Local Parametric Estimation in High Frequency Data

Abstract

We give a general time-varying parameter model, where the multidimensional parameter possibly includes jumps. The quantity of interest is defined as the integrated value over time of the parameter process $\Theta =T^{-1}{\int }_0^T{\theta }_t^{*}dt$. We provide a local parametric estimator (LPE) of Θ and conditions under which we can show the central limit theorem. Roughly speaking those conditions correspond to some uniform limit theory in the parametric version of the problem. The framework is restricted to the specific convergence rate n^1∕2. Several examples of LPE are studied: estimation of volatility, powers of volatility, volatility when incorporating trading information and time-varying MA(1).

Key Words:

• Integrated volatility
• Market microstructure noise
• Powers of volatility
• Quasi-maximum likelihood estimator

ACKNOWLEDGMENTS

We are indebted to the editor, Todd Clark, two anonymous referees, and an anonymous associate editor, Simon Clinet, Takaki Hayashi, Dacheng Xiu, participants of the seminars in Berlin and Tokyo and conferences in Osaka, Toyama, the SoFie annual meeting in Hong Kong, the PIMS meeting in Edmonton for valuable comments, which helped in improving the quality of the paper.

Notes

1 If we set down the asymptotic theory in the same way as in Dahlhaus (1997, p. 3), we conjecture that the results of this article would stay true.

2 Note that the local approach in this article is related to the large-T-based-approach and problem of Giraitis, Kapetanios, and Yates (2014).

3 It is possible to specify the problem with a general rate of convergence, but all the considered examples from this article are with convergence rate n1∕2.

4 One can look at definitions of stable convergence in Rényi (1963), Aldous and Eagleson (1978), Chapter 3 (p. 56) of Hall and Heyde (1980), Rootzén (1980), Section 2 (pp. 169–170) of Jacod and Protter (1998), Definition VIII.5.28 in Jacod and Shiryaev (2003) or Definition 1 in Podolskij and Vetter (2010).

5 The related assumption is that τ is a ${\mathcal{J}}_t$-stopping time.

6 We assume that $F_n(x,y,z)$ is jointly measurable, and that $P_{t,n}$ is taking values on a Borel space. Additionally, we assume that for any $(P_{s,n},U_{i,n},{\theta }_s^{*})$, we have $\mathbb{E}|F_n(P_{s,n},U_n,{\theta }_s^{*})|<\infty$

7 The advised reader will have noticed that Fn is not a function in the ordinary sense. We still abusively refer to it as a “function.”

8 Here and in the following statements, the stable convergence in law is with respect to the filtration considered in Li, Xie, and Zheng (2016).

9 The advised reader will have noticed that a priori, $\text{sign}(R_{i,n})$ and ηt are not independent, so that the assumptions of the LPM do not hold entirely. This problem can be circumvented as the former is actually conditionally independent from the latter.

10 Actually, the estimator considered here slightly differs from the original definition (p. 8) in Robert and Rosenbaum (2012) as it provides smaller theoretical finite sample bias. Asymptotically, both estimators are equivalent and thus all the theory provided in Robert and Rosenbaum (2012) can be used to prove Theorem 7.

11 We can actually show that any time series in state space form can be expressed with a corresponding Fn function.

12 The MLE is always performed on a compact set, so the assumption is trivially satisfied in that case, which corresponds to Examples 4.1–4.3. Moreover, the estimator of η in Example 4.4 is bounded by definition, but one would need to bound the volatility estimator to apply the technique.

13 Details about the model can be found in a previous version of the manuscript circulated under the name “Estimating the Integrated Parameter of the Locally Parametric Model in High-Frequency Data.”