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Estimating the quadratic covariation matrix from noisy observations: local method of moments and efficiency. (English) Zbl 1302.62190
Summary: An efficient estimator is constructed for the quadratic covariation or integrated co-volatility matrix of a multivariate continuous martingale based on noisy and nonsynchronous observations under high-frequency asymptotics. Our approach relies on an asymptotically equivalent continuous-time observation model where a local generalised method of moments in the spectral domain turns out to be optimal. Asymptotic semiparametric efficiency is established in the Cramér-Rao sense. Main findings are that nonsynchronicity of observation times has no impact on the asymptotics and that major efficiency gains are possible under correlation. Simulations illustrate the finite-sample behaviour.

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G05 Nonparametric estimation
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[1] Aït-Sahalia, Y., Fan, J. and Xiu, D. (2010). High-frequency covariance estimates with noisy and asynchronous financial data. J. Amer. Statist. Assoc. 105 1504-1517. · Zbl 1388.62303 · doi:10.1198/jasa.2010.tm10163
[2] Altmeyer, R. and Bibinger, M. (2014). Functional stable limit theorems for efficient spectral covolatility estimators. Preprint. Available at . · Zbl 1327.62188 · arxiv.org
[3] Andersen, T. and Bollerslev, T. (1997). Intraday perdiodicity and volatility persistence in financial markets. J. Empir. Financ. 4 115-158.
[4] Andersen, T. G., Bollerslev, T. and Diebold, F. X. (2010). Parametric and nonparametric volatility measurement. In Handbook of Financial Econometrics (Y. Aït-Sahalia and L. P. Hansen, eds.) 67-137. Elsevier, Amsterdam.
[5] Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A. and Shephard, N. (2011). Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and nonsynchronous trading. J. Econometrics 162 149-169. · Zbl 1441.62599 · doi:10.1016/j.jeconom.2010.07.009
[6] Barndorff-Nielsen, O. E. and Shephard, N. (2004). Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics. Econometrica 72 885-925. · Zbl 1141.91634 · doi:10.1111/j.1468-0262.2004.00515.x
[7] Bibinger, M., Hautsch, N., Malec, P. and Reiß, M. (2014). Supplement to “Estimating the quadratic covariation matrix from noisy observations: Local method of moments and efficiency.” . · Zbl 1302.62190 · doi:10.1214/14-AOS1224 · dx.doi.org
[8] Bibinger, M. and Reiß, M. (2014). Spectral estimation of covolatility from noisy observations using local weights. Scand. J. Stat. 41 23-50. · Zbl 1349.62441 · doi:10.1111/sjos.12019
[9] Christensen, K., Podolskij, M. and Vetter, M. (2013). On covariation estimation for multivariate continuous Itô semimartingales with noise in nonsynchronous observation schemes. J. Multivariate Anal. 120 59-84. · Zbl 1293.62172 · doi:10.1016/j.jmva.2013.05.002
[10] Ciesielski, Z., Kerkyacharian, G. and Roynette, B. (1993). Quelques espaces fonctionnels associés à des processus gaussiens. Studia Math. 107 171-204. · Zbl 0809.60004 · eudml:216028
[11] Cohen, A. (2003). Numerical Analysis of Wavelet Methods. Studies in Mathematics and Its Applications 32 . North-Holland, Amsterdam. · Zbl 1038.65151
[12] Fackler, P. L. (2005). Notes on matrix calculus. Lecture notes, North Carolina State Univ. Available at . · www4.ncsu.edu
[13] Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica 50 1029-1054. · Zbl 0502.62098 · doi:10.2307/1912775
[14] Hayashi, T. and Yoshida, N. (2011). Nonsynchronous covariation process and limit theorems. Stochastic Process. Appl. 121 2416-2454. · Zbl 1233.60014 · doi:10.1016/j.spa.2010.12.005
[15] Jacod, J. and Rosenbaum, M. (2013). Quarticity and other functionals of volatility: Efficient estimation. Ann. Statist. 41 1462-1484. · Zbl 1292.60033 · doi:10.1214/13-AOS1115
[16] Kallenberg, O. (2002). Foundations of Modern Probability , 2nd ed. Probability and Its Applications ( New York ). Springer, New York. · Zbl 0996.60001
[17] Kittaneh, F. (1985). On Lipschitz functions of normal operators. Proc. Amer. Math. Soc. 94 416-418. · Zbl 0549.47006 · doi:10.2307/2045225
[18] Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation , 2nd ed. Springer, New York. · Zbl 0916.62017
[19] Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics : Some Basic Concepts , 2nd ed. Springer, New York. · Zbl 0952.62002
[20] Li, Y., Mykland, P. A., Renault, E., Zhang, L. and Zheng, X. (2014). Realized volatility when sampling times are possibly endogenous. Econometric Theory 30 580-605. · Zbl 1296.91290 · doi:10.1017/S0266466613000418
[21] Liu, C. and Tang, C. Y. (2014). A quasi-maximum likelihood approach for integrated covariance matrix estimation with high frequency data. J. Econometrics 180 217-232. · Zbl 1293.91196 · doi:10.1016/j.jeconom.2014.01.008
[22] Reiß, M. (2011). Asymptotic equivalence for inference on the volatility from noisy observations. Ann. Statist. 39 772-802. · Zbl 1215.62113 · doi:10.1214/10-AOS855
[23] Shephard, N. and Xiu, D. (2012). Econometric analysis of multivariate realised QML: Efficient positive semi-definite estimators of the covariation of equity prices. · Zbl 1391.62295
[24] Zhang, L. (2011). Estimating covariation: Epps effect, microstructure noise. J. Econometrics 160 33-47. · Zbl 1441.62911 · doi:10.1016/j.jeconom.2010.03.012
[25] 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. · Zbl 1117.62461 · doi:10.1198/016214505000000169 · miranda.asa.catchword.org
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