Spatial regression models with spatially correlated errors

Abstract

This thesis studies the spatial regression models with lattice data, with emphasis on models with spatially correlated errors. For the large scale variation of the data, the non-parametric, additive nonparametric and semi-parametric structure are adopted; while for the small scale variation, the errors are assumed to satisfy the torus, separable or unilateral SGAR model. Following Martins-Filho & Yao (2009), we propose to estimate the large scale variation with a two-step fitting procedure, which firstly forms a new process with the same conditional mean as the original one and i.i.d. errors, and secondly applies the estimation to the new process. Such approach takes both nonstationary mean/trend effects and spatial dependencies into account, hence overmatches the traditional estimations. Asymptotic properties of both first-and second-step estimators are investigated. For the first-step estimators of the unknown regression function, the convergence rate with all three types of errors is considered, and when errors satisfy the separable or unilateral SGAR model, the asymptotic normality is established. For the second-step estimators of the unknown regression function, the asymptotic normality with three types of error structures is established. In the semi-parametric model, we also establish the asymptotic normality of the first-and second-step estimators of the linear parameters. For all the models, simulations are conducted to assess the performance of our fitting. Under the condition that spatially correlated errors exist, the results show that our estimation works better than the traditional methods. The improvement of our estimation is significant when the volatility of the errors is large. As an illustration of our approach, a case study of the housing price in Hong Kong is given. It is shown that our approach improves the estimation, especially when some key factor is absent in the modelling.