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

• Published in: Journal of geographical systems 2012, Vol.14 (1), p.91-124
• Main Authors: Smith, Tony E., Lee, Ka Lok
• Format: Article
• Language: English
• Subjects: Areal geology; Areal geology. Maps; Autocorrelation; Beta; Computer Appl. in Social and Behavioral Sciences; Dependent variables; Earth sciences; Earth, ocean, space; Econometrics; Economics; Economics and Finance; Errors; Exact sciences and technology; Geographic information systems; Geographical Information Systems/Cartography; Geologic maps, cartography; Geometry; Hypotheses; Landscape/Regional and Urban Planning; Least squares method; Mathematical models; Original Article; Population density; Regional/Spatial Science; Regression; Statistics; Studies; Urban Economics; Variables
• ISSN: 1435-5930; 1435-5949
• DOI: 10.1007/s10109-011-0152-x

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.