  Understanding the heterogeneity over spatial locations is an important
problem that has been widely studied in many applications such as economics and
environmental science. In this paper, we focus on regression models for spatial
panel data analysis, where repeated measurements are collected over time at
various spatial locations. We propose a novel class of nonparametric priors
that combines Markov random field (MRF) with the product partition model (PPM),
and show that the resulting prior, called by MRF-PPM, is capable of identifying
the latent group structure among the spatial locations while efficiently
utilizing the spatial dependence information. We derive a closed-form
conditional distribution for the proposed prior and introduce a new way to
compute the marginal likelihood that renders efficient Bayesian inference. We
further study the theoretical properties of the proposed MRF-PPM prior and show
a clustering consistency result for the posterior distribution. We demonstrate
the excellent empirical performance of our method via extensive simulation
studies and applications to a US precipitation data and a California median
household income data study.
