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Likelihood inference for a nonstationary fractional autoregressive model

  • Johansen, Søren
  • Nielsen, Morten Ørregaard

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 X1,...,XT given the initial values , as is usually done. The initial values are not modeled but assumed to be bounded. This represents a considerable generalization relative to 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 these results to prove consistency of the maximum likelihood estimator for d,b in a large compact subset of {1/2

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Article provided by Elsevier in its journal Journal of Econometrics.

Volume (Year): 158 (2010)
Issue (Month): 1 (September)
Pages: 51-66

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Handle: RePEc:eee:econom:v:158:y:2010:i:1:p:51-66
Contact details of provider: Web page: http://www.elsevier.com/locate/jeconom

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  1. Newey, Whitney K, 1991. "Uniform Convergence in Probability and Stochastic Equicontinuity," Econometrica, Econometric Society, vol. 59(4), pages 1161-67, July.
  2. 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.
  3. 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.
  4. Sowell, Fallaw, 1990. "The Fractional Unit Root Distribution," Econometrica, Econometric Society, vol. 58(2), pages 495-505, March.
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  6. 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.
  7. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
  8. Donald W.K. Andrews, 1990. "Generic Uniform Convergence," Cowles Foundation Discussion Papers 940, Cowles Foundation for Research in Economics, Yale University.
  9. 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.
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  12. Nielsen, Morten rregaard, 2004. "Efficient Likelihood Inference In Nonstationary Univariate Models," Econometric Theory, Cambridge University Press, vol. 20(01), pages 116-146, February.
  13. 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.
  14. Peter M. Robinson & Javier Hualde, 2002. "Cointegration in Fractional Systems with Unknown Integration Orders," Faculty Working Papers 07/02, School of Economics and Business Administration, University of Navarra.
  15. 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.
  16. 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.
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