To date, spatial econometric methods are not found in the main commercial econometric and statistical software packages, although macro and scripting facilities may be used to implement some estimators (Anselin and Hudak, 1992). The only comprehensive software to handle both estimation and specification testing of spatial regression models is the special-purpose SpaceStat package (Anselin, 1992b, 1998). A narrower set of techniques, such as maximum likelihood estimation of spatial models is included in the Matlab routines of Pace and Barry (1998), and estimation of spatial error models is part of the S+Spatialstats add-on to S-Plus (MathSoft, 1996).32
In contrast to maximum likelihood estimation, method of moments and 2SLS can easily be implemented with standard software, provided that spatial lags can be computed. This requires the construction of a spatial weights matrix, which must often be derived from information in a geographic information system. Similarly, once a spatial lag can be computed, the LM/RS statistics are straightforward to implement.
The main practical problem is encountered in maximum likelihood estimation where the Jacobian determinant must be evaluated for every iteration in a nonlinear optimization procedure. The original solution to this problem was suggested by Ord (1975), who showed how the log Jacobian can be decomposed in terms that contain the eigenvalues of the weights matrix
ln|I – pW| = £ ln(1 – pro,). (14.33)
This is easy to implement in a standard optimization routine by treating the individual elements in the sum as observations on an auxiliary term in the log – likelihood (see Anselin and Hudak, 1992). However, the computation of the eigenvalues quickly becomes numerically unstable for matrices of more than
1,0 observations. In addition, for large data sets this approach is inefficient in that it does not exploit the high degree of sparsity of the spatial weights matrix. Recently suggested solutions to this problem fall into two categories. Approximate solutions avoid the computation of the Jacobian determinant, but instead approximate it by a polynomial function or by means of simulation methods (e. g. Barry and Pace, 1999). Exact solutions are based on Cholesky or LU decomposition methods that exploit the sparsity of the weights (Pace and Barry, 1997a, 1997b), or use a characteristic polynomial approach (Smirnov and Anselin, 2000). While much progress has been made, considerable work remains to be done to develop efficient algorithms and data structures to allow for the analysis of very large spatial data sets.
3 Concluding Remarks
This review chapter has been an attempt to present the salient issues pertaining to the methodology of spatial econometrics. It is by no means complete, but it is hoped that sufficient guidance is provided to pursue interesting research directions. Many challenging problems remain, both methodological in nature as well as in terms of applying the new techniques to meaningful empirical problems. Particularly in dealing with spatial effects in models other than the standard linear regression, much needs to be done to complete the spatial econometric toolbox. It is hoped that the review presented here will stimulate statisticians and econometricians to tackle these interesting and challenging problems.
* This paper benefited greatly from comments by Wim Vijverberg and two anonymous referees. A more comprehensive version of this paper is available as Anselin (1999).
1 A more extensive review is given in Anselin and Bera (1998) and Anselin (1999).
2 An extensive collection of recent applications of spatial econometric methods in economics can be found in Anselin and Florax (2000).
3 In this chapter, I will use the terms spatial dependence and spatial autocorrelation interchangeably. Obviously, the two are not identical, but typically, the weaker form is used, in the sense of a moment of a joint distribution. Only seldom is the focus on the complete joint density (a recent exception can be found in Brett and Pinkse (1997)).
4 See Anselin (1988a), for a more extensive discussion.
5 One would still need to establish the class of spatial stochastic processes that would allow for the consistent estimation of the covariance; see Frees (1995) for a discussion of the general principles.
6 See Anselin and Bera (1998) for an extensive and technical discussion.
7 On a square grid, one could envisage using North, South, East and West as spatial shifts, but in general, for irregular spatial units such as counties, this is impractical, since the number of neighbors for each county is not constant.
8 In Anselin (1988a), the term spatial lag is introduced to refer to this new variable, to emphasize the similarity to a distributed lag term rather than a spatial shift.
9 By convention, wu = 0, i. e. a location is never a neighbor of itself. This is arbitrary, but can be assumed without loss of generality. For a more extensive discussion of spatial weights, see Anselin (1988a, ch. 3), Cliff and Ord (1981), Upton and Fingleton (1985).
10 See Anselin and Bera (1998) for further details.
11 See McMillen (1992) for an illustration.
12 The specification of spatial covariance functions is not arbitrary, and a number of conditions must be satisfied in order for the model to be "valid" (details are given in Cressie (1993, pp. 61-3, 67-8 and 84-6)).
13 Specifically, this may limit the applicability of GMM estimators that are based on a central limit theorem for stationary mixing random fields such as the one by Bolthausen (1982), used by Conley (1996).
14 Cressie (1993, pp. 100-1).
15 See Kelejian and Prucha (1999a, 1999b).
16 For ease of exposition, the error term is assumed to be iid, although various forms of heteroskedasticity can be incorporated in a straightforward way (Anselin, 1988a, ch. 6).
17 Details and a review of alternative specifications are given in Anselin and Bera (1998).
18 For further details, see Anselin (1988a, 1988b). A recent application is Baltagi and Li (2000b).
19 Methodological issues associated with spatial probit models are considered in Case (1992), McMillen (1992), Pinkse and Slade (1998) and Beron and Vijverberg (2000).
20 For an extensive discussion, see Beron and Vijverberg (2000).
21 Other classic treatments of ML estimation in spatial models can be found in Whittle (1954), Besag (1974), and Mardia and Marshall (1984).
22 For a formal demonstration, see Anselin (1988a) and Kelejian and Prucha (1997).
23 For details, see, e. g. McMillen (1992), Pinkse and Slade (1998), Beron and Vijverberg (2000), and also, for general principles, Poirier and Ruud (1988).
24 For a careful consideration of these issues, see Kelejian and Prucha (1999a).
25 For technical details, see, e. g. Kelejian and Robinson (1993), Kelejian and Prucha (1998).
26 A recent application of this method is given in Bell and Bockstael (2000). An extension of this idea to the residuals of a spatial 2SLS estimation is provided in Kelejian and Prucha (1998).
27 See also Case (1992) and McMillen (1992) for a similar focus on heteroskedasticity in the spatial probit model.
28 See also the discussion in Haining (1990, pp. 131-3).
29 For example, for row-standardized weights, S0 = N, and I = e’We/e’e. See Anselin and Bera (1998) for an extensive discussion.
30 Moran’s I is not based on an explicit alternative and has power against both (see Anselin and Rey, 1991).
31 As shown in Anselin and Kelejian (1997) these tests are asymptotically equivalent.
32 Neither of these toolboxes include specification tests. Furthermore, S+Spatialstats has no routines to handle the spatial lag model.
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