Abstract
Networks with a very large number of nodes appear in many application areas and pose challenges to the traditional Gaussian graphical modelling approaches. In this paper we focus on the estimation of a Gaussian graphical model when the dependence between variables has a block-wise structure. We propose a penalised likelihood estimation of the inverse covariance matrix, also called Graphical LASSO, applied to block averages of observations, and derive its asymptotic properties. Monte Carlo experiments, comparing the properties of our estimator with those of the conventional Graphical LASSO, show that the proposed approach works well in the presence of block-wise dependence structure and is also robust to possible model misspecification. We conclude the paper with an empirical study on economic growth and convergence of 1,088 European small regions in the years 1980 to 2012. While requiring a-priori information on the block structure, for example given by the hierarchical structure of data, our approach can be adopted for estimation and prediction using very large panel data sets. Also, it is particularly useful when there is a problem of missing values and outliers or when the focus of the analysis is on out-of-sample prediction. This article is protected by copyright. All rights reserved
Highlights
Estimation of large covariance matrices and their inverse has several applications in various areas, from economics and finance to health, biology, computer science and engineering
The F1 score and area under the curve (AUC) show that block-Graphical LASSO (GLASSO) has higher true positive rates and substantially lower false positive rates, while the entropy loss (EL) and Frobenius loss (FL) are always lower for block-GLASSO, indicating that the latter provides a better estimation of the precision matrix
F1 is the F1 score, AUC is the area under the receiver operating characteristic (ROC), EL is the average EL in (4.6) and FL is the average FL in (4.7)
Summary
Estimation of large covariance matrices and their inverse has several applications in various areas, from economics and finance to health, biology, computer science and engineering. We consider the case of networks with a very large number of nodes and we focus on the estimation of Gaussian graphical models when the dependency between variables has a block-wise structure. Lee and Yu (2007) considered the identification and estimation of interaction effects in the context of a spatial autoregressive model where the spatial weights matrix (and the associated precision matrix) has such a block diagonal structure with equal entries Note that this is a more restrictive assumption to that used in this paper, as it does not allow for dependences between groups. N ; Sc is used to denote the complement of a set S
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