Summary: The identification and analysis of interactions among multiple simultaneously recorded time series is an important problem in many scientific areas. Of particular interest are directed interactions that describe the dynamics of the systems and thus help to determine the causal driving mechanisms of the underlying system. The dynamic relationships among multiple series intuitively can be visualized by a path diagram (or graph), in which the variables are represented by vertices or nodes, and directed edges between the vertices indicate the dynamic or causal influences among the variables. We review recent results on the properties of such graphical representations, which show that path diagrams provide an ideal basis for discussing and investigating causal relationships in multivariate time series. The key role in this graphical approach is played by the so-called global Markov properties, which provide graphical conditions for the (in-)dependences that may be observed if only subprocesses instead of the full process are considered. Such considerations are, for example, central for the discussion of systems that may contain latent variables. The empirical analysis of dynamic interactions is commonly based on the concept of Granger causality. While this concept is well understood in the time domain, the time series of interest often are characterized in terms of their spectral properties. Therefore, particular emphasis will be given to the frequency-domain interpretation of Granger causality and the graphical concepts discussed in this chapter.
For the entire collection see [Zbl 1104.62328].