The Endogenous Analysis of Flows, with Applications to Migrations, Social Mobility and Opinion Shifts

Abstract

Two major traditions coexist in the statistical analysis of spatial and social change: mobility analysis (e.g., Boudon, 1973) initiated by Prais (1953) on one hand, and gravity modelling for geographic flows (e.g., Fotheringham and O'Kelly, 1989; Sen and Smith, 1995), initiated by Reilly (1931) on the other hand. This paper focuses on the formal content of both traditions and explores their relationship. Three small data sets (migrations, occupational mobility and opinion shifts) illustrate the issues. In particular, we show how the sojourn times and jump matrices of mobility analysis are related to the accessibilities and expansivities of gravity modelling. We also address in detail the issues of time-dependence and categories aggregation. Two well-behaved indices of mobility are defined and discussed, namely the average sojourn time, which measures short-time mobility, and the second eigenvalue of the transition matrix measuring long-time mobility.

Keywords:

• accessibilities
• aggregation
• Courgeau index
• expansivities
• gravity model
• jump matrix
• Markov embedding
• mobility indices
• quasi-symmetry
• relative entropy
• reversibility
• Shorrocks index
• sojourn times
• stayers-movers

Notes

¹The parameters in (4) are in one-to-one correspondence with the parameters of the log-linear formulation (see, e.g., Bishop et al., 1975) with ν _jk  = ν _kj , λ_• = 0, μ_• = 0 and ν_•k  = 0; however, the correspondence is not particularly enlightening. Although formally convenient, log-linear parameters possess no easy interpretation nor good aggregation properties.

²Defining (respectively ) as the left-hand side of (10) with weights a _j (respectively b _j ) and substituting (4) yields and , that is , as required from (4).

³Note that flows resulting from the aggregation of independent flows are not independent either. This well know “defect” actually justifies factorial approaches such that correspondence analysis, aiming at decomposing dependent joint counts into a sum of independent components.

⁴ h(t) is the entropy rate of the dynamical process governed by W(t) (see e.g. Cover and Thomas, 1991).