Time Series Gaussian Chain Graph Models

2026-04-08

AI Summary

This study addresses the challenge of jointly modeling undirected intra-block dependencies and directed inter-block lagged causal relationships in multivariate time series. It introduces, for the first time, a Gaussian chain graph model for time series, decomposing the inverse spectral density matrix in the frequency domain into a group-sparse component—capturing directed inter-block edges—and a group-low-rank component—representing undirected intra-block connections. A three-stage estimation procedure based on regularized Whittle likelihood is developed, integrating group Lasso with a novel tensor unfolding nuclear norm penalty to enable efficient graph structure learning. Theoretical analysis establishes model identifiability and graph estimation consistency. Simulations and empirical analysis of U.S. macroeconomic data demonstrate that the proposed method effectively uncovers complex mechanisms such as monetary policy transmission.

Abstract

Time series graphical models have recently received considerable attention for characterizing (conditional) dependence structures in multivariate time series. In many applications, the multivariate series exhibit variable-partitioned blockwise dependence, with distinct patterns within and across blocks. In this paper, we introduce a new class of time series Gaussian chain graph models that represent contemporaneous and lagged causal relations via directed edges across blocks, while capturing within-block conditional dependencies through undirected edges. In the frequency domain, this formulation induces a cross-frequency shared group sparse plus group low-rank decomposition of the inverse spectral density matrices, which we exploit to establish identifiability of the time series chain graph structure. Building on this, we then propose a three-stage learning procedure for estimating the undirected and directed edge sets, which involves optimizing a regularized Whittle likelihood with a group lasso penalty to encourage group sparsity and a novel tensor-unfolding nuclear norm penalty to enforce group low-rank structure. We investigate the asymptotic properties of the proposed method, ensuring its consistency for exact recovery of the chain graph structure. The superior empirical performance of the proposed method is demonstrated through both extensive simulation studies and an application to U.S. macroeconomic data that highlights key monetary policy transmission mechanisms.

Problem

• time series
• chain graph models
• blockwise dependence
• conditional dependence
• causal relations

Innovation

• time series chain graph
• group sparsity
• group low-rank
• inverse spectral density
• regularized Whittle likelihood