Sparse Estimation of Huge Networks with a Block-Wise Structure

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 misspeci cation. 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.