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The average treatment effect on the treated is $\mathbb{E}[Y(1)-Y(0)\mid A=1]$, averaging over those who actually receive treatment. Their untreated potential outcomes are counterfactual. This can differ from the average treatment effect when effects vary and treatment uptake changes the population mix. Under conditional exchangeability, consistency in causal inference and adequate untreated overlap, it equals $\mathbb{E}[Y\mid A=1]-\mathbb{E}[\mathbb{E}(Y\mid A=0,X)\mid A=1]$. The second term standardizes untreated outcomes to the treated covariate distribution rather than using the crude untreated mean.

Caliper matching permits a candidate matching pair only when a prespecified distance is at most a tolerance $c$. For propensity score $e(X)$, possible rules are $|e(X_i)-e(X_j)|\le c$ or $|\mathrm{logit}e(X_i)-\mathrm{logit}e(X_j)|\le c$. These are different distances; the numeric tolerance must be interpreted on the selected scale.^[ref]

For a treated unit with propensity score $0.40$, a raw-score caliper $c=0.03$ admits a control with score $0.42$ and excludes one with score $0.46$. The matching algorithm still has to choose among admissible controls and decide whether controls may be reused. A tighter caliper excludes distant pairs but can leave more treated units unmatched. Excluding those units changes the population represented by the matched estimate; a caliper is not a guarantee of an unchanged average treatment effect on the treated. Check covariate balance after matching, and account for the paired or reused-control structure in statistical inference.

The no unmeasured confounding assumption is the conditional exchangeability statementAlso assume consistency of potential outcomes, no interference between units, and positivity in causal inference, in particular $\mathbb{P}(A=0\mid X)>0$ on the covariate support of treated units. ThenThe first treated potential outcome equals the observed treated mean by consistency. Moreover,Subtracting proves the displayed identification formula for the average treatment effect on the treated.

The average treatment effect is more relevant than the average treatment effect on the treated because the contemplated policy changes treatment status for schools that are presently non-UTC, not merely the 43 schools that selected into existing UTC status. More precisely, the ideal target would be the average effect among the candidate schools the minister might convert; if that population is represented by all eligible schools, it is the ATE. An ATT from existing UTCs answers a narrower historical question and may not transport to the policy targets.