An Note on Why Geographically Weighted Regression Overcomes Multidimensional-Kernel-Based Varying-Coefficient Model

TL;DR

This paper demonstrates that geographically weighted regression (GWR) is more efficient than multidimensional-kernel-based varying-coefficient models, providing theoretical and simulation evidence for its superiority in local estimation.

Contribution

The paper offers a theoretical comparison showing GWR's asymptotic efficiency over MLWE and explores the relationship between bandwidth selection and scale parameter design.

Findings

GWR is asymptotically more efficient than MLWE.

Distance-based weighting in GWR outperforms multidimensional kernels.

Optimal bandwidth selection relates to scale parameter design.

Abstract

It is widely known that geographically weighted regression(GWR) is essentially same as varying-coefficient model. In the former research about varying-coefficient model, scholars tend to use multidimensional-kernel-based locally weighted estimation(MLWE) so that information of both distance and direction is considered. However, when we construct the local weight matrix of geographically weighted estimation, distance among the locations in the neighbor is the only factor controlling the value of entries of weight matrix. In other word, estimation of GWR is distance-kernel-based. Thus, in this paper, under stationary and limited dependent data with multidimensional subscripts, we analyze the local mean squared properties of without any assumption of the form of coefficient functions and compare it with MLWE. According to the theoretical and simulation results, geographically-weighted locally linear estimation(GWLE) is asymptotically more efficient than MLWE. Furthermore, a relationship between optimal bandwith selection and design of scale parameters is also obtained.