Community network auto-regression for high-dimensional time series

Modeling responses on the nodes of a large-scale network is an important task that arises commonly in practice. This paper proposes a community network vector autoregressive (CNAR) model, which utilizes the network structure to characterize the dependence and intra-community homogeneity of the high-dimensional time series. The CNAR model greatly increases the flexibility and generality of the network vector autoregressive (NAR) model proposed by Zhu et al. (2017) by allowing heterogeneous network effects across different network communities. In addition, the non-community-related latent factors are included to account for unknown cross-sectional dependence. The number of network communities can diverge as the network expands, which leads to estimating a diverging number of model parameters. We obtain a set of stationary conditions and develop an efficient two-step weighted least-squares estimator. The consistency and asymptotic normality properties of the estimators are established. Theoretical results show that the two-step estimator can further improve the efficiency of one-step estimator when the error admits a factor structure. The advantages of the CNAR model are illustrated on a variety of synthetic and real datasets.