The Annals of Statistics

Nonparametric change-point analysis of volatility

Markus Bibinger, Moritz Jirak, and Mathias Vetter

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In this work, we develop change-point methods for statistics of high-frequency data. The main interest is in the volatility of an Itô semimartingale, the latter being discretely observed over a fixed time horizon. We construct a minimax-optimal test to discriminate continuous paths from paths with volatility jumps, and it is shown that the test can be embedded into a more general theory to infer the smoothness of volatilities. In a high-frequency setting, we prove weak convergence of the test statistic under the hypothesis to an extreme value distribution. Moreover, we develop methods to infer changes in the Hurst parameters of fractional volatility processes. A simulation study is conducted to demonstrate the performance of our methods in finite-sample applications.

Article information

Ann. Statist., Volume 45, Number 4 (2017), 1542-1578.

Received: February 2016
Revised: June 2016
First available in Project Euclid: 28 June 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62G10: Hypothesis testing

High-frequency data nonparametric change-point test minimax-optimal test stochastic volatility volatility jumps


Bibinger, Markus; Jirak, Moritz; Vetter, Mathias. Nonparametric change-point analysis of volatility. Ann. Statist. 45 (2017), no. 4, 1542--1578. doi:10.1214/16-AOS1499.

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  • Aït-Sahalia, Y. and Jacod, J. (2009). Testing for jumps in a discretely observed process. Ann. Statist. 37 184–222.
  • Alvarez, A., Panloup, F., Pontier, M. and Savy, N. (2012). Estimation of the instantaneous volatility. Stat. Inference Stoch. Process. 15 27–59.
  • Andersen, T. G. and Bollerslev, T. (1998). Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. Internat. Econom. Rev. 39 885–905.
  • Andrews, D. W. K. (1993). Tests for parameter instability and structural change with unknown change point. Econometrica 61 821–856.
  • Arcones, M. A. (1994). Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 2242–2274.
  • Aue, A., Hörmann, S., Horváth, L. and Reimherr, M. (2009). Break detection in the covariance structure of multivariate time series models. Ann. Statist. 37 4046–4087.
  • Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66 47–78.
  • Barndorff-Nielsen, O. E. and Shephard, N. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 64 253–280.
  • Bibinger, M., Jirak, M. and Vetter, M. (2016a). Supplement to “Nonparametric change-point analysis of volatility.” DOI:10.1214/16-AOS1499SUPPA.
  • Bibinger, M., Jirak, M. and Vetter, M. (2016b). Supplement to “Nonparametric change-point analysis of volatility.” DOI:10.1214/16-AOS1499SUPPB.
  • Bobkov, S. G., Chistyakov, G. P. and Götze, F. (2013). Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem. Ann. Probab. 41 2479–2512.
  • Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384–2398.
  • Coeurjolly, J.-F. (2000). Simulation and identification of the fractional Brownian motion: A bibliographical and comparative study. J. Stat. Softw. 50 1–53.
  • Comte, F. and Renault, E. (1998). Long memory in continuous-time stochastic volatility models. Math. Finance 8 291–323.
  • Csörgő, M. and Horváth, L. (1997). Limit Theorems in Change-Point Analysis. Wiley, Chichester.
  • Embrechts, P. and Maejima, M. (2000). An introduction to the theory of self-similar stochastic processes. In Proceedings of the Summer School on Mathematical Physics 1999: Methods of Renormalization Group (Tokyo) 14 1399–1420. Internat. J. Modern Phys. B, 12-13.
  • Gatheral, J., Jaisson, T. and Rosenbaum, M. (2014). Volatility is rough. Preprint. Available at arXiv:1410.3394.
  • Hall, P. (1991). On convergence rates of suprema. Probab. Theory Related Fields 89 447–455.
  • Hinkley, D. V. (1971). Inference about the change-point from cumulative sum tests. Biometrika 58 509–523.
  • Hoffmann, M. and Nickl, R. (2011). On adaptive inference and confidence bands. Ann. Statist. 39 2383–2409.
  • Iacus, S. M. and Yoshida, N. (2012). Estimation for the change point of volatility in a stochastic differential equation. Stochastic Process. Appl. 122 1068–1092.
  • Ingster, Y. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I. Math. Methods Statist. 2 85–114.
  • Ingster, Y. I. and Suslina, I. A. (2003). Nonparametric Goodness-of-Fit Testing Under Gaussian Models. Lecture Notes in Statistics 169. Springer, New York.
  • Jacod, J. (1997). On continuous conditional Gaussian martingales and stable convergence in law. In Séminaire de Probabilités, XXXI. Lecture Notes in Math. 1655 232–246. Springer, Berlin.
  • Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517–559.
  • Jacod, J. and Rosenbaum, M. (2013). Quarticity and other functionals of volatility: Efficient estimation. Ann. Statist. 41 1462–1484.
  • Jacod, J. and Todorov, V. (2010). Do price and volatility jump together? Ann. Appl. Probab. 20 1425–1469.
  • Mancini, C. (2009). Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps. Scand. J. Stat. 36 270–296.
  • Marsaglia, G., Tsang, Wan, W. and Wang, J. (2003). Evaluating Kolmogorov’s distribution. J. Stat. Softw. 8 1–4.
  • Móricz, F. A., Serfling, R. J. and Stout, W. F. (1982). Moment and probability bounds with quasisuperadditive structure for the maximum partial sum. Ann. Probab. 10 1032–1040.
  • Müller, H.-G. (1992). Change-points in nonparametric regression analysis. Ann. Statist. 20 737–761.
  • Müller, H.-G. and Stadtmüller, U. (1999). Discontinuous versus smooth regression. Ann. Statist. 27 299–337.
  • Mykland, P. A. (2012). A Gaussian calculus for inference from high frequency data. Ann. Finance 8 235–258.
  • Mykland, P. A. and Zhang, L. (2009). Inference for continuous semimartingales observed at high frequency. Econometrica 77 1403–1445.
  • Page, E. S. (1955). A test for a change in a parameter occurring at an unknown point. Biometrika 42 523–527.
  • Pettitt, A. N. (1980). A simple cumulative sum type statistic for the change-point problem with zero-one observations. Biometrika 67 79–84.
  • Phillips, P. C. B. (1987). Time series regression with a unit root. Econometrica 55 277–301.
  • Spokoiny, V. G. (1998). Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice. Ann. Statist. 26 1356–1378.
  • Spokoiny, V. (2009). Multiscale local change point detection with applications to value-at-risk. Ann. Statist. 37 1405–1436.
  • Todorov, V. and Tauchen, G. (2011). Volatility jumps. J. Bus. Econom. Statist. 29 356–371.
  • Wu, W. B. (2007). Strong invariance principles for dependent random variables. Ann. Probab. 35 2294–2320.
  • Wu, W. B. and Zhao, Z. (2007). Inference of trends in time series. J. R. Stat. Soc. Ser. B. Stat. Methodol. 69 391–410.

Supplemental materials

  • Complete proofs. We provide all remaining proofs for the results from Sections 3, 4 and 5.
  • Application and simulations. We present complementary simulations for different sample sizes accompanied by a sensitivity analysis of the dependence on $k_{n}$ and a discussion of data applications.