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A new mobility index for transition matrices

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Abstract

In this work we construct a mobility index able to grasp the prevailing direction in the evolution of a given set of statistical units. We consider the case of dynamics ruled by a transition matrix, whose states are based on an ordered economic variable (firm size or income, among others) such that the future position of an individual can be better or worse than the current one. The existing indices measure only the absolute value of mobility, without providing information about the main direction in the dynamics. We propose here a whole family of directional indices defined as functions of the transition matrix, so that their absolute value measures the intensity of mobility, and their sign (\(+/-\)) represents the prevailing direction towards improvement/worsening in the dynamics under study.

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Notes

  1. The \(p_{ij}\)’s may depend on time without affecting the results, since we are considering a single-step transition matrix.

  2. The naive idea consists in calculating the linear transformation \(I\in [m_1,m_2]\rightarrow I^{\prime }\in [-1,+1]\) given by the formula \(I^{\prime }=\frac{2}{m_1+m_2}I+\frac{m_1-m_2}{m_1+m_2}\). Unfortunately this choice has a relevant drawback when the interval \([m_1,m_2]\) is not symmetric respect to 0, that is when \(m_2\ne -m_1\). Indeed it happens that matrices with the original index value \(I^{\omega ,v}(P)<0\) result to have positive normalized value \(I^{\prime }(P)\) and viceversa.

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Acknowledgments

The authors are grateful to Provincia di Prato for the ASIA-ISTAT dataset, and to CRISP (Milano) for the C.OBB. dataset. They also wish to thank E. Fabrizi and the anonymous referees for their constructive comments and suggestions.

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Authors and Affiliations

  1. Department of Economic and Social Sciences, Università Cattolica del Sacro Cuore, Piacenza, Italy

    C. Ferretti

  2. Department of Industrial Engineering, University of Parma, Parma, Italy

    P. Ganugi

Authors
  1. C. Ferretti
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  2. P. Ganugi
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Correspondence to C. Ferretti.

Additional information

The paper reflects the opinions and results of both the authors. Nevertheless Sects. 2, 3, 5.1 and 6.1 are attributable to C. Ferretti and Sects. 4, 5.2, 6.2 and 6.3 to P. Ganugi.

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Ferretti, C., Ganugi, P. A new mobility index for transition matrices. Stat Methods Appl 22, 403–425 (2013). https://doi.org/10.1007/s10260-013-0232-9

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