Skip to main content


  • Ulrich Hounyo (a1)

This article introduces a local Gaussian bootstrap method useful for the estimation of the asymptotic distribution of high-frequency data-based statistics such as functions of realized multivariate volatility measures as well as their asymptotic variances. The new approach consists of dividing the original data into nonoverlapping blocks of M consecutive returns sampled at frequency h (where h−1 denotes the sample size) and then generating the bootstrap observations at each frequency within a block by drawing them randomly from a mean zero Gaussian distribution with a variance given by the realized variance computed over the corresponding block.

Our main contributions are as follows. First, we show that the local Gaussian bootstrap is first-order consistent when used to estimate the distributions of realized volatility and realized betas under assumptions on the log-price process which follows a continuous Brownian semimartingale process. Second, we show that the local Gaussian bootstrap matches accurately the first four cumulants of realized volatility up to o(h), implying that this method provides third-order refinements. This is in contrast with the wild bootstrap of Gonçalves and Meddahi (2009, Econometrica 77(1), 283–306), which is only second-order correct. Third, we show that the local Gaussian bootstrap is able to provide second-order refinements for the realized beta, which is also an improvement of the existing bootstrap results in Dovonon, Gonçalves, and Meddahi (2013, Journal of Econometrics 172, 49–65) (where the pairs bootstrap was shown not to be second-order correct under general stochastic volatility). In addition, we highlight the connection between the local Gaussian bootstrap and the local Gaussianity approximation of continuous semimartingales established by Mykland and Zhang (2009, Econometrica 77, 1403–1455) and show the suitability of this bootstrap method to deal with the new class of estimators introduced in that article. Lastly, we provide Monte Carlo simulations and use empirical data to compare the finite sample accuracy of our new bootstrap confidence intervals for integrated volatility with the existing results.

Corresponding author
*Address correspondence to Ulrich Hounyo, Department of Economics, University at Albany, SUNY, Building 25, Room 103, 1400 Washington Ave, Albany, NY 12222, USA; e-mail:
Hide All

I would like to thank Nour Meddahi for very stimulating discussions, which led to the idea of this article. I am especially indebted to Sílvia Gonçalves for her valuable comments. I am also grateful to Asger Lunde, Peter Exterkate, Matthew Webb, Prosper Dovonon, Rasmus T. Varneskov, Kim Christensen, the co-editor Eric Renault, the editor Peter C. B. Phillips, and anonymous referees for helpful advice, comments, and suggestions. Note that the original draft of the article has been circulated under the title “Bootstrapping realized volatility and realized beta under a local Gaussianity assumption”. I acknowledge support from CREATES—Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation, as well as support from the Oxford-Man Institute of Quantitative Finance.

Hide All
Andersen, T.G., & Bollerslev, T. (1998) Answering skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39(4): 885905.
Andersen, T.G., Dobrev, D., & Schaumburg, E. (2014) A robust neighborhood truncation approach to estimation of integrated quarticity. Econometric Theory 30, 359.
Barndorff-Nielsen, O., Graversen, S.E., Jacod, J., Podolskij, M., & Shephard, N. (2006) A central limit theorem for realised power and bipower variations of continuous semimartingales. In Kabanov, Y., Lipster, R., & Stoyanov, J. (eds.), From Stochastic Analysis to Mathematical Finance, Festschrift for Albert Shiryaev, pp. 3368. Springer.
Barndorff-Nielsen, O., Hansen, P., Lunde, A., & Shephard, N. (2008) Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica 76, 14811536.
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., & Shephard, N. (2009) Realised kernels in practice: Trades and quotes. Econometrics Journal 12, C1C32.
Barndorff-Nielsen, O. & Shephard, N. (2002) Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society, Series B 64, 253280.
Barndorff-Nielsen, O. & Shephard, N. (2003) Realised power variation and stochastic volatility models. Bernoulli 9, 243265.
Barndorff-Nielsen, O. & Shephard, N. (2004a) Econometric analysis of realised covariation: High frequency based covariance, regression and correlation in financial economics. Econometrica 72, 885925.
Barndorff-Nielsen, O. & Shephard, N. (2004b) Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics 2, 148.
Davidson, R. & Flachaire, E. (2008) The Wild Bootstrap, Tamed at Last. Journal of Econometrics 146(1), 162169.
Dovonon, P., Gonçalves, S., & Meddahi, N. (2013) Bootstrapping realized multivariate volatility measures. Journal of Econometrics 172, 4965.
Dovonon, P., Gonçalves, S., Hounyo, U., & Meddahi, N. (2018) Bootstrapping high-frequency jump tests. Journal of the American Statistical Association, first published online 02 April 2018. doi:10.1080/01621459.2018.1447485.
Gonçalves, S., Hounyo, U., & Meddahi, N. (2014) Bootstrap inference for pre-averaged realized volatility based on non-overlapping returns. Journal of Financial Econometrics 12(4), 679707.
Gonçalves, S. & Meddahi, N. (2009) Bootstrapping realized volatility. Econometrica 77(1), 283306.
Hall, P. (1992) The Bootstrap and Edgeworth Expansion. Springer-Verlag.
Hall, P., Horowitz, J.L., & Jing, B.-Y. (1995) On blocking rules for the bootstrap with dependent data. Biometrika 82, 561574.
Hansen, P.R. & Lunde, A. (2006) Realized variance and market microstructure noise. Journal of Business and Economics Statistics 24, 127161.
Hayashi, T. & Yoshida, N. (2005) On covariance estimation of non-synchronously observed diffusion processes. Bernoulli 11, 359379.
Heston, S. (1993) Closed-form solution for options with stochastic volatility with applications to bonds and currency options. Review of Financial Studies 6, 327343.
Hounyo, U. (2017) Bootstrapping integrated covariance matrix estimators in noisy jump-diffusion models with non-synchronous trading. Journal of Econometrics 197(1), 130152.
Hounyo, U. (2018) Online Supplementary Material to “A local Gaussian bootstrap method for realized volatility and realized beta”. Econometric Theory Supplementary Material.
Hounyo, U., Gonçalves, S., & Meddahi, N. (2017) Bootstrapping pre-averaged realized volatility under market microstructure noise. Econometric Theory 33(4), 791838.
Hounyo, U. & Varneskov, R.T. (2017) A local stable bootstrap for power variations of pure-jump semimartingales and activity index estimation. Journal of Econometrics 198(1), 1028.
Hounyo, U. & Veliyev, B. (2016) Validity of Edgeworth expansions for realized volatility estimators. Econometrics Journal 19(1), 132.
Huang, X. & Tauchen, G. (2005) The relative contribution of jumps to total price variance. Journal of Financial Econometrics 3, 456499.
Jacod, J., Li, Y., Podolskij, M., & Vetter, M. (2009) Microstructure noise in the continuous case: the pre-averaging approach. Stochastic Processes and Their Approach 119, 22492276.
Jacod, J. & Protter, P. (2012) Discretization of Processes. Springer-Verlag.
Jacod, J. & Rosenbaum, M. (2013) Quarticity and other functionals of volatility efficient estimation. The Annals of Statistics 41, 14621484.
Katz, M.L. (1963) Note on the Berry–Esseen theorem. Annals of Mathematical Statistics 34, 11071108.
Kristensen, D. (2010) Nonparametric filtering of the realized spot volatility: A kernel-based approach. Econometric Theory 26, 6093.
Li, J., Todorov, V., & Tauchen, G. (2017) Adaptive estimation of continuous-time regression models using high-frequency data. Journal of Econometrics 200(1), 3647.
Mammen, E. (1993) Bootstrap and wild bootstrap for high dimensional linear models. Annals of Statistics 21, 255285.
Mykland, P.A. & Zhang, L. (2009) Inference for continous semimartingales observed at high frequency. Econometrica 77, 14031455.
Mykland, P.A. & Zhang, L. (2011) The double Gaussian approximation for high frequency data. Scandinavian Journal of Statistics 38, 215236.
Mykland, P.A., Shephard, N., & Shepphard, K. (2012) Efficient and Feasible Inference for the Components of Financial Variation using Blocked Multipower Variation. Working paper, Oxford University.
Podolskij, M. & Vetter, M. (2009) Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli 15(3), 634658.
Politis, D.N., Romano, J.P., & Wolf, M. (1999) Subsampling. Springer-Verlag.
Renault, E., Sarisoy, C., & Werker, B.J. (2017) Efficient estimation of integrated volatility and related processes. Econometric Theory 33, 439478.
Shao, J. & Tu, D. (1995) The Jackknife and Bootstrap. Springer Verlag.
Zhang, L, Mykland, P.A., & Aït-Sahalia, Y. (2005) A tale of two time-scales: Determining integrated volatility with noisy high frequency data. Journal of the American Statistical Association 100, 13941411.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
Type Description Title
Supplementary materials

Hounyo supplementary material
Hounyo supplementary material 1

 PDF (366 KB)
366 KB


Full text views

Total number of HTML views: 0
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 74 *
Loading metrics...

* Views captured on Cambridge Core between 25th April 2018 - 23rd May 2018. This data will be updated every 24 hours.