Spatial Autoregressive Model with Mixture of Gaussian Distribution for the Random Effect: Formulation, Estimation and Application

Spatial econometrics is pivotal in understanding spatial dependencies across diverse fields like urban economics, environmental economics, and disease spread.This study highlights the significance of spatial grouping for data management and pattern detection, particularly in epidemiological analysis and policy planning.The Spatial Autoregressive random effect (SAR-RE) model is a classical model for analysing datasets with repeated observations across units over time, particularly when these units are situated in a spatial context.The mixture effect models account for the presence of different sub-groups within the overall population, each of which has a unique response pattern.In this paper, the proposed methodology integrates the SAR-RE model into a mixture framework, allowing for the consideration of diverse spatial patterns and class-specific coefficients.By incorporating class-specific coefficients, the model accommodates heterogeneous spatial structures within the data, providing a more nuanced understanding of spatial dependencies.The Spatial autoregressive model along with the assumption that the random effect follows a mixture of Gaussian distributions is developed to analyse panel data with spatial dependency and unobserved heterogeneity.The parameters of the model are estimated using the Limited-Memory BFGS (L-BFGS) quasi-Newton methodbased EM algorithm for good convergence of the estimated.The classification of subjects into different latent classes is carried out based on their posterior probabilities.The model is applied to state-wise COVID-19 confirmed rates, revealing insightful patterns.The analysis employs the estimated model for the interpretation and comprehensive understanding of spatially dependent panel data with unobserved heterogeneity.The results of the empirical study show that the proposed model outperforms the existing model based on performance metrics criteria.