Abstract
The SIML (abbreviation of Separating Information Maximal Likelihood) method, has been introduced by N. Kunitomo and S. Sato and their collaborators to estimate the integrated volatility of high-frequency data that is assumed to be an Itô process but with so-called microstructure noise. The SIML estimator turned out to share many properties with the estimator introduced by P. Malliavin and M. E. Mancino. The present paper establishes the consistency and the asymptotic normality under a general sampling scheme but without microstructure noise. Specifically, a fast convergence shown for Malliavin–Mancino estimator by E. Clement and A. Gloter is also established for the SIML estimator.
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Notes
It has been pointed out that the “bias” \( \textbf{E} [ V^{j,j'} - \widehat{\Sigma ^{j,j'}_{n,m_n}}(0)] \) converges to zero and the mean square error \( \textbf{E} [ (V^{j,j'} - \widehat{\Sigma ^{j,j'}_{n,m_n}}(q) )^2 \) does not diverge when \( m_n = o (n) \) as \( n \rightarrow \infty \), which is not the case with the realized volatility.
To be precise, they used the IBP technique to prove the convergence of \( \textrm{Res}_t \) in (3.1) and \( \langle M, W\rangle \), but used another approach to prove the convergence of \( I^2 \) for which the Malliavin differentiability for b is not required.
Here we avoid using the commonly used notation D for the derivative so as not to mix it up with the Dirichlet kernel.
Misaki and Kunitomo (2015) studied a random sampling scheme for SIML in a mild situation.
References
Clément, E., & Gloter, A. (2011). Limit theorems in the Fourier transform method for the estimation of multivariate volatility. Stochastic Processes and Their Applications, 121, 1097–1124.
Fukasawa, M. (2010). Realized volatility with stochastic sampling. Stochastic Processes and Their Applications, 120, 829–852.
Jacod, J. (1997). On continuous conditional gaussian martingales and stable convergence in law, Seminaire de Probabilites, XXXI. Lecture Notes in Mathematics (Vol. 1655, pp. 232–246). Springer.
Jacod, J., & Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Annals of Probability, 26(1), 267–307.
Kunitomo, N., & Kurisu, D. (2021). Detecting factors of quadratic variation in the presence of market microstructure noise. Japanese Journal of Statistics and Data Science, 4(1), 601–641.
Kunitomo, N., & Sato, S. (2008a). Separating information maximum likelihood estimation of realized volatility and covariance with micro-market noise. Discussion Paper CIRJE-F-581, Graduate School of Economics, University of Tokyo.
Kunitomo, N., & Sato, S. (2008b). Realized Volatility, Covariance and Hedging Coefficient of Nikkei 225 Futures with Micro-Market Noise. Discussion Paper CIRJE-F-601, Graduate School of Economics, University of Tokyo.
Kunitomo, N., & Sato, S. (2010). Robustness of the separating information maximum likelihood estimation of realized volatility with micro-market noise. CIRJE Discussion Paper F-733, University of Tokyo.
Kunitomo, N., & Sato, S. (2011). The SIML estimation of realized volatility of the Nikkei-225 futures and hedging coefficient with micro-market noise. Mathematics and Computers in Simulation, 81, 1272–1289.
Kunitomo, N., & Sato, S. (2013). Separating information maximum likelihood estimation of realized volatility and covariance with micro-market noise. North American Journal of Economics and Finance, 26, 282–309.
Kunitomo, N., & Sato, S. (2022). Local SIML estimation of some Brownian and jump functionals under market micro-structure noise. Japanese Journal of Statistics and Data Science, 5(2), 831–870.
Kunitomo, N., Sato, S., & Kurisu, D. (2018). Separating Information Maximum Likelihood Method for High-Frequency Financial Data, Springer Briefs in Statistics. JSS Research Series in Statistics. Springer.
Malliavin, P., & Mancino, M. E. (2002). Fourier series method for measurement of multivariate volatilities. Finance and Stochastics, 6, 49–61.
Malliavin, P., & Mancino, M. E. (2009). A Fourier transform method for nonparametric estimation of multivariate volatility. Annals of Statistics, 37, 1983–2010.
Mancino, M. E., Recchioni, M. C., & Sanfelici, S. (2017). Fourier-Malliavin Volatility Estimation, Theory and Practice. Springer Briefs in Quantitative Finance. Springer.
Mancino, M. E., & Sanfelici, S. (2008). Robustness of Fourier estimator of integrated volatility in the presence of microstructure noise. Computational Statistics and Data Analysis, 52, 2966–2989.
Mancino, M. E., & Sanfelici, S. (2012). Estimation of quarticity with high-frequency data. Quantitative Finance, 12, 607–622.
Misaki, H., & Kunitomo, N. (2015). On robust properties of the SIML estimation of volatility under micro-market noise and random sampling. International Review of Economics and Finance, 40, 265–281.
Nualart, D. (2006). The Malliavin Calculus and Related Topics. Probability and its Applications (New York) (2nd ed.). Springer.
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A Appendices
A Appendices
1.1 A.1 A proof of Lemma 3.3
Let us put
By extending \( \varphi ^j \) and \( \varphi ^{j'} \) periodically, \( \mathcal {D}^{j,j'}_m (u,s) = D_m(\varphi ^j(u) + \varphi ^{j'} (s)) + D_m(\varphi ^j(u) - \varphi ^{j'}(s)) \) is periodic in both u and s with the period 2. Therefore, we have
Hence, it is sufficient to show that
We put \(a:=\limsup _{m_n, n \rightarrow \infty } m_n \rho _n \) which is in \( [ 0, \infty ) \) by assumption. Moreover, let
and
For \(m_n\) and n large enough, we see that
Therefore, we obtain
Since it holds that
for \( x \in \textbf{R} \), it follows from (A.3) that
Since (A.2) implies, for sufficiently large n, \(|u-c|>\frac{2a+2}{m_n} \Rightarrow \left| \varphi ^{j}(u)-c\right| \ge |u-c|-\frac{2a}{m_n}\), one has
(by changing variables with \( w = \frac{m_n}{2} |u-c|-a\))
This establishes (A.1) since clearly one has
by (A.4). \(\square \)
1.2 A.2 A proof of Proposition 3.1
Under the condition that \( \rho _n m_n^2 \rightarrow 0 \), we have
by a similar argument as the one we did for the proof of Lemma 2.1. Therefore, it suffices to prove that
We note that
holds by extending f(s, u) from \( \{ (s,u): s\le u \} \) to [0, 1] symmetrically. Then, by letting
we have
On the other hand, it follows from the expression (2.9) that
Since it holds that
we have
Then, by Lemma 3.3, we see that
Finally we shall prove the convergence of the first term in (A.7). Recalling (2.5), we have
We rely on the uniformly continuity of f. For arbitrary sufficiently small \( \varepsilon > 0 \), we can take \( \delta > 0 \) such that
Let
and
Then clearly \( (s,u) \in A^{\pm }_\delta \) satisfies \( |s-u|< \delta \) and therefore \( |f(s,u)-f(s,s)| < \varepsilon \). Then, we can bound the right-hand-side of (A.8) by
Since
and
we have
which shows the convergence to zero (as \( m \rightarrow \infty \)) of the first term in (A.7). \(\square \)
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Akahori, J., Namba, R. & Watanabe, A. The SIML method without microstructure noise. Jpn J Stat Data Sci (2024). https://doi.org/10.1007/s42081-024-00249-y
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DOI: https://doi.org/10.1007/s42081-024-00249-y
Keywords
- SIML method
- Malliavin–Mancino’s Fourier estimator
- Non-parametric estimation
- Consistency
- Asymptotic normality