Journal of Geographical Systems

, Volume 15, Issue 3, pp 291–317 | Cite as

Constrained variants of the gravity model and spatial dependence: model specification and estimation issues

  • Daniel A. Griffith
  • Manfred M. FischerEmail author
Original Article


In this paper, we distinguish three constrained variants of the gravity model of spatial interaction: doubly constrained, production constrained and attraction constrained exponential gravity models. These model variants include origin- and/or destination-specific balancing factors that act as constraints to ensure that the estimated rows and columns of the flow data matrix sum to the observed row and column totals. Because flows are typically counts, the Poisson rather than the normal probability model specification furnishes the appropriate statistical distribution, and parameter estimation can be achieved via Poisson regression. This probability model specification motivates the use of origin and/or destination fixed effects or—under certain conditions—the use of origin- and/or destination-specific random effects for model estimation. The paper establishes theoretical connections between balancing factors, fixed effects represented by binary indicator variables and random effects. The results pertaining to both the doubly and singly constrained cases of spatial interaction are illustrated with an empirical example while accounting for spatial dependence between flows from locations neighbouring both the origins and destinations during estimation.


Constrained gravity models Count data Patent citation flows Poisson Spatial dependence in origin–destination flows Spatial filtering Spatial econometrics 

JEL Classification

C13 C31 R15 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.University of Texas at DallasRichardsonUSA
  2. 2.Vienna University of Economics and BusinessViennaAustria

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