Abstract
To facilitate the reading of the later chapters, I present an initial discussion, in fairly general terms, of the notion of connectedness in space, and the main operational tools by which spatial effects are encompassed in econometric work: the spatial weight matrices and spatial lag operators. I also briefly discuss some complicating factors related to the notion of space implied by the various techniques.
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Notes on Chapter 3
A further motivation for keeping this discussion brief is that the issue of spatial connectivity has received extensive attention in the texts of Cliff and Ord (1981) and Upton and Fingleton (1985). Therefore, I will concentrate on issues which have received less attention and on viewpoints that differ from the approach taken in these texts.
For a more rigorous discussion, see Besag (1974), Bartlett (1978), and Haining (1979).
For a more extensive discussion, see Upton and Fingleton (1985, Chapter 3).
The shortest path on a network obtained from connecting all the points in the system is not the only way in which contiguity can be defined. Alternatively, the notion of Gabriel connectivity could be used. According to this concept, two points are considered contiguous if all other points in the system are outside a circle on which circumference the points are at opposite ends (the so—called least squares adjacency criterion). For details, see Matula and Sokal (1980).
The best—known types of spatial tessellations are the Thiessen polygons (also called Dirichlet tessellation or Voronoi polygons), which are constructed from the perpendicular bisectors of the lines that connect the irregularly spaced points (for an overview, see Ripley 1981, pp. 38–44, Amrhein, Guevara and Griffith 1983, and Upton and Fingleton 1985, pp. 96–104). These polygons are often related to notions of spatial market areas, and can easily be used to obtain measures of contiguity. For example, in Figure 3.1, the square grids form Thiessen polygons for their centroids (points connected by the dashed lines). Other tessellations, such as the Delaunay triangularization (the dual of the Thiessen polygons), and various other mosaics are less directly relevant for applications in regional science.
See Cliff and Ord (1973, p. 10), for an example.
For overviews and applications, see, e.g., Pool and Kochen (1978), Burt (1980), White, Burton, and Dow (1981), Loftin and Ward (1983), and the papers by Doreian: Doreian (1974, 1980, 1981, 1982), and Doreian, Teuter and Wang (1984).
An exception to the a priori choice of weights is the approach suggested by Kooijman (1976). There, an alternative teat statistic for spatial autocorrelation is constructed from a constrained maximisation process in function of the contiguity weights, somewhat similar to the technique of kriging, which is widely used in geological sciences. For an overview of kriging, see Matheron (1971), Clark (1979), Ripley (1981, pp. 44–50), and Nipper and Streit (1982). Hanham, Hohn and Bohland (1984) present an application to a demographic phenomenon. However, most applications of kriging are in the physical sciences, without much relevance for the types of problems considered in regional science. Therefore, this technique will not be further considered and the interested reader is referred to the sources listed above.
A wide range of functional forms can be considered. Weibull (1976) outlines an axiomatic approach which restricts this choice to specific types of functions. See also Anselin ( 1980, Chapter 8), for a more extensive discussion.
The precise effects on estimation are presented in detail in Blommestein (1985).
i This view is discussed at greater length in Anselin (1986b).
The level of aggregation varied from 99 counties to 6 zones.
See also the comments in Johnston (1984), and Openshaw (1984).
This issue is a familiar one in the regional science literature, and known as the regional homogeneity problem. For recent overviews, see, e.g., Johnson (1975), Schulze (1977, 1987), and Lin (1985). It is further discussed in Section 10.1.
For an overview of the identification problem in econometrics, see Fisher (1968) and Hausman (1983).
For examples and more detailed discussion, see, e.g., Whittle (1954), Bartlett (1975), and Griffith (1987).
Since an extensive discussion of these primarily geometric considerations are tangential to the main focus of the book, the interested reader is referred to the literature for more detailed treatments. The main ideas on this issue go back to Garrison and Marble (1964), Gould (1967), and Tinkler (1972). More recent discussions of the use of the principal eigenvalue and associated eigenvectors to characterize spatial configuration can be found in Boots (1982, 1984, 1985 ), Boots and Tinkler (1983), and Griffith (1984). A related issue of the information content in binary maps is discussed in Gatrell (1977).
The Yan and Ames index consists of determining for each element of the contiguity matrix (which converts the weight matrix to a boolean matrix) how many powers are needed in order for it to become nonzero. An associated order matrix (which gives an indication of path length in the network) then forms the basis for the derivation of various summary indices, such as row, column, and overall means. A similar approach consists of using the weight matrix itself and considering how many powers are needed in order for an element of the matrix to become less than a pre—set small positive value. Details and a more extensive overview are given in Hamilton and Jensen (1983) and Szyrmer (1985).
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© 1988 Springer Science+Business Media Dordrecht
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Anselin, L. (1988). The Formal Expression of Spatial Effects. In: Spatial Econometrics: Methods and Models. Studies in Operational Regional Science, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7799-1_3
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