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.

中文翻译：

---

使用L 1,2范数估计图形套索

高斯图形模型最近在经济学中用于获得代理商之间的依赖关系网络。广泛使用的估计器是图形套索（GLASSO），它相当于使用精度矩阵$ \ Omega $上的$ L_ {1,1} $矩阵范数进行正则化的最大似然估计。$ L_ {1,1} $范数是套索惩罚，可控制稀疏性或$ \ Omega $中零的数目。我们提出了一种新的估算器，称为结构化图形套索（SGLASSO），它使用$ L_ {1,2} $混合范数。使用$ L_ {1,2} $惩罚控件来控制$ \ Omega $中稀疏性的结构。我们表明，当网络大小固定时，SGLASSO渐近等效于一个不可行的GLASSO问题，该问题优先考虑了高度节点的稀疏性恢复。蒙特卡洛模拟显示，在估计总体精度矩阵和估计图形模型的结构方面，SGLASSO优于GLASSO。在使用经典企业投资数据集进行的实证说明中，我们获得了具有核心外围结构的企业依赖网络，通用汽车，通用电气和美国钢铁公司构成了企业核心组。