Backfitting Estimation for Geographically Weighted Regression Models with Spatial Autocorrelation in the Response

Geographically weighted regression model with a spatially autoregressive term of the response variable (GWR-Lag model for short) is a useful tool to simultaneously model spatial autocorrelation in the response variable and spatial heterogeneity in the regression relationship. The profile maximum likelihood estimation has been developed to calibrate the model, where a single bandwidth is used for all of the spatially varying coefficients. However, due to different spatial scales at which the explanatory variables might operate, the coefficients may have different levels of spatial heterogeneity and cannot be effectively estimated by a single bandwidth. To deal with such multiscale problem, a backfitting method is proposed to calibrate the GWR-Lag model, where the optimal bandwidth size is separately selected for each coefficient and the autoregressive parameter is estimated by maximizing the log-likelihood function. A simulation study with a comparison of the GWR-based profile maximum likelihood (GWR-ML) estimation is conducted to assess the performance of the backfitting approach. The empirical results show that the backfitting method not only provides useful scale information for each explanatory variable, but also yields more accurate estimators of the coefficients. Furthermore, a real-life data set is analyzed to demonstrate the effectiveness of the backfitting approach.