Simultaneous Spatial Systems with Incommensurate Partitions

This paper addresses the problem of incommensurate spatial partitions in simultaneous equations systems by developing a framework that allows for explicit feedback between variables defined on distinct spatial supports. Building on Kelejian’s foundational work on spatial simultaneity, we represent cross-partition relationships using a bipartite graph from which both bipartite and induced uni-partite spatial lags are derived and embedded within a simultaneous equations model. The framework nests commonly used empirical approaches as special cases and clarifies the sources of bias that arise when simultaneity and induced spatial dependence are ignored. Monte Carlo experiments show that reciprocal spillovers across partitions are the primary driver of bias in spatial feedback parameters, and that neither large samples nor perfect spatial alignment are sufficient to eliminate this bias. The results highlight the need to explicitly model simultaneity in spatial systems defined over misaligned partitions and extend Kelejian’s methodological legacy to a broader class of spatial processes.