  We introduce graphical time series models for the analysis of dynamic
relationships among variables in multivariate time series. The modelling
approach is based on the notion of strong Granger causality and can be applied
to time series with non-linear dependencies. The models are derived from
ordinary time series models by imposing constraints that are encoded by mixed
graphs. In these graphs each component series is represented by a single vertex
and directed edges indicate possible Granger-causal relationships between
variables while undirected edges are used to map the contemporaneous dependence
structure. We introduce various notions of Granger-causal Markov properties and
discuss the relationships among them and to other Markov properties that can be
applied in this context.
