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Uniform Consistency for Nonparametric Estimators in Null Recurrent Time Series

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  • Jiti Gao
  • Degui Li
  • Dag Tjøstheim

Abstract

This paper establishes a suite of uniform consistency results for nonparametric kernel density and regression estimators when the time series regressors concerned are nonstationary null-recurrent Markov chains. Under suitable conditions, certain rates of convergence are also obtained for the proposed estimators. Our results can be viewed as an extension of some well-known uniform consistency results for the stationary time series case to the nonstationary time series case.

Suggested Citation

  • Jiti Gao & Degui Li & Dag Tjøstheim, 2011. "Uniform Consistency for Nonparametric Estimators in Null Recurrent Time Series," Monash Econometrics and Business Statistics Working Papers 13/11, Monash University, Department of Econometrics and Business Statistics.
    • Handle: RePEc:msh:ebswps:2011-13
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    References listed on IDEAS

    as
    1. Jia Chen & Jiti Gao & Degui Li, 2009. "Semiparametric Regression Estimation in Null Recurrent Nonlinear Time Series," School of Economics and Public Policy Working Papers 2009-02, University of Adelaide, School of Economics and Public Policy.
    2. 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.
    3. Chen, Jia & Li, Degui & Zhang, Lixin, 2010. "Robust estimation in a nonlinear cointegration model," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 706-717, March.
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    5. Bandi, Federico M. & Moloche, Guillermo, 2018. "On The Functional Estimation Of Multivariate Diffusion Processes," Econometric Theory, Cambridge University Press, vol. 34(4), pages 896-946, August.
    6. Wang, Qiying & Phillips, Peter C.B., 2009. "Asymptotic Theory For Local Time Density Estimation And Nonparametric Cointegrating Regression," Econometric Theory, Cambridge University Press, vol. 25(3), pages 710-738, June.
    7. Peter C.B. Phillips & Joon Y. Park, 1998. "Nonstationary Density Estimation and Kernel Autoregression," Cowles Foundation Discussion Papers 1181, Cowles Foundation for Research in Economics, Yale University.
    8. Gao, Jiti & King, Maxwell & Lu, Zudi & Tjøstheim, Dag, 2009. "Nonparametric Specification Testing For Nonlinear Time Series With Nonstationarity," Econometric Theory, Cambridge University Press, vol. 25(6), pages 1869-1892, December.
    9. Cai, Zongwu & Li, Qi & Park, Joon Y., 2009. "Functional-coefficient models for nonstationary time series data," Journal of Econometrics, Elsevier, vol. 148(2), pages 101-113, February.
    10. 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.
    11. Roussas, George G., 1990. "Nonparametric regression estimation under mixing conditions," Stochastic Processes and their Applications, Elsevier, vol. 36(1), pages 107-116, October.
    12. Gao, Jiti, 2007. "Nonlinear time series: semiparametric and nonparametric methods," MPRA Paper 39563, University Library of Munich, Germany, revised 01 Sep 2007.
    13. Gao, Jiti & Tjøstheim, Dag & Yin, Jiying, 2013. "Estimation in threshold autoregressive models with a stationary and a unit root regime," Journal of Econometrics, Elsevier, vol. 172(1), pages 1-13.
    14. Qiying Wang & Peter C. B. Phillips, 2009. "Structural Nonparametric Cointegrating Regression," Econometrica, Econometric Society, vol. 77(6), pages 1901-1948, November.
    15. 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.
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    Citations

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    Cited by:

    1. Kanaya, Shin & Kristensen, Dennis, 2016. "Estimation Of Stochastic Volatility Models By Nonparametric Filtering," Econometric Theory, Cambridge University Press, vol. 32(4), pages 861-916, August.
    2. Federico M Bandi & Valentina Corradi & Daniel Wilhelm, 2016. "Possibly Nonstationary Cross-Validation," CeMMAP working papers CWP11/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Yuliana Linke & Igor Borisov & Pavel Ruzankin & Vladimir Kutsenko & Elena Yarovaya & Svetlana Shalnova, 2024. "Multivariate Universal Local Linear Kernel Estimators in Nonparametric Regression: Uniform Consistency," Mathematics, MDPI, vol. 12(12), pages 1-23, June.
    4. Bravo, Francesco & Li, Degui & Tjøstheim, Dag, 2021. "Robust nonlinear regression estimation in null recurrent time series," Journal of Econometrics, Elsevier, vol. 224(2), pages 416-438.
    5. Kim, Jihyun & Park, Joon Y., 2017. "Asymptotics for recurrent diffusions with application to high frequency regression," Journal of Econometrics, Elsevier, vol. 196(1), pages 37-54.
    6. 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.
    7. Li, Degui & Li, Runze, 2016. "Local composite quantile regression smoothing for Harris recurrent Markov processes," Journal of Econometrics, Elsevier, vol. 194(1), pages 44-56.
    8. Ruijun Bu & Jihyun Kim & Bin Wang, 2020. "Uniform and Lp Convergences of Nonparametric Estimation for Diffusion Models," Working Papers 202021, University of Liverpool, Department of Economics.
    9. Li, Degui & Phillips, Peter C. B. & Gao, Jiti, 2016. "Uniform Consistency Of Nonstationary Kernel-Weighted Sample Covariances For Nonparametric Regression," Econometric Theory, Cambridge University Press, vol. 32(3), pages 655-685, June.
    10. Degui Li & Dag Tjøstheim & Jiti Gao, 2012. "Nonlinear Regression with Harris Recurrent Markov Chains," Monash Econometrics and Business Statistics Working Papers 14/12, Monash University, Department of Econometrics and Business Statistics.
    11. Biqing Cai & Chaohua Dong & Jiti Gao, 2015. "Orthogonal Series Estimation in Nonlinear Cointegrating Models with Endogeneity," Monash Econometrics and Business Statistics Working Papers 18/15, Monash University, Department of Econometrics and Business Statistics.
    12. 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.
    13. Yayi Yan & Jiti Gao & Bin Peng, 2021. "Asymptotics for Time-Varying Vector MA(∞) Processes," Monash Econometrics and Business Statistics Working Papers 22/21, Monash University, Department of Econometrics and Business Statistics.
    14. James A. Duffy, 2015. "Uniform Convergence Rates over Maximal Domains in Structural Nonparametric Cointegrating Regression," Economics Papers 2015-W03, Economics Group, Nuffield College, University of Oxford.

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    More about this item

    Keywords

    β-null recurrent Markov chain; nonparametric estimation; rate of convergence; uniform consistency;
    All these keywords.

    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
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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