Markov properties for graphical time series models

This paper deals with the Markov properties of a new class of graphical time series models which focus on the dynamic interrelationships between the components of multivariate time series. The modelling approach is based on the concept of strong Granger-causality and thus can also be applied to nonlinear models. The constraints defining the models are encoded by mixed graphs in which each component series is represented by one vertex and directed edges indicates possible Granger-causal relationships between the variables while undirected edges are used to map the contemporaneous dependence structure of the time series. We discuss various Markov properties for time series models and how these are related to each other. In particular we introduce the notion of causal Markov property. We show that global Markov properties can be formulated in terms of a moralization criterion based on separation in undirected graphs or alternatively using a pathwise separation criterion called p-separation. As an example of graphical time series models we consider multivariate ARCH models which satisfy the causal Markov property with respect to a given graph.