# Time Series Gaussian Chain Graph Models

arXiv: https://arxiv.org/abs/2604.07018 · HTML: https://mlsanity.com/paper/2604.07018/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 50 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Builds on (most cited first, 9 of 9)

- [On Causal Discovery with Equal Variance Assumption](https://mlsanity.com/paper/1807.03419/) (2018)
- [On Low Rank Directed Acyclic Graphs and Causal Structure Learning](https://mlsanity.com/paper/2006.05691/) (2020)
- [Graphical LASSO Based Model Selection for Time Series](https://mlsanity.com/paper/1410.1184/) (2014)
- [On Sparse High-Dimensional Graphical Model Learning For Dependent Time Series](https://mlsanity.com/paper/2111.07897/) (2021)
- [Regularized Estimation and Testing for High-Dimensional Multi-Block Vector-Autoregressive Models](https://mlsanity.com/paper/1708.05879/) (2017)
- [Banded Spatio-Temporal Autoregressions](https://mlsanity.com/paper/1803.01699/) (2018)
- [FNETS: Factor-adjusted network estimation and forecasting for high-dimensional time series](https://mlsanity.com/paper/2201.06110/) (2022)
- [Identifiability and Consistent Estimation for Gaussian Chain Graph Models](https://mlsanity.com/paper/2303.01031/) (2023)
- [Linear Regression with Weak Exogeneity](https://mlsanity.com/paper/2308.08958/) (2023)

## Abstract (arXiv)

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.