From Potential Values to an Observed Record
An instrumental-variable study distinguishes the instrument or encouragement from the treatment a person actually receives. For example, a program may offer a consultation (\(Z=1\)) intended to increase use of a service (\(D=1\)), while some people offered the consultation do not use the service and some not offered it do. To connect this setup to data, we need to specify how each person's observed treatment and outcome relate to their potential values under the instrument value they actually received.
Let \(Z_i\in\{0,1\}\) be the observed instrument for person \(i\), \(D_i\in\{0,1\}\) the observed treatment receipt, and \(Y_i\) the observed outcome. The potential treatment \(D_i(z)\) is the treatment person \(i\) would receive if the instrument were set to \(z\). To allow for an outcome that could depend directly on the instrument as well as on treatment, write \(Y_i(d,z)\) for the outcome person \(i\) would have if treatment were set to \(d\) and the instrument to \(z\).
For example, if \(Z_i=1\) and the person receives treatment, then \(D_i(1)=1\), and the observed outcome is \(Y_i(1,1)\). If that person does not receive treatment, the observed outcome is instead \(Y_i(0,1)\). In either case the data contain only the potential outcome for the realized treatment-instrument pair. The other potential outcomes are counterfactual for that person.
These equalities describe how observed data correspond to potential outcomes; they do not say that the instrument was randomly assigned or that it affects the outcome only through treatment. Those are distinct assumptions. In particular, consistency by itself does not justify dropping \(Z_i\) from the outcome notation.
When the Outcome Can Be Written as \(Y_i(D_i)\)
A common instrumental-variable restriction is that, once treatment is fixed, changing the instrument does not change the outcome. This is the exclusion restriction. It is useful here because it permits a shorter outcome notation, but it is an additional assumption, not part of consistency.
The distinction matters in a study of encouragement. If an offer provides extra information that can affect an outcome even when a person does not take up the service, the outcome might depend on \(Z_i\) at a fixed \(D_i\). In that case, the more general notation \(Y_i(d,z)\) keeps the possibility visible. Whether exclusion is a credible scientific assumption depends on how the encouragement works; observing a dataset does not establish it.
Worked Example: Reading One Person's Record
Suppose a hypothetical encouragement study records whether a person was offered a consultation, whether they used a service, and a follow-up score in points. For person A, suppose \(Z_A=1\), \(D_A(1)=1\), and \(Y_A(1,1)=24\). Their observed record is \(Z_A=1\), \(D_A=1\), and \(Y_A=24\).
The consistency substitutions are $$ D_A=D_A(Z_A)=D_A(1)=1, \qquad Y_A=Y_A(D_A,Z_A)=Y_A(1,1)=24\text{ points}. $$
The observed values agree with the potential values for the realized instrument and treatment. This does not reveal \(D_A(0)\), \(Y_A(0,0)\), or \(Y_A(1,0)\). If the exclusion restriction were also assumed, \(Y_A(1,1)=Y_A(1,0)\), and the observed outcome could be denoted \(Y_A(1)\). Consistency alone does not imply that equality.
How Consistency Maps a Study Dataset
In an encouragement study, one row of data typically records a person's observed assignment, treatment receipt, and outcome. The potential-value notation describes the larger set of values that could have been observed for that person under different assignments. Consistency selects the entries corresponding to the realized assignment and receipt; it does not fill in the other entries.
Worked Example: Mapping Several Hypothetical Participants
Consider three hypothetical participants in an encouragement study. The outcome is a follow-up score in points. Suppose their potential treatment values and the relevant potential outcomes are as shown. Only the potential values matching each person's realized \(Z_i\) and \(D_i\) are observed.
| Person | \(Z_i\) | \(D_i(0)\) | \(D_i(1)\) | Potential outcome at realized pair | Observed \(D_i\) | Observed \(Y_i\) |
|---|---|---|---|---|---|---|
| A | 1 | 0 | 1 | \(Y_A(1,1)=24\) | 1 | 24 |
| B | 1 | 0 | 0 | \(Y_B(0,1)=18\) | 0 | 18 |
| C | 0 | 1 | 1 | \(Y_C(1,0)=21\) | 1 | 21 |
For A, consistency gives \(D_A=D_A(1)=1\) and \(Y_A=Y_A(1,1)=24\) points. For B, it gives \(D_B=D_B(1)=0\) and \(Y_B=Y_B(0,1)=18\) points. For C, it gives \(D_C=D_C(0)=1\) and \(Y_C=Y_C(1,0)=21\) points. Each observed treatment and outcome matches the potential value at the realized instrument.
Notice that the table does not provide all potential outcomes. For instance, B's \(Y_B(0,0)\) is not observed. Nor does the fact that B's \(D_B(1)=0\) tell us what B's outcome would have been under a different instrument assignment. The observed dataset gives one realized row per person, not a record of every possible state.
A Short Derivation and the Assumptions Behind It
The observed-outcome expression follows by applying consistency in two linked steps. First, the realized instrument selects the person's potential treatment. Second, the realized treatment and instrument select the corresponding potential outcome.
Given the realized \(Z_i\), consistency gives \(D_i=D_i(Z_i)\).
Given realized treatment and instrument, consistency gives \(Y_i=Y_i(D_i,Z_i)\).
Therefore \(Y_i=Y_i(D_i(Z_i),Z_i)\). If exclusion also holds, the instrument argument can be removed, yielding \(Y_i=Y_i(D_i(Z_i))\).
The substitution is justified when the recorded instrument, treatment, and outcome correspond to the stated assignments and treatment definition. The treatment and outcome must be well defined: for example, “receives treatment” should not combine materially different versions if those versions have different effects. The usual potential-outcomes formulation also assumes no interference between participants, so one person's assignment does not change another person's potential treatment or outcome. These conditions make the individual-indexed potential values coherent.
Random assignment or independence between the instrument and potential values is not required for this algebraic link. Such conditions matter for interpreting comparisons across instrument groups, but they are separate from consistency. Similarly, exclusion is not needed to write \(Y_i(D_i,Z_i)\); it is needed for the shorter expression \(Y_i(D_i)\).
Worked Example: Why Exclusion Changes the Notation
Suppose, hypothetically, a person has \(Z_i=1\), does not receive treatment, and has observed outcome \(Y_i=18\) points. A possible set of potential outcomes is \(Y_i(0,1)=18\) and \(Y_i(0,0)=15\).
Consistency uses the realized pair: $$ Y_i=Y_i(D_i,Z_i)=Y_i(0,1)=18\text{ points}. $$
Because \(Y_i(0,1)\ne Y_i(0,0)\), this set of potential values does not satisfy exclusion at treatment level \(d=0\). It would therefore be incorrect to replace the observed value with a single \(Y_i(0)\) under these values. If instead the hypothetical values were \(Y_i(0,1)=Y_i(0,0)=18\), exclusion would hold at \(d=0\), and the observed outcome could be written \(Y_i(0)\). This comparison illustrates the notation; an observed record alone cannot reveal both potential outcomes and so cannot verify the equality.
Diagnostics, Comparisons, and Common Errors
Consistency is a mapping rule, not a statistical test. A dataset can be checked for coding errors—for example, that the recorded treatment agrees with the treatment status actually received under the recorded encouragement—but the counterfactual values needed to test the full potential-outcome relation are not observed. Diagnostics therefore focus on whether the study definitions and records make the mapping credible.
- Check timing and definitions. The instrument should be recorded as assigned or received according to the study's definition, and treatment receipt should refer to the relevant period. A treatment measure taken before the encouragement cannot be its realized response.
- Check outcome measurement. Outcome values should use the same measurement rule across instrument and treatment conditions. Differential recording or ambiguous outcome timing can make the observed value a poor match to the stated potential outcome.
- Check treatment versions and interference. If the treatment label hides substantively different versions, or if participants affect each other's outcomes, the simple individual-level potential values may not describe the study adequately.
- Do not diagnose exclusion from consistency. The fact that \(Y_i\) equals the recorded outcome at the realized pair does not show that \(Y_i(d,1)=Y_i(d,0)\). The latter concerns alternative instrument values at fixed treatment.
A useful comparison is between analyzing the effect of assignment and analyzing the effect of treatment received. Assignment-based analyses keep \(Z_i\) as the exposure of interest and do not require rewriting the outcome as \(Y_i(D_i)\). A direct comparison by \(D_i\) instead groups people by actual treatment receipt, which may differ from assignment. The potential-outcome equations clarify why these are not automatically the same comparison: \(D_i\) is selected by \(Z_i\), while \(Y_i\) is observed at the realized treatment-instrument pair. Identifying causal effects from those comparisons requires assumptions beyond the consistency relations taught here.
A frequent mistake is to write \(Y_i=Y_i(D_i)\) immediately and treat this as the definition of consistency. That shortcut silently rules out any direct effect of the instrument. The correction is to begin with \(Y_i=Y_i(D_i,Z_i)\), then simplify only when the exclusion restriction is explicitly adopted. A related error is to suppose that observing \(Y_i\) reveals what the same person would have experienced under another assignment. It reveals only the value at the realized pair.
Check Your Understanding
Answer each question using the distinction between consistency and exclusion. Solutions are included below.
- If \(Z_i=0\), \(D_i(0)=1\), and \(Y_i(1,0)=13\) points, what are the observed treatment and outcome under consistency?
- What extra assumption permits writing \(Y_i=Y_i(D_i)\) rather than \(Y_i=Y_i(D_i,Z_i)\)?
- A person has \(Z_i=1\), \(D_i=0\), and \(Y_i=9\). Which potential outcome is observed under consistency?
- Does observing a person's outcome at \(Z_i=1,D_i=0\) show that \(Y_i(0,1)=Y_i(0,0)\)? Explain.
Solutions. (1) \(D_i=D_i(0)=1\), and \(Y_i=Y_i(1,0)=13\) points. (2) The exclusion restriction, which requires \(Y_i(d,1)=Y_i(d,0)\) for each treatment level \(d\). (3) \(Y_i(0,1)=9\), the outcome at the realized treatment-instrument pair. (4) No. Only \(Y_i(0,1)\) is observed for that person; \(Y_i(0,0)\) is counterfactual, so the equality cannot be checked from this record.