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Type I and Type II Fractional Brownian Motions: a Reconsideration

  • James Davidson

    (Department of Economics, University of Exeter)

  • Nigar Hashimzade

    (University of Reading)

The so-called type I and type II fractional Brownian motions are limit distributions associated with the fractional integration model in which pre-sample shocks are either included in the lag structure, or suppressed. There can be substantial differences between the distributions of these two processes and of functionals derived from them, so that it becomes an important issue to decide which model to use as a basis for inference. Alternative methods for simulating the type I case are contrasted, and for models close to the nonstationarity boundary, truncating infinite sums is shown to result in a significant distortion of the distribution. A simple simulation method that overcomes this problem is described and implemented. The approach also has implications for the estimation of type I ARFIMA models, and a new conditional ML estimator is proposed, using the annual Nile minima series for illustration.

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File URL: http://people.exeter.ac.uk/cc371/RePEc/dpapers/DP0816.pdf
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Paper provided by Exeter University, Department of Economics in its series Discussion Papers with number 0816.

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Date of creation: 2008
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Handle: RePEc:exe:wpaper:0816
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  1. Luis A. Gil-Alana & Guglielmo M. Caporale, 2008. "Modelling the US, the UK and Japanese unemployment rates. Fractional integrationand structural breaks," Faculty Working Papers 11/08, School of Economics and Business Administration, University of Navarra.
  2. Granger, C. W. J., 1980. "Long memory relationships and the aggregation of dynamic models," Journal of Econometrics, Elsevier, vol. 14(2), pages 227-238, October.
  3. Haldrup, Niels & Nielsen, Morten Orregaard, 2007. "Estimation of fractional integration in the presence of data noise," Computational Statistics & Data Analysis, Elsevier, vol. 51(6), pages 3100-3114, March.
  4. Søren Johansen & Morten Ørregaard Nielsen, 2010. "Likelihood inference for a nonstationary fractional autoregressive model," Working Papers 1172, Queen's University, Department of Economics.
  5. Enriquez, Nathanaël, 2004. "A simple construction of the fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 109(2), pages 203-223, February.
  6. Yin-Wong Cheung & Francis X. Diebold, 1993. "On maximum-likelihood estimation of the differencing parameter of fractionally integrated noise with unknown mean," Working Papers 93-5, Federal Reserve Bank of Philadelphia.
  7. Juan J. Dolado & Jesus Gonzalo & Laura Mayoral, 2002. "A Fractional Dickey-Fuller Test for Unit Roots," Econometrica, Econometric Society, vol. 70(5), pages 1963-2006, September.
  8. David Byers & James Davidson & David Peel, 2002. "Modelling political popularity: a correction," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 165(1), pages 187-189.
  9. 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.
  10. David Byers & James Davidson & David Peel, 1997. "Modelling Political Popularity: an Analysis of Long-range Dependence in Opinion Poll Series," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 160(3), pages 471-490.
  11. Coakley, Jerry & Dollery, Jian & Kellard, Neil, 2008. "The role of long memory in hedging effectiveness," Computational Statistics & Data Analysis, Elsevier, vol. 52(6), pages 3075-3082, February.
  12. Marinucci, D. & Robinson, P. M., 2000. "Weak convergence of multivariate fractional processes," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 103-120, March.
  13. Davidson, James & Sibbertsen, Philipp, 2002. "Generating schemes for long memory processes: Regimes, aggregation and linearity," Technical Reports 2002,46, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
  14. Davidson, James & Hashimzade, Nigar, 2008. "Alternative Frequency And Time Domain Versions Of Fractional Brownian Motion," Econometric Theory, Cambridge University Press, vol. 24(01), pages 256-293, February.
  15. Caporale, Guglielmo Maria & Gil-Alana, Luis A., 2008. "Modelling the US, UK and Japanese unemployment rates: Fractional integration and structural breaks," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4998-5013, July.
  16. Davidson, James, 2006. "Alternative bootstrap procedures for testing cointegration in fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 133(2), pages 741-777, August.
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