State-Space Modeling of Time-Varying Spillovers on Networks

Abstract

Crime counts in city neighbourhoods, disease counts in counties, and sales at firms joined by trade are naturally represented as counts on the nodes of a network. In each case a high count at one node can raise the counts at the nodes linked to it next period. The strength of that spillover changes over time, and standard network autoregressions hold it fixed. We therefore use a network state-space model, in which the spillover is a coefficient that drifts and a filter estimates its value in each period. What the data reveal about that coefficient depends on the network. It is learned by contrasting nodes whose neighbours have high counts with nodes whose neighbours have low ones. If every node is linked to every other, all nodes share the same neighbours, the contrasts vanish, and the spillover is not identified. Robustness is often checked by refitting with the links spread evenly, and a stable coefficient is read as reassurance. That refit is the same model with the spillover rescaled, so it cannot disagree. Forecasts carry a second warning: beyond two steps ahead, simulation averages a quantity with no finite mean, and the output gives no sign of it. We give a measure of what a network and a data set reveal about the spillover, the accuracy the filter can reach, and an exact test of whether the network matters. Burglaries in Chicago, COVID-19 cases in Texas counties, and measles cases in the Weser--Ems districts illustrate all three. On both disease datasets the model outperforms every competing forecast in the comparison.