  The paper develops a general flexible framework for Network Autoregressive
Processes (NAR), wherein the response of each node 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,
while the framework can accommodate heavier than Gaussian errors with both
spatial-autorgressive and factor based 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 generalized least
squares estimators for both a fixed, as well as a diverging number of network
nodes, and also provide their ridge regularized counterparts that exhibit
better performance in large network settings, together with their asymptotic
distributions. 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.
