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The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing

Author

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  • Kim Christensen

    (Aarhus University and CREATES)

  • Martin Thyrsgaard

    (Aarhus University and CREATES)

  • Bezirgen Veliyev

    (Aarhus University and CREATES)

Abstract

We propose a nonparametric estimator of the empirical distribution function (EDF) of the latent spot variance of the log-price of a financial asset. We show that over a fixed time span our realized EDF (or REDF)-inferred from noisy high-frequency data-is consistent as the mesh of the observation grid goes to zero. In a double symptotic framework, with time also increasing to infinity, the REDF converges to the cumulative distribution function of volatility, if it exists. We exploit these results to construct some new goodness-of-fit tests for stochastic volatility models. In a Monte Carlo study, the REDF is found to be accurate over the entire support of volatility. This leads to goodness-of-fit tests that are both correctly sized and relatively powerful against common alternatives. In an empirical application, we recover the REDF from stock market high-frequency data. We inspect the goodness-of-fit of several two-parameter marginal distributions that are inherent in standard stochastic volatility models. The inverse Gaussian offers the best overall description of random equity variation, but the fit is less than perfect. This suggests an extra parameter (as available in, e.g., the generalized inverse Gaussian) is required to model stochastic variance.

Suggested Citation

  • Kim Christensen & Martin Thyrsgaard & Bezirgen Veliyev, 2018. "The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing," CREATES Research Papers 2018-19, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2018-19
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    2. Sigurd Emil Rømer & Rolf Poulsen, 2020. "How Does the Volatility of Volatility Depend on Volatility?," Risks, MDPI, vol. 8(2), pages 1-18, June.
    3. Kim Christensen & Mathias Siggaard & Bezirgen Veliyev, 2023. "A Machine Learning Approach to Volatility Forecasting," Journal of Financial Econometrics, Oxford University Press, vol. 21(5), pages 1680-1727.
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    5. Yinfen Tang & Tao Su & Zhiyuan Zhang, 2022. "Distribution-free specification test for volatility function based on high-frequency data with microstructure noise," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(8), pages 977-1022, November.

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    More about this item

    Keywords

    Empirical processes; goodness-of-fit; high-frequency data; microstructure noise; pre-averaging; realized variance; stochastic volatility;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General

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