Testing for Spatial Autocorrelation among the Residuals of the Geographically Weighted Regression

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Abstract
Geographically weighted regression (GWR) is a useful technique for exploring spatial nonstationarity by calibrating, for example, a regression model which allows different relationships to exist at different points in space. In this line of research, many spatial data sets have been successfully analyzed and some statistical tests for spatial variation have been developed. However, an important assumption in these studies is that the disturbance terms of the GWR model are uncorrelated and of common variance. Similar to the case in the ordinary linear regression, spatial autocorrelation can invalidate the standard assumption of homoscedasticity of the disturbances and mislead the results of statistical inference. Therefore, developing some statistical methods to test for spatial autocorrelation is a very important issue. In this paper, two kinds of the statistical tests for spatial autocorrelation among the residuals of the GWR model are suggested. Also, an efficient approximation method for calculating the p-values of the test statistics is proposed. Some simulations are run to examine the performances of the proposed methods and the results are encouraging. The study not only makes it possible to test for spatial autocorrelation among the GWR residuals in a conventional statistical manner, but also provides a useful means for model validation.