Journal of Geographical Systems

, Volume 14, Issue 1, pp 91–124 | Cite as

The effects of spatial autoregressive dependencies on inference in ordinary least squares: a geometric approach

  • Tony E. SmithEmail author
  • Ka Lok Lee
Original Article


There is a common belief that the presence of residual spatial autocorrelation in ordinary least squares (OLS) regression leads to inflated significance levels in beta coefficients and, in particular, inflated levels relative to the more efficient spatial error model (SEM). However, our simulations show that this is not always the case. Hence, the purpose of this paper is to examine this question from a geometric viewpoint. The key idea is to characterize the OLS test statistic in terms of angle cosines and examine the geometric implications of this characterization. Our first result is to show that if the explanatory variables in the regression exhibit no spatial autocorrelation, then the distribution of test statistics for individual beta coefficients in OLS is independent of any spatial autocorrelation in the error term. Hence, inferences about betas exhibit all the optimality properties of the classic uncorrelated error case. However, a second more important series of results show that if spatial autocorrelation is present in both the dependent and explanatory variables, then the conventional wisdom is correct. In particular, even when an explanatory variable is statistically independent of the dependent variable, such joint spatial dependencies tend to produce “spurious correlation” that results in over-rejection of the null hypothesis. The underlying geometric nature of this problem is clarified by illustrative examples. The paper concludes with a brief discussion of some possible remedies for this problem.


Spatial dependence Spatial autocorrelation Spatial error model OLS regression Geometric approach 

JEL Classification

C12 C31 



The authors are indebted to Federico Martellosio for valuable comments and suggestions on an earlier draft of this paper. We are also grateful to the two referees for their constructive comments.

Supplementary material

10109_2011_152_MOESM1_ESM.doc (744 kb)
Supplementary material 1 (DOC 744 kb)


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Electrical and Systems EngineeringUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.The Wharton SchoolUniversity of PennsylvaniaPhiladelphiaUSA

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