Estimation of parameters and hypothesis testing of multivariate spatial autoregressive model.

Spatial dependence plays a critical role in modeling multivariate response variables, particularly in fields such as epidemiology and environmental studies. However, existing spatial regression models, such as the Spatial Autoregressive (SAR) model, are designed for univariate responses and are insufficient when multiple response variables are influenced by spatial location. To address this gap, we introduce a Multivariate Spatial Autoregressive (MSAR) model. While previous research has focused primarily on parameter estimation for the proposed model, limited attention has been given to the statistical significance of these parameters. Moreover, existing estimation methods often rely on pseudo-distributions, which may not accurately reflect the underlying data characteristics. This study employs Maximum Likelihood Estimation (MLE), optimized using a concentrated log-likelihood approach, under the assumption of normally distributed data. To assess parameter significance, we apply both the Maximum Likelihood Ratio Test (MLRT) for joint hypotheses and the Wald Test for individual parameters. The findings confirm that the proposed model yields unbiased and consistent parameter estimates. Furthermore, the significance tests reveal key predictor variables associated with pneumonia and diarrhea cases among toddlers. The proposed model achieves a Root Mean Square Error of 5 and an R-squared value of 60 %, demonstrating its effectiveness in capturing spatial dependence in multivariate settings. The main contributions of this study include:•Development of a MSAR model estimated using MLE to capture spatial dependencies among multiple response variables.•Implementation of formal hypothesis testing procedures for model parameters using the Likelihood Ratio and Wald tests.•Application of the proposed model to spatial health data at the village level in Tuban District, East Java, Indonesia, focusing on health problems among children under five.