A Hierarchical and Geographically Weighted Regression Model and Its Backfitting Maximum Likelihood Estimator

Spatial heterogeneity is a typical and common form of spatial effect. Geographically weighted regression (GWR) and its extensions are important local modeling techniques for exploring spatial heterogeneity. However, when dealing with spatial data sampled at a micro-level but the geographical locations of them are only known at a higher level, GWR-based models encounter several problems, such as difficulty in establishing the bandwidth. Because data with this characteristic exhibit spatial hierarchical structures, such data can be suitably handled using hierarchical linear modeling (HLM). This model calibrates random effects for sample-level variables in each group to address spatial heterogeneity. However, it does not work when exploring spatial heterogeneity in some group-level variables when there is insufficient variance in each group. In this study, we therefore propose a hierarchical and geographically weighted regression (HGWR) model, together with a back-fitting maximum likelihood estimator, that can be applied to examine spatial heterogeneity in the regression relationships of data where observations nest into high-order groupings and share the same or very close coordinates within those groups. The HGWR model divides coefficients into three types: local fixed effects, global fixed effects, and random effects. Results of a simulation experiment show that HGWR distinguishes local fixed effects from others and also global effects from random effects. Spatial heterogeneity is reflected in the estimates of local fixed effects, along with the spatial hierarchical structure. Compared with GWR and HLM, HGWR produces estimates with the lowest deviations of coefficient estimates. Thus, the ability of HGWR to tackle both spatial and group-level heterogeneity simultaneously suggests its potential as a promising data modeling tool for handling the increasingly common occurrence where data, in secure settings for example, remove the specific geographic identifiers of individuals and release their locations only at a group level.