Engle-Granger Representation in Spatial and Spatio-Temporal Models

The literature on panel models has made considerable progress in the last few decades, integrating non-stationary data both in the time and spatial domain. However, there remains a gap in the literature that simultaneously models non-stationarity and cointegration in both the time and spatial dimensions. This paper develops Granger representation theorems for spatial and spatio-temporal dynamics. In a panel setting, this provides a way to represent both spatial and temporal equilibria and dynamics as error correction models. This requires potentially two different processes for modelling spatial (or network) dynamics, both of which can be expressed in terms of spatial weights matrices. The first captures strong cross-sectional dependence, so that a spatial difference, suitably defined, is weakly cross-section dependent (granular) but can be nonstationary. The second is a conventional weights matrix that captures short-run spatio-temporal dynamics as stationary and granular processes. In large samples, cross-section averages serve the first purpose and we propose the mean group, common correlated effects estimator together with multiple testing of cross-correlations to provide the short-run spatial weights. We apply this model to house prices in the 375 MSAs of the US. We show that our approach is useful for capturing both weak and strong cross-section dependence, and partial adjustment to two long-run equilibrium relationships in terms of time and space.