Block Bootstrap for Spatiotemporal Data in Generalized Space Time Autoregressive (GSTAR)

Abstract

Generalized Space-Time Autoregressive is a model that can be used for data with spatial and temporal dependence. The GSTAR model is widely used in various phenomena such as rainfall, temperature, inflation, and others. GSTAR assumes normality of errors and non-autocorrelation. If the assumption of normality of errors is not met, then inference on parameters cannot be made. One solution to this problem is to use bootstrapping. However, bootstrapping for spatiotemporal data in the GSTAR model has not been developed. Therefore, this study aims to develop a bootstrapping method for spatiotemporal data in the GSTAR model. This development is done by adapting bootstrapping methods for time series data, namely, the non-overlapping block bootstrap (NBB) and the moving block bootstrap (MBB). This research continued with a series of simulations to evaluate the performance of the block bootstrap method as the number of observations, block length, and number of bootstrap replications were varied. Furthermore, the method’s effectiveness was tested using rainfall data from Malang Regency. Simulation results show that both resampling schemes satisfy the asymptotic condition, where the bias decreases monotonically with increasing sample size (T) and block length. MBB consistently produces lower bias than NBB due to its more intensive use of overlapping data, which effectively reduces boundary effects. Although inference on autoregressive parameters can be accurate, inference on spatial autoregressive parameters yields less satisfactory results, indicating the limitations of time blocks in capturing complex spatial dependencies. Increasing the number of replications above B=100 does not significantly improve the precision of the variance estimate, indicating computational efficiency at that threshold. The t-test results confirm that there is no statistically significant difference in performance between NBB and MBB. Nevertheless, MBB is more recommended for practical applications due to its higher information density and better estimation stability.