Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data. (English) Zbl 1431.62472
Summary: We show how pre-averaging can be applied to the problem of measuring the ex-post covariance of financial asset returns under microstructure noise and non-synchronous trading. A pre-averaged realised covariance is proposed, and we present an asymptotic theory for this new estimator, which can be configured to possess an optimal convergence rate or to ensure positive semi-definite covariance matrix estimates. We also derive a noise-robust Hayashi-Yoshida estimator that can be implemented on the original data without prior alignment of prices. We uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high-frequency equity data.
MSC:
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 62M05 | Markov processes: estimation; hidden Markov models |
| 60F05 | Central limit and other weak theorems |
Keywords:
central limit theorem; diffusion models; high-frequency data; market microstructure noise; non-synchronous trading; pre-averaging; realised covarianceReferences:
| [1] | Aït-Sahalia, Y.; Mykland, P. A.; Zhang, L., How often to sample a continuous-time process in the presence of market microstructure noise, Review of Financial Studies, 18, 2, 351-416 (2005) |
| [2] | Aldous, D. J.; Eagleson, G. K., On mixing and stability of limit theorems, Annals of Probability, 6, 2, 325-331 (1978) · Zbl 0376.60026 |
| [3] | Andersen, T. G.; Bollerslev, T.; Diebold, F. X., Parametric and nonparametric volatility measurement, (Hansen, L. P.; Ait-Sahalia, Y., Handbook of Financial Econometrics (2010), North-Holland), pp. 67-138 |
| [4] | Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; Labys, P., The distribution of realized exchange rate volatility, Journal of the American Statistical Association, 96, 453, 42-55 (2001) · Zbl 1015.62107 |
| [5] | Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; Labys, P., Modeling and forecasting realized volatility, Econometrica, 71, 2, 579-625 (2003) · Zbl 1142.91712 |
| [6] | Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; Wu, J., Realized beta: persistence and predictability, (Fomby, T.; Terrell, D., Advances in Econometrics: Econometric Analysis of Economic and Financial Time Series in Honor of R.F. Engle, and C.W.J. Granger, vol. B (2006)), 1-40 · Zbl 1190.91145 |
| [7] | Back, K., Asset prices for general processes, Journal of Mathematical Economics, 20, 4, 371-395 (1991) · Zbl 0727.90014 |
| [8] | Bandi, F.M., Russell, J.R., 2005. Realized covariation, realized beta and microstructure noise. Working Paper. Graduate School of Business, University of Chicago.; Bandi, F.M., Russell, J.R., 2005. Realized covariation, realized beta and microstructure noise. Working Paper. Graduate School of Business, University of Chicago. |
| [9] | Bandi, F. M.; Russell, J. R., Separating microstructure noise from volatility, Journal of Financial Economics, 79, 3, 655-692 (2006) |
| [10] | Bandi, F. M.; Russell, J. R., Microstructure noise, realized variance, and optimal sampling, Review of Economic Studies, 75, 2, 339-369 (2008) · Zbl 1138.91394 |
| [11] | Barndorff-Nielsen, O. E.; Graversen, S. E.; Jacod, J.; Podolskij, M.; Shephard, N., A central limit theorem for realized power and bipower variations of continuous semimartingales, (Kabanov, Y.; Lipster, R.; Stoyanov, J., From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift (2006), Springer-Verlag), 33-68 · Zbl 1106.60037 |
| [12] | Barndorff-Nielsen, O. E.; Hansen, P. R.; Lunde, A.; Shephard, N., Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise, Econometrica, 76, 6, 1481-1536 (2008) · Zbl 1153.91416 |
| [13] | Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., Shephard, N., 2008b. Multivariate realised kernels: consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading. Working Paper. University of Aarhus.; Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., Shephard, N., 2008b. Multivariate realised kernels: consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading. Working Paper. University of Aarhus. · Zbl 1441.62599 |
| [14] | Barndorff-Nielsen, O. E.; Shephard, N., Econometric analysis of realized volatility and its use in estimating stochastic volatility models, Journal of the Royal Statistical Society: Series B, 64, 2, 253-280 (2002) · Zbl 1059.62107 |
| [15] | Barndorff-Nielsen, O. E.; Shephard, N., Econometric analysis of realized covariation: high frequency based covariance, regression, and correlation in financial economics, Econometrica, 72, 3, 885-925 (2004) · Zbl 1141.91634 |
| [16] | Barndorff-Nielsen, O. E.; Shephard, N., Variation, jumps, market frictions and high frequency data in financial econometrics, (Blundell, R.; Torsten, P.; Newey, W. K., Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress (2007), Cambridge University Press) · Zbl 1095.60023 |
| [17] | Boudt, K., Croux, C., Laurent, S., 2008. Outlyingness weighted quadratic covariation. Working Paper. Leuven, K.U.; Boudt, K., Croux, C., Laurent, S., 2008. Outlyingness weighted quadratic covariation. Working Paper. Leuven, K.U. |
| [18] | Christensen, K., Oomen, R.C.A., Podolskij, M., 2010. Realised quantile-based estimation of the integrated variance. Journal of Econometrics, forthcoming (doi:10.1016/j.jeconom.2010.04.008).; Christensen, K., Oomen, R.C.A., Podolskij, M., 2010. Realised quantile-based estimation of the integrated variance. Journal of Econometrics, forthcoming (doi:10.1016/j.jeconom.2010.04.008). · Zbl 1431.62473 |
| [19] | Christensen, K., Podolskij, M., Vetter, M., 2010. A central limit theorem for the Hayashi-Yoshida estimator under microstructure noise and non-synchronous trading. Working Paper. University of Aarhus.; Christensen, K., Podolskij, M., Vetter, M., 2010. A central limit theorem for the Hayashi-Yoshida estimator under microstructure noise and non-synchronous trading. Working Paper. University of Aarhus. |
| [20] | Corsi, F., Audrino, F., 2007. Realised correlation tick-by-tick. Working Paper. Available at: http://ssrn.com/abstract=957997.; Corsi, F., Audrino, F., 2007. Realised correlation tick-by-tick. Working Paper. Available at: http://ssrn.com/abstract=957997. · Zbl 1284.91465 |
| [21] | de Jong, F.; Nijman, T., High frequency analysis of lead-lag relationships between financial markets, Journal of Empirical Finance, 4, 2-3, 259-277 (1997) |
| [22] | Delbaen, F.; Schachermayer, W., A general version of the fundamental theorem of asset pricing, Mathematische Annalen, 300, 463-520 (1994) · Zbl 0865.90014 |
| [23] | Diebold, F.X., Strasser, G.H., 2008. On the correlation structure of microstructure noise in theory and practice. Working Paper. University of Pennsylvania.; Diebold, F.X., Strasser, G.H., 2008. On the correlation structure of microstructure noise in theory and practice. Working Paper. University of Pennsylvania. |
| [24] | Dimson, E., Risk measurement when shares are subject to infrequent trading, Journal of Financial Economics, 7, 2, 197-226 (1979) |
| [25] | Epps, T. W., Comovements in stock prices in the very short run, Journal of the American Statistical Association, 74, 366, 291-298 (1979) |
| [26] | Fan, J., Zhang, J., Yu, K., 2009. Asset allocation and risk assessment with gross exposure constraints for vast portfolios. Working Paper. Princeton University.; Fan, J., Zhang, J., Yu, K., 2009. Asset allocation and risk assessment with gross exposure constraints for vast portfolios. Working Paper. Princeton University. |
| [27] | Fisher, L., Some new stock-market indexes, Journal of Business, 39, 1-2, 191-225 (1966) |
| [28] | Fleming, J.; Kirby, C.; Ostdiek, B., The economic value of volatility timing, Journal of Finance, 56, 1, 329-352 (2001) |
| [29] | Fleming, J.; Kirby, C.; Ostdiek, B., The economic value of volatility timing using “realized” volatility, Journal of Financial Economics, 67, 3, 473-509 (2002) |
| [30] | Ghysels, E.; Harvey, A. C.; Renault, E., Stochastic volatility, (Maddala, G. S.; Rao, C. R., Handbook of Statistics, vol. 14 (1996), North-Holland), 119-191 |
| [31] | Glassermann, P., Monte Carlo Methods in Financial Engineering (2004), Springer-Verlag · Zbl 1038.91045 |
| [32] | Gloter, A.; Jacod, J., Diffusions with measurement errors. I — local asymptotic normality, ESAIM: Probability and Statistics, 5, 225-242 (2001) · Zbl 1008.60089 |
| [33] | Gloter, A.; Jacod, J., Diffusions with measurement errors. II — measurement errors, ESAIM: Probability and Statistics, 5, 243-260 (2001) · Zbl 1009.60065 |
| [34] | Griffin, J.E., Oomen, R.C.A., 2006. Covariance measurement in the presence of non-synchronous trading and market microstructure noise. Working Paper. University of Warwick.; Griffin, J.E., Oomen, R.C.A., 2006. Covariance measurement in the presence of non-synchronous trading and market microstructure noise. Working Paper. University of Warwick. · Zbl 1441.62710 |
| [35] | Hansen, P. R.; Lunde, A., Realized variance and market microstructure noise, Journal of Business and Economic Statistics, 24, 2, 127-161 (2006) |
| [36] | Hayashi, T.; Yoshida, N., On covariance estimation of non-synchronously observed diffusion processes, Bernoulli, 11, 2, 359-379 (2005) · Zbl 1064.62091 |
| [37] | Hayashi, T.; Yoshida, N., Asymptotic normality of a covariance estimator for nonsynchronously observed diffusion processes, Annals of the Institute of Statistical Mathematics, 60, 2, 367-406 (2008) · Zbl 1332.62284 |
| [38] | Jacod, J.; Li, Y.; Mykland, P. A.; Podolskij, M.; Vetter, M., Microstructure noise in the continuous case: the pre-averaging approach, Stochastic Processes and their Applications, 119, 7, 2249-2276 (2009) · Zbl 1166.62078 |
| [39] | Jacod, J.; Shiryaev, A. N., Limit Theorems for Stochastic Processes (2003), Springer-Verlag · Zbl 0830.60025 |
| [40] | Jagannathan, R.; Wang, Z., The conditional CAPM and the cross-section of expected returns, Journal of Finance, 51, 1, 3-53 (1996) |
| [41] | Kinnebrock, S., Podolskij, M., 2008. An econometric analysis of modulated realised covariance, regression and correlation in noisy diffusion models. Working Paper. CREATES, University of Aarhus.; Kinnebrock, S., Podolskij, M., 2008. An econometric analysis of modulated realised covariance, regression and correlation in noisy diffusion models. Working Paper. CREATES, University of Aarhus. · Zbl 1139.60310 |
| [42] | Large, J., 2007. Accounting for the Epps effect: realized covariation, cointegration and common factors. Working Paper. Oxford-Man Institute, University of Oxford.; Large, J., 2007. Accounting for the Epps effect: realized covariation, cointegration and common factors. Working Paper. Oxford-Man Institute, University of Oxford. |
| [43] | Lettau, M.; Ludvigson, S., Resurrecting the (C)CAPM: a cross-sectional test when risk premia are time-varying, Journal of Political Economy, 109, 6, 1238-1287 (2001) |
| [44] | Lo, A.; MacKinlay, C., An econometric analysis of nonsynchronous trading, Journal of Econometrics, 45, 1-2, 181-211 (1990) · Zbl 0712.62102 |
| [45] | Malliavin, P.; Mancino, M. E., Fourier series method for measurement of multivariate volatilities, Finance and Stochastics, 6, 1, 49-61 (2002) · Zbl 1008.62091 |
| [46] | Martens, M., 2003. Estimating unbiased and precise realized covariances. Working Paper. Erasmus School of Economics, Rotterdam.; Martens, M., 2003. Estimating unbiased and precise realized covariances. Working Paper. Erasmus School of Economics, Rotterdam. |
| [47] | Mykland, P. A.; Zhang, L., ANOVA for diffusions and Itô processes, Annals of Statistics, 34, 4, 1931-1963 (2006) · Zbl 1246.91110 |
| [48] | Mykland, P. A.; Zhang, L., Inference for continuous semimartingales observed at high frequency: A general approach, Econometrica, 77, 5, 1403-1445 (2009) · Zbl 1182.62216 |
| [49] | Oomen, R. C.A., Properties of realized variance under alternative sampling schemes, Journal of Business and Economic Statistics, 24, 2, 219-237 (2006) |
| [50] | Palandri, A., 2006. Consistent realized covariance for asynchronous observations contaminated by market microstructure noise. Working Paper. University of Copenhagen.; Palandri, A., 2006. Consistent realized covariance for asynchronous observations contaminated by market microstructure noise. Working Paper. University of Copenhagen. |
| [51] | Podolskij, M.; Vetter, M., Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps, Bernoulli, 15, 3, 634-658 (2009) · Zbl 1200.62131 |
| [52] | Renò, R., A closer look at the Epps effect, International Journal of Theoretical and Applied Finance, 6, 1, 87-102 (2003) · Zbl 1079.91537 |
| [53] | Rényi, A., On stable sequences of events, Sankhya: The Indian Journal of Statistics; Series A, 25, 3, 293-302 (1963) · Zbl 0141.16401 |
| [54] | Scholes, M.; Williams, J., Estimating betas from nonsynchronous data, Journal of Financial Economics, 5, 3, 309-327 (1977) |
| [55] | Voev, V.; Lunde, A., Integrated covariance estimation using high-frequency data in the presence of noise, Journal of Financial Econometrics, 5, 1, 68-104 (2007) |
| [56] | Zhang, L., Efficient estimation of stochastic volatility using noisy observations: a multi-scale approach, Bernoulli, 12, 6, 1019-1043 (2006) · Zbl 1117.62119 |
| [57] | Zhang, L., 2008. Estimating covariation: Epps effect, microstructure noise. Working Paper. University of Illinois at Chicago.; Zhang, L., 2008. Estimating covariation: Epps effect, microstructure noise. Working Paper. University of Illinois at Chicago. |
| [58] | Zhang, L.; Mykland, P. A.; Aït-Sahalia, Y., A tale of two time scales: determining integrated volatility with noisy high-frequency data, Journal of the American Statistical Association, 100, 472, 1394-1411 (2005) · Zbl 1117.62461 |
| [59] | Zhou, B., High-frequency data and volatility in foreign-exchange rates, Journal of Business and Economic Statistics, 14, 1, 45-52 (1996) |
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