Skip to main content
SpringerLink
Log in

Bootstrapping continuous-time autoregressive processes

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Barndorff-Nielsen, O. E., Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their applications in financial economics. Journal of the Royal Statistical Society Series B, 63, 167–241.

    Google Scholar 

  • Brockwell, P. J. (1994). A note on the embedding of discrete-time ARMA processes. Journal of Time Series Analysis, 16, 451–460.

    Article  MathSciNet  Google Scholar 

  • Brockwell, P. J. (2001). Continuous-time ARMA processes. Handbook of Statistics (Vol. 19, pp. 249–276). Amsterdam: Elsevier.

  • Brockwell, P. J. (2012). Lévy-driven CARMA processes. In T. G. Andersen, R. A. Davis, J. P. Kreiss, Th Mikosch (Eds.), Handbook of Financial Time Series (pp. 457–480). Berlin: Springer.

  • Brockwell, P. J., Davis, R. A. (1991). Time Series: Theory and Methods (2nd ed.). New York: Springer.

  • Brockwell, P. J., Lindner, A. (2009). Existence and uniqueness of stationary Lévy-driven CARMA processes. Stochastic Processs and their Applications, 119, 2660–2681.

    Google Scholar 

  • Brockwell, P. J., Lindner, A. (2012). Lévy-driven time series models for financial data. In T. Subba Rao, C. R. Rao (Eds.), Handbook of Statistics (Vol. 30, pp. 543–563). Amsterdam: Elsevier.

  • Brockwell, P. J., Marquardt, T. (2005). Lévy-driven and fractionally integrated ARMA processes with continuous time parameter. Statistica Sinica, 15, 477–494.

    Google Scholar 

  • Brockwell, P. J., Schlemm, E. (2011). Parametric estimation of the driving Lévy process of multivariate CARMA processes from discrete observations. Journal of Multivariate Analysis, 115, 217–251.

    Google Scholar 

  • Brockwell, P. J., Davis, R. A., Yang, Y. (2010). Estimation for non-negative Lévy-driven CARMA processes. Journal of Business and Economic Statistics, 29, 250–259.

    Google Scholar 

  • Brockwell, P. J., Ferrazzano, V., Klüppelberg, C. (2012). High-frequency sampling of a continuous-time ARMA process. Journal of Time Series Analysis, 33, 152–160.

    Google Scholar 

  • Bühlmann, P. (1997). Sieve bootstrap for time series. Bernoulli, 3, 123–148.

    Article  MATH  MathSciNet  Google Scholar 

  • Bühlmann, P., Künsch, H. R. (1995). The blockwise bootstrap for general parameters of a stationary time series. Scandinavian Journal of Statistics, 22, 35–54.

    Google Scholar 

  • Chan, K. S., Tong, H. (1987). A note on embedding a discrete parameter ARMA model in a continuous parameter ARMA model. Journal of Time Series Analysis, 8, 272–281.

    Google Scholar 

  • Cohen, S., Lindner, A. (2012). A Central Limit Theorem for the Sample Autocorrelations of a Lévy Driven Continuous Time Moving Average Process with Finite Fourth Moment. Preprint available on: https://www.tu-braunschweig.de/stochastik/team/lindner/publications

  • Dahlhaus, R. (1985). Asymptotic normality of spectral estimates. Journal of Multivariate Analysis, 16, 412–431.

    Article  MATH  MathSciNet  Google Scholar 

  • Doob, T. L. (1944). The elementary Gaussian processes. The Annals of Mathematical Statistics, 15, 229–282.

    Article  MATH  MathSciNet  Google Scholar 

  • He, S. W., Wang, J. G. (1989). On embedding a discrete-parameter ARMA model in a continuous-parameter ARMA model. Journal of Time Series Analysis, 10, 315–323.

    Google Scholar 

  • Huzii, M. (2001). Embedding a Gaussian discrete-time autoregressive moving average process in a Gaussian continuous-time autoregressive moving average process. Journal of Time Series Analysis, 28, 498–520.

    Article  MathSciNet  Google Scholar 

  • Jentsch, C., Kreiss, J.-P. (2010). The multiple hybrid bootstrap—Resampling multivariate linear processes. Journal of Multivariate Analysis, 101, 2320–2345.

    Google Scholar 

  • Jones, R.H. (1980). Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics, 22(3), 389–395.

    Google Scholar 

  • Kreiss J.-P. (1997). Asymptotical Properties of Residual Bootstrap for Autoregression. Preprint available on: https://www.tu-braunschweig.de/stochastik/team/kreiss/forschung/publications

  • Kreiss, J.-P., Franke, J. (1992). Bootstrapping stationary autoregressive moving average models. Journal of Time Series Analysis, 13, 297–319.

    Google Scholar 

  • Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. The Annals of Statistics, 17, 1217–1241.

    Article  MATH  MathSciNet  Google Scholar 

  • Niebuhr, T., Kreiss, J.-P. (2012). Asymptotics for Autocovariances and Integrated Periodograms for Linear Processes Observed at Lower Frequencies. Preprint available on: https://www.tu-braunschweig.de/stochastik/team/niebuhr

  • Nordman, D., Lahiri, S., Fridley, B. (2007). Optimal block size for variance estimation by a spatial block bootstrap method. The Indian Journal of Statistics, 69, 468–493.

    Google Scholar 

  • Paparoditis, E. (1996). Bootstrapping autoregressive and moving average parameter estimates of infinite order vector autoregressive processes. Journal of Multivariate Analysis, 57, 277–296.

    Article  MATH  MathSciNet  Google Scholar 

  • Phillips, A. W. (1959). The estimation of parameters in systems of stochastic differential equations. Biometrika, 46, 67–76.

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The financial support of the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged as is the support of PJB by NSF Grant DMS-1107031. The authors are also indebted to an anonymous referee, whose comments led to an improved version of the paper.

Author information

Authors and Affiliations

  1. Department of Statistics, Colorado State University, Fort Collins, CO, 80523-1877, USA

    Peter J. Brockwell

  2. Institut für Mathematische Stochastik, Technische Universität Braunschweig, 38092 , Braunschweig, Germany

    Jens-Peter Kreiss & Tobias Niebuhr

Authors
  1. Peter J. Brockwell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jens-Peter Kreiss
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tobias Niebuhr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tobias Niebuhr.

About this article

Cite this article

Brockwell, P.J., Kreiss, JP. & Niebuhr, T. Bootstrapping continuous-time autoregressive processes. Ann Inst Stat Math 66, 75–92 (2014). https://doi.org/10.1007/s10463-013-0406-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-013-0406-0

Keywords

Search

Navigation