A General Modeling Framework for Network Autoregressive Processes

A general flexible framework for Network Autoregressive Processes (NAR) is developed, wherein the response of each node in the network linearly depends on its past values, a prespecified linear combination of neighboring nodes and a set of node-specific covariates. The corresponding coefficients are node-specific, and the framework can accommodate heavier than Gaussian errors with spatial-autoregressive, factor-based, or in certain settings general covariance structures. We provide a sufficient condition that ensures the stability (stationarity) of the underlying NAR that is significantly weaker than its counterparts in previous work in the literature. Further, we develop ordinary and (estimated) generalized least squares estimators for both fixed, as well as diverging numbers of network nodes, and also provide their ridge regularized counterparts that exhibit better performance in large network settings, together with their asymptotic distributions. We derive their asymptotic distributions that can be used for testing various hypotheses of interest to practitioners. We also address the issue of misspecifying the network connectivity and its impact on the aforementioned asymptotic distributions of the various NAR parameter estimators. The framework is illustrated on both synthetic and real air pollution data.

本文构建了一种通用且灵活的网络自回归过程（Network Autoregressive Processes，NAR）框架，网络中各节点的响应线性依赖于其自身历史取值、邻居节点的预设线性组合，以及一组节点专属协变量。各系数均为节点专属，该框架可支持比高斯分布更重尾的误差项，且兼容空间自回归、因子型，或特定场景下的一般协方差结构。本文给出了确保所提NAR框架平稳性的充分条件，该条件相较于现有文献中的同类结论显著更宽松。进一步，本文针对固定节点数与发散节点数两类场景，分别推导了普通最小二乘与（估计）广义最小二乘估计量，并给出了适配大规模网络场景、性能更优的岭正则化变体，同时推导了各类估计量的渐近分布。这些渐近分布可用于检验行业从业者关注的各类研究假设。本文还探讨了网络连接结构误设问题，及其对前述各类NAR参数估计量渐近分布的影响。本文通过合成数据集与真实空气污染数据集对所提框架进行了实例演示与验证。