  Contemporary time series data often feature objects connected by a social
network that naturally induces temporal dependence involving connected
neighbours. The network vector autoregressive model is useful for describing
the influence of linked neighbours, while recent generalizations aim to
separate influence and homophily. Existing approaches, however, require either
correct specification of a time series model or accurate estimation of a
network model or both, and rely exclusively on least-squares for parameter
estimation. This paper proposes a new autoregressive model incorporating a
flexible form for latent variables used to depict homophily. We develop a
first-order differencing method for the estimation of influence requiring only
the influence part of the model to be correctly specified. When the part
including homophily is correctly specified admitting a semiparametric form, we
leverage and generalize the recent notion of neighbour smoothing for parameter
estimation, bypassing the need to specify the generative mechanism of the
network. We develop new theory to show that all the estimated parameters are
consistent and asymptotically normal. The efficacy of our approach is confirmed
via extensive simulations and an analysis of a social media dataset.
