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Identifiability of VAR(1) model in a stationary setting

Bixuan Liu

Abstract

We consider a classical First-order Vector AutoRegressive (VAR(1)) model, where we interpret the autoregressive interaction matrix as influence relationships among the components of the VAR(1) process that can be encoded by a weighted directed graph. A majority of previous work studies the structural identifiability of the graph based on time series observations and therefore relies on dynamical information. In this work we assume that an equilibrium exists, and study instead the identifiability of the graph from the stationary distribution, meaning that we seek a way to reconstruct the influence graph underlying the dynamic network using only static information. We use an approach from algebraic statistics that characterizes models using the Jacobian matroids associated with the parametrization of the models, and we introduce sufficient graphical conditions under which different graphs yield distinct steady-state distributions. Additionally, we illustrate how our results could be applied to characterize networks inspired by ecological research.

Identifiability of VAR(1) model in a stationary setting

Abstract

We consider a classical First-order Vector AutoRegressive (VAR(1)) model, where we interpret the autoregressive interaction matrix as influence relationships among the components of the VAR(1) process that can be encoded by a weighted directed graph. A majority of previous work studies the structural identifiability of the graph based on time series observations and therefore relies on dynamical information. In this work we assume that an equilibrium exists, and study instead the identifiability of the graph from the stationary distribution, meaning that we seek a way to reconstruct the influence graph underlying the dynamic network using only static information. We use an approach from algebraic statistics that characterizes models using the Jacobian matroids associated with the parametrization of the models, and we introduce sufficient graphical conditions under which different graphs yield distinct steady-state distributions. Additionally, we illustrate how our results could be applied to characterize networks inspired by ecological research.

Paper Structure

This paper contains 34 sections, 30 theorems, 165 equations, 7 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. VAR(1) model.
  3. Problem statement.
  4. Motivation.
  5. Related work.
  6. Our contribution.
  7. Outline.
  8. Identifiability of the VAR(1) model in a stationary setting
  9. Problem settings
  10. Global and generic identifiability
  11. Identifiability criteria
  12. Main results
  13. Jacobian Structure of VAR(1) in the Stationary Setting
  14. Maximal classes
  15. Definition and properties
  16. ...and 19 more sections

Key Result

Proposition 1

Let $\mathbf{M}_1$ and $\mathbf{M}_2$ be two stationary VAR(1) models. If then these two models are generically identifiable.

Figures (7)

  • Figure 1: Example of a graphical VAR(1) model and the encoded graph $G$
  • Figure 2: Example of a directed graph
  • Figure 3: The graph associated with maximal classes $\left\{\{1,2\},\{2,3\}\right\}$.
  • Figure 4: Identifiable graphs using Theorem \ref{['thm:iden_maxc']}
  • Figure 5: A family of identifiable graphs
  • ...and 2 more figures

Theorems & Definitions (93)

  • Definition 1
  • Remark 1: Structural Properties
  • Definition 2
  • Example 1
  • Definition 3
  • Definition 4
  • Proposition 1
  • Definition 5
  • Definition 6
  • Definition 7
  • ...and 83 more