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Relational Graph in Vector Autoregression: A Case Study on the Effect of the Great Recession on Connectivity of Economic Indicators

Arkaprava Roy, Anindya Roy, Subhashis Ghosal

TL;DR

This work develops a Bayesian framework to learn the stationary (contemporaneous) graph of a high-dimensional $VAR(p)$ process under sparsity and causality constraints, enabling a before-versus-after analysis of interconnected economic indicators around the Great Recession. A novel reduced-rank parameterization separates the graph structure from temporal dynamics, allowing efficient likelihood computation and scalable inference in large systems. The authors establish posterior contraction results and demonstrate with FRED-QD data that the recession induced substantial, group-specific changes in interdependencies, while simulations and comparisons with GGM/GCGM corroborate the value of explicitly modeling VAR dependence. The methodology is broadly applicable to other prolonged shocks and extreme events where understanding editing connectivity in dynamic networks is crucial.

Abstract

Under a high-dimensional vector autoregressive (VAR) model, we propose a way of efficiently estimating both the stationary graph structure between the nodal time series and their temporal dynamics. The framework is then used to make inferences on the change in interdependencies between several economic indicators due to the impact of the Great Recession, the financial crisis that lasted from 2007 through 2009. There are several key advantages of the proposed framework; (1) it develops a reparametrized VAR likelihood that can be used in general high-dimensional VAR problems, (2) it strictly maintains causality of the estimated process, making inference on stationary features more meaningful and (3) it is computationally efficient due to the reduced rank structure of the parameterization. We apply the methodology to the seasonally adjusted quarterly economic indicators available in the FRED-QD database of the Federal Reserve. The analysis essentially confirms much of the prevailing knowledge about the impact of the Great Recession on different economic indicators. At the same time, it provides deeper insight into the nature and extent of the impact on the interplay of the different indicators. We also contribute to the theory of Bayesian VAR by showing the consistency of the posterior under sparse priors for the parameters of the reduced rank formulation of the VAR process.

Relational Graph in Vector Autoregression: A Case Study on the Effect of the Great Recession on Connectivity of Economic Indicators

TL;DR

This work develops a Bayesian framework to learn the stationary (contemporaneous) graph of a high-dimensional process under sparsity and causality constraints, enabling a before-versus-after analysis of interconnected economic indicators around the Great Recession. A novel reduced-rank parameterization separates the graph structure from temporal dynamics, allowing efficient likelihood computation and scalable inference in large systems. The authors establish posterior contraction results and demonstrate with FRED-QD data that the recession induced substantial, group-specific changes in interdependencies, while simulations and comparisons with GGM/GCGM corroborate the value of explicitly modeling VAR dependence. The methodology is broadly applicable to other prolonged shocks and extreme events where understanding editing connectivity in dynamic networks is crucial.

Abstract

Under a high-dimensional vector autoregressive (VAR) model, we propose a way of efficiently estimating both the stationary graph structure between the nodal time series and their temporal dynamics. The framework is then used to make inferences on the change in interdependencies between several economic indicators due to the impact of the Great Recession, the financial crisis that lasted from 2007 through 2009. There are several key advantages of the proposed framework; (1) it develops a reparametrized VAR likelihood that can be used in general high-dimensional VAR problems, (2) it strictly maintains causality of the estimated process, making inference on stationary features more meaningful and (3) it is computationally efficient due to the reduced rank structure of the parameterization. We apply the methodology to the seasonally adjusted quarterly economic indicators available in the FRED-QD database of the Federal Reserve. The analysis essentially confirms much of the prevailing knowledge about the impact of the Great Recession on different economic indicators. At the same time, it provides deeper insight into the nature and extent of the impact on the interplay of the different indicators. We also contribute to the theory of Bayesian VAR by showing the consistency of the posterior under sparse priors for the parameters of the reduced rank formulation of the VAR process.

Paper Structure

This paper contains 20 sections, 16 theorems, 95 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. FRED-QD data and Exploratory Analysis
  3. Partial Correlation Graph Under Autoregression
  4. Reduced Rank Parameterization and Priors
  5. Efficient Computation of the Likelihood
  6. Rank-one Updates
  7. Prior Specification
  8. Posterior Computation
  9. Posterior Contraction
  10. Differential Connectivity due to the Great Recession Using FRED-QD data
  11. Simulation
  12. Discussion
  13. Proof of the Main Theorems
  14. Prior concentration: pre-rate
  15. Test construction
  16. ...and 5 more sections

Key Result

Proposition 1

Let $\boldsymbol{X}_t$ satisfying eq:var be a stationary vector autoregressive time series with stationary variance ${\rm Var}(\boldsymbol{X}_t)=\boldsymbol{\Omega}^{-1}$, a positive definite matrix and error covariance matrix $\boldsymbol{\Sigma}$. Then $\boldsymbol{X}_t$ is causal if and only if where for any $j \geq 1,$$\boldsymbol{C}_j = {\rm Var}(\boldsymbol{X}_{j+1} | \boldsymbol{X}_{j}, \l

Figures (7)

  • Figure 1: Estimated sample precisions for some example groups
  • Figure 2: Estimated change networks. The node numbers correspond to the variable names as described in mccracken2020fred.
  • Figure 3: ROC comparison for different cases: Black = GCGM, Red = GGM, Green = Causal VAR when the true data is generated using VAR(1) model.
  • Figure :
  • Figure :
  • ...and 2 more figures

Theorems & Definitions (28)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Proposition 2
  • proof
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • ...and 18 more