Electronic Journal of Statistics

Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise

Axel Munk and Johannes Schmidt-Hieber

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We consider the models Yi,n=0i/nσ(s)dWs+τ(i/n)εi,n, and i,n=σ(i/n)Wi/n+τ(i/n)εi,n, i=1,,n, where (Wt)t[0,1] denotes a standard Brownian motion and εi,n are centered i.i.d. random variables with E (εi,n2)=1 and finite fourth moment. Furthermore, σ and τ are unknown deterministic functions and (Wt)t[0,1] and (ε1,n,,εn,n) are assumed to be independent processes. Based on a spectral decomposition of the covariance structures we derive series estimators for σ2 and τ2 and investigate their rate of convergence of the MISE  in dependence of their smoothness. To this end specific basis functions and their corresponding Sobolev ellipsoids are introduced and we show that our estimators are optimal in minimax sense. Our work is motivated by microstructure noise models. A major finding is that the microstructure noise εi,n introduces an additionally degree of ill-posedness of 1/2; irrespectively of the tail behavior of εi,n. The performance of the estimates is illustrated by a small numerical study.

Article information

Electron. J. Statist., Volume 4 (2010), 781-821.

First available in Project Euclid: 8 September 2010

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Zentralblatt MATH identifier

Primary: 62M09: Non-Markovian processes: estimation 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62G08: Nonparametric regression 62G20: Asymptotic properties

Brownian motion variance estimation minimax rate microstructure noise Sobolev embedding


Munk, Axel; Schmidt-Hieber, Johannes. Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise. Electron. J. Statist. 4 (2010), 781--821. doi:10.1214/10-EJS568. https://projecteuclid.org/euclid.ejs/1283952132

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