Comparative study of GSTAR and QSTAR spatio-temporal models: an application to rainfall in Morocco

This article presents a comparative study of the Generalized Space-Time Autoregressive (GSTAR) and Quadrant Spatio-Temporal Autoregressive (QSTAR) models, which differ in their structure and stationarity assumptions. GSTAR captures global spatio-temporal dependence through spatial weights, whereas QSTAR represents local directional spatio-temporal dependence. To assess their ability to capture spatio-temporal dynamics, a two-step evaluation is adopted. First, Monte Carlo simulations are conducted on a 1000 × 10 grid to perform a rigorous comparative analysis under different GSTAR spatial weights. Second, an application to rainfall data from 11 spatial locations is carried out for comparative forecasting purposes. The simulation results, based on MSE, show that both models tend to fit the training data closely. However, GSTAR tends to overfit when the spectral radius approaches the stationarity boundary. In contrast, QSTAR, despite its parsimonious structure with only three parameters compared with twenty parameters for GSTAR, exhibits a smaller training-testing error gap and comparable testing performance across simulation settings. The empirical analysis confirms these findings, with QSTAR showing more stable behavior, smaller training-testing RMSE gaps, and better testing performance across folds and lags. These findings highlight the advantages of the QSTAR model in terms of parsimony, stability, and accuracy, making it well suited for spatio-temporal rainfall data.