  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 (Zhu et al, 2017, NAR) model 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. The
theoretical results show that the two-step estimator improves the one-step
estimator by an order of magnitude when the error admits a factor structure.
The advantages of the CNAR model are further illustrated on a variety of
synthetic and real datasets.
