Gravity‐Type Interactive Markov Models—Part II: Lyapunov Stability of Steady States

Smith, Tony; Hsieh, Shang-Hsing

doi:10.1111/0022-4146.00075

Cited by 6 publications

(1 citation statement)

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“…Quasi-symmetry amounts to the reversibility of weights w jk (theorem 1 below; see also Bavaud, 1998), that is, to the phenomenological identity between the Markov process and its time-inverted associated process. Analogous results also hold for more general interactive' Markov processes (Smith and Hsieh, 1997a), where the stability of the stationary flows is guaranteed again by quasi-symmetry (Smith and Hsieh, 1997b). Also, quasi-symmetric weights possess real eigenvalues, which considerably simplifies the study of their time evolution, not addressed here.…”

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The Quasi-Symmetric Side of Gravity Modelling

Bavaud

2002

Environ Plan A

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323) have observed various spatial flows, mobility tables, or opinion-shift data n jk to be well described by the quasi-symmetric model, n jk a j b k g jk , with g jk g kj .As a matter of fact (theorem 1 below), quasi-symmetry precisely enables us to parameterize flows as fng fr, g, ug, where frg are the trivial size effects, fgg are the size-independent accessibilities quantifying the distance-deterrence effect, the symmetric component of migration, and fug are the size-independent utilities quantifying the places' attractivities or push^pull drive, the antisymmetric component of migration.The above ingredients are those of gravity modelling, or more precisely constitute its endogenous aspect (the quasi-symmetry condition). Its exogenous aspect relates (typically by a multiple regression) the accessibilities fgg to travel costs between places, on one hand, and the utilities fug to local socioeconomic conditions, on the other hand.In this paper, I seek to derive or to reveal utility and accessibility parameters from the observation of flows, in the same sense that the sole observation of purchases for a given budget level reveals consumers' utility. As such, this approach is thus also of potential relevance for other nonspatial domains such as the study of mobility tables or opinion shifts. My study is endogenous in that I limit myself to study the theoretical properties of the parameters fg, ug so defined. I do not address the predictive, exogenous part of the program, namely the confrontation of the parameters with socioeconomic and effective distance data.More general, non-quasi-symmetric gravity models can indeed be considered, as, for example, in the work of Sen and Smith (1995), where the symmetry assumption on the accessibilities fgg is dropped. Although perfectly acceptable in a traditional predictive approach, and in particular testable (Smith, 1984), such an enlargement of the class of models under consideration is not feasible in the revealed approach: without quasi-symmetry, the`revealed' parameterization fr, g, ug of flows does not hold any more, and the question of how to construct a mathematically rigorous, behaviorally interpretable parameterization in the general, non-quasi-symmetric case remains open.Abstract. Many properties of gravity models are sole consequences of the quasi-symmetry condition or its avatars. We investigate here quasi-symmetry per se, in contrast to geographical tradition, which has been more focused on the exogenous socioeconomic and spatial conditions. In particular, thè size^utility^accessibility' parameterization of migration counts turn out to rely exclusively upon the quasi-symmetry of flows. Various facets of quasi-symmetry are presented and put in correspondence with Markov chains theory, Bradley^Terry^Luce decision theory, the Weidlich^Haag model, and alternative classical statistical models (marginal homogeneity, symmetry, independence). Existing as well as presumably new estimation and model selection procedures (maximum likelihood, minimum discrimination informati...

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