Abstract: Hidden structures indicative of additional patterns relevant to the scientific inquiry are generally ignored and thus, the classical spatial regression analysis could miss important information carried by the latent variables. We develop novel methodology for uncovering some of such possible structures and patterns in spatial regression analysis. Our approach is to simultaneously model regression terms and hidden jump sets that occur abruptly across space in the presence of spatial dependence. An inequality for the homogeneity measure is derived by which we establish the consistency of jump-set selection. We devise a three-step computational algorithm based on a quasi-likelihood function and homogeneity measure to uncover patterns related to jump coefficients. Under suitable regularity conditions, we prove that the identification procedure is consistent when the hidden jump sets, covariates, and spatial correlation are incorporated into the model from the outset. The simulation study also shows sound finite-sample properties. In a case study, we examine closely county-based poverty rates in relation to industrial and racial compositions prior to the decline of manufacturing in the Upper Midwest of the U.S. Our case study reveals important socio-economic factors on poverty and additionally interesting structures and patterns not detected in classical spatial regression.

Key words and phrases: Homogeneity measure, quasi-likelihood ratio, spatial statistics.