Clearly, if the bandwidth is such as to include a large number of observations, there will be relatively little or no spatial variation in the coefficients, and if the bandwidth is small, there will potentially be large amounts of variation. A natural concern emerges that some variation or smoothness in the pattern of estimated coefficients may be artificially introduced by the technique and may not represent true regression effects. This situation is at the heart of the discussion about the utility of GWR for inference on regression coefficients and is not answered by existing statistical (Leung et al. 2000a) or Monte Carlo (Fother- ingham et al. 2002) tests for significant variation of GWR coefficients because these tests do not consider the source of the variation. This is important because one source of regression coefficient variability in GWR can come from collinear- ity, or dependence in the kernel-weighted design matrix. Collinearity is known in linear models to inflate the variances of regression coefficients (Neter et al. 1996), and GWR is no exception (Griffith 2008). Collinearity has been found in empiri- cal work to be an issue in GWR models at the local level when it is not present in the global linear regression model using the same data (Wheeler 2007). In addition to large variation of estimated regression coefficients, there can be strong depend- ence in GWR coefficients for different regression terms, including the intercept, at least partly attributable to collinearity. Wheeler and Tiefelsdorf (2005) show in a simulation study that while GWR coefficients can be correlated when there is no explanatory variable correlation, the coefficient correlation increases systemati- cally with increasingly more collinearity.
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