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
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 \({{\mathcal {H}}}_t = {{\mathcal {F}}}_t \vee \sigma \{Q_{t_i}, 0\le i\le N\}\).
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).
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.
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.
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 \({\mathbf {G}}\).
Here and in the other theorems, we mean that \(B_t\) is independent of the underlying \(\sigma\)-field \({\mathbf {F}}\).
Note that the definition of \({\widetilde{c}}_{i}\) slightly diverges from the previous section.
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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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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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DOI: https://doi.org/10.1007/s10463-020-00762-3