Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T10:25:38.442Z Has data issue: false hasContentIssue false
  • English
  • Français

ESTIMATION OF INTEGRATED COVARIANCES IN THE SIMULTANEOUS PRESENCE OF NONSYNCHRONICITY, MICROSTRUCTURE NOISE AND JUMPS

Published online by Cambridge University Press:  08 April 2015

Yuta Koike
Yuta Koike*
Affiliation:
The Institute of Statistical Mathematics
*
*Address correspondence to Yuta Koike, Risk Analysis Research Center, The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan; e-mail: kyuta@ism.ac.jp.
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a new estimator for the integrated covariance of two Itô semimartingales observed at a high frequency. This new estimator, which we call the pre-averaged truncated Hayashi–Yoshida estimator, enables us to separate the sum of the co-jumps from the total quadratic covariation even in the case that the sampling schemes of two processes are nonsynchronous and the observation data are polluted by some noise. We also show the asymptotic mixed normality of this estimator under some mild conditions allowing infinite activity jump processes with finite variations, some dependency between the sampling times and the observed processes as well as a kind of endogenous observation error. We examine the finite sample performance of this estimator using a Monte Carlo study and we apply our estimators to empirical data, highlighting the importance of accounting for jumps even in an ultra-high frequency framework.

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 3 , June 2016 , pp. 533 - 611
Copyright
Copyright © Cambridge University Press 2015 

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.)

References

REFERENCES

Aït-Sahalia, Y., Fan, J., & Xiu, D. (2010) High-frequency covariance estimates with noisy and asynchronous financial data. Journal of the American Statistical Association 105, 1504–1517.Google Scholar
Aït-Sahalia, Y., Jacod, J., & Li, J. (2012) Testing for jumps in noisy high frequency data. Journal of Econometrics 168, 207–222.CrossRefGoogle Scholar
Aït-Sahalia, Y. & Yu, J. (2009) High frequency market microstructure noise estimates and liquidity measures. Annals of Applied Statistics 3, 422–457.Google Scholar
Andersen, T.G., Bollerslev, T., & Diebold, F.X. (2007) Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility. The Review of Economics and Statistics 89, 701–720.Google Scholar
Bandi, F.M. & Russell, J.R. (2006) Separating microstructure noise from volatility. Journal of Financial Economics 79, 655–692.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E., Graversen, S.E.G., 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. Springer-Verlag, pp. 33–69.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., & Shephard, N. (2009) Realized kernels in practice: Trades and quotes. Econometrics Journal 12, C1–C32.Google Scholar
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., & Shephard, N. (2011) Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading. Journal of Econometrics 162, 149–169.Google Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2004a) Measuring the Impact of Jumps in Multivariate Price Processes using Bipower Covariation. Discussion paper, Nuffield College, Oxford University.Google Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2004b) Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics 2, 1–37.Google Scholar
Beveridge, S. & Nelson, C.R. (1981) A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘buisiness cycle’. Journal of Monetary Economics 7, 151–174.Google Scholar
Bibinger, M. (2011) Efficient covariance estimation for asynchronous noisy high-frequency data. Scandinavian Journal of Statistics 38, 23–45.Google Scholar
Bibinger, M. (2012) An estimator for the quadratic covariation of asynchronously observed Itô processes with noise: Asymptotic distribution theory. Stochastic Processes and their Applications 122, 2411–2453.Google Scholar
Bos, C.S., Janus, P., & Koopman, S.J. (2012) Spot variance path estimation and its application to high-frequency jump testing. Journal of Financial Econometrics 10, 354–389.CrossRefGoogle Scholar
Boudt, K., Croux, C., & Laurent, S. (2011) Outlyingness weighted covariation. Journal of Financial Econometrics 9, 657–684.Google Scholar
Christensen, K., Kinnebrock, S., & Podolskij, M. (2010) Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data. Jornal of Econometrics 159, 116–133.Google Scholar
Christensen, K., Oomen, R., & Podolskij, M. (2011a) Fact or friction: Jumps at ultra high frequency. CREATES Research Paper 2011–19, Aarhus University.Google Scholar
Christensen, K., Podolskij, M., & Vetter, M. (2011b) On covariation estimation for multivariate continuous Itô semimartingales with noise in non-synchronous observation schemes. CREATES Research Paper 2011–53, Aarhus University.Google Scholar
Cont, R. & Kan, Y.H. (2011) Dynamic hedging of portfolio credit derivatives. SIAM Journal on Financial Mathematics 2, 112–140.CrossRefGoogle Scholar
Dahlhaus, R. (1997) Fitting time series models to nonstationary processes. Annals of Statistics 25, 1–37.CrossRefGoogle Scholar
Delbaen, F. & Schachermayer, W. (1994) A general version of the fundamental theorem of asset pricing. Mathematische Annalen 300, 463–520.Google Scholar
Diebold, F.X. (2006) On market microstructure noise and realized volatility. Journal of Business and Economic Statistics 24, 181–183. Discussion of Hansen & Lunde (2006).CrossRefGoogle Scholar
Donoho, D.L. & Johnstone, I.M. (1994) Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 232–246.Google Scholar
Epps, T.W. (1979) Comovements in stock prices in the very short run. Journal of the American Statistical Association 74, 291–298.Google Scholar
Freedman, D.A. (1975) On tail probabilities for martingales. Annals of Probability 3, 100–118.Google Scholar
Fukasawa, M. (2010) Realized volatility with stochastic sampling. Stochastic Processes and their Applications 120, 829–852.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 122, 3901–3920.Google Scholar
Hansen, P.R. & Lunde, A. (2006) Realized variance and market microstructure noise. Journal of Business & Economic Statistics 24, 127–161.Google Scholar
Hayashi, T., Jacod, J., & Yoshida, N. (2011) Irregular sampling and central limit theorems for power variations: The continuous case. Annales de l’Institut Henri Poincaré-Probabilités et Statistiques 47, 1197–1218.Google Scholar
Hayashi, T. & Yoshida, N. (2005) On covariance estimation of non-synchronously observed diffusion processes. Bernoulli 11, 359–379.Google Scholar
Hayashi, T. & Yoshida, N. (2011) Nonsynchronous covariation process and limit theorems. Stochastic Processes and their Applications 121, 2416–2454.Google Scholar
Jacod, J. (1997) On continuous conditional Gaussian martingales and stable convergence in law. In Azéma, Jacques, Yor, Marc and Emery, Michel (eds.), Séminaire de probabilitiés xxxi. Lecture Notes in Mathematics, vol. 1655, pp. 232–246. Springer.Google Scholar
Jacod, J. (2008) Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Processes and their Applications 118, 517–559.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., Podolskij, M., & Vetter, M. (2010) Limit theorems for moving averages of discretized processes plus noise. Annals of Statistics 38, 1478–1545.Google Scholar
Jacod, J. & Protter, P. (2012) Discretization of processes. Stochastic Modelling and Applied Probability, vol. 67. Springer.Google Scholar
Jacod, J. & Shiryaev, A.N. (2003) Limit theorems for stochastic processes, 2nd ed.Springer.Google Scholar
Jing, B.-Y., Li, C.-X., & Liu, Z. (2011) On Estimating the Integrated Co-Volatility using Noisy High Frequency Data with Jumps. Working paper.Google Scholar
Kalnina, I. (2011) Subsampling high frequency data. Journal of Econometrics 161, 262–283.Google Scholar
Kalnina, I. & Linton, O. (2008) Estimating quadratic variation consistently in the presence of endogenous and diurnal measurement error. Journal of Econometrics 147, 47–59.Google Scholar
Koike, Y. (2013) An estimator for the cumulative co-volatility of asynchronously observed semimartingales with jumps. Scandinavian Journal of Statistics 41, 460–481.CrossRefGoogle Scholar
Li, Y., Mykland, P.A., Renault, E., Zhang, L., & Zheng, X. (2014) Realized volatility when sampling times are possibly endogenous. Econometric theory 30, 580–605.Google Scholar
Li, Y., Zhang, Z., & Zheng, X. (2013) Volatility inference in the presence of both endogenous time and microstructure noise. Stochastic Processes and their Applications 123, 2696–2727.Google Scholar
Mancini, C. (2001) Disentangling the jumps of the diffuson in a geometric jumping Brownian motion. Giornale dell’Istituto Italiano degli Attuari 64, 19–47.Google Scholar
Mancini, C. & Gobbi, F. (2012) Identifying the Brownian covariation from the co-jumps given discrete observations. Econometric Theory 28, 249–273.Google Scholar
Pickands, J. III (1967) Maxima of stationary Gaussian processes. Probability Theory and Related Fields 7, 190–223.Google Scholar
Podolskij, M. & Vetter, M. (2009a) Bipower-type estimation in a noisy diffusion setting. Stochastic Processes and their Applications 119, 2803–2831.Google Scholar
Podolskij, M. & Vetter, M. (2009b) Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli 15, 634–658.Google Scholar
Podolskij, M. & Ziggel, D. (2010) New tests for jumps in semimartingale models. Statistical Inference for Stochastic Processes 13, 15–41.Google Scholar
Renault, E. & Werker, B.J. (2011) Causality effects in return volatility measures with random times. Jornal of Econometrics 160, 272–279.Google Scholar
Robert, C.Y. & Rosenbaum, M. (2012) Volatility and covariation estimation when microstructure noise and trading times are endogenous. Mathematical Finance 22, 133–164.Google Scholar
Shephard, N. & Xiu, D. (2012) Econometric Analysis of Multivariate Realised QML: Estimation of the Covariation of Equity Prices under Asynchronous Trading. Technical report, University of Oxford and University of Chicago.Google Scholar
Shimizu, Y. (2003) Estimation of diffusion processes with jumps from discrete observations. Master’s thesis, University of Tokyo.Google Scholar
Shimizu, Y. (2010) Threshold selection in jump-discriminant filter for discretely observed jump processes. Statistical Methods & Applications 19, 355–378.Google Scholar
Todorov, V. & Bollerslev, T. (2010) Jumps and betas: A new framework for disentangling and estimating systematic risks. Journal of Econometrics 157, 220–235.Google Scholar
Ubukata, M. & Oya, K. (2009) Estimation and testing for dependence in market microstructure noise. Journal of Financial Econometrics 7, 106–151.Google Scholar
Veraart, A.E. (2010) Inference for the jump part of quadratic variation of Itô semimartingales. Econometric Theory 26, 331–368.Google Scholar
Voev, V. & Lunde, A. (2007) Integrated covariance estimation using high-frequency data in the presence of noise. Journal of Financial Econometrics 5, 68–104.Google Scholar
Wang, K., Liu, J., & Liu, Z. (2013) Disentangling the effect of jumps on systematic risk using a new estimator of integrated co-volatility. Journal of Banking & Finance 37, 1777–1786.Google Scholar
Zhang, L. (2011) Estimating covariation: Epps effect, microstructure noise. Journal of Econometrics 160, 33–47.CrossRefGoogle Scholar