IDEAS home Printed from https://ideas.repec.org/p/ehl/lserod/58160.html
   My bibliography  Save this paper

Loch linear fitting under near epoch dependence: uniform consistency with convergence rate

Author

Listed:
  • Li, Degui
  • Lu, Zudi
  • Linton, Oliver

Abstract

Local linear fitting is a popular nonparametric method in nonlinear statistical and econometric modelling. Lu and Linton (2007) established the point wise asymptotic distribution (central limit theorem) for the local linear estimator of nonparametric regression function under the condition of near epoch dependence. We further investigate the uniform consistency of this estimator. The uniformly strong and weak consistencies with convergence rates for the local linear fitting are established under mild conditions. Furthermore, general results of uniform convergence rates for nonparametric kernel-based estimators are provided. Applications of our results to conditional variance function estimation and some economic time series models are also discussed. The results of this paper will be of widely potential interest in time series semiparametric modelling.

Suggested Citation

  • Li, Degui & Lu, Zudi & Linton, Oliver, 2010. "Loch linear fitting under near epoch dependence: uniform consistency with convergence rate," LSE Research Online Documents on Economics 58160, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:58160
    as

    Download full text from publisher

    File URL: http://eprints.lse.ac.uk/58160/
    File Function: Open access version.
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Drost, Feike C & Nijman, Theo E, 1993. "Temporal Aggregation of GARCH Processes," Econometrica, Econometric Society, vol. 61(4), pages 909-927, July.
    2. O. Linton & E. Mammen, 2005. "Estimating Semiparametric ARCH(∞) Models by Kernel Smoothing Methods," Econometrica, Econometric Society, vol. 73(3), pages 771-836, May.
    3. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
    4. Kristensen, Dennis, 2009. "Uniform Convergence Rates Of Kernel Estimators With Heterogeneous Dependent Data," Econometric Theory, Cambridge University Press, vol. 25(5), pages 1433-1445, October.
    5. Pagan, Adrian R. & Schwert, G. William, 1990. "Alternative models for conditional stock volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 267-290.
    6. Xiaohong Chen & Oliver Linton & Ingrid Van Keilegom, 2003. "Estimation of Semiparametric Models when the Criterion Function Is Not Smooth," Econometrica, Econometric Society, vol. 71(5), pages 1591-1608, September.
    7. Lee, Sang-Won & Hansen, Bruce E., 1994. "Asymptotic Theory for the Garch(1,1) Quasi-Maximum Likelihood Estimator," Econometric Theory, Cambridge University Press, vol. 10(1), pages 29-52, March.
    8. Andrews, Donald W.K., 1995. "Nonparametric Kernel Estimation for Semiparametric Models," Econometric Theory, Cambridge University Press, vol. 11(3), pages 560-586, June.
    9. Lu, Zudi, 1996. "A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model," Statistics & Probability Letters, Elsevier, vol. 30(4), pages 305-311, November.
    10. Fan, Jianqing & Yao, Qiwei, 1998. "Efficient estimation of conditional variance functions in stochastic regression," LSE Research Online Documents on Economics 6635, London School of Economics and Political Science, LSE Library.
    11. Barnett,William A. & Powell,James & Tauchen,George E. (ed.), 1991. "Nonparametric and Semiparametric Methods in Econometrics and Statistics," Cambridge Books, Cambridge University Press, number 9780521370905, September.
    12. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    13. Hansen, Bruce E., 2008. "Uniform Convergence Rates For Kernel Estimation With Dependent Data," Econometric Theory, Cambridge University Press, vol. 24(3), pages 726-748, June.
    14. Zudi Lu, 2001. "Asymptotic Normality of Kernel Density Estimators under Dependence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 447-468, September.
    15. Hansen, Bruce E., 1991. "GARCH(1, 1) processes are near epoch dependent," Economics Letters, Elsevier, vol. 36(2), pages 181-186, June.
    16. Barnett,William A. & Powell,James & Tauchen,George E. (ed.), 1991. "Nonparametric and Semiparametric Methods in Econometrics and Statistics," Cambridge Books, Cambridge University Press, number 9780521424318, September.
    17. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, volume 1, number 8355.
    18. Nze, Patrick Ango & Doukhan, Paul, 2004. "Weak Dependence: Models And Applications To Econometrics," Econometric Theory, Cambridge University Press, vol. 20(6), pages 995-1045, December.
    19. Yongmiao Hong, 2000. "Generalized spectral tests for serial dependence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(3), pages 557-574.
    20. Linton, Oliver & Sancetta, Alessio, 2009. "Consistent estimation of a general nonparametric regression function in time series," Journal of Econometrics, Elsevier, vol. 152(1), pages 70-78, September.
    21. Yu, K. & Jones, M.C., 2004. "Likelihood-Based Local Linear Estimation of the Conditional Variance Function," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 139-144, January.
    22. Elias Masry, 1996. "Multivariate Local Polynomial Regression For Time Series:Uniform Strong Consistency And Rates," Journal of Time Series Analysis, Wiley Blackwell, vol. 17(6), pages 571-599, November.
    23. Carrasco, Marine & Chen, Xiaohong, 2002. "Mixing And Moment Properties Of Various Garch And Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 18(1), pages 17-39, February.
    24. Lu, Zudi & Linton, Oliver, 2007. "Local Linear Fitting Under Near Epoch Dependence," Econometric Theory, Cambridge University Press, vol. 23(1), pages 37-70, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Li, Degui & Lu, Zudi & Linton, Oliver, 2012. "Local Linear Fitting Under Near Epoch Dependence: Uniform Consistency With Convergence Rates," Econometric Theory, Cambridge University Press, vol. 28(5), pages 935-958, October.
    2. Linton, Oliver & Mammen, Enno, 2003. "Estimating semiparametric ARCH (8) models by kernel smoothing methods," LSE Research Online Documents on Economics 2187, London School of Economics and Political Science, LSE Library.
    3. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, volume 1, number 8355.
    4. O. Linton & E. Mammen, 2005. "Estimating Semiparametric ARCH(∞) Models by Kernel Smoothing Methods," Econometrica, Econometric Society, vol. 73(3), pages 771-836, May.
    5. Xu, Ke-Li & Phillips, Peter C. B., 2011. "Tilted Nonparametric Estimation of Volatility Functions With Empirical Applications," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(4), pages 518-528.
    6. Peter C.B. Phillips & Ke-Li Xu, 2007. "Tilted Nonparametric Estimation of Volatility Functions," Cowles Foundation Discussion Papers 1612, Cowles Foundation for Research in Economics, Yale University, revised Jul 2010.
    7. Degui Li & Oliver Linton & Zudi Lu, 2012. "A Flexible Semiparametric Model for Time Series," Monash Econometrics and Business Statistics Working Papers 17/12, Monash University, Department of Econometrics and Business Statistics.
    8. Li, Degui & Linton, Oliver & Lu, Zudi, 2015. "A flexible semiparametric forecasting model for time series," Journal of Econometrics, Elsevier, vol. 187(1), pages 345-357.
    9. Pagan, Adrian, 1996. "The econometrics of financial markets," Journal of Empirical Finance, Elsevier, vol. 3(1), pages 15-102, May.
    10. Lu, Zudi & Linton, Oliver, 2007. "Local Linear Fitting Under Near Epoch Dependence," Econometric Theory, Cambridge University Press, vol. 23(1), pages 37-70, February.
    11. Sadikoglu, Serhan, 2019. "Essays in econometric theory," Other publications TiSEM 99d83644-f9dc-49e3-a4e1-5, Tilburg University, School of Economics and Management.
    12. repec:awi:wpaper:0473 is not listed on IDEAS
    13. Peter Malec, 2016. "A Semiparametric Intraday GARCH Model," Cambridge Working Papers in Economics 1633, Faculty of Economics, University of Cambridge.
    14. Kristensen, Dennis & Shin, Yongseok, 2012. "Estimation of dynamic models with nonparametric simulated maximum likelihood," Journal of Econometrics, Elsevier, vol. 167(1), pages 76-94.
    15. Juan Carlos Escanciano, 2020. "Uniform Rates for Kernel Estimators of Weakly Dependent Data," Papers 2005.09951, arXiv.org.
    16. Bu, Ruijun & Kim, Jihyun & Wang, Bin, 2023. "Uniform and Lp convergences for nonparametric continuous time regressions with semiparametric applications," Journal of Econometrics, Elsevier, vol. 235(2), pages 1934-1954.
    17. Degiannakis, Stavros & Xekalaki, Evdokia, 2004. "Autoregressive Conditional Heteroskedasticity (ARCH) Models: A Review," MPRA Paper 80487, University Library of Munich, Germany.
    18. Kanaya, Shin, 2017. "Uniform Convergence Rates Of Kernel-Based Nonparametric Estimators For Continuous Time Diffusion Processes: A Damping Function Approach," Econometric Theory, Cambridge University Press, vol. 33(4), pages 874-914, August.
    19. Escanciano, Juan Carlos & Jacho-Chávez, David T. & Lewbel, Arthur, 2014. "Uniform convergence of weighted sums of non and semiparametric residuals for estimation and testing," Journal of Econometrics, Elsevier, vol. 178(P3), pages 426-443.
    20. Ke Zhu, 2018. "Statistical inference for autoregressive models under heteroscedasticity of unknown form," Papers 1804.02348, arXiv.org, revised Aug 2018.
    21. Bollerslev, Tim & Engle, Robert F. & Nelson, Daniel B., 1986. "Arch models," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 49, pages 2959-3038, Elsevier.

    More about this item

    Keywords

    α − mixing; local linear fitting; near epoch dependence; convergence rates; uniform consistency;
    All these keywords.

    JEL classification:

    • J1 - Labor and Demographic Economics - - Demographic Economics

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:ehl:lserod:58160. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: LSERO Manager (email available below). General contact details of provider: https://edirc.repec.org/data/lsepsuk.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.