GRAPHICAL MODELLING OF MULTIVARIATE TIME SERIES WITH LATENT VARIABLES
Michael D. Eichler · 2006
Abstract. In time series analysis, inference about cause-e®ect relationships among multiple times series is commonly based on the concept of Granger causality, which exploits temporal structure to achieve causal ordering of dependent variables. One major problem in the application of Granger causality for the identi¯cation of causal relationships is the possible presence of latent variables that a®ect the mea-sured components and thus lead to so-called spurious causalities. In this paper, we describe a new graphical approach for modelling the dependence structure of mul-tivariate stationary time series that are a®ected by latent variables. Is is based on mixed graphs in which directed edges represent direct in°uences among the variables while dashed edges|directed or undirected|indicate associations that are induced by latent variables. For Gaussian processes, this approach leads to vector autoregressive processes with errors that are not independent but correlated according to the dashed edges in the graph. We show that these models can be viewed as graphical ARMA models that satisfy the Granger causality restrictions encoded by general mixed graphs. We discuss identi¯ability of the parameters and illustrate the approach by an example. 1.