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Small Sample Properties of Tests for Spatial Dependence in Regression Models: Some Further Results

  • Luc Anselin
  • Raymond J. G. M. Florax
Part of the Advances in Spatial Science book series (ADVSPATIAL)

Abstract

It has now been more than two decades since Cliff and Ord (1972) and Hordijk (1974) applied the principle of Moran’s Itest for spatial autocorrelation to the residuals of regression models for cross-sectional data. To date, Moran’sIstatistic is still the most widely applied diagnostic for spatial dependence in regression models [e.g., Johnston (1984), King (1987), Case (1991)]. However, in spite of the well known consequences of ignoring spatial dependence for inference and estimation [for a review, see Anselin (1988a)], testing for this type of misspecification remains rare in applied empirical work, as illustrated in the surveys of Anselin and Griffith (1988) and Anselin and Hudak (1992). In part, this may be due to the rather complex expressions for the moments of Moran’s I, and the difficulties encountered in implementing them in econometric Software [for detailed discussion, see Cliff and Ord (1981), Anselin (1992), Tiefelsdorf and Boots (1994)]. Recently, a number of alternatives to Moran’s Ihave been developed, such as the tests of Burridge (1980) and Anselin (1988b, 1994), which are based on the Lagrange Multiplier (LM) principle, and the robust tests of Bera and Yoon (1992) and Kelejian and Robinson (1992). These tests are all asymptotic and distributed as X 2variates. Since they do not require the computation of specific moments of the statistic, they are easy to implement and straightforward to interpret. However, they are all large sample tests and evidence on their finite sample properties is still limited.

Keywords

Lagrange Multiplier Spatial Dependence Lagrange Multiplier Test Error Dependence Small Sample Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Anselin, L., A Note on Small Sample Properties of Estimators in a First-Order Spatial Autoregressive Model, Environment and Planning A, 14, 1023–1030, 1982.CrossRefGoogle Scholar
  2. Anselin, L., Some Further Notes on Spatial Models and Regional Science, Journal of Regional Science, 26, 799–802, 1986.CrossRefGoogle Scholar
  3. Anselin, L., Spatial Econometrics: Methods and Models, Dordrecht: Kluwer Academic Publishers, 1988a.Google Scholar
  4. Anselin, L., Lagrange Multiplier Test Diagnostics for Spatial Dependence and Spatial Heterogeneity, Geographical Analysis, 20, 1–17, 1988b.CrossRefGoogle Scholar
  5. Anselin, L., Space Stat: A Program for the Analysis of Spatial Data, Santa Barbara: National Center for Geographie Information and Analysis, University of California, 1992.Google Scholar
  6. Anselin, L., Testing for Spatial Dependence in Linear Regression Models: A Review, Morgantown: West Virginia University, Regional Research Institute Research Paper, 94–16, 1994.Google Scholar
  7. Anselin, L. and R.J.G.M. Florax, Small Sample Properties of Tests for Spatial Dependence in Regression Models: Some Further Results, Morgantown: West Virginia University, Regional Research University Research Paper, 94–14, 1994.Google Scholar
  8. Anselin, L. and D.A. Griffith, Do Spatial Effects Really Matter in Regression Analysis?, Papers of the Regional Science Association, 65, 11–34, 1988.Google Scholar
  9. Anselin, L. and S. Hudak, Spatial Econometrics in Practice, a Review of Software Options, Regional Science and Urban Economics, 22, 509–536, 1992.CrossRefGoogle Scholar
  10. Anselin, L. and S. Rey, Properties of Tests for Spatial Dependence in Linear Regression Models, Geographical Analysis, 23, 112–131, 1991.CrossRefGoogle Scholar
  11. Bartels, C.P.A. and L. Hordijk, On the Power of the Generalized Moran Contiguity Coefficient in Testing for Spatial Autocorrelation Among Regression Disturbances, Regional Science and Urban Economics, 7, 83–101, 1977.CrossRefGoogle Scholar
  12. Bera, A.K. and C.M. Jarque, Model Specification Tests, A Simultaneous Approach, Journal of Econometrics, 20, 59–82, 1982.CrossRefGoogle Scholar
  13. Bera, A.K. and A. Ullah, Rao’s Score Test in Econometrics, Journal of Quantitative Economics, 7, 189–220, 1991.Google Scholar
  14. Bera, A.K. and M.J. Yoon, Simple Diagnostic Tests for Spatial Dependence, Champaign: University of Illinois, Department of Economics, 1992 (mimeo).Google Scholar
  15. Boots, B., Evaluating Principal Eigenvalues as Measures of Network Structure, Geographical Analysis, 16, 270–275, 1984.CrossRefGoogle Scholar
  16. Boots, B.N. and G.F. Royle, A Conjecture on the Maximum Value of the Principal Eigenvalue of a Planar Graph, Geographical Analysis, 23, 276–282, 1991.CrossRefGoogle Scholar
  17. Brandsma, A.S. and R.H. Ketellapper, Further Evidence on Alternative Procedures for Testing of Spatial Autocorrelation Among Regression Disturbances, in: C.P.A. Bartels and R.H. Ketellapper (eds.), Exploratory and Explanatory Statistical Analysis of Spatial Data, Boston: Martinus Nijhoff, 1979.Google Scholar
  18. Burridge, P., On the Cliff-Ord Test for Spatial Autocorrelation, Journal ofthe Royal Statistical Society B, 42, 107–108, 1980.Google Scholar
  19. Case, A., Spatial Patterns in Household Demand, Econometrica, 59, 953–965, 1991.CrossRefGoogle Scholar
  20. Cliff, A. and J.K. Ord, Testing for Spatial Autocorrelation Among Regression Residuals, Geographical Analysis, 4, 267–284, 1972.CrossRefGoogle Scholar
  21. Cliff, A. and J.K. Ord, Spatial Processes: Models and Applications, London: Pion, 1981.Google Scholar
  22. Davidson, R. and J.G. MacKinnon, Estimation andlnference in Econometrics, New York: Oxford University Press, 1993.Google Scholar
  23. Florax, R. and H. Folmer, Specification and Estimation of Spatial Linear Regression Models: Monte Carlo Evaluation of Pre-Test Estimators, Regional Science and Urban Economics, 22, 405–432, 1992.CrossRefGoogle Scholar
  24. Florax, R. and H. Folmer, The Relevance of Hendry’s Econometric Methodology in Linear Spatial Process Modeling: Experimental Simulation Results for ML and IV estimators, Working Paper, Department of General Economics, Wageningen Agricultural University, 1994.Google Scholar
  25. Haining, R., Spatial Models and Regional Science: A Comment on Anselin’s Paper and Research Directions, Journal of Regional Science, 26, 793–798, 1986.CrossRefGoogle Scholar
  26. Hordijk, L., Spatial Correlation in the Disturbances of a Linear Interregional Model, Regional and Urban Economics, 4, 117–140, 1974.CrossRefGoogle Scholar
  27. Huang, J.S., The Autoregressive Moving Average Model for Spatial Analysis, Australian Journal of Statistics, 26, 169–178, 1984.CrossRefGoogle Scholar
  28. Jarque, C.M. and A.K. Bera, Efficient Tests for Normality, Homoscedasticity and Serial Independence in Regression Residuals, Economics Letters, 6, 255–259, 1980.CrossRefGoogle Scholar
  29. Johnston, J., Econometric Methods, New York: McGraw-Hill, 1984.Google Scholar
  30. Kelejian, H.H. and D.P. Robinson, Spatial Autocorrelation: A New Computationally Simple Test with an Application to Per Capita County Policy Expenditures, Regional Science and Urban Economics, 22, 317–331, 1992.CrossRefGoogle Scholar
  31. Kelejian, H.H. and D.P. Robinson, A Suggested Method of Estimation for Spatial Interdependent Models with Autocorrelated Errors, and an Application to a County Expenditure Model, Papers in Regional Science, 12, 297–312, 1993.CrossRefGoogle Scholar
  32. Kelejian, H.H. and D.P. Robinson, Spatial Correlation: The Cliff and Ord Model and a Suggested Alternative, 1995 (this issue).Google Scholar
  33. King, M.L., A Small Sample Property of the Cliff-Ord Test for Spatial Correlation, Journal ofthe Royal Statistical Society B, 43, 263–264, 1981.Google Scholar
  34. King, M.L., Testing for Autocorrelation in Linear Regression Models: A Survey, in: M. King and D. Giles (eds.), Specification Analysis in the Linear Model, London: Routledge and Kegan Paul, 1987.Google Scholar
  35. Ord, J.K., Estimation Methods for Models of Spatial Interaction, Journal of the American Statistical Association, 70, 120–126, 1975.CrossRefGoogle Scholar
  36. Tiefelsdorf, M. and B. Boots, The Exact Distribution of Moran’s I, Environment and Planning A, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Luc Anselin
    • 1
  • Raymond J. G. M. Florax
    • 2
  1. 1.West Virginia UniversityMorgantownUSA
  2. 2.Wageningen Agricultural UniversityWageningenThe Netherlands

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