Mixed orthogonality graphs for continuous-time stationary processes

TL;DR

This paper develops a graph-based framework to model dependencies in multivariate continuous-time stationary processes, introducing new concepts of causality and correlation with explicit characterizations for AR processes.

Contribution

It introduces mixed orthogonality graphs for continuous-time processes, linking graph structures to dependence and causality concepts, with explicit characterizations for AR models.

Findings

Orthogonality graphs represent Granger causality and correlation.

Sufficient criteria for Markov properties in the graphs.

Explicit edge characterization for multivariate AR processes.

Abstract

In this paper, we introduce different concepts of Granger causality and contemporaneous correlation for multivariate stationary continuous-time processes to model different dependencies between the component processes. Several equivalent characterisations are given for the different definitions, in particular by orthogonal projections. We then define two mixed graphs based on different definitions of Granger causality and contemporaneous correlation, the (mixed) orthogonality graph and the local (mixed) orthogonality graph. In these graphs, the components of the process are represented by vertices, directed edges between the vertices visualise Granger causal influences and undirected edges visualise contemporaneous correlation between the component processes. Further, we introduce various notions of Markov properties in analogy to Eichler (2012), which relate paths in the graphs to different dependence structures of subprocesses, and we derive sufficient criteria for the (local) orthogonality graph to satisfy them. Finally, as an example, for the popular multivariate continuous-time AR (MCAR) processes, we explicitly characterise the edges in the (local) orthogonality graph by the model parameters.