Vector autoregressive moving average models: A review

Vector AutoRegressive Moving Average (VARMA) models form a powerful and general model class for analyzing dynamics among multiple time series. While VARMA models encompass the Vector AutoRegressive (VAR) models, their popularity in empirical applications is dominated by the latter. Can this phenomenon be explained fully by the simplicity of VAR models? Perhaps many users of VAR models have not fully appreciated what VARMA models can provide. The goal of this review is to provide a comprehensive resource for researchers and practitioners seeking insights into the advantages and capabilities of VARMA models. We start by reviewing the identification challenges inherent to VARMA models thereby encompassing classical and modern identification schemes and we continue along the same lines regarding estimation, specification, and diagnosis of VARMA models. We then highlight the practical utility of VARMA models in terms of Granger Causality analysis, forecasting and structural analysis as well as recent advances and extensions of VARMA models to further facilitate their adoption in practice. Finally, we discuss some interesting future research directions where VARMA models can fulfill their potentials in applications as compared to their subclass of VAR models.