Estimation of graphical models using the L 1,2 norm

Khai Xiang Chiong, Hyungsik Roger Moon

Research output: Contribution to journalArticle

Abstract

Gaussian graphical models are recently used in economics to obtain networks of dependence among agents. A widely used estimator is the graphical least absolute shrinkage and selection operator (GLASSO), which amounts to a maximum likelihood estimation regularized using the L 1,1 matrix norm on the precision matrix Ω. The L 1,1 norm is a LASSO penalty that controls for sparsity, or the number of zeros in Ω. We propose a new estimator called structured GLASSO (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 Ω. 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 data set, we obtain a network of firms' dependence that exhibits the core–periphery structure, with General Motors, General Electric and US Steel forming the core group of firms.

Original languageEnglish
Pages (from-to)247-263
Number of pages17
JournalEconometrics Journal
Volume21
Issue number3
DOIs
Publication statusPublished - 2018 Oct

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Graphical models
Shrinkage
Operator
Penalty
Estimator
General Motors
Firm investment
Steel
Monte Carlo simulation
Maximum likelihood estimation
General Electric
Economics
Node

All Science Journal Classification (ASJC) codes

  • Economics and Econometrics

Cite this

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abstract = "Gaussian graphical models are recently used in economics to obtain networks of dependence among agents. A widely used estimator is the graphical least absolute shrinkage and selection operator (GLASSO), which amounts to a maximum likelihood estimation regularized using the L 1,1 matrix norm on the precision matrix Ω. The L 1,1 norm is a LASSO penalty that controls for sparsity, or the number of zeros in Ω. We propose a new estimator called structured GLASSO (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 Ω. 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 data set, we obtain a network of firms' dependence that exhibits the core–periphery structure, with General Motors, General Electric and US Steel forming the core group of firms.",
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Estimation of graphical models using the L 1,2 norm . / Chiong, Khai Xiang; Moon, Hyungsik Roger.

In: Econometrics Journal, Vol. 21, No. 3, 10.2018, p. 247-263.

Research output: Contribution to journalArticle

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