High‐frequency sampling and kernel estimation for continuous‐time moving average processes. (17th March 2013)

• Record Type: Journal Article
• Authors: Brockwell, Peter J.; Ferrazzano, Vincenzo; Klüppelberg, Claudia
• Is Part Of: Journal of time series analysis. Volume 34:Number 3(2013:May)
• Page Start: 385
• Page End: 404
• Publication Date: 2013-03-17
• Subjects: Time-series analysis -- Periodicals; 519.232
• DOI: 10.1111/jtsa.12022
• Languages: English
• ISSNs: 0143-9782

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>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 <italic>g</italic> of a second‐order stationary Lévy‐driven continuous‐time moving average (CMA) process <italic>Y</italic> based on observations of the discrete‐time process <italic>Y</italic><sup><italic>Δ</italic></sup> obtained by sampling <italic>Y</italic> at <italic>Δ</italic>, 2<italic>Δ</italic>, …, <italic>nΔ</italic> for small <italic>Δ</italic>. We approximate <italic>g</italic> by <italic>g</italic><sup><italic>Δ</italic></sup> based on the Wold representation and prove its pointwise convergence to <italic>g</italic> as <italic>Δ</italic> → 0 for continuous‐time autoregressive moving average (CARMA) processes. Two non‐parametric estimators of <italic>g</italic><sup><italic>Δ</italic></sup>, on the basis of the innovations algorithm and the Durbin–Levinson algorithm, are proposed to estimate <italic>g</italic>. For a Gaussian CARMA process, we give conditions on the sample size <italic>n</italic> and the grid spacing <italic>Δ</italic>(<italic>n</italic>) under which the innovations estimator is consistent and asymptotically normal as <italic>n</italic> → <italic>∞</italic>. The estimators can be calculated from sampled observations of <italic>any</italic> 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 <italic>et al</italic>. (2012) for sampled CARMA processes to a much wider class of CMA processes.</p> </abstract>