Bootstrapping Realized Volatility

Authors

  • Sílvia Gonçalves,

    1. Département de Sciences Économiques, CIREQ and CIRANO, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, QC, H3C 3J7, Canada; silvia.goncalves@umontreal.ca
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  • Nour Meddahi

    1. Finance and Accounting Group, Tanaka Business School, Imperial College London, Exhibition Road, London SW7 2AZ, U.K.; n.meddahi@imperial.ac.uk
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    • We would like to thank participants at the 2005 North American Winter Meeting of the Econometric Society, the SBFSIF II conference, Québec (April 2005), the CIREQ Montréal Financial Econometrics (May 2005), the SETA conference, Taipei (May 2005), the 2005 CEA meetings, the Princeton–Chicago High Frequency Conference (June 2005), and the NBER Summer Institute 2005, as well as seminar participants at Concordia University, Université de Toulouse I, the St. Louis Fed, and Universidade Nova de Lisboa. We also thank Torben Andersen, António Antunes, Christian Brownlees, Rui Castro, Valentina Corradi, Peter Hansen, Emma Iglesias, Atsushi Inoue, Lutz Kilian, and especially Per Mykland and Neil Shephard for helpful comments on the first version of the paper. In addition, we are grateful to three anonymous referees and a co-editor for many valuable suggestions. This work was supported by grants from FQRSC, SSHRC, MITACS, NSERC, and Jean-Marie Dufour's Econometrics Chair of Canada. Parts of this paper were completed while Gonçalves was visiting the Banco de Portugal, Lisboa, and the Finance Department at Stern Business School and Meddahi was visiting Toulouse University and CREST–Paris.


Abstract

We propose bootstrap methods for a general class of nonlinear transformations of realized volatility which includes the raw version of realized volatility and its logarithmic transformation as special cases. We consider the independent and identically distributed (i.i.d.) bootstrap and the wild bootstrap (WB), and prove their first-order asymptotic validity under general assumptions on the log-price process that allow for drift and leverage effects. We derive Edgeworth expansions in a simpler model that rules out these effects. The i.i.d. bootstrap provides a second-order asymptotic refinement when volatility is constant, but not otherwise. The WB yields a second-order asymptotic refinement under stochastic volatility provided we choose the external random variable used to construct the WB data appropriately. None of these methods provides third-order asymptotic refinements. Both methods improve upon the first-order asymptotic theory in finite samples.

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