Second-order asymptotic expansion for a non-synchronous covariation estimator
Dalalyan, Arnak ; Yoshida, Nakahiro
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011), p. 748-789 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Dans cet article, nous considérons le problème d'estimation de la covariation de deux processus de diffusion observés de façon asynchrone. Nous nous plaçons dans le cadre présenté dans [Bernoulli 11 (2005) 359-379, Ann. Inst. Statist. Math. 60 (2008) 367-406] et établissons un développement asymptotique au second ordre de la loi de l'estimateur de Hayashi-Yoshida. Ce développement est valable pour les drifts aléatoires non-anticipatifs et pour des pas d'échantillonnage irréguliers, éventuellement aléatoires, mais indépendant des processus observés. L'approche utilisée pour obtenir les principaux résultats peut être décomposée en trois étapes. La première consiste à établir un développement au second-ordre de la loi de l'estimateur dans le cadre gaussien. La deuxième est l'obtention d'une décomposition stochastique de l'estimateur lui-même et la dernière est l'évaluation de la covariance de Malliavin. A titre d'exemple, nous calculons les constantes du développement au second ordre dans le cas où l'échantillonnage est obtenu par deux processus de Poisson indépendants.

In this paper, we consider the problem of estimating the covariation of two diffusion processes when observations are subject to non-synchronicity. Building on recent papers [Bernoulli 11 (2005) 359-379, Ann. Inst. Statist. Math. 60 (2008) 367-406], we derive second-order asymptotic expansions for the distribution of the Hayashi-Yoshida estimator in a fairly general setup including random sampling schemes and non-anticipative random drifts. The key steps leading to our results are a second-order decomposition of the estimator's distribution in the gaussian set-up, a stochastic decomposition of the estimator itself and an accurate evaluation of the Malliavin covariance. To give a concrete example, we compute the constants involved in the resulting expansions for the particular case of sampling scheme generated by two independent Poisson processes.

Publié le : 2011-01-01
DOI : https://doi.org/10.1214/10-AIHP383
Classification:  60G44,  62M09 
@article{AIHPB_2011__47_3_748_0,
     author = {Dalalyan, Arnak and Yoshida, Nakahiro},
     title = {Second-order asymptotic expansion for a non-synchronous covariation estimator},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {47},
     year = {2011},
     pages = {748-789},
     doi = {10.1214/10-AIHP383},
     mrnumber = {2841074},
     zbl = {pre05956526},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2011__47_3_748_0}
}

Dalalyan, Arnak; Yoshida, Nakahiro. Second-order asymptotic expansion for a non-synchronous covariation estimator. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) pp. 748-789. doi : 10.1214/10-AIHP383. http://gdmltest.u-ga.fr/item/AIHPB_2011__47_3_748_0/

[1] T. G. Andersen and T. Bollerslev. Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. Int. Econ. Rev. 39 (1998) 885-905.

[2] T. G. Andersen, T. Bollerslev, F. X. Diebold and H. Ebens. The distribution of realized stock return volatility. J. Fin. Econ. 61 (2001) 43-76.

[3] T. G. Andersen, T. Bollerslev, F. X. Diebold and P. Labys. The distribution of realized exchange rate volatility. J. Amer. Statist. Assoc. 96 (2001) 42-55. | MR 1952727 | Zbl 1015.62107

[4] G. J. Babu and K. Singh. On one term Edgeworth correction by Efron's bootstrap. Sankhyā A 46 (1984) 219-232. | MR 778872 | Zbl 0568.62019

[5] O. E. Barndorff-Nielsen, P. R. Hansen, A. Lunde and N. Shephard. Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading. Manuscript. Available at http://www.nuffield.ox.ac.uk/economics/papers/index2007and2008.aspx.

[6] O. E. Barndorff-Nielsen and N. Shephard. Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 (2002) 253-280. | MR 1904704 | Zbl 1059.62107

[7] P. Bertail and S. Clémencon. Edgeworth expansions of suitably normalized sample mean statistics for atomic Markov chains. Probab. Theory Related Fields 130 (2004) 388-414. | MR 2095936 | Zbl 1075.62075

[8] A. Bose. Edgeworth correction by bootstrap in autoregressions. Ann. Statist. 16 (1988) 1709-1722. | MR 964948 | Zbl 0653.62016

[9] F. Comte and E. Renault. Long memory in continuous-time stochastic volatility models. Math. Finance 8 (1998) 291-323. | MR 1645101 | Zbl 1020.91021

[10] D. Dacunha-Castelle and D. Florens-Zmirou. Estimation of the coefficients of diffusion from discrete observations. Stochastics 19 (1986) 263-284. | MR 872464 | Zbl 0626.62085

[11] T. Epps. Comovements in stock prices in the very short run. J. Amer. Statist. Assoc. 74 (1979) 291-298.

[12] D. Florens-Zmirou. On estimating the diffusion coefficient from discrete observations. J. Appl. Probab. 30 (1993) 790-804. | MR 1242012 | Zbl 0796.62070

[13] M. Fukasawa. Edgeworth expansion for ergodic diffusions. Probab. Theory Related Fields 142 (2008) 1-20. | MR 2413265 | Zbl 1154.60018

[14] V. Genon-Catalot and J. Jacod. On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann. Inst. Henri Poincaré 29 (1993) 119-151. | Numdam | MR 1204521 | Zbl 0770.62070

[15] J. E. Griffin and R. C. A. Oomen. Covariance measurement in the presence of non-synchronous trading and market microstructure noise. Preprint, 2006. Available at http://ssrn.com/abstract=912541. | MR 2745867

[16] P. Hall. The Bootstrap and Edgeworth Expansion. Springer, New York, 1992. | MR 1145237 | Zbl 0744.62026

[17] T. Hayashi and S. Kusuoka. Consistent estimation of covariation under non-synchronicity. Stat. Inference Stoch. Process. 11 (2008) 93-106. | MR 2357555 | Zbl 1148.62070

[18] T. Hayashi and N. Yoshida. On covariance estimation of non-synchronously observed diffusion processes. Bernoulli 11 (2005) 359-379. | MR 2132731 | Zbl 1064.62091

[19] T. Hayashi and N. Yoshida. Nonsynchronous covariance estimator and limit theorem. Preprint, 2006.

[20] T. Hayashi and N. Yoshida. Asymptotic normality of a covariance estimator for non-synchronously observed diffusion processes. Ann. Inst. Statist. Math. 60 (2008) 367-406. | MR 2403524 | Zbl pre05560793

[21] T. Hayashi and N. Yoshida. Nonsynchronous covariance estimator and limit theorem II. Preprint, 2008. | MR 2403524

[22] T. Hoshikawa, T. Kanatani, K. Nagai and Y. Nishiyama. Nonparametric estimation methods of integrated multivariate volatilities. Working paper, 2006. | Zbl pre05368899

[23] J. Jacod. On processes with conditional independent increments and stable convergence in law. Semin. Probab. Strasbourg 36 (2002) 383-401. | Numdam | MR 1971599 | Zbl 1034.60035

[24] H. Koul and D. Surgailis. Asymptotic expansion of M-estimators with long-memory errors. Ann. Statist. 25 (1997) 818-850. | MR 1439325 | Zbl 0885.62101

[25] M. Kessler. Estimation of diffusion processes from discrete observations. Scand. J. Statist. 24 (1997) 211-229. | MR 1455868 | Zbl 0879.60058

[26] A. W. Lo and A. C. Mackinlay. An econometric analysis of non-synchronous trading. J. Econometrics 45 (1990) 181-211. | MR 1067232 | Zbl 0712.62102

[27] P. Malliavin and M. E. Mancino. Fourier series method for measurement of multivariate volatilities. Finance Stoch. 6 (2002) 49-61. | MR 1885583 | Zbl 1008.62091

[28] P. A. Mykland. Asymptotic expansions for martingales. Ann. Probab. 21 (1993) 800-818. | MR 1217566 | Zbl 0776.60047

[29] P. A. Mykland. A Gaussian calculus for inference from high frequency data. Technical Report 563, Dept. Statistics, Univ. Chicago.

[30] P. A. Mykland and L. Zhang. Anova for diffusions and Ito processes. Ann. Statist. 34 (2006) 1931-1963. | MR 2283722

[31] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Springer, Berlin, 2006. | MR 2200233 | Zbl 0837.60050

[32] A. Palandri. Consistent realized covariance for asynchronous observations contaminated by market microstructure noise. Manuscript. Available at http://www.palandri.eu/research.html.

[33] B. L. S. Prakasa-Rao. Asymptotic theory for non-linear least square estimator for diffusion processes. Math. Oper. Statist. Ser. Stat. 14 (1983) 195-209. | MR 704787 | Zbl 0532.62060

[34] B. L. S. Prakasa-Rao. Statistical inference from sampled data for stochastic processes. Contemp. Math. 80 (1988) 249-284. | MR 999016 | Zbl 0687.62069

[35] C. Robert and M. Rosenbaum. Ultra high frequency volatility and co-volatility estimation in a microstructure model with uncertainty zones. Submitted.

[36] Y. Sakamoto and N. Yoshida. Asymptotic expansion under degeneracy. J. Japan Stat. Soc. 33 (2003) 145-156. | MR 2039891 | Zbl 1063.62115

[37] J. Shanken. Nonsynchronous data and the covariance-factor structure of returns. J. Finance 42 (1987) 221-231.

[38] A. N. Shiryaev. Probability, 2nd edition. Graduate Texts in Mathematics 95. Springer, New York, 1996. | MR 1368405 | Zbl 0835.60002

[39] M. Scholes and J. Williams. Estimating betas from non-synchronous data. J. Fin. Econ. 5 (1977) 309-328.

[40] T. J. Sweeting. Speeds of convergence for the multidimensional central limit theorem. Ann. Probab. 5 (1977) 28-41. | MR 428400 | Zbl 0362.60041

[41] V. Voev and A. Lunde. Integrated covariance estimation using high-frequency data in the presence of noise. Working paper. Presented at CIREQ Conference on Realized Volatility, 2006.

[42] N. Yoshida. Estimation for diffusion processes from discrete observation. J. Multivariate Anal. 41 (1992) 220-242. | MR 1172898 | Zbl 0811.62083

[43] N. Yoshida. Malliavin calculus and asymptotic expansion for martingales. Probab. Theory Related Fields 109 (1997) 301-342. | MR 1481124 | Zbl 0888.60020

[44] L. Zhang. Estimating covariation: Epps effect, microstructure noise. J. Econometrics (2010). To appear. | MR 2745865

[45] L. Zhang, P. A. Mykland and Y. Ait-Sahalia. Edgeworth expansions for realized volatility and related estimators. J. Econometrics (2010). To appear. | MR 2745877