What Does It Mean to Identify a Causal Effect?
A causal question concerns outcomes under alternative treatment assignments, including the assignment each person did not receive. That missing counterfactual can matter even when the observed data are extensive: the data may support one value of the causal effect, or they may support several. The distinction is between point identification and set identification.
Let \(A\in\{0,1\}\) denote treatment, and let \(Y(1)\) and \(Y(0)\) denote the potential outcomes that would be observed under treatment and control, respectively. The observed outcome is \(Y=A Y(1)+(1-A)Y(0)\). For a population mean effect, the target is
In each individual, only one potential outcome is observed. Identification asks what values of \(\tau\) are compatible with the distribution of the observed data and the stated assumptions. It is a question about what the data-generating model implies, not about how precisely a statistic has been calculated from a finite sample.
An identified set can be an interval, a collection of separate values, or—in a finite-population problem—a discrete set. It is not automatically a confidence interval. The identified set represents ambiguity that remains even with complete knowledge of the observed-data distribution under the assumptions. A confidence interval, by contrast, addresses sampling uncertainty in estimating a target or identified set from a finite sample.
How Assumptions Can Deliver a Point
A standard route to point identification uses consistency, conditional exchangeability, and positivity. Consistency means that a person's observed outcome equals the potential outcome corresponding to the treatment actually received. Conditional exchangeability means \((Y(1),Y(0))\) is independent of \(A\) given measured covariates \(X\). Positivity means that for every covariate value in the target population, both treatment levels have positive probability: \(0<P(A=a\mid X=x)<1\) for \(a=0,1\).
Together, these conditions give an identifying formula. First average the potential outcome over covariate values. Exchangeability permits replacing the unobserved potential-outcome mean within each covariate group with the mean among people who received that treatment. Consistency then relates that mean to the observed outcome.
The first equality is iterated expectation. The second uses conditional exchangeability; the third uses consistency. Positivity is needed because the conditional mean among people with \(A=a\) must be defined and supported by observed data at the covariate values being averaged. Subtracting the expression for \(a=0\) from the expression for \(a=1\) identifies \(\tau\).
Worked Example: A Point-Identified Mean Effect Under Conditional Exchangeability
Suppose a hypothetical target population has two covariate strata. The first stratum contains 60% of the population, and the second contains 40%. In each stratum, assume the treatment and control outcome means describe the population conditional means. The values are:
| Stratum | Population share | \(P(A=1\mid X)\) | \(E[Y\mid A=1,X]\) | \(E[Y\mid A=0,X]\) |
|---|---|---|---|---|
| 1 | 0.60 | 0.50 | 0.80 | 0.50 |
| 2 | 0.40 | 0.25 | 0.60 | 0.20 |
Assume consistency, conditional exchangeability given stratum, and positivity. Positivity holds because the treatment probabilities are 0.50 and 0.25, both strictly between zero and one; control probabilities are therefore 0.50 and 0.75. Under these assumptions, the conditional observed means identify the corresponding potential-outcome means. The mean effect in stratum 1 is \(0.80-0.50=0.30\), and in stratum 2 it is \(0.60-0.20=0.40\). Weight by the population shares:
Thus the population mean effect is point identified as 0.34, or 34 percentage points if \(Y\) is a binary outcome coded 0 or 1. This conclusion depends on the stated assumptions. The calculation does not establish conditional exchangeability; that assumption is substantive and cannot generally be verified from the observed outcome means alone.
When the Data Support More Than One Effect
Without a sufficient identifying assumption, the unobserved potential outcomes can vary while leaving every observed treatment and outcome unchanged. If those different completions imply different causal effects, the observed data do not point identify the effect. Restrictions such as a known outcome scale can still rule out some possibilities, producing an identified set rather than a single value.
Worked Example: A Single Treated Person and an Outcome Range
Consider a hypothetical finite-population target consisting of one person. The person receives treatment, and the observed outcome is \(Y(1)=7\). Suppose the outcome scale is known to be continuous from 0 to 10, so the unobserved \(Y(0)\) must lie in \([0,10]\). Assume consistency and no further restriction on the untreated potential outcome.
The individual causal effect is \(Y(1)-Y(0)\). Substituting the observed value and the allowed range gives:
Every value in this interval is compatible with the observed treated outcome and the stated scale restriction: for any effect \(d\in[-3,7]\), setting the unobserved \(Y(0)=7-d\) gives a value between 0 and 10. The effect is therefore set identified as \([-3,7]\), not point identified. The interval is the strongest conclusion supported by the information specified here; choosing, for example, an effect of 2 would require additional information or assumptions.
Worked Example: Two Compatible Potential-Outcome Worlds
Consider four hypothetical people. Two receive treatment and have observed outcomes 1 and 0. Two receive control and have observed outcomes 1 and 0. Outcomes are binary. The observed data are fixed, but the missing potential outcomes are not observed.
In one possible completion, assign the treated people \(Y(0)=1,1\), and the control people \(Y(1)=0,0\). The four individual effects \(Y(1)-Y(0)\) are then \(0,-1,-1,0\). The finite-population average effect is
In a second possible completion, assign the treated people \(Y(0)=0,0\), and the control people \(Y(1)=1,1\). The effects are \(1,0,0,1\), giving
Both completions preserve every observed treatment and outcome, and both respect the binary outcome scale. They imply different average effects. Therefore, the observed data and the scale restriction alone do not point identify the effect. In this finite example, effects need not fill a continuous interval: the set of compatible values can be discrete. These two completions establish that both \(-0.50\) and \(0.50\) are possible; they do not, by themselves, calculate every value in the full identified set.
What a Range Does—and Does Not—Say
A set-valued conclusion is not a failed attempt to estimate a point. It communicates that the assumptions and observed data leave meaningful causal ambiguity. If the identified set includes effects with different signs, for example, those inputs do not determine whether the effect is beneficial or harmful. Reporting only one value would conceal that ambiguity and suggest more information than the analysis provides.
Identification and estimation must also be kept separate. The identification question is what the population data distribution and assumptions determine. Estimation uses a finite sample to approximate the identified quantity or set. Even if assumptions point identify \(\tau\), a sample estimate will usually have sampling uncertainty. Conversely, a very large sample can reduce sampling uncertainty without eliminating ambiguity caused by unobserved counterfactuals or weak assumptions.
A point-identified result is conditional on its identifying assumptions; it does not mean those assumptions are guaranteed to be true. For example, covariate overlap can be examined in observed data, but conditional exchangeability concerns potential outcomes and is generally not directly testable. Diagnostics can reveal some problems with support or model implementation, but they do not convert an untestable identifying assumption into an observed fact.
The main alternatives can be summarized as follows:
| Conclusion | What the assumptions and data imply | Appropriate interpretation |
|---|---|---|
| Point identification | Exactly one parameter value is compatible. | State the identified value and the assumptions it depends on. |
| Set identification | Multiple values are compatible, but the information restricts them to a set. | Report the set and explain what remains unresolved. |
| No useful identification | The stated information does not support the proposed restriction on the parameter. | Do not present an unsupported point or range as an implication of the analysis. |
Diagnostics, Limitations, and a Common Mistake
Before claiming point identification, check that each assumption needed for the identifying argument has been stated and that the observed data support its use. For conditional identification, inspect whether treatment and control are represented across the covariate values in the target population. Sparse or absent treatment groups in some strata signal a positivity problem: conditional means may be poorly determined or unavailable there. Also verify that the target population and outcome scale are clear. A range derived under an outcome restriction is only as defensible as that restriction.
These checks have limits. Observed data can expose lack of overlap, but good overlap does not prove exchangeability. A model can produce a precise numerical answer even when its causal identifying assumptions are doubtful. Likewise, a narrow identified set is not automatically credible if it is narrow only because of an implausibly strong assumption.
Common mistake: “The estimate is 0.34, so the causal effect is point identified, and a 95% confidence interval around it accounts for everything uncertain.” This reasoning confuses calculation with identification and sampling uncertainty with assumptions about missing potential outcomes. The correction is to state the assumptions that identify the target first. If those assumptions imply one value, sampling methods can quantify uncertainty in estimating it. If they imply multiple values, report the identified set; a confidence procedure may address sampling uncertainty, but it does not erase the set’s identification ambiguity.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If consistency holds but conditional exchangeability does not, has the mean causal effect necessarily been point identified? Explain.
- A hypothetical treated person has \(Y(1)=4\), and the unobserved \(Y(0)\) is known only to lie between 1 and 6. What is the identified set for the individual effect?
- In the two-stratum example, what would happen to the stated point-identification argument if one stratum had no treated individuals?
- Why is an identified set not the same thing as a 95% confidence interval?
Solutions. (1) No. Consistency links observed outcomes to received treatment, but by itself it does not determine the unobserved counterfactual means. (2) The effect is \(4-Y(0)\), so \(4-6\leq 4-Y(0)\leq 4-1\), giving the set \([-2,3]\). (3) Positivity would fail in that stratum, so the observed treated mean there would not identify the treated potential-outcome mean without additional assumptions or information. (4) The identified set describes multiple parameter values compatible with the observed-data distribution and assumptions; a confidence interval describes sampling uncertainty under a statistical procedure. One concerns identification ambiguity, the other finite-sample uncertainty.