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Asymptotics for recurrent diffusions with application to high frequency regression

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  • Kim, Jihyun
  • Park, Joon Y.

Abstract

We provide the asymptotic theory for functionals of recurrent diffusions. Our asymptotics are completely general and applicable for all cases, including positive and null recurrent diffusions, and diffusions with and without the integrability conditions. They are established directly from the representation of diffusion as time-changed Brownian motion. Our approach provides a unified framework, and combines all existing theories of diffusion asymptotics with new results that appear to be particularly relevant in many practical applications. For an illustration of our asymptotics, we employ them to analyze a class of high frequency regressions that is commonly used in empirical economics and finance.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:econom:v:196:y:2017:i:1:p:37-54
    DOI: 10.1016/j.jeconom.2015.12.019
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    References listed on IDEAS

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    1. Tang, Cheng Yong & Chen, Song Xi, 2009. "Parameter estimation and bias correction for diffusion processes," Journal of Econometrics, Elsevier, vol. 149(1), pages 65-81, April.
    2. Qiying Wang & Peter C. B. Phillips, 2009. "Structural Nonparametric Cointegrating Regression," Econometrica, Econometric Society, vol. 77(6), pages 1901-1948, November.
    3. 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.
    4. Davis, Richard A., 1982. "Maximum and minimum of one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 13(1), pages 1-9, July.
    5. Federico M. Bandi & Peter C. B. Phillips, 2003. "Fully Nonparametric Estimation of Scalar Diffusion Models," Econometrica, Econometric Society, vol. 71(1), pages 241-283, January.
    6. Harry Van Zanten, 2003. "On Uniform Laws of Large Numbers for Ergodic Diffusions and Consistency of Estimators," Statistical Inference for Stochastic Processes, Springer, vol. 6(2), pages 199-213, May.
    7. Aït-Sahalia, Yacine & Park, Joon Y., 2012. "Stationarity-based specification tests for diffusions when the process is nonstationary," Journal of Econometrics, Elsevier, vol. 169(2), pages 279-292.
    8. Park, Joon Y & Phillips, Peter C B, 2001. "Nonlinear Regressions with Integrated Time Series," Econometrica, Econometric Society, vol. 69(1), pages 117-161, January.
    9. Myklebust, Terje & Karlsen, Hans Arnfinn & Tjøstheim, Dag, 2012. "Null Recurrent Unit Root Processes," Econometric Theory, Cambridge University Press, vol. 28(1), pages 1-41, February.
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    Citations

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    Cited by:

    1. Chang, Yoosoon & Lu, Ye & Park, Joon Y., 2018. "Understanding Regressions with Observations Collected at High Frequency over Long Span," Working Papers 2018-10, University of Sydney, School of Economics.
    2. Jiang, Bibo & Lu, Ye & Park, Joon Y., 2020. "Testing for Stationarity at High Frequency," Journal of Econometrics, Elsevier, vol. 215(2), pages 341-374.
    3. Wang, Bin & Zheng, Xu, 2022. "Testing for the presence of jump components in jump diffusion models," Journal of Econometrics, Elsevier, vol. 230(2), pages 483-509.
    4. Park, Joon Y. & Wang, Bin, 2021. "Nonparametric estimation of jump diffusion models," Journal of Econometrics, Elsevier, vol. 222(1), pages 688-715.
    5. Jin Seo Cho & Peter C. B. Phillips & Juwon Seo, 2019. "Parametric Inference on the Mean of Functional Data Applied to Lifetime Income Curves," Working papers 2019rwp-153, Yonsei University, Yonsei Economics Research Institute.
    6. Jin Seo Cho & Peter C. B. Phillips & Juwon Seo, 2022. "Parametric Conditional Mean Inference With Functional Data Applied To Lifetime Income Curves," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 63(1), pages 391-456, February.
    7. Kim, Jihyun & Meddahi, Nour, 2020. "Volatility regressions with fat tails," Journal of Econometrics, Elsevier, vol. 218(2), pages 690-713.
    8. Pellatt, Daniel F. & Sun, Yixiao, 2023. "Asymptotic F test in regressions with observations collected at high frequency over long span," Journal of Econometrics, Elsevier, vol. 235(2), pages 1281-1309.
    9. Jeong, Minsoo, 2022. "Modelling persistent stationary processes in continuous time," Economic Modelling, Elsevier, vol. 109(C).
    10. 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.
    11. Kim, Jihyun & Park, Joon & Wang, Bin, 2020. "Estimation of Volatility Functions in Jump Diffusions Using Truncated Bipower Increments," TSE Working Papers 20-1096, Toulouse School of Economics (TSE).
    12. Kim, Jihyun & Meddahi, Nour, 2020. "Volatility Regressions with Fat Tails," TSE Working Papers 20-1097, Toulouse School of Economics (TSE).
    13. Jihyun Kim & Nour Meddahi, 2020. "Volatility Regressions with Fat Tails," Post-Print hal-03142647, HAL.
    14. Jeong, Minsoo, 2018. "Consistent estimator of nonparametric structural spurious regression model for high frequency data," Economics Letters, Elsevier, vol. 162(C), pages 18-21.
    15. 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.

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    More about this item

    Keywords

    Diffusion; Positive and null recurrences; Asymptotics; Limit distribution; Continuous time regression;
    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
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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