A Novel Structural AR Modeling Approach for a Continuous Time Linear Markov System

We often use a discrete time vector autoregressive (DVAR) model to analyse continuous time, multivariate, linear Markov systems through their time series data sampled at discrete time steps. However, the DVAR model has been considered not to be structural representation and hence not to have bijective correspondence with system dynamics in general. In this paper, we characterize the relationships of the DVAR model with its corresponding structural vector AR (SVAR) and continuous time vector AR (CVAR) models through finite difference approximation of time differentials. Our analysis shows that the DVAR model of a continuous time, multivariate, linear Markov system bijectively corresponds to the system dynamics. Further we clarify that the SVAR and the CVAR models are uniquely reproduced from their DVAR model under a highly generic condition. Based on these results, we propose a novel Continuous time and Structural Vector AutoRegressive (CSVAR) modeling approach for continuous time, linear Markov systems to derive the SVAR and the CVAR models from their DVAR model empirically derived from the observed time series. We demonstrate its superior performance through some numerical experiments on both artificial and real world data.