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Uniform Consistency of Nonstationary Kernel-Weighted Sample Covariances for Nonparametric Regression

Author

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  • Degui Li
  • Peter C. B. Phillips
  • Jiti Gao

Abstract

We obtain uniform consistency results for kernel-weighted sample covariances in a nonstationary multiple regression framework that allows for both fixed design and random design coefficient variation. In the fixed design case these nonparametric sample covariances have different uniform convergence rates depending on direction, a result that differs fundamentally from the random design and stationary cases. The uniform convergence rates derived are faster than the corresponding rates in the stationary case and confirm the existence of uniform super-consistency. The modelling framework and convergence rates allow for endogeneity and thus broaden the practical econometric import of these results. As a specific application, we establish uniform consistency of nonparametric kernel estimators of the coefficient functions in nonlinear cointegration models with time varying coefficients and provide sharp convergence rates in that case. For the fixed design models, in particular, there are two uniform convergence rates that apply in two different directions, both rates exceeding the usual rate in the stationary case.

Suggested Citation

  • Degui Li & Peter C. B. Phillips & Jiti Gao, 2013. "Uniform Consistency of Nonstationary Kernel-Weighted Sample Covariances for Nonparametric Regression," Monash Econometrics and Business Statistics Working Papers 27/13, Monash University, Department of Econometrics and Business Statistics.
  • Handle: RePEc:msh:ebswps:2013-27
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    References listed on IDEAS

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    1. Peter C. B. Phillips & Bruce E. Hansen, 1990. "Statistical Inference in Instrumental Variables Regression with I(1) Processes," Review of Economic Studies, Oxford University Press, vol. 57(1), pages 99-125.
    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.
    4. Phillips, Peter C.B. & Li, Degui & Gao, Jiti, 2017. "Estimating smooth structural change in cointegration models," Journal of Econometrics, Elsevier, vol. 196(1), pages 180-195.
    5. P. C. B. Phillips & S. N. Durlauf, 1986. "Multiple Time Series Regression with Integrated Processes," Review of Economic Studies, Oxford University Press, vol. 53(4), pages 473-495.
    6. Gao, Jiti & Kanaya, Shin & Li, Degui & Tjøstheim, Dag, 2015. "Uniform Consistency For Nonparametric Estimators In Null Recurrent Time Series," Econometric Theory, Cambridge University Press, vol. 31(5), pages 911-952, October.
    7. 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.
    8. Kunpeng Li & Degui Li & Zhongwen Liang & Cheng Hsiao, 2017. "Estimation of semi-varying coefficient models with nonstationary regressors," Econometric Reviews, Taylor & Francis Journals, vol. 36(1-3), pages 354-369, March.
    9. 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.
    10. Qiying Wang & Peter C. B. Phillips, 2009. "Structural Nonparametric Cointegrating Regression," Econometrica, Econometric Society, vol. 77(6), pages 1901-1948, November.
    11. Wang, Qiying & Xiang Rachel Wang, Ying, 2013. "Nonparametric Cointegrating Regression With Nnh Errors," Econometric Theory, Cambridge University Press, vol. 29(1), pages 1-27, February.
    12. Gao, Jiti & Phillips, Peter C.B., 2013. "Semiparametric estimation in triangular system equations with nonstationarity," Journal of Econometrics, Elsevier, vol. 176(1), pages 59-79.
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    Cited by:

    1. Dong, Chaohua & Linton, Oliver, 2018. "Additive nonparametric models with time variable and both stationary and nonstationary regressors," Journal of Econometrics, Elsevier, vol. 207(1), pages 212-236.
    2. Phillips, Peter C.B. & Li, Degui & Gao, Jiti, 2017. "Estimating smooth structural change in cointegration models," Journal of Econometrics, Elsevier, vol. 196(1), pages 180-195.
    3. Li, Degui & Phillips, Peter C.B. & Gao, Jiti, 2020. "Kernel-based Inference in Time-Varying Coefficient Cointegrating Regression," Journal of Econometrics, Elsevier, vol. 215(2), pages 607-632.
    4. Yayi Yan & Jiti Gao & Bin peng, 2020. "A Class of Time-Varying Vector Moving Average (infinity) Models," Monash Econometrics and Business Statistics Working Papers 39/20, Monash University, Department of Econometrics and Business Statistics.
    5. Dong, Chaohua & Linton, Oliver & Peng, Bin, 2021. "A weighted sieve estimator for nonparametric time series models with nonstationary variables," Journal of Econometrics, Elsevier, vol. 222(2), pages 909-932.

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    More about this item

    Keywords

    and phrases: Cointegration; Functional coefficients; Kernel degeneracy; Nonparametric kernel smoothing; Random coordinate rotation; Super-consistency; Uniform convergence rates; Time varying coefficients.;
    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
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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