Essays on the econometric analysis of spatial data

This thesis addresses issues in the econometric analysis of data observed over regular or irregular lattices, a lattice being a fixed set of observational units that have spatial connotation, in a broad sense. As far as the case of data coming from irregular lattices is concerned, we investigate the correlation structure of spatial autoregression models and we analyse the properties of tests for spatial autocorrelation. As for the case of regular lattices, we focus on statistical issues associated to some matrices, the so-called spatial design matrices, that arise naturally in many inferential problems in the context of isotropic spatial processes defined on uniform grids. Chapter 1, titled ‘The Correlation Structure of Spatial Autoregressions’, proposes a novel method to study the properties of spatial autoregression models defined over irregular lattices. A little graph theory provides simple interpretations of the correlations implied by such models in terms of the walks connecting tow vertices, and reveals the statistical consequences of the presence of symmetries or regularities in the configuration of the observational units. Chapter 2, titled ‘Properties of Invariant Tests for Spatial Autocorrelation in the Linear Regression Model’, sheds some new light on how the power of some popular tests for spatial autocorrelation in the errors of a linear model is affected by the matrix of regressors and by the assumed spatial structure. Conditions for unbiasedness and monotonicity of the power function of the tests are studied. Chapter 3, titled ‘Spatial Design Matrices and Associated Quadratic Forms: Structure and Properties’, provides a complete characterization of the structure of spatial design matrices. The structural results are applied to study the statistical properties of statistics associated to the spatial design matrices, in particular of the classical variogram estimator, under several assumptions about the actual variogram. Chapter 4, titled ‘Circular Approximation to the Design Matrices of Isotropic Spatial Processes’, develops an approximation to the spatial design matrices, with the aim of alleviating the computational effort required to obtain the cumulants of some of the statistics discussed in Chapter 3. The performance of the approximation is discussed in the case of independent data and of second-order stationary and isotropic processes.