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Asymptotic theory for nonparametric regression with spatial data

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  • Robinson, P.M.

Abstract

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.

Suggested Citation

  • Robinson, P.M., 2011. "Asymptotic theory for nonparametric regression with spatial data," Journal of Econometrics, Elsevier, vol. 165(1), pages 5-19.
  • Handle: RePEc:eee:econom:v:165:y:2011:i:1:p:5-19 DOI: 10.1016/j.jeconom.2011.05.002
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    References listed on IDEAS

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    11. Yao, Qiwei & Brockwell, Peter J, 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. Robinson, P.M. & Vidal Sanz, J., 2006. "Modified Whittle estimation of multilateral models on a lattice," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1090-1120, May.
    13. Lu, Zudi & Lundervold, Arvid & Tjøstheim, Dag & Yao, Qiwei, 2007. "Exploring spatial nonlinearity using additive approximation," LSE Research Online Documents on Economics 5401, London School of Economics and Political Science, LSE Library.
    14. Robinson, P. M., 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.
    15. Yao, Qiwei & Brockwell, Peter J, 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.
    16. 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.
    17. Yao, Qiwei & Brockwell, Peter J., 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.
    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. Marc Hallin & Zudi Lu & Lanh T. Tran, 2001. "Density estimation for spatial linear processes," ULB Institutional Repository 2013/2109, ULB -- Universite Libre de Bruxelles.
    20. 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.
    21. Robinson, Peter M., 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.
    22. 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.
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    Cited by:

    1. Jenish, Nazgul, 2012. "Nonparametric spatial regression under near-epoch dependence," Journal of Econometrics, Elsevier, vol. 167(1), pages 224-239.
    2. Hidalgo, Javier & Schafgans, Marcia, 2017. "Inference and testing breaks in large dynamic panels with strong cross sectional dependence," Journal of Econometrics, Elsevier, vol. 196(2), pages 259-274.
    3. Robinson, Peter M. & Thawornkaiwong, Supachoke, 2012. "Statistical inference on regression with spatial dependence," Journal of Econometrics, Elsevier, vol. 167(2), pages 521-542.
    4. Jungyoon Lee & Peter M Robinson, 2013. "Series Estimation under Cross-sectional Dependence," STICERD - Econometrics Paper Series 570, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    5. Feng, Guohua & Gao, Jiti & Peng, Bin & Zhang, Xiaohui, 2017. "A varying-coefficient panel data model with fixed effects: Theory and an application to US commercial banks," Journal of Econometrics, Elsevier, vol. 196(1), pages 68-82.
    6. Javier Hidalgo & Marcia M Schafgans, 2015. "Inference and Testing Breaks in Large Dynamic Panels with Strong Cross Sectional Dependence," STICERD - Econometrics Paper Series /2015/583, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    7. Miguel A. Delgado & Peter M Robinson, 2013. "Non-Nested Testing of Spatial Correlation," STICERD - Econometrics Paper Series 568, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    8. Boneva, Lena & Linton, Oliver & Vogt, Michael, 2015. "A semiparametric model for heterogeneous panel data with fixed effects," Journal of Econometrics, Elsevier, vol. 188(2), pages 327-345.
    9. Lee, Jungyoon & Robinson, Peter M., 2013. "Series estimation under cross-sectional dependence," LSE Research Online Documents on Economics 58188, London School of Economics and Political Science, LSE Library.
    10. Marcos Sanso-Navarro & Maria Vera-Cabello, 2015. "The effects of knowledge and innovation on regional growth: Nonparametric evidence," ERSA conference papers ersa15p949, European Regional Science Association.
    11. Peng, Bin, 2016. "Inference on modelling cross-sectional dependence for a varying-coefficient model," Economics Letters, Elsevier, vol. 145(C), pages 1-5.
    12. Lee, Jungyoon & Robinson, Peter M., 2016. "Series estimation under cross-sectional dependence," Journal of Econometrics, Elsevier, vol. 190(1), pages 1-17.
    13. repec:spr:stpapr:v:58:y:2017:i:4:d:10.1007_s00362-015-0735-6 is not listed on IDEAS
    14. repec:cep:stiecm:/2013/570 is not listed on IDEAS
    15. Jesùs Mur, 2013. "Causality, Uncertainty and Identification: Three Issues on the Spatial Econometrics Agenda," SCIENZE REGIONALI, FrancoAngeli Editore, vol. 2013(1), pages 5-27.
    16. repec:kap:jecgro:v:22:y:2017:i:2:d:10.1007_s10887-016-9139-2 is not listed on IDEAS
    17. Eduardo A. Souza-Rodrigues, 2016. "Nonparametric Regression with Common Shocks," Econometrics, MDPI, Open Access Journal, vol. 4(3), pages 1-17, September.
    18. Benhenni, Karim & Su, Yingcai, 2016. "Optimal sampling designs for nonparametric estimation of spatial averages of random fields," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 341-351.
    19. Jungyoon Lee & Peter Robinson, 2016. "Series estimation under cross-sectional dependence," LSE Research Online Documents on Economics 63380, London School of Economics and Political Science, LSE Library.
    20. repec:cep:stiecm:/2013/568 is not listed on IDEAS
    21. Delgado, Miguel A. & Robinson, Peter M., 2013. "Non-nested testing of spatial correlation," LSE Research Online Documents on Economics 58169, London School of Economics and Political Science, LSE Library.
    22. Delgado, Miguel A. & Robinson, Peter, 2015. "Non-nested testing of spatial correlation," LSE Research Online Documents on Economics 61433, London School of Economics and Political Science, LSE Library.
    23. Al-Sulami, Dawlah & Jiang, Zhenyu & Lu, Zudi & Zhu, Jun, 2017. "Estimation for semiparametric nonlinear regression of irregularly located spatial time-series data," Econometrics and Statistics, Elsevier, vol. 2(C), pages 22-35.
    24. Delgado, Miguel A. & Robinson, Peter M., 2015. "Non-nested testing of spatial correlation," Journal of Econometrics, Elsevier, vol. 187(1), pages 385-401.

    More about this item

    Keywords

    Nonparametric regression; Spatial data; Weak dependence; Long range dependence; Heterogeneity; Consistency; Central limit theorem;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models

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