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Higher-Order Improvements of the Sieve Bootstrap for Fractionally Integrated Processes

  • D.S. Poskitt

    ()

  • Simone D. Grose

    ()

  • Gael M. Martin

    ()

This paper investigates the accuracy of bootstrap-based inference in the case of long memory fractionally integrated processes. The re-sampling method is based on the semi-parametric sieve approach, whereby the dynamics in the process used to produce the bootstrap draws are captured by an autoregressive approximation. Application of the sieve method to data pre-filtered by a semi-parametric estimate of the long memory parameter is also explored. Higher-order improvements yielded by both forms of re-sampling are demonstrated using Edgeworth expansions for a broad class of statistics that includes first- and second-order moments, the discrete Fourier transform and regression coefficients. The methods are then applied to the problem of estimating the sampling distributions of the sample mean and of selected sample autocorrelation coefficients, in experimental settings. In the case of the sample mean, the pre-filtered version of the bootstrap is shown to avoid the distinct underestimation of the sampling variance of the mean which the raw sieve method demonstrates in finite samples, higher order accuracy of the latter notwithstanding. Pre-filtering also produces gains in terms of the accuracy with which the sampling distributions of the sample autocorrelations are reproduced, most notably in the part of the parameter space in which asymptotic normality does not obtain. Most importantly, the sieve bootstrap is shown to reproduce the (empirically infeasible) Edgeworth expansion of the sampling distribution of the autocorrelation coefficients, in the part of the parameter space in which the expansion is valid.

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File URL: http://business.monash.edu/econometrics-and-business-statistics/research/publications/ebs/wp25-13.pdf
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Paper provided by Monash University, Department of Econometrics and Business Statistics in its series Monash Econometrics and Business Statistics Working Papers with number 25/13.

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Date of creation: 2013
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Handle: RePEc:msh:ebswps:2013-25
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  1. Offer Lieberman & Peter C.B. Phillips, 2004. "Expansions for Approximate Maximum Likelihood Estimators of the Fractional Difference Parameter," Cowles Foundation Discussion Papers 1474, Cowles Foundation for Research in Economics, Yale University.
  2. S. D. Grose & D. S. Poskitt, 2006. "The Finite-Sample Properties of Autoregressive Approximations of Fractionally-Integrated and Non-Invertible Processes," Monash Econometrics and Business Statistics Working Papers 15/06, Monash University, Department of Econometrics and Business Statistics.
  3. D. Poskitt, 2007. "Autoregressive approximation in nonstandard situations: the fractionally integrated and non-invertible cases," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(4), pages 697-725, December.
  4. Lieberman, Offer & Rousseau, Judith & Zucker, David M., 2001. "Valid Edgeworth Expansion For The Sample Autocorrelation Function Under Long Range Dependence," Econometric Theory, Cambridge University Press, vol. 17(01), pages 257-275, February.
  5. Andrews, Donald W.K. & Lieberman, Offer, 2005. "Valid Edgeworth Expansions For The Whittle Maximum Likelihood Estimator For Stationary Long-Memory Gaussian Time Series," Econometric Theory, Cambridge University Press, vol. 21(04), pages 710-734, August.
  6. Lieberman, Offer & Phillips, Peter C.B., 2004. "Expansions For The Distribution Of The Maximum Likelihood Estimator Of The Fractional Difference Parameter," Econometric Theory, Cambridge University Press, vol. 20(03), pages 464-484, June.
  7. Morten Ørregaard Nielsen & Per Frederiksen, 2005. "Finite Sample Comparison of Parametric, Semiparametric, and Wavelet Estimators of Fractional Integration," Working Papers 1189, Queen's University, Department of Economics.
  8. Sowell, Fallaw, 1992. "Maximum likelihood estimation of stationary univariate fractionally integrated time series models," Journal of Econometrics, Elsevier, vol. 53(1-3), pages 165-188.
  9. Donald W. K. Andrews & Yixiao Sun, 2004. "Adaptive Local Polynomial Whittle Estimation of Long-range Dependence," Econometrica, Econometric Society, vol. 72(2), pages 569-614, 03.
  10. D.S. Poskitt & Gael M. Martin & Simone D. Grose, 2012. "Bias Reduction of Long Memory Parameter Estimators via the Pre-filtered Sieve Bootstrap," Monash Econometrics and Business Statistics Working Papers 8/12, Monash University, Department of Econometrics and Business Statistics.
  11. Magnus, J.R., 1986. "The exact moments of a ratio of quadratic forms in normal variables," Other publications TiSEM c6725407-ac3c-44fd-b6d1-5, Tilburg University, School of Economics and Management.
  12. D. S. Poskitt, 2006. "Properties of the Sieve Bootstrap for Fractionally Integrated and Non-Invertible Processes," Monash Econometrics and Business Statistics Working Papers 12/06, Monash University, Department of Econometrics and Business Statistics.
  13. Edwin Choi & Peter Hall, 2000. "Bootstrap confidence regions computed from autoregressions of arbitrary order," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(2), pages 461-477.
  14. Faÿ, Gilles, 2010. "Moment bounds for non-linear functionals of the periodogram," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 983-1009, June.
  15. L. Giraitis & P.M. Robinson, 2003. "Edgeworth expansions for semiparametric Whittle estimation of long memory," LSE Research Online Documents on Economics 291, London School of Economics and Political Science, LSE Library.
  16. Jurgen A. Doornik & Marius Ooms, 2001. "Computational Aspects of Maximum Likelihood Estimation of Autoregressive Fractionally Integrated Moving Average Models," Economics Papers 2001-W27, Economics Group, Nuffield College, University of Oxford.
  17. repec:adr:anecst:y:1986:i:4:p:05 is not listed on IDEAS
  18. Donald W.K. Andrews & Offer Lieberman, 2002. "Higher-order Improvements of the Parametric Bootstrap for Long-memory Gaussian Processes," Cowles Foundation Discussion Papers 1378, Cowles Foundation for Research in Economics, Yale University.
  19. Faÿ, Gilles & Moulines, Eric & Soulier, Philippe, 2004. "Edgeworth expansions for linear statistics of possibly long-range-dependent linear processes," Statistics & Probability Letters, Elsevier, vol. 66(3), pages 275-288, February.
  20. Poskitt, D.S., 1994. "A Note on Autoregressive Modeling," Econometric Theory, Cambridge University Press, vol. 10(05), pages 884-899, December.
  21. Hosking, Jonathan R. M., 1996. "Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series," Journal of Econometrics, Elsevier, vol. 73(1), pages 261-284, July.
  22. Smith, Murray D., 1989. "On the expectation of a ratio of quadratic forms in normal variables," Journal of Multivariate Analysis, Elsevier, vol. 31(2), pages 244-257, November.
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