[size=13.3333px]Modelling spatial processes : the identification and analysis of spatial relationships in regression residuals by means of Moran's I
ganization of the Monograp
I Underlying Concepts of Spatial Processes
3
T h e
2.1
2.2
2.3
2.3.1
2.3.2
Regression M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
A Review of Ordinary Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . 13
Ordinary Least Squares with Correlated Disturbances . . . . . . . . . . 17
The Impact of Unaccounted Variables . . . . . . . . . . . . . . . . . . . . . . . 18
The Two Stage Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . 19
The Misspecified Regression Model . . . . . . . . . . . . . . . . . . . . . . . . . . 20
T h e
3.1
3.1.1
3.1.2
3.1.3
3.1.4
3.1.5
3.2
3.2.1
3.2.2
3.2.3
3.2.4
Specification of Spatial Relations . . . . . . . . . . . . . . . . . . 23
Implementation of Spatial Structures . . . . . . . . . . . . . . . . . . . . . . . . 24
Typology of Spatial Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Relations between Spatial Objects . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Similarity Versus Dissimilarity Relations . . . . . . . . . . . . . . . . . . . . . 25
Functional Relationships in Relative Space . . . . . . . . . . . . . . . . . . . 28
Coding Schemes of Spatial Structure Matrices . . . . . . . . . . . . . . . . 29
Some Special Spatial Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Regular Spatial Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Higher Order Link Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
A Link between the Contiguity Matrices and Spatial Configura
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Local Spatial Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 G a u s s i a n Spatial Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.1.1
4.1.2
4.1.3
4.2
4.2.1
4.2.2
4.2.3
4.3
4.3.1
4.3.2
37
Blending the Disturbances with a Spatial Structure . . . . . . . . . . . . 37
The Internal Composition of Stochastic Spatial Processes . . . . . . 38
Constraining Assumptions on Spatial Stochastic Processes . . . . . 40
Excursion: Non-Hierarchical Aggregation of Spatial Objects . . . . 41
Definition and Properties of Gaussian Spatial Processes . . . . . . . . 42
The Autoregressive Spatial Process . . . . . . . . . . . . . . . . . . . . . . . . . . 43
The Moving Average Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Excursion: The Common Factor Constraint and the Spatial Lag
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Testing for Gaussian Spatial Processes . . . . . . . . . . . . . . . . . . . . . . . 47
Test Principles for Spatial Autocorrelation in Regression Residuals 49
Statistical Properties of the Moran's I Test . . . . . . . . . . . . . . . . . . 51