Causality for VARMA processes with instantaneous effects: The global Markov property, faithfulness and instrumental variables

Abstract: Causal reasoning has gained great attention over the last half century as it allows (or at least intends) to answer questions which go above those within the capabilities of classical inferential statistics using just observational data. So far, causal research has been focused mostly on the i.i.d. setting. However, many are the situations where there exists a non-trivial dependence structure between sequential observations. Motivated by this fact, the main purpose of this work is to study causal properties of time series under the structural assumption of a VARMA model with instantaneous effects. First, the global Markov property is studied, building on existing work for VAR processes without instantaneous effects. Infinite graphs which represent the dependencies of the process are defined so that separation statements translate to conditional independencies in the stationary distribution of the process. Second, faithfulness is examined as a counterpart of this Markov property. Conditions are given so that the stationary distribution of the process is almost surely faithful to said infinite graphs. In addition, an instrumental variable regression framework is developed for VARMA models with instantaneous effects. This allows to identify and consistently estimate total causal effects.