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Two state–space representations, also known as state–space models (SSMs), are proposed to estimate dynamic spatial relationships from time series data. At each time step, the weight matrix, which captures the latent state, is updated in the context of a spatial autoregressive process. Specifically, two types of SSMs are considered: the first one identifies the spatial effects in the form of multivariate regression, where the higher orders of the spatial matrix are introduced to the regression coefficient, while the second one updates the spatial matrix taking advantage of the likelihood function of a spatial autoregressive (SAR) model. Different filtering algorithms are proposed to estimate the latent state. The simulation results show that the first state–space representation performs better in both lower-dimensional and higher-dimensional cases, while the performance of the second representation is sensitive to the state dimension. In a real-world case study, the time-varying weight matrices are estimated with weekly credit default swap (CDS) data for 16 banks, and the results show that the methods can identify communities that are consistent with the country-driven partition.

Estimating time-varying proximity with a state–space model