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Conventionally, the construction of a pair-matched sample selects treated andcontrol units and pairs them in a single step with a view to balancing observedcovariates $\mathbf{x}$ and reducing the heterogeneity or dispersion oftreated-minus-control response differences, $Y$. In contrast, the method ofcardinality matching developed here first selects the maximum number of unitssubject to covariate balance constraints and, with a balanced sample for$\mathbf{x}$ in hand, then separately pairs the units to minimize heterogeneityin $Y$. Reduced heterogeneity of pair differences in responses $Y$ is known toreduce sensitivity to unmeasured biases, so one might hope that cardinalitymatching would succeed at both tasks, balancing $\mathbf{x}$, stabilizing $Y$.We use cardinality matching in an observational study of the effectiveness offor-profit and not-for-profit private high schools in Chile - a controversialsubject in Chile - focusing on students who were in government run primaryschools in 2004 but then switched to private high schools. By pairing tominimize heterogeneity in a cardinality match that has balanced covariates, ameaningful reduction in sensitivity to unmeasured biases is obtained.

annals of applied statistics/the annals of applied statistics

Matching for balance, pairing for heterogeneity in an observational  study of the effectiveness of for-profit and not-for-profit high schools in  Chile