Applications of Functional Dependence to Spatial Econometrics

In this paper, we generalize the concept of functional dependence (FD) from time series (see Wu 2005, Proceedings of the National Academy of Sciences 102, 14150–14154) and stationary random fields (see El Machkouri, Volný, and Wu 2013, Stochastic Processes and Their Applications 123, 1–14) to nonstationary spatial processes. Within conventional settings in spatial econometrics, we define the concept of spatial FD measure and establish a moment inequality, an exponential inequality, a Nagaev-type inequality, a law of large numbers, and a central limit theorem. We show that the dependent variables generated by some common spatial econometric models, including spatial autoregressive (SAR) models, threshold SAR models, and spatial panel data models, are functionally dependent under regular conditions. Furthermore, we investigate the properties of FD measures under various transformations, which are useful in applications. Moreover, we compare spatial FD with the spatial mixing and spatial near-epoch dependence proposed in Jenish and Prucha (2009, Journal of Econometrics 150, 86–98, 2012, Journal of Econometrics 170, 178–190), and we illustrate its advantages.