Krzysztof Podgórski

In univariate spatial stochastic models, parameter space dimension is reduced through structural models with a known adjacency matrix. This structural reduction is also applied in multivariate spatial settings, where matrix-valued observations represent locations along one coordinate and multivariate variables along the other. However, such reduction often goes too far, omitting parameters that capture natural and important dependencies. Widely used models, including the spatial error and spatial lag models, lack parameters for intra-location dependencies. In a spatial econometric context, for example, while parameters link inflation and interest rates across economies, there is no explicit way to represent the effect of inflation on interest rates within a given economy. Through examples and analytical arguments, it is shown that when intralocation feedback exists in the data, standard models fail to capture it, leading to serious misrepresentation of other effects. As a remedy, this paper develops multivariate spatial models that incorporate feedback between variables at the same location. Given the high-dimensional nature of structural models, the challenge is to introduce such effects without substantially enlarging the parameter space, thereby avoiding overparameterization or non-identifiability. This is achieved by adding a single parameter that accounts for intralocation feedback. The proposed models are well-defined under a general second-order framework, accommodating non-Gaussian distributions. Dimensions of the parameter space, model identification, and other fundamental properties are established. Statistical inference is discussed using both empirical precision matrix methods and maximum likelihood. While the main contribution lies in static models, extensions to time-dependent data are also formulated, showing that dynamic generalizations are straightforward.