High‐Frequency Sampling and Kernel Estimation for Continuous‐Time Moving Average Processes

20 Pages Posted: 26 Apr 2013

See all articles by Peter Brockwell

Peter Brockwell

Colorado State University, Fort Collins

Vincenzo Ferrazzano

Technische Universität München (TUM)

C. Klüppelberg

Technische Universität München (TUM)

Date Written: May 2013


Interest in continuous‐time processes has increased rapidly in recent years, largely because of high‐frequency data available in many applications. We develop a method for estimating the kernel function g of a second‐order stationary Lévy‐driven continuous‐time moving average (CMA) process Y based on observations of the discrete‐time process YΔ obtained by sampling Y at Δ, 2Δ, …, nΔ for small Δ. We approximate g by gΔ based on the Wold representation and prove its pointwise convergence to g as Δ → 0 for continuous‐time autoregressive moving average (CARMA) processes. Two non‐parametric estimators of gΔ, on the basis of the innovations algorithm and the Durbin–Levinson algorithm, are proposed to estimate g. For a Gaussian CARMA process, we give conditions on the sample size n and the grid spacing Δ(n) under which the innovations estimator is consistent and asymptotically normal as n → ∞. The estimators can be calculated from sampled observations of any CMA process, and simulations suggest that they perform well even outside the class of CARMA processes. We illustrate their performance for simulated data and apply them to the Brookhaven turbulent wind speed data. Finally, we extend results of Brockwell et al. (2012) for sampled CARMA processes to a much wider class of CMA processes.

Keywords: CARMA process, continuous‐time moving average process, FICARMA process, high‐frequency data, kernel estimation, regular variation, spectral theory, turbulence, Wold representation

Suggested Citation

Brockwell, Peter and Ferrazzano, Vincenzo and Kluppelberg, Claudia, High‐Frequency Sampling and Kernel Estimation for Continuous‐Time Moving Average Processes (May 2013). Journal of Time Series Analysis, Vol. 34, Issue 3, pp. 385-404, 2013, Available at SSRN: https://ssrn.com/abstract=2256851 or http://dx.doi.org/10.1111/jtsa.12022

Peter Brockwell (Contact Author)

Colorado State University, Fort Collins ( email )

Fort Collins, CO 80523
United States
970-491-3481 (Phone)
970-491-7895 (Fax)

Vincenzo Ferrazzano

Technische Universität München (TUM)

Arcisstrasse 21
Munich, DE 80333

Claudia Kluppelberg

Technische Universität München (TUM) ( email )

Center for Mathematical Sciences
D-80290 Munich

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