Spatial Autoregressive Modeling on Linear Mixed Models for Dependency Between Regions

This study develops a linear mixed model (LMM) that includes spatial effects between regions with a spatial autoregressive model (SAR model). Between observations (regions) on that LMM are usually assumed to be independent. However, these assumptions are not always fulfilled due to dependency between regions. There are two important parts in spatial modeling: spatial dependence and spatial heterogeneity. In this study, we are concerned with the spatial lag or SAR models because dependency between variables of interest is easier to predict. On the other hand, all observations are real and can be directly seen from the data patterns. In addition, as a challenge for researchers to find all estimators while the values of the spatial dependence, sampling variance, and component variance are all unknown. This study aims to find all parameter estimators using a numerical approach and exact solutions. All exact estimators obtained are consistent estimators.