Dovonon, Prosper and Goncalves, Silvia and Meddahi, Nour (2010): Bootstrapping realized multivariate volatility measures.

PDF
MPRA_paper_40123.pdf Download (321Kb)  Preview 
Abstract
We study bootstrap methods for statistics that are a function of multivariate high frequency returns such as realized regression coefficients and realized covariances and correlations. For these measures of covariation, the Monte Carlo simulation results of BarndorffNielsen and Shephard (2004) show that finite sample distortions associated with their feasible asymptotic theory approach may arise if sampling is not too frequent. This motivates our use of the bootstrap as an alternative tool of inference for covariation measures. We consider an i.i.d. bootstrap applied to the vector of returns. We show that the finite sample performance of the bootstrap is superior to the existing firstorder asymptotic theory. Nevertheless, and contrary to the existing results in the bootstrap literature for regression models subject to heteroskedasticity in the error term, the Edgeworth expansion for the i.i.d. bootstrap that we develop here shows that this method is not second order accurate. We argue that this is due to the fact that the conditional mean parameters of realized regression models are heterogeneous under stochastic volatility.
Item Type:  MPRA Paper 

Original Title:  Bootstrapping realized multivariate volatility measures 
Language:  English 
Keywords:  Realized regression, realized beta, realized correlation, bootstrap, Edgeworth expansions 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C15  Statistical Simulation Methods: General C  Mathematical and Quantitative Methods > C0  General > C01  Econometrics 
Item ID:  40123 
Depositing User:  Prosper Dovonon 
Date Deposited:  18. Jul 2012 10:04 
Last Modified:  12. Feb 2013 06:13 
References:  [1] Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys, 2003. “Modeling and Forecasting Realized Volatility,” Econometrica, 71, 529626. [2] Andersen, T.G., Bollerslev, T., Diebold, F.X. and J. Wu, 2005. “A Framework for Exploring the Macroeconomic Determinants of Systematic Risk,” American Economic Review, 95, 398404. [3] Andersen, T.G., Bollerslev, T., Diebold, F.X. and J.Wu, 2006. “Realized Beta: Persistence and Predictability,” in T. Fomby (ed.) Advances in Econometrics: Econometric Analysis of Economic and Financial Time Series in Honor of R.F. Engle and C.W.J. Granger , Volume B, 140. [4] BarndorffNielsen, O., S.E. Graversen, J. Jacod, and N. Shephard, 2006, “Limit theorems for bipower variation in financial econometrics,” Econometric Theory, 22, 677719. [5] BarndorffNielsen, O., S. E. Graversen, J. Jacod, M. Podolskij, and N. Shephard, 2006. “A central limit theorem for realised power and bipower variations of continuous semimartingales.” In Y. Kabanov, R. Lipster, and J. Stoyanov (Eds.), From Stochastic Analysis to Mathematical Finance, Festschrift for Albert Shiryaev, 33  68. Springer. [6] BarndorffNielsen, O. and N. Shephard, 2002. “Econometric analysis of realized volatility and its use in estimating stochastic volatility models,” Journal of the Royal Statistical Society, Series B, 64, 253280. [7] BarndorffNielsen, O. and N. Shephard, 2004. “Econometric analysis of realised covariation: high frequency based covariance, regression and correlation in financial economics,” Econometrica, 72, 885925. [8] BarndorffNielsen, O. and N. Shephard, 2006. “Econometrics of testing for jumps in financial economics using bipower variation,” Journal of Financial Econometrics, 4, 130. [9] BarndorffNielsen, O., P. Hansen, A. Lunde, and N. Shephard, 2008a. “Designing realised kernels to measure the expost variation of equity prices in the presence of noise,” Econometrica, 76, 14811536. [10] BarndorffNielsen, O., P. Hansen, A. Lunde, and N. Shephard, 2008b. “Multivariate realised kernels: consistent positive semidefinite estimators of the covariation of equity prices with noise and nonsynchronous trading,” Working paper, Oxford University. [11] Bollerslev, T. and V. Todorov, 2010. “Jumps and betas: a new theoretical framework for disentangling and estimating systematic risks,” Journal of Econometrics, 157, 220235. [12] Campbell, J., A. Sunderam, and L. Viceira, 2009. “Inflation bets or deflation hedges? The changing risks of nominal bonds,” Working paper, Harvard University. [13] Freedman, D. A., 1981. “Bootstrapping regression models,” Annals of Statistics, Vol. 9, No. 6, 12181228. [14] Griffin, J., and R. Oomen, 2006. “Covariance measurement in the presence of nonsynchronous trading and market microstructure noise,” working paper, Warwick University. [15] Gobbi, F. and C. Mancini, 2008. “Identifying the diffusion covariation and the cojumps given discrete observations”, Working paper, University of Firenze. [16] Goncalves, S. and N. Meddahi, 2008. “Edgeworth Corrections for Realized Volatility”, Econometric Reviews, 27 (1), 139162. [17] Goncalves, S. and N. Meddahi, 2009. “Bootstrapping realized volatility,” Econometrica, 77, 283306. [18] Hall, P., 1992. The bootstrap and Edgeworth expansion. SpringerVerlag, New York. [19] Hayashi, T., and N. Yoshida, 2005. “On covariance estimation of nonsynchronously observed diffusion processes,” Bernoulli 11, 359–379. [20] Jacod, J., 1994. “Limit of random measures associated with the increments of a Brownian semimartingale,” Preprint number 120, Laboratoire de Probabilitit´es, Universite Pierre et Marie Curie, Paris. [21] Jacod, J. and P. Protter, 1998. “Asymptotic error distributions for the Euler method for stochastic differential equations,” Annals of Probability 26, 267307. [22] Jacod, J. and V. Todorov, 2008. “Testing for Common Arrivals of Jumps for Discretely Observed Multidimensional Processes,” Annals of Statistics, forthcoming. [23] Katz, M.L., 1963. “Note on the BerryEsseen theorem,” Annals of Mathematical Statistics 34, 11071108. [24] Mammen, E., 1993. “Bootstrap and wild bootstrap for high dimensional linear models,” Annals of Statistics 21, 255285. [25] Meddahi, N., 2002. “A theoretical comparison between integrated and realized volatility,” Journal of Applied Econometrics 17, 475508. [26] Mykland, P., 2010. “A Gaussian calculus for inference from high frequency data,” Annals of Finance, Forthcoming. [27] Mykland, P.A., and L. Zhang, 2006. “ANOVA for diffusions and Ito processes,” Annals of Statistics, 34 19311963. [28] Podolskij , M. and M. Vetter, 2010. “Understanding limit theorems for semimartingales: a short survey,” Statistica Nederlandica, Forthcoming. [29] Serfling, R.J., 1980. Approximation theorems of mathematical statistics, Wiley, New York. [30] Viceira, L.M., 2007. “Bond Risk, Bond Return Volatility, and the Term Structure of Interest Rates,” working paper, Harvard Business School. [31] Voev, V. and A. Lunde, 2007. “Integrated covariance estimation using highfrequency data in the presence of noise,” Journal of Financial Econometrics, 5, 68  104. [32] Zhang, L., 2010. “Estimating covariation: Epps effect, microstructure noise,” Journal of Econometrics, Forthcoming. [33] Zhang, L, P.A. Mykland, and Y. AitSahalia, 2005. “A tale of two timescales: determining integrated volatility with noisy high frequency data,” Journal of the American Statistical Association, 100, 1394 1411. [34] Zhang, L., Mykland, P. and Y. AıtSahalia, 2010. “Edgeworth expansions for realized volatility and related estimators,” Journal of Econometrics, Forthcoming. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/40123 