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A LOCAL GAUSSIAN BOOTSTRAP METHOD FOR REALIZED VOLATILITY AND REALIZED BETA

Published online by Cambridge University Press:  25 April 2018

Ulrich Hounyo
Ulrich Hounyo*
Affiliation:
University at Albany, SUNY and CREATES
*
*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: [email protected].
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Abstract

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.

Type
ARTICLES
Information
Econometric Theory , Volume 35 , Issue 2 , April 2019 , pp. 360 - 416
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

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.

References

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