Dynamic Spatial Autoregressive Models with Time-Varying Spatial Weighting Matrices

Abstract

We propose a new spatio-temporal model with time-varying spatial weighting matrices. We allow for a general parameterization of the spatial matrix, such as: (i) a function of the inverse distances among pairs of units in space to the power of an unknown time-varying distance decay parameter, and (ii) a negative exponential function of the time-varying parameter as in (i). The filtering procedure of the time-varying unknown parameters is performed using the information contained in the score of the conditional distribution of the observables. We provide conditions for the stationarity and ergodicity of the filtered sequence of the spatial matrices as well as for the consistency and asymptotic normality of the maximum likelihood estimator (MLE). An extensive Monte Carlo simulation study to investigate the finite sample properties of the maximum likelihood estimator is reported. In the empirical part of the paper we analyze the association between eight European countries' perceived risk. Our findings suggest that the economically strong countries have their perceived risk increased due to their spatial connection with the economically weaker countries. A second empirical analysis investigates the evolution of the spatial connection between the house prices in different areas of the UK. In this case we identify periods when the usually adopted sparse weighting matrix is not sufficient to describe the underlying spatial process.

The Importance of Heterogeneity in Dynamic Network Models Applied to European Systemic Risk

“…A similar score-driven approach is found in Blasques et al (2016), Catania and Billé (2017), and Billé et al (2019). As mentioned before, however, our work differs from theirs in that we allow for more dynamic heterogeneity across network players, which proves empirically highly important.…”

“…This brings us to the second challenge: time-variation in the network connections. Blasques et al (2016), Catania and Billé (2017), and Billé et al (2019) show that the overall strength of network connections can change substantially over time in stock markets, debt markets, and real estate markets. These papers introduce dynamic extensions of the baseline spatial model that allow for additional dynamics in network propagation strength on top of what is captured by simple observable variables.…”

Heterogeneity and dynamics in network models

“…Finally, methodologically, our work relates to the literature on score-driven time-varying parameter models as in Creal et al (2013). A similar score-driven approach is found in Blasques et al (2016), Catania and Billé (2017), and Billé et al (2019). Our work differs from theirs in that we allow for more dynamic heterogeneity across network players, which proves empirically highly important.…”

“…The second challenge relates to time‐variation in network connections. Blasques et al (2016), Catania and Billé (2017), and Billé et al (2019) show that the overall strength of network connections can change substantially over time in stock markets, debt markets, and real estate markets. These papers introduce dynamic extensions of the baseline spatial model that allow for additional dynamics in network propagation strength on top of what is captured by simple observables.…”

Spatial Limited Dependent Variable Models: A Review Focused on Specification, Estimation, and Health Economics Applications

“…The problems of endogeneity raised by such an approach are considered in, for example, Kelejian and Piras (2014) and Qu and Lee (2015). Alternatively, one can take the advantage of time information, see, for example, Billé and Catania (2018).…”