Graphical modelling of multivariate time series with latent variables

In time series analysis, inference about cause-effect relationships is commonly based on the concept of Granger-causality, which exploits temporal structure to achieve causal ordering of dependent variables. One major problem of the application of Granger-causality for the identification of causal relationships is the possible presence of latent variables that af-fect the measured components and thus lead to so-called spurious causalities. In this pa-per, we describe a new graphical approach for modelling the dependence structure of mul-tivariate stationary time series that are af-fected by latent variables. Is is based on mixed graphs in which dashed edges indicate associations that are induced by latent vari-ables. For Gaussian processes, we present a new class of models that satisfy the Granger-causality restrictions imposed by such graphs and show that these models can be viewed as suitably constrained ARMA models. Fur-thermore, we discuss maximum likelihood es-timation based on Whittle’s approximation of the log-likelihood and propose an iterative algorithm for computing the associated esti-mates.