Journal of Geographical Systems

, Volume 15, Issue 3, pp 265–289 | Cite as

A spatial interaction model with spatially structured origin and destination effects

  • James P. LeSageEmail author
  • Carlos Llano
Original Article


We introduce a Bayesian hierarchical regression model that extends the traditional least-squares regression model used to estimate gravity or spatial interaction relations involving origin-destination flows. Spatial interaction models attempt to explain variation in flows from n origin regions to n destination regions resulting in a sample of N = n 2 observations that reflect an n by n flow matrix converted to a vector. Explanatory variables typically include origin and destination characteristics as well as distance between each region and all other regions. Our extension introduces latent spatial effects parameters structured to follow a spatial autoregressive process. Individual effects parameters are included in the model to reflect latent or unobservable influences at work that are unique to each region treated as an origin and destination. That is, we estimate 2n individual effects parameters using the sample of N = n 2 observations. We illustrate the method using a sample of commodity flows between 18 Spanish regions during the 2002 period.


Commodity flows Spatial autoregressive random effects Bayesian hierarchical models Spatial connectivity of Origin-destination flows 

JEL Classification

C21 R11 R32 


  1. Ballester C, Calvó-Armengol A, Zenou Y (2006) Who’s who in networks. Wanted: the key player. Econometrica 74:1403–1417CrossRefGoogle Scholar
  2. Banerjee S, Gelfand AE, Polasek W (2000) Geostatistical modelling for spatial interaction data with application to postal service performance. J Stat Plan Inference 90:87–105CrossRefGoogle Scholar
  3. Barry RP, Pace RK (1997) Kriging with large data sets using sparse matrix techniques. Commun Stat Comput Simul 26:619–629Google Scholar
  4. Besag JE, York JC, Mollie A (1991) Bayesian image restoration, with two applications in spatial statistics. (with discussion). Ann Inst Stat Math 43:1–59CrossRefGoogle Scholar
  5. Besag JE, Kooperberg CL (1995) On conditional and intrinsic autoregressions. Biometrika 82:733–746Google Scholar
  6. Bonacich PB (1987) Power and centrality: a family of measures. Am J Sociol 92:1170–1182CrossRefGoogle Scholar
  7. Cressie N (1995) Bayesian smoothing of rates in small geographic areas. J Reg Sci 35:659–673CrossRefGoogle Scholar
  8. Fischer MM, Scherngell T, Jansenberger E (2006) The geography of knowledge spillovers between high-technology firms in Europe evidence from a spatial interaction modelling perspective. Geogr Anal 38(3):288–309Google Scholar
  9. Gelfand, Baerjee (2004) Hierarchical modeling and analysis for spatial data. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  10. Gelfand AE, Sahu S, Carlin BP (1995) Efficient parameterizations for normal linear mixed models. Biometrika 82(3):479-488CrossRefGoogle Scholar
  11. LeSage JP, Fischer MM, Scherngell T (2007) Knowledge spillovers across Europe, evidence from a poisson spatial interaction model with spatial effects. Pap Reg Sci 86(3):393–421CrossRefGoogle Scholar
  12. LeSage JP, Pace RK (2008) Spatial econometric modeling of origin-destination flows. J Reg Sci 48(5):941–967CrossRefGoogle Scholar
  13. Llano C (2004) Economía espacial y sectorial: el comercio interregional en el marco input-output. Instituto de Estudios Fiscales. Investigaciones 1-2004Google Scholar
  14. Llano C, Esteban A, Pulido A, Prez J (2010) Opening the interregional trade black box: the C-intereg database for the Spanish economy (1995–2005). Int Reg Sci Rev 33:302–337CrossRefGoogle Scholar
  15. Oliver J, Luria J, Roca A, Prez J (2003) en La apertura exterior de las regiones en España: Evolución del comercio interregional e internacional de las Comunidades Autónomas. 1995–1998. Institut dÉstudis Autonòmics. Generalitat de Catalunya. Ed Tirant lo Blanch. ValenciaGoogle Scholar
  16. Sen A, Smith TE (1995) Gravity models of spatial interaction behavior. Springer, HeidelbergCrossRefGoogle Scholar
  17. Smith TE, LeSage JP (2004) A Bayesian probit model with spatial dependencies. In: LeSage JP, Kelley Pace R (eds) Advances in econometrics: volume 18: spatial and spatiotemporal econometrics. Elsevier Ltd, Oxford, pp 127–160Google Scholar
  18. Sun D, Tsutakawa RK, Speckman PL (1999) Posterior distribution of hierarchical models using car(1) distributions. Biometrika 86:341-350CrossRefGoogle Scholar
  19. Sun D, Tsutakawa RK, Kim H, He Z (2000) Bayesian analysis of mortality rates with disease maps. Stat Med 19:2015–2035CrossRefGoogle Scholar
  20. Tiefelsdorf M (2003) Misspecifications in interaction model distance decay relations: a spatial structure effect. J Geogr Syst 5:25–50Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Fields Endowed Chair of Urban and Regional Economics, Department of Finance and EconomicsMcCoy College of Business Administration, Texas State UniversitySan MarcosUSA
  2. 2.Departamento de Análisis Económico, and CEPREDEUniversidad Autónoma de MadridMadridSpain

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