Estimation of Graphical Models using the $L_{1,2}$ Norm

TL;DR

This paper introduces SGLASSO, a new estimator for Gaussian graphical models using the $L_{1,2}$ norm, which improves structure recovery, especially for high-degree nodes, over the traditional GLASSO method.

Contribution

The paper proposes the Structured Graphical Lasso (SGLASSO) using the $L_{1,2}$ norm, offering better sparsity structure control and asymptotic properties compared to existing methods.

Findings

SGLASSO outperforms GLASSO in simulation studies.

SGLASSO better estimates the network structure, especially for high-degree nodes.

Empirical application reveals a core-periphery network among firms.

Abstract

Gaussian graphical models are recently used in economics to obtain networks of dependence among agents. A widely-used estimator is the Graphical Lasso (GLASSO), which amounts to a maximum likelihood estimation regularized using the $L_{1,1}$ matrix norm on the precision matrix $\Omega$. The $L_{1,1}$ norm is a lasso penalty that controls for sparsity, or the number of zeros in $\Omega$. We propose a new estimator called Structured Graphical Lasso (SGLASSO) that uses the $L_{1,2}$ mixed norm. The use of the $L_{1,2}$ penalty controls for the structure of the sparsity in $\Omega$. We show that when the network size is fixed, SGLASSO is asymptotically equivalent to an infeasible GLASSO problem which prioritizes the sparsity-recovery of high-degree nodes. Monte Carlo simulation shows that SGLASSO outperforms GLASSO in terms of estimating the overall precision matrix and in terms of estimating the structure of the graphical model. In an empirical illustration using a classic firms' investment dataset, we obtain a network of firms' dependence that exhibits the core-periphery structure, with General Motors, General Electric and U.S. Steel forming the core group of firms.