Estimation of Graphical Models Using the L₁,₂ Norm

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 (Formula presented.) matrix norm on the precision matrix Ω. The (Formula presented.) 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 (Formula presented.) mixed norm. The use of the (Formula presented.) 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.