Identification of causal effects from observational data rests on three logically distinct conditions: observed outcomes correspond to the specified intervention, treatment groups are exchangeable after adjustment, and every relevant covariate stratum contains the treatment alternatives being compared.
Conditional exchangeability
Definition. Conditional exchangeability for binary treatment is \[ \{Y(0),Y(1)\}\perp\!\!\!\perp A\mid X. \] Mean exchangeability, sufficient for the ATE mean, is the weaker condition \[ \mathbb E\{Y(a)\mid A,X\}= \mathbb E\{Y(a)\mid X\},\qquad a=0,1. \]
Exchangeability is not a testable independence in the observed law because one potential outcome is missing for every unit. Subject-matter knowledge, design, negative controls, and sensitivity analysis are needed to defend or probe it.
Positivity and overlap
Definition. Positivity requires \[ 0<e(X)<1\quad\text{almost surely on the target population}, \qquad e(x)=P(A=1\mid X=x). \] Strong overlap assumes \(\varepsilon\le e(X)\le1-\varepsilon\) for some \(\varepsilon>0\).
Structural nonpositivity occurs when treatment is impossible in part of the target population. Practical nonpositivity occurs when estimated probabilities are extremely close to zero or one in finite samples. The former changes what is identifiable; the latter creates large variance and extrapolation.
The adjustment formula
Theorem. Under consistency, conditional mean exchangeability, and positivity, \[ \mathbb EY(a)= \mathbb E\{\mathbb E(Y\mid A=a,X)\}. \] Consequently, \[ \operatorname{ATE} =\mathbb E\{m_1(X)-m_0(X)\}, \qquad m_a(x)=\mathbb E(Y\mid A=a,X=x). \]
Proof. By iterated expectation, \[ \mathbb EY(a)=\mathbb E[\mathbb E\{Y(a)\mid X\}]. \] Mean exchangeability replaces the inner mean by \(\mathbb E\{Y(a)\mid A=a,X\}\). Positivity makes this observed conditional mean well defined on the target support, and consistency replaces \(Y(a)\) with \(Y\).
Inverse-probability representation
Theorem. Under the same assumptions, \[ \mathbb EY(1)=\mathbb E\left\{\frac{AY}{e(X)}\right\}, \qquad \mathbb EY(0)=\mathbb E\left\{\frac{(1-A)Y}{1-e(X)}\right\}. \]
Proof. Condition on \(X\): \[ \mathbb E\left(\frac{AY}{e(X)}\middle|X\right) =\frac{P(A=1\mid X)}{e(X)} \mathbb E(Y\mid A=1,X)=m_1(X). \] Integrate and use the adjustment theorem. The control identity is identical.
Why positivity is an identification condition
Counterexample. Suppose \(A=X\) with \(X\in\{0,1\}\). Then no controls are observed at \(X=1\) and no treated units at \(X=0\). The observed law identifies \(m_1(1)\) and \(m_0(0)\), but not \(m_0(1)\) or \(m_1(0)\). Even if conditional exchangeability is asserted, the ATE over both covariate strata remains unidentified without extrapolation restrictions.
Minimal sufficient adjustment sets
Proposition. If \(\{Y(0),Y(1)\}\perp\!\!\!\perp A\mid X\) and a statistic \(b(X)\) satisfies \(A\perp\!\!\!\perp X\mid b(X)\), then \[ \{Y(0),Y(1)\}\perp\!\!\!\perp A\mid b(X). \]
Proof. For any potential-outcome event \(B\), \[ P\{Y(a)\in B\mid A,b(X)\} =\int P\{Y(a)\in B\mid A,X\}\, P(\mathrm{d}X\mid A,b(X)). \] Exchangeability removes \(A\) from the first factor; the balancing property removes it from the second. The resulting integral depends only on \(b(X)\).
Conditioning on the wrong variables
Collider bias. Let \(A\) and \(Y\) be independent causes of \(C\). Marginally \(A\perp Y\), but conditioning on \(C\) generally associates them. Thus “adjust for every observed variable” is not a valid strategy. Variables affected by treatment can also block part of the causal effect or induce selection bias.
Consistency as intervention semantics
When treatment has multiple versions, write \(Y(a,v)\). The coarse statement \(Y=Y(a)\) when \(A=a\) assumes either that versions have identical effects or that their distribution is part of the intervention definition. Similarly, measurement error in \(A\) can break consistency even when treatment assignment itself is unconfounded.
Diagnostics do not verify exchangeability
Overlap plots and covariate balance diagnose the observed treatment mechanism. They can reveal practical positivity failure or failure of a weighting procedure to balance measured covariates. They cannot establish balance of unmeasured confounders. A defensible analysis states the measured adjustment set, the causal rationale for it, the target population on which positivity is plausible, and a sensitivity analysis for residual confounding.