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A General Limit Theory for Nonlinear Functionals of Nonstationary Time Series

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Abstract

Limit theory is provided for a wide class of covariance functionals of a nonstationary process and stationary time series. The results are relevant to estimation and inference in nonlinear nonstationary regressions that involve unit root, local unit root or fractional processes and they include both parametric and nonparametric regressions. Self normalized versions of these statistics are considered that are useful in inference. Numerical evidence reveals a strong bimodality in the ?nite sample distributions that persists for very large sample sizes although the limit theory is Gaussian. New self normalized versions are introduced that deliver improved approximations.

Suggested Citation

  • Qiying Wang & Peter C. B. Phillips, 2022. "A General Limit Theory for Nonlinear Functionals of Nonstationary Time Series," Cowles Foundation Discussion Papers 2337, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:2337
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    References listed on IDEAS

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    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.
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    3. Peng, Jiangyan & Wang, Qiying, 2018. "Weak Convergence To Stochastic Integrals Under Primitive Conditions In Nonlinear Econometric Models," Econometric Theory, Cambridge University Press, vol. 34(5), pages 1132-1157, October.
    4. 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.
    5. Qiying Wang & Peter C. B. Phillips, 2009. "Structural Nonparametric Cointegrating Regression," Econometrica, Econometric Society, vol. 77(6), pages 1901-1948, November.
    6. Park, Joon Y. & Phillips, Peter C.B., 1999. "Asymptotics For Nonlinear Transformations Of Integrated Time Series," Econometric Theory, Cambridge University Press, vol. 15(3), pages 269-298, June.
    7. Qiying Wang & Yan-Xia Lin & Chandra M. Gulati, 2003. "Strong Approximation for Long Memory Processes with Applications," Journal of Theoretical Probability, Springer, vol. 16(2), pages 377-389, April.
    8. Carlo V. Fiorio & Vassilis A. Hajivassiliou & Peter C. B. Phillips, 2010. "Bimodal t-ratios: the impact of thick tails on inference," Econometrics Journal, Royal Economic Society, vol. 13(2), pages 271-289, July.
    9. Chan, Nigel & Wang, Qiying, 2015. "Nonlinear regressions with nonstationary time series," Journal of Econometrics, Elsevier, vol. 185(1), pages 182-195.
    10. P. Jeganathan, 2008. "Limit Theorems for Functionals of Sums that Converge to Fractional Brownian and Stable Motions," Cowles Foundation Discussion Papers 1649, Cowles Foundation for Research in Economics, Yale University.
    11. Phillips, Peter C.B. & Wang, Ying, 2023. "When bias contributes to variance: True limit theory in functional coefficient cointegrating regression," Journal of Econometrics, Elsevier, vol. 232(2), pages 469-489.
    12. Duffy, James A., 2020. "Asymptotic Theory For Kernel Estimators Under Moderate Deviations From A Unit Root, With An Application To The Asymptotic Size Of Nonparametric Tests," Econometric Theory, Cambridge University Press, vol. 36(4), pages 559-582, August.
    13. Park, Joon Y., 2014. "Nonstationary Nonlinearity: A Survey On Peter Phillips’S Contributions With A New Perspective," Econometric Theory, Cambridge University Press, vol. 30(4), pages 894-922, August.
    14. Wang, Qiying & Phillips, Peter C.B. & Kasparis, Ioannis, 2021. "Latent Variable Nonparametric Cointegrating Regression," Econometric Theory, Cambridge University Press, vol. 37(1), pages 138-168, February.
    15. Wang, Qiying & Phillips, Peter C.B., 2011. "Asymptotic Theory For Zero Energy Functionals With Nonparametric Regression Applications," Econometric Theory, Cambridge University Press, vol. 27(2), pages 235-259, April.
    16. 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.
    17. 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.
    18. Park, Joon Y & Phillips, Peter C B, 2001. "Nonlinear Regressions with Integrated Time Series," Econometrica, Econometric Society, vol. 69(1), pages 117-161, January.
    19. Hu, Zhishui & Phillips, Peter C.B. & Wang, Qiying, 2021. "Nonlinear Cointegrating Power Function Regression With Endogeneity," Econometric Theory, Cambridge University Press, vol. 37(6), pages 1173-1213, December.
    20. Wang, Qiying & Phillips, Peter C. B., 2016. "Nonparametric Cointegrating Regression With Endogeneity And Long Memory," Econometric Theory, Cambridge University Press, vol. 32(2), pages 359-401, April.
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    Cited by:

    1. Giraitis, Liudas & Li, Yufei & Phillips, Peter C.B., 2024. "Robust inference on correlation under general heterogeneity," Journal of Econometrics, Elsevier, vol. 240(1).
    2. Qiying Wang & Peter C. B. Phillips & Ying Wang, 2023. "New asymptotics applied to functional coefficient regression and climate sensitivity analysis," Cowles Foundation Discussion Papers 2365, Cowles Foundation for Research in Economics, Yale University.

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    More about this item

    Keywords

    Endogeneity; Limit theory; Local time; Nonlinear functional; Nonstationarity; Sample covariance; Zero energy;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Spatio-temporal Models

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