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On the Consistency of Bootstrap Testing for a Parameter on the Boundary of the Parameter Space

Author

Listed:
  • Giuseppe Cavaliere

    (Università di Bologna)

  • Heino Bohn Nielsen

    (University of Copenhagen)

  • Anders Rahbek

    (University of Copenhagen)

Abstract
It is well-known that with a parameter on the boundary of the parameter space, such as in the classic cases of testing for a zero location parameter or no ARCH effects, the classic nonparametric bootstrap - based on unrestricted parameter estimates - leads to inconsistent testing. In contrast, we show here that for the two aforementioned cases a nonparametric bootstrap test based on parameter estimates obtained under the null - referred to as 'restricted bootstrap' - is indeed consistent. While the restricted bootstrap is simple to implement in practice, novel theoretical arguments are required in order to establish consistency. In particular, since the bootstrap is analyzed both under the null hypothesis and under the alternative, non-standard asymptotic expansions are required to deal with parameters on the boundary. Detailed proofs of the asymptotic validity of the restricted bootstrap are given and, for the leading case of testing for no ARCH, a Monte Carlo study demonstrates that the bootstrap quasi-likelihood ratio statistic performs extremely well in terms of empirical size and power for even remarkably small samples, outperforming the standard and bootstrap Lagrange multiplier tests as well as the asymptotic quasi-likelihood ratio test.

Suggested Citation

  • Giuseppe Cavaliere & Heino Bohn Nielsen & Anders Rahbek, 2016. "On the Consistency of Bootstrap Testing for a Parameter on the Boundary of the Parameter Space," Quaderni di Dipartimento 6, Department of Statistics, University of Bologna.
    • Handle: RePEc:bot:quadip:wpaper:136
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    File URL: http://amsacta.unibo.it/id/eprint/5418
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    References listed on IDEAS

    as
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    Cited by:

    1. Gregory Fletcher Cox, 2024. "A Simple and Adaptive Confidence Interval when Nuisance Parameters Satisfy an Inequality," Papers 2409.09962, arXiv.org.
    2. Paul M. Beaumont & Aaron D. Smallwood, 2024. "Conditional sum of squares estimation of k-factor GARMA models," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 108(3), pages 501-543, September.
    3. Boswijk, H. Peter & Cavaliere, Giuseppe & Georgiev, Iliyan & Rahbek, Anders, 2021. "Bootstrapping non-stationary stochastic volatility," Journal of Econometrics, Elsevier, vol. 224(1), pages 161-180.
    4. Giuseppe Cavaliere & Rasmus Søndergaard Pedersen & Anders Rahbek, 2018. "The Fixed Volatility Bootstrap for a Class of Arch(q) Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(6), pages 920-941, November.
    5. Feiyu Jiang & Dong Li & Ke Zhu, 2019. "Non-standard inference for augmented double autoregressive models with null volatility coefficients," Papers 1905.01798, arXiv.org.
    6. Giovanni Angelini & Giuseppe Cavaliere & Luca Fanelli, 2022. "Bootstrap inference and diagnostics in state space models: With applications to dynamic macro models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 37(1), pages 3-22, January.
    7. Jiang, Feiyu & Li, Dong & Zhu, Ke, 2020. "Non-standard inference for augmented double autoregressive models with null volatility coefficients," Journal of Econometrics, Elsevier, vol. 215(1), pages 165-183.
    8. Cavaliere, Giuseppe & Nielsen, Heino Bohn & Pedersen, Rasmus Søndergaard & Rahbek, Anders, 2022. "Bootstrap inference on the boundary of the parameter space, with application to conditional volatility models," Journal of Econometrics, Elsevier, vol. 227(1), pages 241-263.
    9. Heino Bohn Nielsen & Anders Rahbek, 2023. "Penalized Quasi-likelihood Estimation and Model Selection in Time Series Models with Parameters on the Boundary," Papers 2302.02867, arXiv.org.
    10. Luiza S. C. Piancastelli & Wagner Barreto‐Souza & Hernando Ombao, 2023. "Flexible bivariate INGARCH process with a broad range of contemporaneous correlation," Journal of Time Series Analysis, Wiley Blackwell, vol. 44(2), pages 206-222, March.
    11. Giannerini, Simone & Goracci, Greta & Rahbek, Anders, 2024. "The validity of bootstrap testing for threshold autoregression," Journal of Econometrics, Elsevier, vol. 239(1).

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    Keywords

    Bootstrap; Boundary; ARCH; Location model;
    All these keywords.

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