Space-time Autoregressive-moving-average model with exogenous inputs (STARMAX) model models how a time-space target process depends on its own and its neighbors' past, and on other time-space exogenous inputs. Time-space data arise in many physical science studies. The interpretability of STARMAX model makes it highly desirable. There is not a ready to use program that will fit this model. This paper extends the model by introducing the similar time-space lag structure to exogenous inputs. This paper also develops and implements a fast Kalman filter parameter estimation and prediction procedure that allows missing values in target process, and can be applied on big data.