New estimation approaches for graphical models with elastic net penalty

In the context of undirected Gaussian graphical models, three estimators based on elastic net penalty for the underlying dependence graph are introduced. The aim is to estimate a sparse precision matrix, from which to retrieve both the underlying conditional dependence graph and the partial correlations. The first estimator is derived from the direct penalization of the precision matrix in the likelihood function, while the second uses penalized regressions to estimate the precision matrix. Finally, the third estimator relies on a two stage procedure that estimates the edge set first and then the precision matrix elements. Through simulations the performances of the proposed methods are investigated on a set of well-known network structures. Results on simulated data show that in high-dimensional situations the second estimator performs relatively well, while in low-dimensional settings the two stage procedure outperforms most estimators as the sample size grows. Nonetheless, there are situations where the first estimator is also a good choice. Mixed results suggest that the elastic net penalty is not always the best choice when compared to the LASSO penalty, i.e. pure ℓ1 penalty, even if elastic net penalty tends to outperform LASSO in presence of highly correlated data from the cluster structure. Finally, using real-world data on U.S. economic sectors, dependencies are estimated and the impact of Covid-19 pandemic on the network strength is studied.