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POWER PROPERTIES OF INVARIANT TESTS FOR SPATIAL AUTOCORRELATION IN LINEAR REGRESSION

Published online by Cambridge University Press:  13 August 2009

Federico Martellosio
Federico Martellosio*
Affiliation:
University of Reading
*
*Address correspondence to Federico Martellosio, School of Economics, University of Reading, URS Building, Whiteknights PO Box 219, Reading RG6 6AW, UK; e-mail: [email protected].
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Abstract

This paper derives some exact power properties of tests for spatial autocorrelation in the context of a linear regression model. In particular, we characterize the circumstances in which the power vanishes as the autocorrelation increases, thus extending the work of Krämer (2005). More generally, the analysis in the paper sheds new light on how the power of tests for spatial autocorrelation is affected by the matrix of regressors and by the spatial structure. We mainly focus on the problem of residual spatial autocorrelation, in which case it is appropriate to restrict attention to the class of invariant tests, but we also consider the case when the autocorrelation is due to the presence of a spatially lagged dependent variable among the regressors. A numerical study aimed at assessing the practical relevance of the theoretical results is included.

Type
ARTICLES
Information
Econometric Theory , Volume 26 , Issue 1 , February 2010 , pp. 152 - 186
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Anderson, T.W. (1948) On the theory of testing serial correlation. Skandinavisk Aktuarietidskrift 31, 88–116.Google Scholar
Anselin, L. (1988) Spatial Econometrics: Methods and Models. Kluwer.CrossRefGoogle Scholar
Baltagi, B.H. (2006) Random effects and spatial autocorrelation with equal weights. Econometric Theory 22, 973–984.CrossRefGoogle Scholar
Bartels, R. (1992) On the power function of the Durbin-Watson test. Journal of Econometrics 51, 101–112.Google Scholar
Bell, K.P. & Bockstael, N.E. (2000) Applying the generalized-moments estimation approach to spatial problems involving microlevel data. Review of Economics and Statistics 82, 72–82.Google Scholar
Berenblut, I.I. & Webb, G.I. (1973) A new test for autocorrelated errors in the linear regression model. Journal of the Royal Statistical Society B 35, 33–50.Google Scholar
Besag, J.E. (1974) Spatial interaction and the statistical analysis of lattice data. Journal of the Royal Statistical Society B 36, 192–236.Google Scholar
Besag, J.E. & Kooperberg, C. (1995) On conditional and intrinsic autoregression. Biometrika 82, 733–746.Google Scholar
Bhattacharyya, B.B., Richardson, G.D., & Franklin, L.A. (1997) Asymptotic inference for near unit roots in spatial autoregression. Annals of Statistics 25, 1709–1724.Google Scholar
Biggs, N.L. (1993) Algebraic Graph Theory, 2nd ed.Cambridge University Press.Google Scholar
Case, A. (1991) Spatial patterns in household demand. Econometrica 59, 953–966.Google Scholar
Cliff, A.D. & Ord, J.K. (1981) Spatial Processes: Models and Applications. Pion.Google Scholar
Cox, D.R. & Hinkley, D.V. (1974) Theoretical Statistics. Chapman & Hall.CrossRefGoogle Scholar
Cordy, C.B. & Griffith, D.A. (1993) Efficiency of least squares estimators in the presence of spatial autocorrelation. Communications in Statistics: Simulation and Computation 22, 1161–1179.CrossRefGoogle Scholar
Cressie, N. (1993) Statistics for Spatial Data, revised ed. Wiley.CrossRefGoogle Scholar
Cvetković, D.M., Doob, M., & Sachs, H. (1980) Spectra of Graphs. Academic Press.Google Scholar
Dielman, D.E. & Pfaffenberger, R.C. (1989) Efficiency of ordinary least squares for linear models with autocorrelation. Journal of the American Statistical Association 84, 248.CrossRefGoogle Scholar
Durbin, J. (1970) An alternative to the bounds test for testing for serial correlation in least-squares regression. Econometrica 38, 422–29.CrossRefGoogle Scholar
Fingleton, B. (1999) Spurious spatial regression: Some Monte Carlo results with a spatial unit root and spatial cointegration. Journal of Regional Science 39, 1–19.Google Scholar
Florax, R.J.G.M. & de Graaff, T. (2004) The performance of diagnostic tests for spatial dependence in linear regression models: A meta-analysis of simulation studies. In Anselin, L., Florax, R.J.G.M., & Rey, S.J. (eds.), Advances in Spatial Econometrics: Methodology, Tools and Applications, pp. 29–65. Springer.CrossRefGoogle Scholar
Gall, J., Pap, G., & van Zuijlen, M. (2004) Maximum likelihood estimator of the volatility of forward rates driven by geometric spatial AR sheet. Journal of Applied Mathematics 4, 293–309.Google Scholar
Gantmacher, F.R. (1974) The Theory of Matrices, vol. II. Chelsea.Google Scholar
Goldstein, R. (2000) The term structure of interest rates as a random field. Review of Financial Studies 13, 365–384.Google Scholar
Golub, G.H. & Van Loan, C.F. (1996) Matrix Computations, vol. II, 3rd ed.Johns Hopkins University Press.Google Scholar
Hardy, G., Littlewood, J.E., & Pólya, G. (1952) Inequalities, 2nd ed.Cambridge University Press.Google Scholar
Horn, R. & Johnson, C.R. (1985) Matrix Analysis. Cambridge University Press.Google Scholar
Huse, C., (2006) Term Structure Modelling with Spatial Dependence and Observable State Variables. Paper presented at the International Workshop on Spatial Econometrics and Statistics, Rome.Google Scholar
Imhof, J.P. (1961) Computing the distribution of quadratic forms in normal variables. Biometrika 48, 419–26.Google Scholar
James, A.T. (1954) Normal multivariate analysis and the orthogonal group. Annals of Mathematical Statistics 25, 40–75.Google Scholar
Kadiyala, K.R. (1970) Testing for the independence of regression disturbances. Econometrica 38, 97–117.Google Scholar
Kalbfleisch, J.D. & Sprott, D.A. (1970) Application of likelihood methods to models involving large numbers of parameters. Journal of the Royal Statistical Society B 32, 175–208.Google Scholar
Kariya, T. (1980) Locally robust tests for serial correlation in least squares regression. Annals of Statistics 8, 1065–1070.Google Scholar
Kariya, T. (1988) The class of models for which the Durbin-Watson test is locally optimal. International Economic Review 29, 167–175.CrossRefGoogle Scholar
Kelejian, H.H. & Prucha, I.R. (2001) On the asymptotic distribution of the Moran I test statistic with applications. Journal of Econometrics 104, 219–257.Google Scholar
Kelejian, H.H. & Prucha, I.R. (2002) 2SLS and OLS in a spatial autoregressive model with equal spatial weights. Regional Science and Urban Economics 32, 691–707.Google Scholar
Kelejian, H.H. & Prucha, I.R. (2009) Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. Journal of Econometrics, forthcoming.Google Scholar
Kennedy, D. (1994) The term structure of interest rates as a Gaussian random field. Mathematical Finance 4, 247–258.Google Scholar
King, M.L. (1980) Robust tests for spherical symmetry and their application to least squares regression. Annals of Statistics 8, 1265–1271.Google Scholar
King, M.L. (1981) A small sample property of the Cliff-Ord test for spatial autocorrelation. Journal of the Royal Statistical Society B 43, 263–4.Google Scholar
King, M.L. (1988) Towards a theory of point optimal testing. Econometric Reviews 6, 169–255.CrossRefGoogle Scholar
King, M.L. & Hillier, G.H. (1985) Locally best invariant tests of the error covariance matrix of the linear regression model. Journal of the Royal Statistical Society B 47, 98–102.Google Scholar
Kleiber, C. & Krämer, W. (2005) Finite-sample power of the Durbin-Watson test against fractionally integrated disturbances. Econometrics Journal 8, 406–417.CrossRefGoogle Scholar
Krämer, W. (1985) The power of the Durbin-Watson test for regressions without an intercept. Journal of Econometrics 28, 363–370.CrossRefGoogle Scholar
Krämer, W. (2005) Finite sample power of Cliff–Ord-type tests for spatial disturbance correlation in linear regression. Journal of Statistical Planning and Inference 128, 489–496.Google Scholar
Krämer, W. & Donninger, C. (1987) Spatial autocorrelation among errors and relative efficiency of OLS in the linear regression model. Journal of the American Statistical Association 82, 577–579.Google Scholar
Lee, L.F. (2002) Consistency and efficiency of least squares estimation for mixed regressive, spatial autoregressive models. Econometric Theory 18, 252–277.Google Scholar
Lehmann, E.L. & Romano, J. (2005) Testing Statistical Hypotheses, 3rd ed.Springer.Google Scholar
Militino, A.F., Ugarte, M.D., & García-Reinaldos, L. (2004) Alternative models for describing spatial dependence among dwelling selling prices. Journal of Real Estate Finance and Economics 29, 193–209.CrossRefGoogle Scholar
Moran, P.A.P. (1950) Notes on continuos stochastic phenomena. Biometrika 37, 17–23.Google Scholar
Ord, J.K. (1975) Estimation methods for models of spatial interaction. Journal of the American Statistical Association 70, 120–6.Google Scholar
Pace, R.K. & LeSage, J.P. (2002) Semiparametric maximum likelihood estimates of spatial dependence. Geographical Analysis 34, 76–90.Google Scholar
Paulauskas, V. (2007) On unit roots for spatial autoregressive models. Journal of Multivariate Analysis 98, 209–226.Google Scholar
Pinske, J. & Slade, M.E. (1998). Contracting in space: An application of spatial statistics to discrete-choice models. Journal of Econometrics 85, 125–154.Google Scholar
Rahman, S. & King, M.L. (1997) Marginal-likelihood score-based tests of regression disturbances in the presence of nuisance parameters. Journal of Econometrics 82, 81–106.Google Scholar
Small, J.P. (1993) The limiting power of point optimal autocorrelation tests. Communications in Statistics: Theory and Methods 22, 2463–2470.Google Scholar
Tillman, J.A. (1975) The power of the Durbin-Watson test. Econometrica 43, 959–974.Google Scholar
Tunnicliffe Wilson, G. (1989) On the use of marginal likelihood in time series model estimation. Journal of the Royal Statistical Society B 51, 15–27.Google Scholar
Whittle, P. (1954) On stationary processes in the plane. Biometrika 41, 434–449.Google Scholar
Zeisel, H. (1989) On the power of the Durbin-Watson test under high autocorrelation. Communications in Statistics: Theory and Methods 18, 3907–3916.Google Scholar