Unconfoundedness and Overlap
Adjustment for observed covariates identifies causal effects only under assumptions connecting treatment assignment, potential outcomes, and the covariates.
Conditional unconfoundedness
The core assumption is
\[\boxed{(Y(0),Y(1))\perp W\mid X}.\]Within groups that share the same covariates $X$, treatment assignment is independent of both potential outcomes. This is also called conditional ignorability or selection on observables.
Unconfoundedness is not a statement that treatment is marginally random. It states that adjustment for $X$ is sufficient. The set $X$ must include the common causes needed to block confounding paths, and it should consist of pretreatment variables.
Overlap
The positivity or overlap condition is
\[\boxed{0<P(W=1\mid X)<1}.\]At every covariate value relevant to the target population, both treatment states must be possible. If some types of units are always treated or always untreated, their missing counterfactual mean cannot be learned from comparable observed units without extrapolation.
In applications, probabilities very close to zero or one create weak practical overlap even when strict mathematical positivity holds. This leads to unstable weights and estimates driven by a few observations.
Identification of the ATE
Under consistency, unconfoundedness, and overlap,
\[E[Y(w)]=E\left[E[Y^{obs}\mid W=w,X]\right], \qquad w\in\{0,1\}.\]Therefore,
\[ATE= E\left[ E[Y^{obs}\mid W=1,X] - E[Y^{obs}\mid W=0,X] \right].\]This result motivates regression adjustment, stratification, matching, and weighting.
What can be diagnosed
Overlap and observed-covariate balance can be assessed with data. Unconfoundedness itself cannot be verified from observed treatment and outcome data because it concerns missing potential outcomes and unmeasured variables.