A General Framework for Estimation and Inference of Geographically Weighted Regression Models: 1. Location-Specific Kernel Bandwidths and a Test for Locational Heterogeneity

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Abstract
Geographically weighted regression (GWR) has been proposed as a technique to explore spatial parametric nonstationarity. The method has been developed mainly along the lines of local regression and smoothing techniques, a strategy that has led to a number of difficult questions about the regularity conditions of the likelihood function, the effective number of degrees of freedom, and in general the relevance of extending the method to derive inference and model specification tests. In this paper we argue that placing GWR within a different statistical context, as a spatial model of error variance heterogeneity, or what might be termed locational heterogeneity, solves these difficulties. A maximum-likelihood-based framework for estimation and inference of a general geographically weighted regression model is presented that leads to a method to estimate location-specific kernel bandwidths. Moreover, a test for locational heterogeneity is derived and its use exemplified with a case study.

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