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Bootstrapping continuous-time autoregressive processes

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  • Peter Brockwell
  • Jens-Peter Kreiss
  • Tobias Niebuhr

Abstract

We develop a bootstrap procedure for Lévy-driven continuous-time autoregressive (CAR) processes observed at discrete regularly-spaced times. It is well known that a regularly sampled stationary Ornstein–Uhlenbeck process [i.e. a CAR(1) process] has a discrete-time autoregressive representation with i.i.d. noise. Based on this representation a simple bootstrap procedure can be found. Since regularly sampled CAR processes of higher order satisfy ARMA equations with uncorrelated (but in general dependent) noise, a more general bootstrap procedure is needed for such processes. We consider statistics depending on observations of the CAR process at the uniformly-spaced times, together with auxiliary observations on a finer grid, which give approximations to the derivatives of the continuous time process. This enables us to approximate the state-vector of the CAR process which is a vector-valued CAR(1) process, and whose sampled version, on the uniformly-spaced grid, is a multivariate AR(1) process with i.i.d. noise. This leads to a valid residual-based bootstrap which allows replication of CAR $$(p)$$ processes on the underlying discrete time grid. We show that this approach is consistent for empirical autocovariances and autocorrelations. Copyright The Institute of Statistical Mathematics, Tokyo 2014

Suggested Citation

  • Peter Brockwell & Jens-Peter Kreiss & Tobias Niebuhr, 2014. "Bootstrapping continuous-time autoregressive processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(1), pages 75-92, February.
    • Handle: RePEc:spr:aistmt:v:66:y:2014:i:1:p:75-92
    DOI: 10.1007/s10463-013-0406-0
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    References listed on IDEAS

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    1. Jentsch, Carsten & Kreiss, Jens-Peter, 2010. "The multiple hybrid bootstrap -- Resampling multivariate linear processes," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2320-2345, November.
    2. K. S. Chan & H. Tong, 1987. "A Note On Embedding A Discrete Parameter Arma Model In A Continuous Parameter Arma Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 8(3), pages 277-281, May.
    3. Dahlhaus, Rainer, 1985. "Asymptotic normality of spectral estimates," Journal of Multivariate Analysis, Elsevier, vol. 16(3), pages 412-431, June.
    4. Peter J. Brockwell & Vincenzo Ferrazzano & Claudia Klüppelberg, 2012. "High‐frequency sampling of a continuous‐time ARMA process," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(1), pages 152-160, January.
    5. Paparoditis, Efstathios, 1996. "Bootstrapping Autoregressive and Moving Average Parameter Estimates of Infinite Order Vector Autoregressive Processes," Journal of Multivariate Analysis, Elsevier, vol. 57(2), pages 277-296, May.
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    1. P. Brockwell, 2014. "Recent results in the theory and applications of CARMA processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 647-685, August.

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