if the instrument were set to $z$, for $z\in\{0,1\}$. Thus $D_i(0)$ and $D_i(1)$ are the unit’s two treatment-receipt potential outcomes. Each is binary. The observed instrument $Z_i$ selects which potential treatment receipt is observed: $$ D_i=D_i(Z_i). $$

The notation $D_i(z)$ describes treatment receipt under an instrument setting; it does not say that the unit necessarily follows the setting or receives treatment. A unit could have $D_i(0)=D_i(1)=0$, for example, if neither instrument state would lead to treatment receipt. The pair $\{D_i(0),D_i(1)\}$ describes the unit’s response to the two possible instrument values. In observed data, however, only the potential treatment receipt indexed by the realized $Z_i$ is revealed.

The instrument itself is observed as $Z_i$. It is not necessary to define a potential instrument $Z_i(z)$ merely to describe how $Z_i$ affects treatment receipt. A separate upstream assignment mechanism could motivate potential values for $Z_i$, but that is a different layer of the causal description. The notation here begins with the instrument value and asks what treatment receipt and outcome would follow under specified settings.

Worked Example: Selecting the Observed Treatment Receipt

Suppose a hypothetical individual has $D_i(0)=0$ and $D_i(1)=1$. The observed instrument is $Z_i=1$. Use consistency to determine the observed treatment receipt.

The consistency relation is $D_i=D_i(Z_i)$. Substituting the observed instrument value gives $$ D_i=D_i(1)=1. $$ Thus this individual is observed to receive treatment. If the same individual had instead been observed with $Z_i=0$, consistency would give $D_i=D_i(0)=0$. The two calculations use different potential treatment receipts; they do not imply that both are observed for the same individual.

Outcome Potential Outcomes: Two Levels of Indexing

An outcome can be indexed by the instrument setting and the treatment setting. This two-index notation is the most general way to represent outcomes for the pair of interventions in view. Let $Y_i(z,d)$ denote the outcome that would be observed for unit $i$ if the instrument were set to $z$ and treatment receipt were set to $d$, where $z,d\in\{0,1\}$. The outcome may be binary, continuous, or of another type appropriate to the application.

Definition: $Y_i(z,d)$ is the potential outcome for unit $i$ under instrument value $z$ and treatment value $d$. The four potential outcomes are $Y_i(0,0)$, $Y_i(0,1)$, $Y_i(1,0)$, and $Y_i(1,1)$. With observed instrument $Z_i$ and observed treatment $D_i$, consistency gives $$ Y_i=Y_i(Z_i,D_i). $$

This notation allows the potential outcome to differ across instrument values even when treatment is held fixed. It therefore does not, by definition alone, rule out a direct effect of the instrument on the outcome. That possibility matters because the restriction that the outcome depends on the instrument only through treatment is an additional instrumental-variable assumption, not a consequence of writing down $Y_i(z,d)$.

A more compact notation is often used when that additional restriction is imposed. Under the exclusion restriction, outcomes at the same treatment value do not depend on the instrument value: $Y_i(0,d)=Y_i(1,d)$ for each $d$. In that case, define $Y_i(d)$ as the common value. Then observed outcomes can be written $Y_i=Y_i(D_i)$. This compact form is useful, but using it as a maintained model entails the exclusion restriction. It should not be confused with the unrestricted definition of $Y_i(z,d)$.

Worked Example: Mapping an Observed Outcome to Its Potential Outcome

For a hypothetical individual, suppose $Z_i=0$, $D_i(0)=1$, and the relevant outcome potential outcomes include $Y_i(0,1)=12$. The individual’s observed outcome is 12 units. Show which potential outcome consistency selects, and identify another potential outcome that is not selected.

First, consistency for treatment receipt gives $D_i=D_i(Z_i)=D_i(0)=1$. The realized settings are therefore $Z_i=0$ and $D_i=1$. Consistency for the outcome then gives $$ Y_i=Y_i(Z_i,D_i)=Y_i(0,1)=12. $$ For example, $Y_i(1,1)$ is not selected by the observed settings. Its value is counterfactual for this individual, even though it is part of the potential-outcome description. The observed value 12 does not by itself reveal whether $Y_i(1,1)$ equals 12 or differs from it.

Consistency in Indicator Form

The consistency relations can be written as equations that explicitly select the potential outcome corresponding to the realized values. Since $Z_i$ is binary, one of $1-Z_i$ and $Z_i$ equals 1 and the other equals 0. Consequently, the treatment-receipt relation is equivalent to

$$ D_i=(1-Z_i)D_i(0)+Z_iD_i(1). $$

If $Z_i=0$, the right-hand side is $D_i(0)$; if $Z_i=1$, it is $D_i(1)$. For the outcome, the corresponding selection equation is

Y_i=\sum_{z=0}^{1}\sum_{d=0}^{1} \mathbf{1}\{Z_i=z,\ D_i=d\}Y_i(z,d).

Here $\mathbf{1}\{A\}$ equals 1 when condition $A$ is true and 0 otherwise. Exactly one pair, $(z,d)=(Z_i,D_i)$, matches the observed settings, so exactly one term in the sum contributes. The equation therefore reduces to $Y_i(Z_i,D_i)$, as required by consistency.

When exclusion is imposed and $Y_i(d)$ is used instead, the same selection logic gives

$$ Y_i=(1-D_i)Y_i(0)+D_iY_i(1). $$

This last expression selects according to treatment alone because exclusion makes the instrument index unnecessary for the outcome. Without that restriction, the two-index outcome equation is the appropriate consistency statement; the treatment-only equation does not follow merely from the fact that $D_i$ is observed.

Worked Example: Checking Both Consistency Equations

Consider a hypothetical individual with $Z_i=1$, $D_i(0)=0$, and $D_i(1)=1$. Suppose the two-index outcome potential outcomes are $Y_i(0,0)=4$, $Y_i(0,1)=7$, $Y_i(1,0)=5$, and $Y_i(1,1)=9$. Apply the indicator equations to determine the observed treatment and outcome.

For treatment receipt, $$ D_i=(1-1)D_i(0)+1\cdot D_i(1) =0\cdot 0+1\cdot 1=1. $$ The realized pair of settings is therefore $(Z_i,D_i)=(1,1)$. In the outcome sum, the indicator for $(z,d)=(1,1)$ equals 1; the other three indicators equal 0. Hence $$ Y_i=0\cdot4+0\cdot7+0\cdot5+1\cdot9=9. $$ The value $Y_i(0,1)=7$, for example, is not the observed outcome because it corresponds to a different instrument setting.

The four specified outcome values are allowed under the unrestricted notation. In particular, $Y_i(0,1)=7$ and $Y_i(1,1)=9$ differ at the same treatment value, so this hypothetical specification does not satisfy exclusion. The calculation of the observed outcome remains valid: consistency selects $Y_i(1,1)$ given the realized pair of settings.

What the Observed Data Reveal

For each unit, the observed record contains $Z_i$, $D_i$, and $Y_i$, along with any recorded covariates. Consistency identifies the potential treatment receipt at the observed instrument value and the potential outcome at the observed instrument-treatment pair. It does not reveal every counterfactual value for that unit. For instance, when $Z_i=1$, $D_i(1)$ is linked to observed treatment receipt, but $D_i(0)$ is not observed for that same unit. The outcome $Y_i(1,D_i)$ is likewise selected, while outcomes indexed by other settings are not observed.

This distinction is central to causal analysis. Potential outcomes provide a well-defined language for comparisons across possible settings, while the data show only the setting that occurred for each unit. Estimation and identification require additional structure to learn about unobserved counterfactuals from observed data. Consistency is the link between the two, not a guarantee that the missing values can be recovered.

In applications, the intervention labels must also be sufficiently well defined for the potential outcomes to have a clear meaning. If “instrument value 1” bundles several substantively different versions of an offer, or if treatment receipt has materially different versions, a single potential outcome for that label may be ambiguous. The notation is only as meaningful as the interventions it represents.

Common Mistake: Treating Notation as an Assumption

Wrong reasoning: “I wrote $Y_i(d)$, so the instrument cannot affect the outcome directly.” Correction: $Y_i(d)$ is treatment-only notation under exclusion. Begin with $Y_i(z,d)$ when that restriction has not been imposed.
Wrong reasoning: “Because $D_i=D_i(Z_i)$, the instrument is valid.” Correction: This is consistency for treatment receipt. It does not establish instrument independence, exclusion, or any other validity condition.
Wrong reasoning: “The observed outcome reveals the unit’s outcomes under both treatment values.” Correction: Consistency selects the potential outcome under the realized settings only; the other potential outcomes remain unobserved for that unit.
Wrong reasoning: “$D_i(1)$ is the treatment actually received by everyone with $Z_i=1$, regardless of their identity.” Correction: $D_i(1)$ is unit-specific. Consistency equates it to observed $D_i$ only for units whose realized instrument is $Z_i=1$.

Key takeaway: With a binary instrument and treatment, $D_i(z)$ records treatment receipt under instrument setting $z$, while $Y_i(z,d)$ records the outcome under settings $z$ and $d$. Consistency selects $D_i(Z_i)$ and $Y_i(Z_i,D_i)$. Replacing the latter with $Y_i(D_i)$ requires the additional exclusion restriction.

Check Your Understanding

Answer each question using the potential-outcome notation defined above.

If $Z_i=0$, $D_i(0)=1$, and $D_i(1)=0$, what is the observed treatment receipt under consistency?
Write the observed outcome when $Z_i=1$ and $D_i=0$, using the unrestricted outcome notation.
What additional restriction permits replacing $Y_i(z,d)$ with $Y_i(d)$?
For a unit observed with $Z_i=0$, which of $D_i(0)$ and $D_i(1)$ is linked to observed treatment receipt by consistency? Is the other value observed for that unit?
Why does the equation $D_i=D_i(Z_i)$ not establish that the instrument is valid?