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A discrete model for bootstrap iteration

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  • Russell Davidson

    (Institute for Fiscal Studies and McGIll)

Abstract

In an attempt to free bootstrap theory from the shackles of asymptotic considerations, this paper studies the possibility of justifying, or validating, the bootstrap, not by letting the sample size tend to infinity, but by considering the sequence of bootstrap P values obtained by iterating the bootstrap. The main idea of the paper is that, if this sequence converges to a random variable that follows the uniform U(0; 1) distribution, then the bootstrap is valid. The idea is studied by making the model under test discrete and finite, so that it is characterised by a finite three-dimensional array of probabilities. This device, when available, renders bootstrap iteration to any desired order feasible. It is used for studying a unit-root test for a process driven by a stationary MA(1) process, where it is known that the unit-root test, even when bootstrapped, becomes quite unreliable when the MA(1) parameter is in the vicinity of -1. Iteration of the bootstrap P value to convergence achieves reliable inference except for a parameter value very close to -1. The paper then endeavours to see these specific results in a wider context, and tries to cast new light on where bootstrap theory may be going.

Suggested Citation

  • Russell Davidson, 2015. "A discrete model for bootstrap iteration," CeMMAP working papers CWP38/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:38/15
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    References listed on IDEAS

    as
    1. Russell Davidson, 2007. "Bootstrapping Econometric Models," Departmental Working Papers 2007-13, McGill University, Department of Economics.
    2. Y.F. Chan, Kenny & M.S. Lee, Stephen, 2001. "An exact iterated bootstrap algorithm for small-sample bias reduction," Computational Statistics & Data Analysis, Elsevier, vol. 36(1), pages 1-13, March.
    3. Davidson, Russell & MacKinnon, James G., 1999. "The Size Distortion Of Bootstrap Tests," Econometric Theory, Cambridge University Press, vol. 15(3), pages 361-376, June.
    4. Davidson, Russell & MacKinnon, James G., 2007. "Improving the reliability of bootstrap tests with the fast double bootstrap," Computational Statistics & Data Analysis, Elsevier, vol. 51(7), pages 3259-3281, April.
    5. Stephen M. S. Lee, 2003. "Prepivoting by weighted bootstrap iteration," Biometrika, Biometrika Trust, vol. 90(2), pages 393-410, June.
    6. Russell Davidson, 2007. "Bootstrapping econometric models (in Russian)," Quantile, Quantile, issue 3, pages 13-36, September.
    7. Davidson, Russell & MacKinnon, James G, 1998. "Graphical Methods for Investigating the Size and Power of Hypothesis Tests," The Manchester School of Economic & Social Studies, University of Manchester, vol. 66(1), pages 1-26, January.
    8. Horowitz, Joel L. & Savin, N. E., 2000. "Empirically relevant critical values for hypothesis tests: A bootstrap approach," Journal of Econometrics, Elsevier, vol. 95(2), pages 375-389, April.
    9. Russell Davidson, 2010. "Size Distortion of Bootstrap Tests: an Example from Unit Root Testing," Review of Economic Analysis, Digital Initiatives at the University of Waterloo Library, vol. 2(2), pages 169-193, June.
    10. MacKinnon, James G, 1996. "Numerical Distribution Functions for Unit Root and Cointegration Tests," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(6), pages 601-618, Nov.-Dec..
    11. MacKinnon, James G, 1994. "Approximate Asymptotic Distribution Functions for Unit-Root and Cointegration Tests," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(2), pages 167-176, April.
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    Cited by:

    1. Russell Davidson, 2015. "Computing, the bootstrap and economics," Canadian Journal of Economics, Canadian Economics Association, vol. 48(4), pages 1195-1214, November.

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    More about this item

    Keywords

    Bootstrap; bootstrap iteration;

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General

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