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.

翻译：我们在大规模VAR模型框架下研究高维局部平稳时间序列的时变网络，其中转移矩阵和（误差）精度矩阵随时间平滑演化。本文探讨了两类时变图：一类包含格兰杰因果关系的边，另一类包含偏相关系数的边。在稀疏结构假设下，我们提出了一种带时变加权组LASSO的惩罚局部线性方法，用于联合估计转移矩阵并识别其显著条目，以及一种时变CLIME方法来估计精度矩阵。利用估计得到的转移矩阵和精度矩阵，我们进一步确定时变网络结构。在温和条件下，我们推导了所提估计量的理论性质，包括一致性和Oracle性质。此外，我们将方法和理论扩展到高相关大规模时间序列场景——此时稀疏假设不再成立，我们允许在估计因子调整后的时变网络前引入公共因子。我们通过大量模拟研究以及美国宏观经济数据集的应用实例，验证了所提方法的有限样本性能。