Constrained Variants of the Gravity Model and Spatial Dependence: Model Specification and Estimation Issues

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.