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Published online by Cambridge University Press:  09 September 2016

Ulrich Hounyo*
Aarhus University
Sílvia Gonçalves
University of Western Ontario
Nour Meddahi
Toulouse School of Economics
*Address correspondence to Ulrich Hounyo, Department of Economics and Business Economics, Aarhus University, 8210 Aarhus V., Denmark; e-mail:


The main contribution of this paper is to propose a bootstrap method for inference on integrated volatility based on the pre-averaging approach, where the pre-averaging is done over all possible overlapping blocks of consecutive observations. The overlapping nature of the pre-averaged returns implies that the leading martingale part in the pre-averaged returns are kn-dependent with kn growing slowly with the sample size n. This motivates the application of a blockwise bootstrap method. We show that the “blocks of blocks” bootstrap method is not valid when volatility is time-varying. The failure of the blocks of blocks bootstrap is due to the heterogeneity of the squared pre-averaged returns when volatility is stochastic. To preserve both the dependence and the heterogeneity of squared pre-averaged returns, we propose a novel procedure that combines the wild bootstrap with the blocks of blocks bootstrap. We provide a proof of the first order asymptotic validity of this method for percentile and percentile-t intervals. Our Monte Carlo simulations show that the wild blocks of blocks bootstrap improves the finite sample properties of the existing first order asymptotic theory. We use empirical work to illustrate its use in practice.

Copyright © Cambridge University Press 2016 

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We would like to thank Ilze Kalnina, Kevin Sheppard and Neil Shephard for many useful comments and discussions. This work was supported by grants FQRSC-ANR and SSHRC. In addition, Ulrich Hounyo acknowledges 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. Finally, Nour Meddahi benefited from the financial support of the chair “Marché des risques et création de valeur” Fondation du risque/SCOR.



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.Google Scholar
Barndorff-Nielsen, O. & Shephard, N. (2006) Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics 4, 130.CrossRefGoogle Scholar
Bühlmann, P. & Künsch, H.R. (1995) The blockwise bootstrap for general parameters of a stationary time series. Scandinavian Journal of Statistics 22(1), 3554.Google Scholar
Christensen, K., Podolskij, M., & Vetter, M. (2013) On covariation estimation for multivariate continuous Ito semimartingales with noise in non-synchronous observation schemes. Journal of Multivariate Analysis 120, 5984.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Gonçalves, S. & Meddahi, N. (2008) Edgeworth corrections for realized volatility. Econometric Reviews 27, 139162.CrossRefGoogle Scholar
Gonçalves, S. & Meddahi, N. (2009) Bootstrapping realized volatility. Econometrica 77(1), 283306.Google Scholar
Gonçalves, S. & White, H. (2002) The bootstrap of the mean for dependent heterogeneous arrays. Econometric Theory 18, 13671384.CrossRefGoogle Scholar
Hansen, P.R. & Lunde, A. (2006) Realized variance and market microstructure noise. Journal of Business and Economics Statistics 24, 127161.CrossRefGoogle Scholar
Hautsch, N. & Podolskij, M. (2013) Pre-averaging based estimation of quadratic variation in the presence of noise and jumps: Theory, implementation, and empirical evidence. Journal of Business and Economic Statistics 31(2), 165183.CrossRefGoogle Scholar
Heston, S. (1993) Closed-form solution for options with stochastic volatility with applications to bonds and currency options. Review of Financial Studies 6, 327343.CrossRefGoogle Scholar
Hounyo, U. (2016) Bootstrapping integrated covariance matrix estimators in noisy jump-diffusion models with non-synchronous trading. Manuscript, Aarhus University.
Hounyo, U. & Veliyev, B. (2016) Validity of edgeworth expansions for realized volatility estimators. Econometrics Journal 19(1), 132.CrossRefGoogle Scholar
Huang, X. & Tauchen, G. (2005) The relative contribution of jumps to total price variance. Journal of Financial Econometrics 3, 456499.CrossRefGoogle Scholar
Jacod, J. (2008) Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Processes and Their Applications 118, 517559.CrossRefGoogle Scholar
Jacod, J., Li, Y., Mykland, P., Podolskij, M., & Vetter, M. (2009) Microstructure noise in the continuous case: the pre-averaging approach. Stochastic Processes and Their Applications 119, 22492276.CrossRefGoogle Scholar
Jacod, J., Podolskij, M., & Vetter, M. (2010) Limit theorems for moving averages of discretized processes plus noise. Annals of Statistics 38, 14781545.CrossRefGoogle Scholar
Jing, B., Liu, Z., & Kong, X. (2014) On the estimation of integrated volatility with jumps and microstructure noise. Journal of Business and Economic Statistics 32(3), 457467.CrossRefGoogle Scholar
Koike, Y. (2016) Estimation of integrated covariances in the simultaneous presence of nonsynchronicity, microstructure noise and jumps. Econometric Theory 32, 533611.CrossRefGoogle Scholar
Künsch, H.R. (1989) The jackknife and the bootstrap for general stationary observations. Annals of Statistics 17, 12171241.CrossRefGoogle Scholar
Liu, R.Y. (1988) Bootstrap procedure under some non-i.i.d. models. Annals of Statistics 16, 16961708.CrossRefGoogle Scholar
Mykland, P. & Zhang, L. (2014) Assessment of uncertainty in high frequency data: The observed asymptotic variance. Manuscript, The University of Chicago.
Paparoditis, E. & Politis, D.N. (2002) Local block bootstrap. Rendering Accounts of the Academy of Science, Series I (Mathematics) 335, 959962 (In French).Google Scholar
Phillips, P.C.B. & Moon, H.R. (1999) Linear regression limit theory for nonstationary panel data. Econometrica 67, 10571111.CrossRefGoogle Scholar
Podolskij, M. & Vetter, M. (2009) Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli 15(3), 634658.CrossRefGoogle Scholar
Politis, D.N. & Romano, J.P. (1992) A general resampling scheme for triangular arrays of α-mixing random variables. Annals of Statistics 20, 19852007.CrossRefGoogle Scholar
Politis, D.N., Romano, J.P., & Wolf, M. (1999) Subsampling. Springer-Verlag.CrossRefGoogle Scholar
Wu, C.F.J. (1986) Jackknife, bootstrap and other resampling methods in regression analysis. Annals of Statistics 14, 12611295.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Zhang, L., Mykland, P. & Aït-Sahalia, Y. (2011) Edgeworth expansions for realized volatility and related estimators. Journal of Econometrics 160, 190203.CrossRefGoogle Scholar
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