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Central limit theorems for long range dependent spatial linear processes

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  • Lahiri, S.N.
  • Robinson, Peter M.

Abstract

Central limit theorems are established for the sum, over a spatial region, of observations from a linear process on a d d-dimensional lattice. This region need not be rectangular, but can be irregularly-shaped. Separate results are established for the cases of positive strong dependence, short range dependence, and negative dependence. We provide approximations to asymptotic variances that reveal differential rates of convergence under the three types of dependence. Further, in contrast to the one dimensional (i.e., the time series) case, it is shown that the form of the asymptotic variance in dimensions d > 1 critically depends on the geometry of the sampling region under positive strong dependence and under negative dependence and that there can be non-trivial edge-effects under negative dependence for d > 1. Precise conditions for the presence of edge effects are also given.

Suggested Citation

  • Lahiri, S.N. & Robinson, Peter M., 2016. "Central limit theorems for long range dependent spatial linear processes," LSE Research Online Documents on Economics 65331, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:65331
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    File URL: http://eprints.lse.ac.uk/65331/
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    References listed on IDEAS

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    1. Paul Doukhan & Patrice Bertail & Philippe Soulier, 2006. "Dependence in Probability and Statistics," Post-Print hal-00268232, HAL.
    2. El Machkouri, Mohamed & Volný, Dalibor & Wu, Wei Biao, 2013. "A central limit theorem for stationary random fields," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 1-14.
    3. Frédéric Lavancier, 2007. "Invariance principles for non-isotropic long memory random fields," Statistical Inference for Stochastic Processes, Springer, vol. 10(3), pages 255-282, October.
    4. Paul Doukhan & Patrice Bertail & Philippe Soulier, 2006. "Dependence in Probability and Statistics," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00268232, HAL.
    5. Beran, Jan & Ghosh, Sucharita & Schell, Dieter, 2009. "On least squares estimation for long-memory lattice processes," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2178-2194, November.
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    Citations

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    Cited by:

    1. Michael Pollmann, 2020. "Causal Inference for Spatial Treatments," Papers 2011.00373, arXiv.org, revised Jan 2023.
    2. Ulrich K. Müller & Mark W. Watson, 2021. "Spatial Correlation Robust Inference," Working Papers 2021-61, Princeton University. Economics Department..
    3. Surgailis, Donatas, 2020. "Scaling transition and edge effects for negatively dependent linear random fields on Z2," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7518-7546.
    4. Pilipauskaitė, Vytautė & Surgailis, Donatas, 2017. "Scaling transition for nonlinear random fields with long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2751-2779.
    5. Zhang, Rongmao & Chan, Ngai Hang & Chi, Changxiong, 2023. "Nonparametric testing for the specification of spatial trend functions," Journal of Multivariate Analysis, Elsevier, vol. 196(C).
    6. Robinson, Peter, 2019. "Spatial long memory," LSE Research Online Documents on Economics 102182, London School of Economics and Political Science, LSE Library.
    7. Ulrich K. Müller & Mark W. Watson, 2022. "Spatial Correlation Robust Inference," Econometrica, Econometric Society, vol. 90(6), pages 2901-2935, November.
    8. Timothy Fortune & Magda Peligrad & Hailin Sang, 2021. "A local limit theorem for linear random fields," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 696-710, September.
    9. Otto, Philipp & Sibbertsen, Philipp, 2023. "Spatial autoregressive fractionally integrated moving average model," Hannover Economic Papers (HEP) dp-712, Leibniz Universität Hannover, Wirtschaftswissenschaftliche Fakultät.
    10. Horváth, Lajos & Kokoszka, Piotr & Wang, Shixuan, 2020. "Testing normality of data on a multivariate grid," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    11. Surgailis, Donatas, 2024. "Scaling limits of nonlinear functions of random grain model, with application to Burgers’ equation," Stochastic Processes and their Applications, Elsevier, vol. 174(C).
    12. Ulrich K. Muller & Mark W. Watson, 2021. "Spatial Correlation Robust Inference," Papers 2102.09353, arXiv.org.
    13. Peligrad, Magda & Sang, Hailin & Xiao, Yimin & Yang, Guangyu, 2022. "Limit theorems for linear random fields with innovations in the domain of attraction of a stable law," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 596-621.

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    More about this item

    Keywords

    central limit theorem; edge effects; increasing domain asymptotics; long memory; negative dependence; positive dependence; sampling region; spatial lattice;
    All these keywords.

    JEL classification:

    • J1 - Labor and Demographic Economics - - Demographic Economics

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