Deriving Identification at a Single Decision Time

From Assumptions to an Observed-Data Mean

At a single decision time, the target is often the mean outcome that would occur if everyone received a specified treatment \(a\), written \(E[Y^a]\). The challenge is that each person receives only one treatment, so the counterfactual outcome \(Y^a\) is not generally observed for everyone. Identification means expressing this target using the distribution of variables that are observed.

Let \(L\) denote covariates measured before treatment \(A\), and let \(Y\) be the observed outcome. The covariates may be categorical, continuous, or mixed. For the derivation below, assume the treatment and outcome are well-defined, the counterfactual outcome is integrable, and the following conditions hold.

These assumptions serve different purposes. Exchangeability lets us use people who received \(a\) to learn about the counterfactual outcome under \(a\), within covariate levels. Consistency connects that counterfactual outcome to the outcome actually recorded. Support prevents the calculation from requiring an unobserved treatment-specific outcome in a covariate stratum where nobody received that treatment.

Deriving the Identification Formula

The formula averages the treatment-specific conditional outcome mean over the covariate distribution in the target population. It does not average over the distribution of \(L\) among people who happened to receive treatment \(a\).

To prove the result, first condition the target mean on \(L\) and use the law of iterated expectation. For almost every \(l\), exchangeability gives the first equality below. Since support makes conditioning on \(A=a,L=l\) available, consistency gives the second.

The middle equality follows from \(Y^a \perp A\mid L\): within each covariate level, learning that \(A=a\) does not change the conditional distribution, and hence does not change the conditional mean, of \(Y^a\). The last equality follows because \(Y=Y^a\) for people with \(A=a\). Integrability ensures these expectations exist. Thus the counterfactual mean is a functional of the observed-data distribution. \(\square\)

Why the Covariate Distribution Matters

The formula is a standardization: it computes a conditional mean for each covariate value under the treatment of interest, then averages those means over the target population’s covariate distribution. This distinction matters when treatment groups differ in their covariate composition. A comparison of the two observed treatment-group means can reflect both treatment and those compositional differences.

A Continuous Covariate and a Weighted Representation

The same argument applies when \(L\) is continuous. The expectation over \(L\) is then an integral, not a sum. For example, a fitted conditional mean must be averaged over the target population’s covariate distribution; evaluating it only at the mean covariate value is not generally equivalent.

There is also an equivalent inverse-probability-weighted representation. Define \(e_a(L)=P(A=a\mid L)\). Under the same assumptions and suitable integrability of the weighted expression,

For the middle equality, conditional on \(L\), the numerator is \(e_a(L)E[Y\mid A=a,L]\); support allows division by \(e_a(L)\). This identity is a second derivation of the same target, not a different causal assumption. Standardization works directly with conditional outcome means, whereas the weighted expression reweights observed outcomes. Each representation can be difficult to estimate when treatment support is weak.

When Identification Fails

This example shows why support is a necessary condition, not merely a technical convenience. In data, warning signs include covariate strata with no treated or no untreated observations, very small treatment probabilities, and estimates that depend heavily on extrapolating a fitted outcome model beyond the observed treatment groups. A model can produce a numerical prediction in an empty stratum, but that prediction does not create information in the data.

Identification also depends on measuring an adequate set of pre-treatment covariates. If an unmeasured common cause of treatment and outcome remains after conditioning on \(L\), conditional exchangeability may fail. Likewise, the derivation requires covariates to precede the decision and the treatment to have a sufficiently well-defined meaning for consistency to apply. These assumptions are substantive; the observed data alone cannot verify them.

Outcome regression and weighting are computational approaches to the same identified quantity when their assumptions and target population agree. The crude comparison \(E[Y\mid A=1]-E[Y\mid A=0]\) does not generally implement the required standardization. Nor does simply adjusting a conditional regression coefficient necessarily report the marginal contrast \(E[Y^1]-E[Y^0]\); the conditional parameter and the population-average target are different quantities.