Advertisement

Topology, Dependency Tests and Estimation Bias in Network Autoregressive Models

  • Steven FarberEmail author
  • Antonio Páez
  • Erik Volz
Chapter
Part of the Advances in Spatial Science book series (ADVSPATIAL)

Abstract

Regression analyses based on spatial datasets often display spatial autocorrelation in the substantive part of the model, or residual pattern in the disturbances. A researcher conducting investigations of a spatial dataset must be able to identify whether this is the case, and if so, what model specification is more appropriate for the data and problem at hand. If autocorrelation is embedded in the dependent variable, the following spatial autoregressive (SAR) model with a spatial lag can be used:
$$\begin{array}{rcl} & & \mathbf{y} = \rho \mathbf{Wy} + \mathbf{X}\beta + \varepsilon , \\ & & \varepsilon \sim N(0,{\sigma }^{2}). \end{array}$$
(1)
On the other hand, when there is residual pattern in the error component of the traditional regression model, the spatial error model (SEM) can be used:
$$\begin{array}{rcl} & & \mathbf{y} = \mathbf{X}\beta + \mathbf{u}, \\ & & \mathbf{u} = \rho \mathbf{Wu} + \varepsilon , \\ & & \varepsilon \sim N(0,{\sigma }^{2}).\end{array}$$
(2)
In the above equations, W is the spatial weight matrix representing the structure of the spatial relationships between observations, ρ is the spatial dependence parameter, u is a vector of autocorrelated disturbances, and all other terms are the elements commonly found in ordinary linear regression analysis.

Keywords

Matrix Density Degree Distribution Cluster Coefficient Estimation Bias Spatial Error Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Anselin L (1986) Some further notes on spatial models and regional science. J Reg Sci 26:799–802CrossRefGoogle Scholar
  2. Anselin L (1988a) Lagrange multiplier test diagnostics for spatial dependence and spatial heterogeneity. Geogr Anal 20:1–17.CrossRefGoogle Scholar
  3. Anselin L (1988b) Spatial econometrics: methods and models. Kluwer, DordrechtGoogle Scholar
  4. Anselin L (2003) Spatial externalities, spatial multipliers, and spatial econometrics. Int Reg Sci Rev 26:153–166CrossRefGoogle Scholar
  5. Anselin L, Florax RJGM (1995) Small sample properties of tests for spatial dependence in regression models: some further results. In: Anselin L, Florax RJGM (eds) New directions in spatial econometrics. Springer, Berlin, pp 21–74CrossRefGoogle Scholar
  6. Anselin L, Rey S (1991) Properties of tests for spatial dependence in linear-regression models. Geogr Anal 23:112–131CrossRefGoogle Scholar
  7. Barabasi AL, Albert R (1999) Emergence of scaling in random networks. Science 286:509–512CrossRefGoogle Scholar
  8. Bartels CPA, Hordijk L (1977) Power of generalized moran contiguity coefficient in testing for spatial autocorrelation among regression disturbances. Reg Sci Urban Econ 7:83–101CrossRefGoogle Scholar
  9. Cliff A, Ord JK (1975) The choice of a test for spatial autocorrelation. In: Davis J, McCullagh M (eds) Display and analysis of spatial data. Wiley, ChichesterGoogle Scholar
  10. Cliff AD, Ord JK (1973) Spatial autocorrelation. Pion, LondonGoogle Scholar
  11. Cliff, A. D. and Ord, J. K. (1981) Spatial Processes: models and applications. Pion, LondonGoogle Scholar
  12. Cordy C, Griffith D (1993) Efficiency of least squares estimators in the presence of spatial autocorrelation. Commun Stat B 22:1161–1179CrossRefGoogle Scholar
  13. Dow MM, Burton ML, White DR (1982) Network auto-correlation – a simulation study of a foundational problem in regression and survey-research. Soc Networks 4:169–200CrossRefGoogle Scholar
  14. Farber S, Páez A, Volz E (2009) Topology and dependency tests in spatial and network autoregressive models. Geogr Anal 41:158–180CrossRefGoogle Scholar
  15. Florax RJGM, de Graaff T (2004) The performance of diagnostic tests for spatial dependence in linear regression models: a meta-analysis of simulation studies. In: Anselin L, Florax RJGM, Rey S (eds) Advances in spatial econometrics: methodology, tools and applications. Springer, Berlin, pp 29–65Google Scholar
  16. Florax RJGM, Rey S (1995) The impact of misspecified spatial structure in linear regression models. In: Anselin L, Florax RJGM (eds) New directions in spatial econometrics, Springer, Berlin, pp 111–135CrossRefGoogle Scholar
  17. Haggett P, Chorly RJ (1970) Network analysis in geography. St. Martin’s Press, New YorkGoogle Scholar
  18. Haining R (1977) Model specification in stationary random fields. Geogr Anal 9:107–129CrossRefGoogle Scholar
  19. Haining R (1978) The moving average model for spatial interaction. Trans Inst Brit Geogr 3:202–225CrossRefGoogle Scholar
  20. Kansky KJ (1963) Structure of transportation networks: relationships between network geometry and regional characteristics. Technical paper, University of ChicagoGoogle Scholar
  21. Kelejian HH, Robinson DP (1998) A suggested test for spatial autocorrelation and/or heteroskedasticity and corresponding Monte Carlo results. Reg Sci Urban Econ 28:389–417CrossRefGoogle Scholar
  22. Leenders RTAJ (2002) Modeling social influence through network autocorrelation: constructing the weight matrix. Soc Networks 24:21–47CrossRefGoogle Scholar
  23. LeSage JP (2009) Spatial econometrics toolbox. http://www.spatial-econometrics.com. 29 Sept. 2009
  24. Mizruchi MS, Neuman EJ (2008) The effect of density on the level of bias in the network autocorrelation model. Soc Networks 30:190–200CrossRefGoogle Scholar
  25. Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167–256CrossRefGoogle Scholar
  26. Páez A, Scott DM (2007) Social influence on travel behavior: a simulation example of the decision to telecommute. Environ Plann 39:647–665CrossRefGoogle Scholar
  27. Páez A, Scott DM, Volz E (2008) Weight matrices for social influence: an investigation of measurement errors and model identification and estimation quality issues. Soc Network 30:309–317CrossRefGoogle Scholar
  28. Smith TE (2009) Estimation bias in spatial models with strongly connected weight matrices. Geogr Anal 41:307–332CrossRefGoogle Scholar
  29. Stetzer F (1982) Specifying weights in spatial forecasting models – the results of some experiments. Environ Plann 14:571–584CrossRefGoogle Scholar
  30. Volz E (2004) Random networks with tunable degree distribution and clustering. Phys Rev E 70:5056115–5056121CrossRefGoogle Scholar
  31. Watts DJ, Strogatz SH (1998) Collective dynamics of “small-world” networks. Nature 393:440–442CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Centre for Spatial Analysis/School of Geography and Earth SciencesMcMaster UniversityHamiltonCanada

Personalised recommendations