Hostname: page-component-5db58dd55d-jhf8m Total loading time: 0 Render date: 2026-05-26T13:25:41.612Z Has data issue: false hasContentIssue false
  • English
  • Français

REALIZED VOLATILITY WHEN SAMPLING TIMES ARE POSSIBLY ENDOGENOUS

Published online by Cambridge University Press:  27 November 2013

Yingying Li ,
Per A. Mykland ,
Eric Renault ,
Lan Zhang  and
Xinghua Zheng
Yingying Li
Affiliation:
Hong Kong University of Science and Technology
Per A. Mykland
Affiliation:
University of Chicago
Eric Renault
Affiliation:
Brown University
Lan Zhang
Affiliation:
University of Illinois at Chicago
Xinghua Zheng*
Affiliation:
Hong Kong University of Science and Technology
*
*Address correspondence to We are very grateful to the editor and anonymous referees for their very valuable comments and suggestions. Xinghua Zheng, Department of ISOM, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; email: xhzheng@ust.hk.
Get access Rights & Permissions [Opens in a new window]

Abstract

When estimating integrated volatilities based on high-frequency data, simplifying assumptions are usually imposed on the relationship between the observation times and the price process. In this paper, we establish a central limit theorem for the realized volatility in a general endogenous time setting. We also establish a central limit theorem for the tricity under the hypothesis that there is no endogeneity, based on which we propose a test and document that this endogeneity is present in financial data.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 30 , Issue 3 , June 2014 , pp. 580 - 605
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Abbring, J.H. (2012) Mixed hitting-time models. Econometrica 80, 783–819.Google Scholar
Aït-Sahalia, Y. & Mykland, P.A. (2003) The effects of random and discrete sampling when estimating continuous-time diffusions. Econometrica 71, 483–549.Google Scholar
Aldous, D.J. & Eagleson, G.K. (1978) On mixing and stability of limit theorems. Annals of Probability 6, 325–331.Google Scholar
Back, K. & Brown, D. (1993) Implied probabilities in GMM estimators. Econometrica 61, 971–975.Google Scholar
Barndorff-Nielsen, O.E., 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., Liptser, R., & Stoyanov, J. (eds.), From Stochastic Calculus to Mathematical Finance, The Shiryaev Festschrift, pp. 33–69. Springer-Verlag.Google Scholar
Barndorff-Nielsen, O.E., Graversen, S.E., Jacod, J., & Shephard, N. (2006) Limit theorems for bipower variation in financial econometrics. Econometric Theory 22, 677–719.Google Scholar
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., & Shephard, N. (2008) Designing realized kernels to measure ex-post variation of equity prices in the presence of noise. Econometrica 76, 1481–1536.Google Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2001) Non-gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society, B 63, 167–241.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2002) Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society, B 64, 253–280.Google Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2004) Power and bipower variation with stochastic volatility and jumps (with discussion). Journal of Financial Econometrics 2, 1–48.Google Scholar
Christensen, K., Podolskij, M., & Vetter, M. (2011) On covariation estimation for multivariate continuous Ito semimartingales with noise in non-synchronous observation schemes. Working paper.Google Scholar
Davison, A.C. & Hinkley, D.V. (1997) Bootstrap Methods and their Application, vol. 1 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, with 1 IBM-PC floppy disk (3.5 inch; HD).Google Scholar
Dellacherie, C. & Meyer, P. (1982) Probabilities and Potential B. North-Holland.Google Scholar
Duffie, D. & Glynn, P. (2004) Estimation of continuous-time Markov processes sampled at random times. Econometrica 72, 1773–1808.Google Scholar
Engle, R.F. (2000) The econometrics of ultra-high frequency data. Econometrica 68, 1–22.Google Scholar
Fan, J., Li, Y., & Yu, K. (2010) Vast volatility Matrix Estimation using High Frequency Data for Portfolio Selection. (to appear in Journal of the American Statistical Association).CrossRefGoogle Scholar
Fukasawa, M. (2010a) Central limit theorem for the realized volatility based on tick time sampling. Finance and Stochastics 14(2010), 209–233.Google Scholar
Fukasawa, M. (2010b) Realized volatility with stochastic sampling. Stochastic Processes and Their Applications 120, 829–552.Google Scholar
Fukasawa, M. & Rosenbaum, M. (2012) Central limit theorems for realized volatility under hitting times of an irregular grid. Stochastic Processes and Their Applications, forthcoming.Google Scholar
Grammig, J. & Wellner, M. (2002) Modeling the interdependence of volatility and inter-transaction duration processes. Journal of Econometrics, 106, 369–400.Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. Academic Press.Google Scholar
Hayashi, T., Jacod, J. & Yoshida, N. (2011) Irregular sampling and central limit theorems for power variations: The continuous case. Annales de L’Institute Henri Poincare, Probability and Statistics 47(4) 1197–1218.Google Scholar
Hayashi, T. & Yoshida, N. (2005) On covariance estimation of non-synchronously observed diffusion processes. Bernoulli 11, 359–379.Google Scholar
Jacod, J. (1994) Limit of Random Measures Associated with the Increments of a Brownian Semimartingale. Technical report, Université de Paris VI.Google Scholar
Jacod, J., Li, Y., Mykland, P.A., Podolskij, M., & Vetter, M. (2009) Microstructure noise in the continuous case: The pre-averaging approach. Stochastic Processes and their Applications 119, 2249–2276.Google Scholar
Jacod, J. & Protter, P. (1998) Asymptotic error distributions for the Euler method for stochastic differential equations. Annals of Probability 26, 267–307.CrossRefGoogle Scholar
Jacod, J. & Shiryaev, A. (2003) Limit Theorems for Stochastic Processes, 2nd ed. Springer-Verlag.Google Scholar
Kinnebrock, S. & Podolskij, M. (2008) A note on the central limit theorem for bipower variation of general functions. Stochastic Processes and their Applications 118, 1056–1070.Google Scholar
Kristensen, D. (2010) Nonparametric filtering of the realized spot volatility: A kernel-based approach. Econometric Theory 26, 60–93.Google Scholar
Lee, S. & Mykland, P.A. (2008) Jumps in financial markets: A new nonparametric test and jump dynamics. Review of Financial Studies 21, 2535–2563.CrossRefGoogle Scholar
Li, Y., Zhang, Z., & Zheng, X. (2013) Volatility Inference in the Presence of Both Endogenous Time and Microstructure Noise. A Special Issue on the Occasion of the 2013 International Year of Statistics (R. Dahlhaus, J. Jacod, P. Mykland, & N. Yoshida eds.), Stochastic Processes and their Applications, 123(7), 2696–2727.Google Scholar
Mancini, C. (2001) Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Giornale dell’Istituto Italiano degli Attuari LXIV, 19–47.Google Scholar
Meddahi, N., Renault, E., & Werker, B. (2006) GARCH and irregularly spaced data. Economics Letters 90, 200–204.Google Scholar
Mykland, P.A. (1994) Bartlett type identities for martingales. Annals of Statistics 22, 21–38.Google Scholar
Mykland, P.A., Shephard, N., & Sheppard, K. (2012) Efficient and feasible inference for the components of financial variation using blocked multipower variation. Working paper, University of Oxford.Google Scholar
Mykland, P.A. & Zhang, L. (2006) ANOVA for diffusions and Itô processes. Annals of Statistics 34, 1931–1963.Google Scholar
Mykland, P.A. & Zhang, L. (2009) Inference for continuous semimartingales observed at high frequency. Econometrica 77, 1403–1455.Google Scholar
Mykland, P.A. & Zhang, L. (2012) The econometrics of high frequency data. In Kessler, M., Lindner, A., & Sørensen, M. (eds.), Statistical Methods for Stochastic Differential Equations, pp. 109–190. Chapman and Hall/CRC Press.Google Scholar
Phillips, P.C.B. & Yu, J. (2007) Information Loss in Volatility Measurement with Flat Price Trading. Working paper.Google Scholar
Protter, P. (2004) Stochastic Integration and Differential Equations: A New Approach, 2nd ed. Springer-Verlag.Google Scholar
Renault, E., Van der Heijden, T., & Werker, B.J. (2012) The Dynamic Mixed Hitting-Time Model for Multiple Transaction Prices and Times. Working paper.Google Scholar
Renault, E. & Werker, B.J. (2011) Causality effects in return volatility measures with random times. Journal of Econometrics 60(1), 272–279.Google Scholar
Renò, R. (2008) Nonparametric estimation of the diffusion coefficient of stochastic volatility models. Econometric Theory 24, 1174–1206.Google Scholar
Rényi, A. (1963) On stable sequences of events. Sankyā Series A 25, 293–302.Google Scholar
Robert, C.Y. & Rosenbaum, M. (2010) On the microstructural hedging error. SIAM Journal of Financial Mathematics 1, 427–453.Google Scholar
Robert, C.Y. & Rosenbaum, M. (2012) Volatility and covariation estimation when microstructure noise and trading times are endogenous. Mathematical Finance 22(1), 133–164.Google Scholar
Rootzén, H. (1980) Limit distributions for the error in approximations of stochastic integrals. Annals of Probability 8, 241–251.Google Scholar
Ross, S. (1996) Stochastic Processes. 2nd ed. Wiley.Google Scholar
Wang, D.C. & Mykland, P.A. (2011) The Estimation of Leverage Effect with High Frequency Data. Working paper, University of Oxford.Google Scholar
Xiu, D. (2010) Quasi-maximum likelihood estimation of volatility with high frequency data. Journal of Econometrics 159, 235–250.Google Scholar
Zhang, L. (2001) From Martingales to ANOVA: Implied and Realized Volatility. Ph.D. thesis, The University of Chicago.Google Scholar
Zhang, L. (2006) Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli 12, 1019–1043.Google Scholar
Zhang, L. (2011) Estimating covariation: Epps effect and microstructure noise. Journal of Econometrics 160, 33–47.Google 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, 1394–1411.CrossRefGoogle Scholar
Supplementary material: PDF

Li et al. Supplementary Material

Supplementary Material

Download Li et al. Supplementary Material(PDF)
PDF 144.5 KB