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Bootstrapping Stationary Autoregressive Moving‐Average Models

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  • Jens‐Peter Kreiss
  • Jürgen Franke

Abstract

. In this paper we develop an asymptotic theory for application of the bootstrap to stationary stochastic processes of autoregressive moving‐average (ARMA) type, with known order (p, q). We give a proof of the asymptotic validity of the bootstrap proposal applied to M estimators for the unknown parameter vector of the process. For this purpose we derive an asymptotic expansion for M estimators in ARMA models and construct an estimate for the unknown distribution function of the residuals which in principle are not observable. A small simulation study is also included.

Suggested Citation

  • Jens‐Peter Kreiss & Jürgen Franke, 1992. "Bootstrapping Stationary Autoregressive Moving‐Average Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 13(4), pages 297-317, July.
    • Handle: RePEc:bla:jtsera:v:13:y:1992:i:4:p:297-317
    DOI: 10.1111/j.1467-9892.1992.tb00109.x
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    Cited by:

    1. Wang Liqiong, 2013. "Bootstrap Point Optimal Unit Root Tests," Journal of Time Series Econometrics, De Gruyter, vol. 6(1), pages 1-31, July.
    2. Moon, Seongman & Velasco, Carlos, 2013. "Tests for m-dependence based on sample splitting methods," Journal of Econometrics, Elsevier, vol. 173(2), pages 143-159.
    3. Lorenzo Pascual & Juan Romo & Esther Ruiz, 2004. "Bootstrap predictive inference for ARIMA processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(4), pages 449-465, July.
    4. Wang, Bin & Zheng, Xu, 2022. "Testing for the presence of jump components in jump diffusion models," Journal of Econometrics, Elsevier, vol. 230(2), pages 483-509.
    5. Jinyong Hahn & Zhipeng Liao, 2021. "Bootstrap Standard Error Estimates and Inference," Econometrica, Econometric Society, vol. 89(4), pages 1963-1977, July.
    6. Michele La Rocca & Cosimo Vitale, 2001. "Parametric bootstrap inference in bilinear models," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3-4), pages 101-116.
    7. F. Jay Breidt & Richard A. Davis & William T. M. Dunsmuir, 1995. "Improved Bootstrap Prediction Intervals For Autoregressions," Journal of Time Series Analysis, Wiley Blackwell, vol. 16(2), pages 177-200, March.
    8. Efstathios Paparoditis, 2005. "Testing the Fit of a Vector Autoregressive Moving Average Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(4), pages 543-568, July.
    9. Shimizu Kenichi, 2013. "The bootstrap does not alwayswork for heteroscedasticmodels," Statistics & Risk Modeling, De Gruyter, vol. 30(3), pages 189-204, August.

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