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

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