  We explore time-varying networks for high-dimensional locally stationary time
series, using the large VAR model framework with both the transition and
(error) precision matrices evolving smoothly over time. Two types of
time-varying graphs are investigated: one containing directed edges of Granger
causality linkages, and the other containing undirected edges of partial
correlation linkages. Under the sparse structural assumption, we propose a
penalised local linear method with time-varying weighted group LASSO to jointly
estimate the transition matrices and identify their significant entries, and a
time-varying CLIME method to estimate the precision matrices. The estimated
transition and precision matrices are then used to determine the time-varying
network structures. Under some mild conditions, we derive the theoretical
properties of the proposed estimates including the consistency and oracle
properties. In addition, we extend the methodology and theory to cover
highly-correlated large-scale time series, for which the sparsity assumption
becomes invalid and we allow for common factors before estimating the
factor-adjusted time-varying networks. We provide extensive simulation studies
and an empirical application to a large U.S. macroeconomic dataset to
illustrate the finite-sample performance of our methods.
