A Kullback-Leibler measure of conditional segregation

In this paper the Kullback-Leibler notion of discrepancy (Kullback and Leibler, 1951) is used to propose a measure of multigroup segregation over a set of organizational units within a multivariate framework. Among the main results of the paper it is established that the Mutual Information index of segregation, M, first proposed by Theil and Finizza (1971), whose ranking has been fully characterized in terms of seven ordinal axioms by Frankel and Volij (2009), can be decomposed to isolate a term which captures segregation conditional on any vector of covariates. Furthermore, consistent estimators for M and the terms in its decomposition are proposed, and their asymptotic properties are obtained. The usefulness of the approach is illustrated by looking at patterns of multiracial segregation across public schools in the U.S. for the academic years 1989-90 and 2005-06. It is found that most within-cities segregation and a significant part of within-districts segregation is accounted for by county-level income per capita and wages per job, and teachers per pupil at school level.