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Estimation for high-frequency data under parametric market microstructure noise

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Abstract

We develop a general class of noise-robust estimators based on the existing estimators in the non-noisy high-frequency data literature. The microstructure noise is a parametric function of the limit order book. The noise-robust estimators are constructed as plug-in versions of their counterparts, where we replace the efficient price, which is non-observable, by an estimator based on the raw price and limit order book data. We show that the technology can be applied to five leading examples where, depending on the problem, price possibly includes infinite jump activity and sampling times encompass asynchronicity and endogeneity.

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Notes

  1. All the defined quantities are implicitly or explicitly indexed by n (except for the integrated parameter which does not depend on n). For example N should be thought and considered as \(N_n\). Consistency and convergence in law refer to the behavior as \(n \rightarrow \infty\). A full specification of the model also involves the stochastic basis \({{\mathcal {B}}}=(\Omega ,{\mathbb {P}},{{\mathcal {F}}},{\mathbf{F}})\), where \({{\mathcal {F}}}\) is a \(\sigma\)-field and \({\mathbf{F}}=({{\mathcal {F}}}_t)_{t\in [0,T]}\) is a filtration, which will be example-specific. We assume that all the processes (including the integrated parameter \(\xi _t\)) are \({\mathbf{F}}\)-adapted (either in a continuous or discrete meaning for \(Q_{t_i}\)) and that the observation times \(t_i\) are \({\mathbf{F}}\)-stopping times. Also, when referring to Itô-semimartingale and stable convergence in law, we automatically mean that the statement is relative to \({\mathbf{F}}\). Finally, we assume in (13) that W is also a Brownian motion under the larger filtration \({{\mathcal {H}}}_t = {{\mathcal {F}}}_t \vee \sigma \{Q_{t_i}, 0\le i\le N\}\).

  2. Note that we do not assume that \(Q_t\) exists for any \(t \in [0,T] - \{t_0, \dots , t_N\}\) as it is often the case in the i.i.d setting, see, e.g., the framework in Jacod et al. (2009).

  3. Here the restriction \(r < 1\) follows from Jacod and Rosenbaum (2013). Indeed, even for the realized volatility problem, (16) may not happen in the case \(r > 1\). Indeed, it yields a different optimal rate of convergence as shown in Jacod and Reiss (2014) (of the form \(N^\kappa {\text {log}}N\) for some \(\kappa >0\)). Moreover, as explained in their Remark 3.4, a CLT is not even achievable in some cases. The case \(r=1\) is let aside. Such bordercase is examined in Vetter (2010) when considering the bipower variation.

  4. Remark 6 (p. 36) in Li et al. (2016) suggests that the threshold RV estimator can be used under endogeneity, but there is no formal proof and this is limited to the case of jumps with finite activity.

  5. i.e. we assume that \(t_i\) are \({\mathbf {G}}\)-stopping times, where \({\mathbf {G}} =({{\mathcal {G}}}_t)_{t \in [0,T]}\) is a sub-filtration of \({\mathbf {F}}\) generated by finitely many Brownian motions, and that \(b_t\), \(\sigma _t\) and \(\delta\) are adapted to \({\mathbf {G}}\).

  6. Here and in the other theorems, we mean that \(B_t\) is independent of the underlying \(\sigma\)-field \({\mathbf {F}}\).

  7. Note that the definition of \({\widetilde{c}}_{i}\) slightly diverges from the previous section.

References

  • Aït-Sahalia, Y., Mykland, P. A., Zhang, L. (2005). How often to sample a continuous-time process in the presence of market microstructure noise. Review of Financial Studies, 18(2), 351–416.

    MATH  Google Scholar 

  • Almgren, R., Chriss, N. (2001). Optimal execution of portfolio transactions. Journal of Risk, 3, 5–40.

    Google Scholar 

  • Altmeyer, R., Bibinger, M. (2015). Functional stable limit theorems for quasi-efficient spectral covolatility estimators. Stochastic Processes and Their Applications, 125(12), 4556–4600.

    MathSciNet  MATH  Google Scholar 

  • Andersen, T. G., Bollerslev, T., Diebold, F. X., Ebens, H. (2001a). The distribution of realized stock return volatility. Journal of Financial Economics, 61(1), 43–76.

    Google Scholar 

  • Andersen, T. G., Bollerslev, T., Diebold, F. X., Labys, P. (2001b). The distribution of realized exchange rate volatility. Journal of the American Statistical Association, 96(453), 42–55.

    MathSciNet  MATH  Google Scholar 

  • Andersen, T. G., Dobrev, D., Schaumburg, E. (2014). A robust neighborhood truncation approach to estimation of integrated quarticity. Econometric Theory, 30(1), 3–59.

    MathSciNet  MATH  Google Scholar 

  • Andersen, T. G., Cebiroglu, G., Hautsch, N. (2017). Volatility, information feedback and market microstructure noise: A tale of two regimes. Retrieved 9 May 2017. Working paper available at https://www.papersssrncom/sol3/paperscfm?abstract_id=2921097.

  • Andersen, T. G., Thyrsgaard, M., Todorov, V. (2019). Time-varying periodicity in intraday volatility. Journal of the American Statistical Association, 114(528), 1695–1707.

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen, O. E., Shephard, N. (2002a). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society Series B (Methodological), 64(2), 253–280.

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen, O. E., Shephard, N. (2002b). Estimating quadratic variation using realized variance. Journal of Applied Econometrics, 17(5), 457–477.

    Google Scholar 

  • Barndorff-Nielsen, O. E., Shephard, N. (2004). Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics, 2(1), 1–37.

    Google Scholar 

  • Barndorff-Nielsen, O. E., Shephard, N. (2006). Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics, 4(1), 1–30.

    Google Scholar 

  • Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J., Podolskij, M., Shephard, N. (2006a). A central limit theorem for realised power and bipower variations of continuous semimartingales. From stochastic calculus to mathematical finance, pp. 33–68. Berlin: Springer.

    Google Scholar 

  • Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J., Shephard, N. (2006b). Limit theorems for bipower variation in financial econometrics. Econometric Theory, 22(4), 677–719.

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen, O. E., Shephard, N., Winkel, M. (2006c). Limit theorems for multipower variation in the presence of jumps. Stochastic Processes and Their Applications, 116(5), 796–806.

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N. (2008). Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica, 76(6), 1481–1536.

    MathSciNet  MATH  Google Scholar 

  • Bibinger, M., Vetter, M. (2015). Estimating the quadratic covariation of an asynchronously observed semimartingale with jumps. Annals of the Institute of Statistical Mathematics, 67(4), 707–743.

    MathSciNet  MATH  Google Scholar 

  • Black, F. (1986). Noise. The Journal of Finance, 41(3), 528–543.

    Google Scholar 

  • Chaker, S. (2017). On high frequency estimation of the frictionless price: The use of observed liquidity variables. Journal of Econometrics, 201(1), 127–143.

    MathSciNet  MATH  Google Scholar 

  • Clinet, S., Potiron, Y. (2018). Efficient asymptotic variance reduction when estimating volatility in high frequency data. Journal of Econometrics, 206(1), 103–142.

    MathSciNet  MATH  Google Scholar 

  • Clinet, S., Potiron, Y. (2019a). Disentangling sources of high frequency market microstructure noise. Journal of Business & Economic Statistics, https://doi.org/10.1080/07350015.2019.1617158.

    Article  MATH  Google Scholar 

  • Clinet, S., Potiron, Y. (2019b). Testing if the market microstructure noise is fully explained by the informational content of some variables from the limit order book. Journal of Econometrics, 209(2), 289–337.

    MathSciNet  MATH  Google Scholar 

  • Cont, R., Kukanov, A., Stoikov, S. (2014). The price impact of order book events. Journal of Financial Econometrics, 12(1), 47–88.

    Google Scholar 

  • Corsi, F., Pirino, D., Reno, R. (2010). Threshold bipower variation and the impact of jumps on volatility forecasting. Journal of Econometrics, 159(2), 276–288.

    MathSciNet  MATH  Google Scholar 

  • Diebold, F. X., Strasser, G. (2013). On the correlation structure of microstructure noise: A financial economic approach. The Review of Economic Studies, 80(4), 1304–1337.

    MathSciNet  MATH  Google Scholar 

  • Glosten, L. R., Harris, L. E. (1988). Estimating the components of the bid/ask spread. Journal of Financial Economics, 21(1), 123–142.

    Google Scholar 

  • Hasbrouck, J. (1993). Assessing the quality of a security market: A new approach to transaction-cost measurement. The Review of Financial Studies, 6(1), 191–212.

    Google Scholar 

  • Hasbrouck, J. (2002). Stalking the “efficient price” in market microstructure specifications: An overview. Journal of Financial Markets, 5(3), 329–339.

    Google Scholar 

  • Hasbrouck, J. (2007). Empirical market microstructure: The institutions, economics, and econometrics of securities trading. New York: Oxford University Press.

    Google Scholar 

  • Hayashi, T., Kusuoka, S. (2008). Consistent estimation of covariation under nonsynchronicity. Statistical Inference for Stochastic Processes, 11(1), 93–106.

    MathSciNet  MATH  Google Scholar 

  • Hayashi, T., Yoshida, N. (2005). On covariance estimation of non-synchronously observed diffusion processes. Bernoulli, 11(2), 359–379.

    MathSciNet  MATH  Google Scholar 

  • Hayashi, T., Yoshida, N. (2008). Asymptotic normality of a covariance estimator for nonsynchronously observed diffusion processes. Annals of the Institute of Statistical Mathematics, 60(2), 367–406.

    MathSciNet  MATH  Google Scholar 

  • Hayashi, T., Yoshida, N. (2011). Nonsynchronous covariation process and limit theorems. Stochastic Processes and Their Applications, 121(10), 2416–2454.

    MathSciNet  MATH  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(7), 2249–2276.

    MathSciNet  MATH  Google Scholar 

  • Jacod, J., Podolskij, M., Vetter, M. (2010). Limit theorems for moving averages of discretized processes plus noise. Annals of Statistics, 38(3), 1478–1545.

    MathSciNet  MATH  Google Scholar 

  • Jacod, J., Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Annals of Probability, 26, 267–307.

    MathSciNet  MATH  Google Scholar 

  • Jacod, J., Protter, P. E. (2011). Discretization of processes. Berlin: Springer.

    MATH  Google Scholar 

  • Jacod, J., Reiss, M. (2014). A remark on the rates of convergence for integrated volatility estimation in the presence of jumps. Annals of Statistics, 42(3), 1131–1144.

    MathSciNet  MATH  Google Scholar 

  • Jacod, J., Rosenbaum, M. (2013). Quarticity and other functionals of volatility: Efficient estimation. Annals of Statistics, 41(3), 1462–1484.

    MathSciNet  MATH  Google Scholar 

  • Kavajecz, K. A. (1999). A specialist’s quoted depth and the limit order book. The Journal of Finance, 54(2), 747–771.

    Google Scholar 

  • Kinnebrock, S., Podolskij, M. (2008). A note on the central limit theorem for bipower variation of general functions. Stochastic Processes and Their Applications, 118(6), 1056–1070.

    MathSciNet  MATH  Google Scholar 

  • Koike, Y. (2014a). An estimator for the cumulative co-volatility of asynchronously observed semimartingales with jumps. Scandinavian Journal of Statistics, 41(2), 460–481.

    MathSciNet  MATH  Google Scholar 

  • Koike, Y. (2014b). Limit theorems for the pre-averaged Hayashi–Yoshida estimator with random sampling. Stochastic Processes and Their Applications, 124(8), 2699–2753.

    MathSciNet  MATH  Google Scholar 

  • Koike, Y. (2016). Estimation of integrated covariances in the simultaneous presence of nonsynchronicity, microstructure noise and jumps. Econometric Theory, 32(3), 533–611.

    MathSciNet  MATH  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(7), 2696–2727.

    MathSciNet  MATH  Google Scholar 

  • Li, Y., Mykland, P. A., Renault, E., Zhang, L., Zheng, X. (2014). Realized volatility when sampling times are possibly endogenous. Econometric Theory, 30(03), 580–605.

    MathSciNet  MATH  Google Scholar 

  • Li, Y., Xie, S., Zheng, X. (2016). Efficient estimation of integrated volatility incorporating trading information. Journal of Econometrics, 195(1), 33–50.

    MathSciNet  MATH  Google Scholar 

  • Madhavan, A. (2000). Market microstructure: A survey. Journal of Financial Markets, 3(3), 205–258.

    Google Scholar 

  • Madhavan, A., Richardson, M., Roomans, M. (1997). Why do security prices change? A transaction-level analysis of NYSE stocks. The Review of Financial Studies, 10(4), 1035–1064.

    Google Scholar 

  • Mancini, C. (2009). Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps. Scandinavian Journal of Statistics, 36(2), 270–296.

    MathSciNet  MATH  Google Scholar 

  • Mancini, C. (2011). The speed of convergence of the threshold estimator of integrated variance. Stochastic Processes and Their Applications, 121(4), 845–855.

    MathSciNet  MATH  Google Scholar 

  • Mancino, M. E., Sanfelici, S. (2012). Estimation of quarticity with high-frequency data. Quantitative Finance, 12(4), 607–622.

    MathSciNet  MATH  Google Scholar 

  • Martin, O., Vetter, M. (2020). The null hypothesis of (common) jumps in case of irregular and asynchronous observations. Scandinavian Journal of Statistics, 47(3), 711–756.

    MathSciNet  MATH  Google Scholar 

  • Meddahi, N. (2002). A theoretical comparison between integrated and realized volatility. Journal of Applied Econometrics, 17(5), 479–508.

    Google Scholar 

  • Mykland, P. A., Zhang, L. (2006). ANOVA for diffusions and Ito processes. Annals of Statistics, 34(4), 1931–1963.

    MathSciNet  MATH  Google Scholar 

  • Mykland, P. A., Zhang, L. (2009). Inference for continuous semimartingales observed at high frequency. Econometrica, 77(5), 1403–1445.

    MathSciNet  MATH  Google Scholar 

  • Mykland, P. A., Zhang, L. (2012). The econometrics of high frequency data. In M. Kessler, A. Lindner, M. Sørensen (Eds.), Statistical methods for stochastic differential equations, pp. 109–190. Boca Raton, FL: Chapman and Hall/CRC Press.

    Google Scholar 

  • Mykland, P. A., Shephard, N., Sheppard, K. (2012). Efficient and feasible inference for the components of financial variation using blocked multipower variation. Technical Report, University of Oxford, available at https://doi.org/10.2139/ssrn2008690.

  • O’hara, M. (1995). Market microstructure theory, Vol. 108. Cambridge, MA: Blackwell.

    Google Scholar 

  • Potiron, Y., Mykland, P. A. (2016). Local parametric estimation in high frequency data. Working paper available at arXiv:160305700.

  • Potiron, Y., Mykland, P. A. (2017). Estimation of integrated quadratic covariation with endogenous sampling times. Journal of Econometrics, 197(1), 20–41.

    MathSciNet  MATH  Google Scholar 

  • Reiss, M. (2011). Asymptotic equivalence for inference on the volatility from noisy observations. Annals of Statistics, 39(2), 772–802.

    MathSciNet  MATH  Google Scholar 

  • Renault, E., Sarisoy, C., Werker, B. J. (2017). Efficient estimation of integrated volatility and related processes. Econometric Theory, 33(2), 439–478.

    MathSciNet  MATH  Google Scholar 

  • Robert, C. Y., Rosenbaum, M. (2010). A new approach for the dynamics of ultra-high-frequency data: The model with uncertainty zones. Journal of Financial Econometrics, 9(2), 344–366.

    Google Scholar 

  • Robert, C. Y., Rosenbaum, M. (2012). Volatility and covariation estimation when microstructure noise and trading times are endogenous. Mathematical Finance, 22(1), 133–164.

    MathSciNet  MATH  Google Scholar 

  • Roll, R. (1984). A simple implicit measure of the effective bid-ask spread in an efficient market. The Journal of Finance, 39(4), 1127–1139.

    Google Scholar 

  • Stoll, H. R. (2000). Presidential address: Friction. The Journal of Finance, 55(4), 1479–1514.

    Google Scholar 

  • Vetter, M. (2010). Limit theorems for bipower variation of semimartingales. Stochastic Processes and Their Applications, 120(1), 22–38.

    MathSciNet  MATH  Google Scholar 

  • Vetter, M. (2015). Estimation of integrated volatility of volatility with applications to goodness-of-fit testing. Bernoulli, 21(4), 2393–2418.

    MathSciNet  MATH  Google Scholar 

  • Xiu, D. (2010). Quasi-maximum likelihood estimation of volatility with high frequency data. Journal of Econometrics, 159(1), 235–250.

    MathSciNet  MATH  Google Scholar 

  • Zhang, L. (2001). From martingales to ANOVA: Implied and realized volatility. PhD thesis, University of Chicago, Department of Statistics, Chicago.

  • Zhang, L. (2006). Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli, 12(6), 1019–1043.

    MathSciNet  MATH  Google Scholar 

  • Zhang, L., Mykland, P. A., Aït-Sahalia, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association, 100(472), 1394–1411.

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank Selma Chaker, Yingying Li, Xinghua Zheng, Manh Cuong Pham, Mathias Vetter, two anonymous referees, the participants of the 2nd International Conference on Econometrics and Statistics in Hong Kong, and the Econometric Society Australasian Meeting 2018 in Auckland for helpful discussions and advice. The research of Yoann Potiron is supported by a special private grant from Keio University and Japanese Society for the Promotion of Science Grant-in-Aid for Young Scientists No. 60781119. The research of Simon Clinet is supported by Japanese Society for the Promotion of Science Grant-in-Aid for Young Scientists No. 19K13671.

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  1. Faculty of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo, 108-8345, Japan

    Simon Clinet

  2. Faculty of Business and Commerce, Keio University, 2-15-45 Mita, Minato-ku, Tokyo, 108-8345, Japan

    Yoann Potiron

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  1. Simon Clinet
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Correspondence to Simon Clinet.

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Clinet, S., Potiron, Y. Estimation for high-frequency data under parametric market microstructure noise. Ann Inst Stat Math 73, 649–669 (2021). https://doi.org/10.1007/s10463-020-00762-3

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