Threshold spatial vector autoregressive with metric exogenous variables (TSpVARX) for regional inflation and money outflow prediction

Currently, some time series models are formed to accommodate several aspects: the reciprocal relationship between variables, the influence of exogenous variables, spatial relationships, and nonlinear relationships between variables. Threshold vector autoregressive with exogenous variables (E-TVAR) is formed to overcome the reciprocal relationship between variables, the influence of exogenous variables, and the nonlinear relationship between variables (Tsagkanos et al. 2018). Spatial vector autoregressive (SpVAR) of Beenstock–Felsenstein (2019) can capture the phenomenon of the relationship between endogenous variables and spatial influence. However, these models cannot consider all four aspects simultaneously. Some economic variables, such as inflation and money outflow, are reciprocally related, influenced by metric exogenous variables, interrelated between regions, and have a nonlinear relationship (Islam−Ahmed 2023, Hendayanti et al. 2017, Yuhan−Sohibien 2018, Suhartono et al. 2018). Therefore, this study aims to propose the TSpVARX that can contain all four of these simultaneously. We conduct a theoretical study to prove the consistent and asymptotically normal properties of the maximum likelihood estimation (MLE) estimator in the g-th regime of the TSpVARX model. In addition, we also conduct a simulation study with some scenarios to evaluate the performance of the MLE estimator in the TSpVARX model. After we conduct theoretical studies and simulations, the TSpVARX model is applied to predict the inflation and money outflow of Yogyakarta, Solo, and Semarang. Our study shows that TSpVARX is better than SpVARX in predicting the inflation of those three cities and the money outflow of Semarang based on the root mean square error (RMSE) and and symmetric mean absolute percentage error (SMAPE).