FNETS: Factor-adjusted network estimation and forecasting for high-dimensional time series

We propose {\tt fnets}, a methodology for network estimation and forecasting of high-dimensional time series exhibiting strong serial- and cross-sectional correlations. We operate under a factor-adjusted vector autoregressive (VAR) model where, after controlling for {\it common} factors accounting for pervasive co-movements of the variables, the remaining {\it idiosyncratic} dependence between the variables is modelled by a sparse VAR process. Network estimation of {\tt fnets} consists of three steps: (i) factor-adjustment via dynamic principal component analysis, (ii) estimation of the parameters of the latent VAR process by means of $\ell_1$-regularised Yule-Walker estimators, and (iii) estimation of partial correlation and long-run partial correlation matrices. In doing so, we learn three networks underpinning the latent VAR process, namely a directed network representing the Granger causal linkages between the variables, an undirected one embedding their contemporaneous relationships and finally, an undirected network that summarises both lead-lag and contemporaneous linkages. In addition, {\tt fnets} provides a suite of methods for separately forecasting the factor-driven and the VAR processes. Under general conditions permitting heavy tails and weak factors, we derive the consistency of {\tt fnets} in both network estimation and forecasting. Simulation studies and real data applications confirm the good performance of {\tt fnets}.