Geographically weighted regression, Multicollinearity, Local regression diagnostics, Spatial eigenvectors, Experimental spatial design,

IntroductionUsually we expect to see some degree of spatial autocorrelation among spatially distributed univariate observations. This autocorrelation originates from (a) missing exogenous factors that exhibit distinctive spatial patterns and thus spatially tie the residuals together or (b) underlying spatial processes that emerge from spatial exchange mechanisms among the observations, and/or (c) an inappropriate spatial aggregation of the underlying observational units (see Anselin, 1988;Tiefelsdorf, 2000). The presence of spatial autocorrelation violates the ordinarily stated assumption of stochastic independence among observations, on which statistical inference from most classical statistical models is based. Thus, ignoring spatial autocorrelation leads to biased standard errors and/or biased parameter estimates, as well as artificially inflated degrees of freedom (see Wakefield, 2003).Common practice, when dealing with spatially distributed observations, is to use either maximum likelihood estimation or Bayesian estimation. These parametric estimation procedures must explicitly specify the distributional characteristics of the underlying models. In contrast, nonparametric methods are distribution free without sacrificing too much information in a sample. However, they may become quite computer intensive. See Hollander and Wolfe (1999) for more details on nonparametric statistical methods. More specifically in the spatial domain, the nonparametric eigenvector filtering procedure does not require restrictive and perhaps unjustified distributional assumptions. The eigenvector spatial filtering procedure is founded on the standard ordinary least squares (OLS) estimator and is, apart from the assumptions of independence and constant variance of the disturbances, distribution free owing to the Gauss^Markov theorem. The spatial filtering estimator is fairly robust to model specification errors compared with a spatial maximum likelihood estimator. The interpretation of its results is straightforward as the different components of a spatial process can be

In analogy to the exact distribution of the Durbin—Watson d statistic for serial autocorrelation of regression residuals, the exact small sample distribution of Moran's I statistic (or alternatively Geary's c) can be derived. Use of algebraic results by Koerts and Abrahamse and theoretical results by Imhof, allows the authors to determine by numerical integration the exact distribution function of Moran's I for normally distributed variables. For the case in which the explanatory variables have been neglected, an upper and a lower bound can be given within which the exact distribution of Moran's I for regression residuals will lie. Furthermore, the proposed methodology is flexible enough to investigate related topics such as the characteristics of the exact distribution for distinct spatial structures as well as their different specifications, the exact power function under different spatial autocorrelation levels, and the distribution of Moran's I for nonnormal random variables.

In spatial statistics and spatial econometrics two coding schemes are used predominately. Except for some initial work, the properties of both coding schemes have not been investigated systematically. In this paper we do so for significant spatial processes specified as either a simultaneous autoregressive or a moving average process. Results show that the C-coding scheme emphasizes spatial objects with relatively large numbers of connections, such as those in the interior of a study region. In contrast, the ^-coding scheme assigns higher leverage to spatial objects with few connections, such as those on the periphery of a study region. To address this topology-induced heterogeneity, we design a novel ^-coding scheme whose properties lie in between those of the C-coding and the JK-coding schemes. To compare these three coding schemes within and across the different spatial processes, we find a set of autocorrelation parameters that makes the processes stochastically homologous via a method based on the exact conditional expectation of Moran's /. In the new S-coding scheme the topology induced heterogeneity can be removed in toto for Moran's/ as well as for moving average processes and it can be substantially alleviated for autoregressive processes.

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