# Implementation Issues

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)

i =1

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.

Notes

* 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.

References

Anselin, L. (1980). Estimation Methods for Spatial Autoregressive Structures. Regional Science Dissertation and Monograph Series 8. Field of Regional Science, Cornell University, Ithaca, N. Y.

Anselin, L. (1988a). Spatial Econometrics: Methods and Models. Kluwer Academic, Dordrecht.

Anselin, L. (1988b). A test for spatial autocorrelation in seemingly unrelated regressions. Economics Letters 28, 335-41.

Anselin, L. (1988c). Lagrange multiplier test diagnostics for spatial dependence and spatial heterogeneity. Geographical Analysis 20, 1-17.

Anselin, L. (1990). Some robust approaches to testing and estimation in spatial econometrics. Regional Science and Urban Economics 20, 141-63.

Anselin, L. (1992a). Space and applied econometrics. Special Issue, Regional Science and Urban Economics 22.

Anselin, L. (1992b). SpaceStat, a Software Program for the Analysis of Spatial Data. National Center for Geographic Information and Analysis, University of California, Santa Barbara, CA.

Anselin, L. (1998). SpaceStat Version 1.90. http://www. spacestat. com.

Anselin, L. (1999). Spatial econometrics, An updated review. Regional Economics Applications Laboratory (REAL), University of Illinois, Urbana-Champaign.

Anselin, L. (2000). Rao’s score test in spatial econometrics. Journal of Statistical Planning and Inference (forthcoming).

Anselin, L., and A. Bera (1998). Spatial dependence in linear regression models with an introduction to spatial econometrics. In A. Ullah and D. E.A. Giles (eds.) Handbook of Applied Economic Statistics, pp. 237-89. New York: Marcel Dekker.

Anselin, L., and R. Florax (1995a). Introduction. In L. Anselin and R. Florax (eds.) New Directions in Spatial Econometrics, pp. 3-18. Berlin: Springer-Verlag.

Anselin, L., and R. Florax (1995b). Small sample properties of tests for spatial dependence in regression models: some further results. In L. Anselin and R. Florax (eds.) New Directions in Spatial Econometrics, pp. 21-74. Berlin: Springer-Verlag.

Anselin, L., and R. Florax (2000). Advances in Spatial Econometrics. Heidelberg: Springer – Verlag.

Anselin, L., and S. Hudak (1992). Spatial econometrics in practice, a review of software options. Regional Science and Urban Economics 22, 509-36.

Anselin, L., and H. H. Kelejian (1997). Testing for spatial error autocorrelation in the presence of endogenous regressors. International Regional Science Review 20, 153-82.

Anselin, L., and S. Rey (1991). Properties of tests for spatial dependence in linear regression models. Geographical Analysis 23, 112-31.

Anselin, L., and S. Rey (1997). Introduction to the special issue on spatial econometrics. International Regional Science Review 20, 1-7.

Anselin, L., A. Bera, R. Florax, and M. Yoon (1996). Simple diagnostic tests for spatial dependence. Regional Science and Urban Economics 26, 77-104.

Baltagi, B., and D. Li (2000a). Double length artificial regressions for testing spatial dependence. Econometric Review, forthcoming.

Baltagi, B., and D. Li (2000b). Prediction in the panel data model with spatial correlation. In L. Anselin and R. Florax (eds.) Advances in Spatial Econometrics. Heidelberg: Springer – Verlag.

Barry, R. P., and R. K. Pace (1999). Monte Carlo estimates of the log determinant of large sparse matrices. Linear Algebra and its Applications 289, 41-54.

Basu, S., and T. G. Thibodeau (1998). Analysis of spatial autocorrelation in housing prices. Journal of Real Estate Finance and Economics 17, 61-85.

Bell, K. P., and N. E. Bockstael (2000). Applying the generalized moments estimation approach to spatial problems involving micro-level data. Review of Economics and Statistics 82, 72-82.

Beron, K. J., and W. P.M. Vijverberg (2000). Probit in a spatial context: a Monte Carlo approach. In L. Anselin and R. Florax (eds.) Advances in Spatial Econometrics. Heidelberg: Springer-Verlag.

Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society B 36, 192-225.

Besag, J., and P. A.P. Moran (1975). On the estimation and testing of spatial interaction in Gaussian lattice processes. Biometrika 62, 555-62.

Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Annals of Probability 10, 1047-50.

Brett, C., and J. Pinkse (1997). Those taxes all over the map! A test for spatial independence of municipal tax rates in British Columbia. International Regional Science Review 20, 131-51.

Brock, W. A., and S. N. Durlauf (1995). Discrete choice with social interactions I: Theory. NBER Working Paper No. W5291. Cambridge, MA: National Bureau of Economic Research.

Burridge, P. (1980). On the Cliff-Ord test for spatial autocorrelation. Journal of the Royal Statistical Society B 42, 107-8.

Case, A. (1992). Neighborhood influence and technological change. Regional Science and Urban Economics 22, 491-508.

Case, A., H. S. Rosen, and J. R. Hines (1993). Budget spillovers and fiscal policy interdependence: evidence from the States. Journal of Public Economics 52, 285-307.

Cliff, A., and J. K. Ord (1973). Spatial Autocorrelation. London: Pion.

Cliff, A., and J. K. Ord (1981). Spatial Processes: Models and Applications. London: Pion.

Conley, T. G. (1996). Econometric modelling of cross-sectional dependence. Ph. D. dissertation. Department of Economics, University of Chicago, Chicago, IL.

Cressie, N. (1993). Statistics for Spatial Data. New York: Wiley.

Davidson, J. (1994). Stochastic Limit Theory. Oxford: Oxford University Press.

Davidson, R. and J. G. MacKinnon (1988). Double-length artificial regressions. Oxford Bulletin of Economics and Statistics 50, 203-17.

Driscoll, J. C., and A. C. Kraay (1998). Consistent covariance matrix estimation with spatially dependent panel data. Review of Economics and Statistics 80, 549-60.

Dubin, R. (1988). Estimation of regression coefficients in the presence of spatially auto – correlated error terms. Review of Economics and Statistics 70, 466-74.

Dubin, R. (1992). Spatial autocorrelation and neighborhood quality. Regional Science and Urban Economics 22, 433-52.

Frees, E. W. (1995). Assessing cross-sectional correlation in panel data. Journal of Econometrics 69, 393-414.

Haining, R. (1990). Spatial Data Analysis in the Social and Environmental Sciences. Cambridge: Cambridge University Press.

Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica 50, 1029-54.

Kelejian, H., and I. Prucha (1997). Estimation of spatial regression models with autoregressive errors by two stage least squares procedures: a serious problem. International Regional Science Review 20, 103-11.

Kelejian, H., and I. Prucha (1998). A generalized spatial two stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances. Journal of Real Estate Finance and Economics 17, 99-121.

Kelejian, H., and I. Prucha (1999a). A generalized moments estimator for the autoregressive parameter in a spatial model. International Economic Review 40, 509-33.

Kelejian, H. H., and I. Prucha (1999b). On the asymptotic distribution of the Moran I test statistic with applications. Working Paper, Department of Economics, University of Maryland, College Park, MD.

Kelejian, H. H., and D. P. Robinson (1993). A suggested method of estimation for spatial interdependent models with autocorrelated errors, and an application to a county expenditure model. Papers in Regional Science 72, 297-312.

Kelejian, H. H., and D. P. Robinson (1998). A suggested test for spatial autocorrelation and/or heteroskedasticity and corresponding Monte Carlo results. Regional Science and Urban Economics 28, 389-417.

Kelejian, H. H., and D. P. Robinson (2000). The influence of spatially correlated hetero – skedasticity on tests for spatial correlation. In L. Anselin and R. Florax (eds.) Advances in Spatial Econometrics. Heidelberg: Springer-Verlag (forthcoming).

King, M. (1981). A small sample property of the Cliff-Ord test for spatial correlation. Journal of the Royal Statistical Society B 43, 263-4.

Lahiri, S. N. (1996). On the inconsistency of estimators under infill asymptotics for spatial data. Sankhya A 58, 403-17.

Magnus, J. (1978). Maximum likelihood estimation of the GLS model with unknown parameters in the disturbance covariance matrix. Journal of Econometrics 7, 281-312. (Corrigenda, Journal of Econometrics 10, 261).

Mardia, K. V., and R. J. Marshall (1984). Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71, 135-46.

MathSoft (1996). S+SpatialStats User’s Manual for Windows and Unix. Seattle, WA: MathSoft, Inc.

McMillen, D. P. (1992). Probit with spatial autocorrelation. Journal of Regional Science 32, 335-48.

Moran, P. A.P. (1948). The interpretation of statistical maps. Biometrika 35, 255-60.

Newey, W. K., and K. D. West (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703-8.

Ord, J. K. (1975). Estimation methods for models of spatial interaction. Journal of the American Statistical Association 70, 120-6.

Pace, R. K., and R. Barry (1997a). Sparse spatial autoregressions. Statistics and Probability Letters 33, 291-7.

Pace, R. K., and R. Barry (1997b). Quick computation of spatial autoregressive estimators. Geographical Analysis 29, 232-46.

Pace, R. K., and R. Barry (1998). Spatial Statistics Toolbox 1.0. Real Estate Research Institute, Lousiana State University, Baton Rouge, LA.

Pace, R. K., R. Barry, C. F. Sirmans (1998). Spatial statistics and real estate. Journal of Real Estate Finance and Economics 17, 5-13.

Paelinck, J., and L. Klaassen (1979). Spatial Econometrics. Farnborough: Saxon House.

Pinkse, J. (1998). Asymptotic properties of the Moran and related tests and a test for spatial correlation in probit models. Working Paper, Department of Economics, University of British Columbia, Vancouver, BC.

Pinkse, J. (2000). Moran-flavored tests with nuisance parameters. In L. Anselin and R. Florax (eds.) Advances in Spatial Econometrics. Heidelberg: Springer-Verlag.

Pinkse, J., and M. E. Slade (1998). Contracting in space: an application of spatial statistics to discrete-choice models. Journal of Econometrics 85, 125-54.

Poirier, D. J., and P. A. Ruud (1988). Probit with dependent observations. Review of Economic Studies 55, 593-614.

Potscher, B. M., and I. R. Prucha (1997). Dynamic Nonlinear Econometric Models. Berlin: Springer.

Rao, C. R. (1973). Linear Statistical Inference and its Applications (2nd edn). New York: Wiley.

Schmidt, P. (1976). Econometrics. New York: Marcel Dekker.

Smirnov, O., and L. Anselin (2000). Fast maximum likelihood estimation of very large spatial autoregressive models: a characteristic polynomial approach. Computational Statistics and Data Analysis.

Upton, G. J., and B. Fingleton (1985). Spatial Data Analysis by Example. Volume 1: Point Pattern and Quantitative Data. New York: Wiley.

Vijverberg, W. (1997). Monte Carlo evaluation of multivariate normal probabilities. Journal of Econometrics 76, 281-307.

Whittle, P. (1954). On stationary processes in the plane. Biometrika 41, 434-49.

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