Quarticity and other functionals of volatility: Efficient estimation

By Jean Jacod, Mathieu Rosenbaum. 2012

128 citations recorded by the source corpus.

We consider a multidimensional Ito semimartingale regularly sampled on [0,t] at high frequency $1/\Delta_n$, with $\Delta_n$ going to zero. The goal of this paper is to provide an estimator for the integral over [0,t] of a given function of the volatility matrix. To approximate the integral, we simply use a Riemann sum based on local estimators of the pointwise volatility. We show that although the accuracy of the pointwise estimation is at most $\Delta_n^{1/4}$, this procedure reaches the parametric rate $\Delta_n^{1/2}$, as it is usually the case in integrated functionals estimation. After a suitable bias correction, we obtain an unbiased central limit theorem for our estimator and show that it is asymptotically efficient within some classes of sub models.

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6cb5e7c3229ad80ffd9725f22eedc175f12bc7fd

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Citations in the corpus

Estimating the Quadratic Covariation Matrix from Noisy Observations: Local Method of Moments and Efficiency (2013, background): “While the univariate case has been studied extensively from both angles (see, e.g., the survey of Andersen et al. [4] or recent work by Reiss [22] and Jacod and Rosenbaum [15]), statistical inference for the quadratic covariation matrix is not yet well understood.”
On the number of common factors with high‐frequency data (2017, methodology): “These are commonly used in the literature (see, e.g., Wang & Zou, 2010; Jing et al., 2012a,b; Jacod & Podolskij, 2013; Jacod & Rosenbaum, 2013).”
Adaptive estimation of continuous-time regression models using high-frequency data (2017, methodology): “The latter provides a central limit theorem for integrated 4 We note that (Jacod and Rosenbaum, 2013) do not need Assumption K or K’ for their result.”
Estimation of integrated volatility of volatility with applications to goodness-of-fit testing (2011, background): “Recently, [19] discussed efficient estimation of ∫ t 0 g(σ(2) s) ds for general functions g.”
Estimating the Spot Covariation of Asset Prices—Statistical Theory and Empirical Evidence (2014, methodology): “419 July 2019, Vol. 37, No. 3 © 2019 Ait-Sahalia and Xiu (2015) showed how to estimate diagonalized integrated covariance matrices exploiting methods from Jacod and Rosenbaum (2013) to deal with functional transformations of volatility.”
Principal Component Analysis of High-Frequency Data (2016, unknown): “Assumption 3 resembles the spacial localization assumption in Li, Todorov, andTauchen (2017) and Li andXiu (2016), which is different from the polynomial growth conditions proposed by Jacod and Rosenbaum (2013).”
Volatility estimation for stochastic PDEs using high-frequency observations (2017, background): “…time, our aim is to extend the well understood statistical theory for semi-martingales, see, for instance, Jacod and Protter (2012), Mykland and Zhang (2009) or Jacod and Rosenbaum (2013), to parabolic SPDEs whose solution can be understood as infinite dimensional stochastic differential equations.”
Generalized Method of Integrated Moments for High‐Frequency Data (2016, unknown): “As shown by Jiang and Tian (2005) and Carr and Wu (2009), the theoretical value of the implied variance is given by”