The measurement of rank mobility

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Abstract

In this paper we investigate the problem of measuring social mobility when the social status of individuals is given by their rank. In order to sensibly represent the rank mobility of subgroups within a given society, we address the problem in terms of partial permutation matrices which include standard (“global”) matrices as a special case. We first provide a characterization of a partial ordering on partial matrices which, in the standard case of global matrices, coincides with the well-known “concordance” ordering. We then provide a characterization of an index of rank mobility based on partial matrices and show that, in the standard case of comparing global matrices, it is equivalent to Spearman's ρ index.

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      Differences in marginal distributions would be fully controlled, however, were analysts to employ the Spearman rank correlation rather than r (because both marginal distributions would be standard uniform distributions), and this would also have the advantage in the intergenerational context of focusing on positional change. Note also D’Agostino and Dardanoni (2009a) who provided an axiomatic characterization of the Spearman rank correlation as an measure of exchange mobility, thereby taking it beyond being a mere “statistical” index. A second question regarding Beta and r is why they should be calculated using log incomes rather than incomes.

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    We would like to thank an anonymous referee for extremely useful suggestions which have significantly contributed to improve the paper.

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