Electronic Journal of Statistics
- Electron. J. Statist.
- Volume 4 (2010), 781-821.
Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise
We consider the models Yi,n=∫0i/nσ(s)dWs+τ(i/n)εi,n, and Ỹi,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.
Electron. J. Statist., Volume 4 (2010), 781-821.
First available in Project Euclid: 8 September 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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
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