Estimation and inference of multi-effect generalized geographically and temporally weighted regression models

Geographically and temporally weighted regression (GTWR) models have been an effective tool for exploring spatiotemporal heterogeneity of regression relationships. However, they cannot effectively model such response variables that follows discrete distributions. In this study, we first extend the distributions of response variables to one-parameter exponential family of distributions and formulate generalized geographically and temporally weighted regression (GGTWR) models with their unilaterally temporally weighted maximum likelihood estimation method. Furthermore, we propose so-called multi-effect GGTWR (MEGGTWR) models in which spatiotemporally varying, constant, temporally varying, and spatially varying coefficients may simultaneously be included to reflect different effects of explanatory variables. A coefficient-average-based estimation method is suggested to calibrate MEGGTWR models and a generalized likelihood ratio statistic based test is formulated to identify the types of coefficients. Simulation studies are then conducted to assess the performance of the proposed estimation and inference methods with the impact of multicollinearity among explanatory variables also examined. The results show that the estimation method for MEGGTWR models can accurately estimate various types of coefficients and the test method is of valid type $I$ error and satisfactory power. Finally, the relationship between childhood hand, foot, and mouth disease cases and climate factors is analyzed by the proposed models with their estimation and inference methods and some interesting spatiotemporal patterns are uncovered.