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

Ignacio González-Pérez

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.

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.
Paper Structure (7 sections, 9 theorems, 4 equations)

This paper contains 7 sections, 9 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Directed acyclic graphs
  4. Acyclic directed mixed graphs
  5. The global Markov property for VAR processes without instantaneous effects
  6. The global Markov property
  7. The global Markov property for VAR processes with instantaneous effects

Key Result

proposition thmcounterproposition

Let $A,B,C$ be pairwise disjoint sets of nodes of a DAG $\mathcal{G}$. Then $A$ and $C$ are $d$-separated by $B$ in $\mathcal{G}$ if and only if $A$ and $C$ are separated by $B$ in $(\mathcal{G}_{AN_{\mathcal{G}}(A\cup B \cup C)})^m$. By $\mathcal{G}_W$ we denote the proper sub-DAG of $\mathcal{G}$

Theorems & Definitions (27)

  • definition thmcounterdefinition: Directed acyclic graph )
  • definition thmcounterdefinition: Descendants, children, ancestors and parents of a node in a DAG )
  • definition thmcounterdefinition: Collider in a path of a DAG )
  • definition thmcounterdefinition: $d$-separation in a DAG )
  • definition thmcounterdefinition: Moralized graph of a DAG )
  • proposition thmcounterproposition: lauritzen1990independence
  • proposition thmcounterproposition
  • corollary thmcountercorollary
  • definition thmcounterdefinition: Acyclic directed mixed graph )
  • remark thmcounterremark
  • ...and 17 more