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The Local Fractional Bootstrap

Author

Listed:
  • Mikkel Bennedsen

    (Aarhus University and CREATES)

  • Ulrich Hounyo

    (Aarhus University and CREATES)

  • Asger Lunde

    (Aarhus University and CREATES)

  • Mikko S. Pakkanen

    (Imperial College London and CREATES)

Abstract

We introduce a bootstrap procedure for high-frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first order validity of the bootstrap method and in simulations we observe that the bootstrap-based hypothesis test provides considerable finite-sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data; we illustrate this by applying the bootstrap method to two empirical data sets: we assess the roughness of a time series of high-frequency asset prices and we test the validity of Kolmogorov's scaling law in atmospheric turbulence data.

Suggested Citation

  • Mikkel Bennedsen & Ulrich Hounyo & Asger Lunde & Mikko S. Pakkanen, 2016. "The Local Fractional Bootstrap," CREATES Research Papers 2016-15, Department of Economics and Business Economics, Aarhus University.
    • Handle: RePEc:aah:create:2016-15
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    References listed on IDEAS

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    1. Xin Huang & George Tauchen, 2005. "The Relative Contribution of Jumps to Total Price Variance," Journal of Financial Econometrics, Oxford University Press, vol. 3(4), pages 456-499.
    2. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2015. "Hybrid scheme for Brownian semistationary processes," CREATES Research Papers 2015-43, Department of Economics and Business Economics, Aarhus University.
    3. Ole E. Barndorff-Nielsen & Peter Reinhard Hansen & Asger Lunde & Neil Shephard, 2008. "Designing Realized Kernels to Measure the ex post Variation of Equity Prices in the Presence of Noise," Econometrica, Econometric Society, vol. 76(6), pages 1481-1536, November.
    4. Davidson, Russell & MacKinnon, James G., 1999. "The Size Distortion Of Bootstrap Tests," Econometric Theory, Cambridge University Press, vol. 15(3), pages 361-376, June.
    5. Mikkel Bennedsen, 2015. "Rough electricity: a new fractal multi-factor model of electricity spot prices," CREATES Research Papers 2015-42, Department of Economics and Business Economics, Aarhus University.
    6. Barndorff-Nielsen, Ole E. & Corcuera, José Manuel & Podolskij, Mark, 2009. "Power variation for Gaussian processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1845-1865, June.
    7. Ole E. Barndorff-Nielsen & Fred Espen Benth & Almut E. D. Veraart, 2013. "Modelling energy spot prices by volatility modulated L\'{e}vy-driven Volterra processes," Papers 1307.6332, arXiv.org.
    8. Nourdin, Ivan & Peccati, Giovanni & Podolskij, Mark, 2011. "Quantitative Breuer-Major theorems," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 793-812, April.
    9. Ole E. Barndorff-Nielsen, 2016. "Assessing Gamma kernels and BSS/LSS processes," CREATES Research Papers 2016-09, Department of Economics and Business Economics, Aarhus University.
    10. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    11. Sílvia Gonçalves & Nour Meddahi, 2009. "Bootstrapping Realized Volatility," Econometrica, Econometric Society, vol. 77(1), pages 283-306, January.
    12. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2014. "Volatility is rough," Papers 1410.3394, arXiv.org.
    13. Gabriel Lang & François Roueff, 2001. "Semi-parametric Estimation of the Hölder Exponent of a Stationary Gaussian Process with Minimax Rates," Statistical Inference for Stochastic Processes, Springer, vol. 4(3), pages 283-306, October.
    14. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2015. "Hybrid scheme for Brownian semistationary processes," Papers 1507.03004, arXiv.org, revised May 2017.
    15. Corcuera, José Manuel & Hedevang, Emil & Pakkanen, Mikko S. & Podolskij, Mark, 2013. "Asymptotic theory for Brownian semi-stationary processes with application to turbulence," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2552-2574.
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    Citations

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    Cited by:

    1. Mikkel Bennedsen, 2016. "Semiparametric inference on the fractal index of Gaussian and conditionally Gaussian time series data," Papers 1608.01895, arXiv.org, revised Mar 2018.
    2. Mikkel Bennedsen, 2016. "Semiparametric inference on the fractal index of Gaussian and conditionally Gaussian time series data," CREATES Research Papers 2016-21, Department of Economics and Business Economics, Aarhus University.
    3. Mikkel Bennedsen & Ulrich Hounyo & Asger Lunde & Mikko S. Pakkanen, 2016. "The Local Fractional Bootstrap," Papers 1605.00868, arXiv.org, revised Oct 2017.

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    More about this item

    Keywords

    Brownian semistationary process; roughness; fractal index; Hölder regularity; fractional Brownian motion; bootstrap; stochastic volatility; turbulence JEL Classification: C12; C22; C63; G12 MSC 2010 Classification: 60G10; 60G15; 60G17; 60G22; 62M07; 62M09; 65C05;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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