Bayesian Estimation of Partially Linear Additive Spatial Autoregressive Models with P-Splines

Zhiyong Chen,Guodong Xing,Minghui Chen,Francisco J Montáns

doi:10.1155/2021/1777469

Abstract

In this paper, we aim to develop a partially linear additive spatial autoregressive model (PLASARM), which is a generalization of the partially linear additive model and spatial autoregressive model. It can be used to simultaneously evaluate the linear and nonlinear effects of the covariates on the response for spatial data. To estimate the unknown parameters and approximate nonparametric functions by Bayesian P-splines, we develop a Bayesian Markov Chain Monte Carlo approach to estimate the PLASARM and design a Gibbs sampler to explore the joint posterior distributions of unknown parameters. Furthermore, we illustrate the performance of the proposed model and estimation method by a simulation study and analysis of Chinese housing price data.

Highlights

Spatial econometrics is a subfield of econometrics dealing with spatial interaction effects among geographical units Elhorst [1]; it has been widely applied in many research fields such as economics, geography, and environmental science
Spatial data are often encountered in economics, geography, and environmental science and can be analyzed by the spatial autoregressive models
To reduce the high risk of misspecification of the traditional spatial autoregressive models and avoid some serious drawbacks of fully nonparametric models, we have developed a partially linear additive spatial autoregressive model (PLASARM) for spatial data, which combines the partially linear additive model and spatial autoregressive model

Summary

Introduction

Spatial econometrics is a subfield of econometrics dealing with spatial interaction effects among geographical units Elhorst [1]; it has been widely applied in many research fields such as economics, geography, and environmental science. Sun and Wu [30] discussed the GMM to estimate the partially linear single-index spatial autoregressive model. Ey contain both linear and nonlinear additive components As they can provide more flexible models than the stringent linear models and mitigate the “curse of dimensionality” phenomenon encountered in nonparametric regression. Combining the partially linear additive models with the spatial autoregressive models, we develop a Bayesian P-splines method and design a Gibbs sampler to explore the joint posterior distributions, along with a Markov chain Monte Carlo algorithm, to estimate the unknown parameters and approximate the nonparametric functions of PLASARM. E rest of the article proceeds as follows: In Section 2, we present PLASARM for spatial dependent data and discuss its identifiability conditions and obtain the likelihood functions by approximating the link function with a Bayesian P-spline method.

Model and Likelihood

Bayesian Estimation

Numerical Illustration

Summary