Annals of Statistics

Limit theorems for moving averages of discretized processes plus noise

Jean Jacod, Mark Podolskij, and Mathias Vetter

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This paper presents some limit theorems for certain functionals of moving averages of semimartingales plus noise which are observed at high frequency. Our method generalizes the pre-averaging approach (see [Bernoulli 15 (2009) 634–658, Stochastic Process. Appl. 119 (2009) 2249–2276]) and provides consistent estimates for various characteristics of general semimartingales. Furthermore, we prove the associated multidimensional (stable) central limit theorems. As expected, we find central limit theorems with a convergence rate n−1/4, if n is the number of observations.

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Ann. Statist., Volume 38, Number 3 (2010), 1478-1545.

First available in Project Euclid: 24 March 2010

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G44: Martingales with continuous parameter 62M09: Non-Markovian processes: estimation
Secondary: 60G42: Martingales with discrete parameter 62G20: Asymptotic properties

Central limit theorem high-frequency observations microstructure noise quadratic variation semimartingale stable convergence


Jacod, Jean; Podolskij, Mark; Vetter, Mathias. Limit theorems for moving averages of discretized processes plus noise. Ann. Statist. 38 (2010), no. 3, 1478--1545. doi:10.1214/09-AOS756.

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  • [1] Aït-Sahalia, Y., Mykland, P. A. and Zhang, L. (2005). How often to sample a continuous-time process in the presence of market microstructure noise. Review of Financial Studies 18 351–416.
  • [2] Aït-Sahalia, Y. and Jacod, J. (2009). Estimating the degree of activity of jumps in high frequency data. Ann. Statist. 37 2202–2244.
  • [3] Aït Sahalia, Y. and Jacod, J. (2009). Is Brownian motion necessary to model high frequency data? Ann. Statist. To appear.
  • [4] Aït Sahalia, Y. and Jacod, J. (2009). Testing for jumps in a discretely observed process. Ann. Statist. 37 184–222.
  • [5] Bandi, F. M. and Russell, J. R. (2006). Separating microstructure noise from volatility. Journal of Financial Economics 79 655–692.
  • [6] Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J., Podolskij, M. and Shephard, N. (2006). A central limit theorem for realised power and bipower variations of continuous semimartingales. In From Stochastic Calculus to Mathematical Finance. Festschrift in Honour of A. N. Shiryaev (Y. Kabanov, R. Liptser and J. Stoyanov, eds.) 33–68. Springer, Heidelberg.
  • [7] Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A. and Shephard, N. (2006). Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica 76 1481–1536.
  • [8] Barndorff-Nielsen, O. E. and Shephard, N. (2002). Econometric analysis of realised volatility and its use in estimating stochastic volatility models. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 253–280.
  • [9] Barndorff-Nielsen, O. E. and Shephard, N. (2004). Power and bipower variation with stochastic volatility and jumps (with discussion). Journal of Financial Econometrics 2 1–48.
  • [10] Cont, R. and Mancini, C. (2009). Nonparametric tests for analysing the fine structure of price fluctuations. Working paper.
  • [11] Gloter, A. and Jacod, J. (2001). Diffusions with measurement errors. II-Optimal estimators. ESAIM Probab. Statist. 5 243–260.
  • [12] Ibragimov, I. A. and Has’minski, R. Z. (1981). Statistical Estimation: Asymptotic Theory. Springer, Berlin.
  • [13] Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517–559.
  • [14] Jacod, J., Li, Y., Mykland, P., Podolskij, M. and Vetter, M. (2009). Microstructure noise in the continuous case: The pre-averaging approach. Stochastic Process. Appl. 119 2249–2276.
  • [15] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
  • [16] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Springer, Berlin.
  • [17] Li, Y. and Mykland, P. (2007). Are volatility estimators robust with respect to modeling assumptions? Bernoulli 13 601–622.
  • [18] Podolskij, M. and Vetter, M. (2009). Bipower-type estimation in a noisy diffusion setting. Stochastic Process. Appl. 119 2803–2831.
  • [19] Podolskij, M. and Vetter, M. (2009). Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli 15 634–658.
  • [20] Tauchen, G. and Todorov, V. (2009). Activity signature functions with application for high-frequency data analysis. J. Econometrics 154 125–138.
  • [21] Zhang, L. (2006). Efficient estimation of volatility using noisy observations. Bernoulli 12 1019–1043.
  • [22] Zhang, L., Mykland, P. A. and Aït-Sahalia, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. J. Amer. Statist. Assoc. 100 1394–1411.