<p>Vector autoregressive (VAR) models are widely used in the analysis of time series and have been extensively studied in the literature. However, in scenarios with a large number of nodes, estimating the transition matrix in VAR models can be challenging. By incorporating the structure of the network into the VAR models, the number of parameters can be significantly reduced. In this paper, we propose a time-varying network vector autoregressive (tvNAR) model. In the tvNAR model, the response of each node at a given time point is assumed to be a linear combination of its previous values and those of its connected neighbors in the network. The coefficients are node-specific and time-varying, allowing the model to capture the unique effect of each node and describe the behavior of non-stationary time series. We propose a locally linear regression estimator of the time-varying nodal coefficients and establish its asymptotic properties. To examine the temporal stability of the coefficients, we propose a Wald-type test. We illustrate the performance of the estimator and the test procedure through simulation studies and empirical analysis of daily Nasdaq stock prices data.</p>