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Asymptotic theory for nonparametric regression with spatial data

  • Robinson, P.M.
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    Nonparametric regression with spatial, or spatio-temporal, data is considered. The conditional mean of a dependent variable, given explanatory ones, is a nonparametric function, while the conditional covariance reflects spatial correlation. Conditional heteroscedasticity is also allowed, as well as non-identically distributed observations. Instead of mixing conditions, a (possibly non-stationary) linear process is assumed for disturbances, allowing for long range, as well as short-range, dependence, while decay in dependence in explanatory variables is described using a measure based on the departure of the joint density from the product of marginal densities. A basic triangular array setting is employed, with the aim of covering various patterns of spatial observation. Sufficient conditions are established for consistency and asymptotic normality of kernel regression estimates. When the cross-sectional dependence is sufficiently mild, the asymptotic variance in the central limit theorem is the same as when observations are independent; otherwise, the rate of convergence is slower. We discuss the application of our conditions to spatial autoregressive models, and models defined on a regular lattice.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0304407611000947
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    Article provided by Elsevier in its journal Journal of Econometrics.

    Volume (Year): 165 (2011)
    Issue (Month): 1 ()
    Pages: 5-19

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    Handle: RePEc:eee:econom:v:165:y:2011:i:1:p:5-19
    DOI: 10.1016/j.jeconom.2011.05.002
    Contact details of provider: Web page: http://www.elsevier.com/locate/jeconom

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    1. Marc Hallin & Zudi Lu & Lanh T. Tran, 2004. "Kernel density estimation for spatial processes: the L1 theory," ULB Institutional Repository 2013/2127, ULB -- Universite Libre de Bruxelles.
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    3. Marc Hallin & Zudi Lu & Lanh T. Tran, 2001. "Density estimation for spatial linear processes," ULB Institutional Repository 2013/2109, ULB -- Universite Libre de Bruxelles.
    4. Peter M. Robinson, 1997. "Large-sample inference for nonparametric regression with dependent errors," LSE Research Online Documents on Economics 302, London School of Economics and Political Science, LSE Library.
    5. Marc Hallin & Zudi Lu & Lanh T. Tran, 2004. "Local linear spatial regression," ULB Institutional Repository 2013/2131, ULB -- Universite Libre de Bruxelles.
    6. Zudi Lu & Arvid Lundervold & Dag Tjøstheim & Qiwei Yao, 2007. "Exploring spatial nonlinearity using additive approximation," LSE Research Online Documents on Economics 5401, London School of Economics and Political Science, LSE Library.
    7. Pinkse, Joris & Shen, Lihong & Slade, Margaret, 2007. "A central limit theorem for endogenous locations and complex spatial interactions," Journal of Econometrics, Elsevier, vol. 140(1), pages 215-225, September.
    8. Donald W.K. Andrews, 2003. "Cross-section Regression with Common Shocks," Cowles Foundation Discussion Papers 1428, Cowles Foundation for Research in Economics, Yale University.
    9. Qiwei Yao & Peter J. Brockwell, 2006. "Gaussian maximum likelihood estimation for ARMA models I: time series," LSE Research Online Documents on Economics 5825, London School of Economics and Political Science, LSE Library.
    10. Peter M. Robinson & J. Vidal Sanz, 2005. "Modified whittle estimation of multilateral models on a lattice," LSE Research Online Documents on Economics 4545, London School of Economics and Political Science, LSE Library.
    11. Qiwei Yao & Peter J Brockwell, 2006. "Gaussian maximum likelihood estimation for ARMA models I: time series," LSE Research Online Documents on Economics 57580, London School of Economics and Political Science, LSE Library.
    12. Castellana, J. V. & Leadbetter, M. R., 1986. "On smoothed probability density estimation for stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 21(2), pages 179-193, February.
    13. Lee, Lung-Fei, 2002. "Consistency And Efficiency Of Least Squares Estimation For Mixed Regressive, Spatial Autoregressive Models," Econometric Theory, Cambridge University Press, vol. 18(02), pages 252-277, April.
    14. Hall, Peter & Hart, Jeffrey D., 1990. "Nonparametric regression with long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 36(2), pages 339-351, December.
    15. Qiwei Yao & Peter J. Brockwell, 2006. "Gaussian Maximum Likelihood Estimation For ARMA Models. I. Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(6), pages 857-875, November.
    16. Harry H. Kelejian & Ingmar R. Prucha, 1995. "A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model," Electronic Working Papers 95-001, University of Maryland, Department of Economics, revised Mar 1997.
    17. Tran, L. T. & Yakowitz, S., 1993. "Nearest Neighbor Estimators for Random Fields," Journal of Multivariate Analysis, Elsevier, vol. 44(1), pages 23-46, January.
    18. Robinson, P.M., 2005. "Robust Covariance Matrix Estimation: Hac Estimates With Long Memory/Antipersistence Correction," Econometric Theory, Cambridge University Press, vol. 21(01), pages 171-180, February.
    19. Qiwei Yao & Peter J Brockwell, 2006. "Gaussian maximum likelihood estimation for ARMA models II: spatial processes," LSE Research Online Documents on Economics 5416, London School of Economics and Political Science, LSE Library.
    20. Robinson, P. M., 1977. "Estimation of a time series model from unequally spaced data," Stochastic Processes and their Applications, Elsevier, vol. 6(1), pages 9-24, November.
    21. P. M. Robinson, 2005. "Robust covariance matrix estimation : 'HAC' estimates with long memory/antipersistence correction," LSE Research Online Documents on Economics 323, London School of Economics and Political Science, LSE Library.
    22. Tran, Lanh Tat, 1990. "Kernel density estimation on random fields," Journal of Multivariate Analysis, Elsevier, vol. 34(1), pages 37-53, July.
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