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

Abstract

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 observedcovariates x and reducing the heterogeneity or dispersion of treatedminus- 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 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 succeedat both tasks, balancing x, stabilizing Y . We use cardinality matching in an observational study of the effectiveness of for-profit and not-for-profit privatehigh schools in Chile—a controversial subject in Chile—focusing on students who were in government run primary schools in 2004 but then switchedto private high schools. By pairing to minimize heterogeneity in a cardinality match that has balanced covariates, a meaningful reduction in sensitivity tounmeasured biases is obtained.