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Likelihood inference for a nonstationary fractional autoregressive model

  • Søren Johansen

    ()

    (University of Copenhagen and CREATES)

  • Morten Ørregaard Nielsen

    ()

    (Queen's University and CREATES)

This paper discusses model-based inference in an autoregressive model for fractional processes which allows the process to be fractional of order d or d-b. Fractional differencing involves infinitely many past values and because we are interested in nonstationary processes we model the data X_{1},...,X_{T} given the initial values X_{-n}, n = 0,1,..., as is usually done. The initial values are not modeled but assumed to be bounded. This represents a considerable generalization relative to all previous work where it is assumed that initial values are zero. For the statistical analysis we assume the conditional Gaussian likelihood and for the probability analysis we also condition on initial values but assume that the errors in the autoregressive model are i.i.d. with suitable moment conditions. We analyze the conditional likelihood and its derivatives as stochastic processes in the parameters, including d and b, and prove that they converge in distribution. We use the results to prove consistency of the maximum likelihood estimator for d,b in a large compact subset of {1/2

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File URL: http://qed.econ.queensu.ca/working_papers/papers/qed_wp_1172.pdf
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Paper provided by Queen's University, Department of Economics in its series Working Papers with number 1172.

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Length: 43 pages
Date of creation: Feb 2010
Date of revision:
Handle: RePEc:qed:wpaper:1172
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  1. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
  2. Donald W.K. Andrews, 1990. "Generic Uniform Convergence," Cowles Foundation Discussion Papers 940, Cowles Foundation for Research in Economics, Yale University.
  3. P. M. Robinson & J. Hualde, 2003. "Cointegration in Fractional Systems with Unknown Integration Orders," Econometrica, Econometric Society, vol. 71(6), pages 1727-1766, November.
  4. repec:cup:cbooks:9780521496032 is not listed on IDEAS
  5. Sowell, Fallaw, 1990. "The Fractional Unit Root Distribution," Econometrica, Econometric Society, vol. 58(2), pages 495-505, March.
  6. Johansen, SØren, 2008. "A Representation Theory For A Class Of Vector Autoregressive Models For Fractional Processes," Econometric Theory, Cambridge University Press, vol. 24(03), pages 651-676, June.
  7. Robinson, P. M., 1991. "Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression," Journal of Econometrics, Elsevier, vol. 47(1), pages 67-84, January.
  8. Ling, Shiqing & Li, W.K., 2001. "Asymptotic Inference For Nonstationary Fractionally Integrated Autoregressive Moving-Average Models," Econometric Theory, Cambridge University Press, vol. 17(04), pages 738-764, August.
  9. Ignacio N. Lobato & Carlos Velasco, 2005. "Efficient Wald Tests For Fractional Unit Roots," Economics Working Papers we056935, Universidad Carlos III, Departamento de Economía.
  10. 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.
  11. 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.
  12. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
  13. Tanaka, Katsuto, 1999. "The Nonstationary Fractional Unit Root," Econometric Theory, Cambridge University Press, vol. 15(04), pages 549-582, August.
  14. Newey, W.K., 1989. "Uniform Convergence In Probability And Stochastic Equicontinuity," Papers 342, Princeton, Department of Economics - Econometric Research Program.
  15. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-72, June.
  16. Morten Oe. Nielsen, . "Efficient Likelihold Inference in Nonstationary Univariate Models," Economics Working Papers 2001-8, School of Economics and Management, University of Aarhus.
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