The Effects of Spatial Autoregressive Dependencies on Inference in Ols: a Geometric Approach*

There is a common belief that the presence of residual spatial autocorrelation in OLS regression leads to inflated significance levels in beta coefficients, and in particular, inflated levels relative to the more efficient Spatial Errors (SE) model. However, 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 variable(s), 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. ______________________________________________________ * 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. Tony E. Smith Department of Electrical and Systems Engineering University of Pennsylvania Ka Lok Lee The Wharton School University of Pennsylvania