A backfitting maximum likelihood estimator for hierarchical and geographically weighted regression modelling, with a case study of house prices in Beijing

Geographically weighted regression (GWR) and its extensions are important local modelling techniques for exploring spatial heterogeneity in regression relationships. However, when dealing with spatial data of overlapping samples – for example, when precise locational information is aggregated to a shared neighbourhood to avoid revealing the addresses of individual survey respondents – GWR-based models can encounter several problems, including obtaining reliable bandwidths. Because data with this characteristic exhibit spatial hierarchical structures, we propose combining hierarchical linear modelling (HLM) with GWR to give a hierarchical and geographically weighted regression (HGWR) model that divides coefficients into sample-level fixed effects, group-level fixed effects, sample-level random effects, and group-level spatially weighted effects. This paper presents a back-fitting likelihood estimator to fit the model, a simulation experiment that suggests that HGWR is better able to capture these effects and the spatial heterogeneity within them than are traditional HLM or GWR models, and a case study looking at predictors of housing price in Beijing, China. The ability of HGWR to tackle both spatial and group-level heterogeneity simultaneously suggests its potential as a promising data modelling tool for handling spatio-temporal big data with spatially hierarchical structures