Local weighting matrices or the necessity of flexibility

The local estimation algorithms are well-known techniques in the current spatial econometric literature. The Geographically Weighted Regressions are very popular to estimate, locally, static models, whereas the SALE or the Zoom approaches are useful solutions in the case of dynamic models. These techniques are well founded from a methodological point of view and present interesting properties. However, Farber and Paez (2008) detect some inconsistencies in the behavior of some of these algorithms that claim for a further analysis. The point that we want to study in this paper refers to the role of the bandwith. This measure defines how many neighbors will be used in the estimation of the local parameters corresponding to each observation. The cross-validation is the most popular criteria to fix the bandwith, although there are several other criteria in the literature. We think that there is a basic problem with this approach. The objective of these algorithms is to relax the restriction of homogeneity of the parameters of the model allowing for local peculiarities; however the definition of local neighborhood is the same. It does not matter if the observation corresponds to an isolated and poorly communicated region or it belongs to a central and highly connected point. According to our view, this is a very restrictive decision that should be avoided. Specifically, we discuss the procedure of specifying the sequence of local weighting matrices that will be used in the analysis. Our purpose is to achieve that these matrices also reflect the local surrounding of each observation. We examine two different strategies in order to construct the local weighting matrices. The first is a parametric approach which involves the J test, as presented by Kelejian (2008), and the second is a nonparametric approach that uses the guidance of the symbolic entropy measures. The first part of the paper presents the overall problem, including a review of the literature; we discuss the solutions in the second part and the third part consists of a Monte Carlo simulation.