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Likelihood inference for a nonstationary fractional autoregressive model

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  • Søren Johansen

    ()
    (University of Copenhagen and CREATES)

  • Morten Ørregaard Nielsen

    ()
    (Queen's University and CREATES)

Abstract

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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Bibliographic Info

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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Keywords: Dickey-Fuller test; fractional unit root; likelihood inference;

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References

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  1. 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.
  2. 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.
  3. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
  4. repec:cup:cbooks:9780521496032 is not listed on IDEAS
  5. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
  6. Andrews, Donald W.K., 1992. "Generic Uniform Convergence," Econometric Theory, Cambridge University Press, vol. 8(02), pages 241-257, June.
  7. Newey, W.K., 1989. "Uniform Convergence In Probability And Stochastic Equicontinuity," Papers 342, Princeton, Department of Economics - Econometric Research Program.
  8. Ignacio N. Lobato & Carlos Velasco, 2005. "Efficient Wald Tests For Fractional Unit Roots," Economics Working Papers we056935, Universidad Carlos III, Departamento de Economía.
  9. Tanaka, Katsuto, 1999. "The Nonstationary Fractional Unit Root," Econometric Theory, Cambridge University Press, vol. 15(04), pages 549-582, August.
  10. Morten Oe. Nielsen, . "Efficient Likelihold Inference in Nonstationary Univariate Models," Economics Working Papers 2001-8, School of Economics and Management, University of Aarhus.
  11. 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.
  12. 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.
  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. 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.
  15. Sowell, Fallaw, 1990. "The Fractional Unit Root Distribution," Econometrica, Econometric Society, vol. 58(2), pages 495-505, March.
  16. 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.
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Citations

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Cited by:
  1. James Davidson & Nigar Hashimzade, 2008. "Type I and Type II Fractional Brownian Motions: a Reconsideration," Discussion Papers 0816, Exeter University, Department of Economics.
  2. Maggie E. C. Jones & Morten Ørregaard Nielsen & Michael Ksawery Popiel, 2014. "A fractionally cointegrated VAR analysis of economic voting and political support," CREATES Research Papers 2014-23, School of Economics and Management, University of Aarhus.
  3. Federico Carlini & Paolo Santucci de Magistris, 2013. "On the identification of fractionally cointegrated VAR models with the F(d) condition," CREATES Research Papers 2013-44, School of Economics and Management, University of Aarhus.
  4. Giuseppe Cavaliere & Morten Ørregaard Nielsen & A. M. Robert Taylor, 2014. "Quasi-Maximum Likelihood Estimation of Heteroskedastic Fractional Time Series Models," Working Papers 1324, Queen's University, Department of Economics.
  5. Søren Johansen, 2011. "An extension of cointegration to fractional autoregressive processes," CREATES Research Papers 2011-06, School of Economics and Management, University of Aarhus.
  6. Luis A. Gil-Alana & Antonio Moreno, 2009. "Fractional Integration and Structural Breaks in U.S. Macro Dynamics," Faculty Working Papers 02/09, School of Economics and Business Administration, University of Navarra.
  7. Søren Johansen & Morten Ørregaard Nielsen, 2012. "The role of initial values in nonstationary fractional time series models," CREATES Research Papers 2012-47, School of Economics and Management, University of Aarhus.
  8. Giuseppe Cavaliere & Morten Ørregaard Nielsen & A.M. Robert Taylor, 2014. "Bootstrap Score Tests for Fractional Integration in Heteroskedastic ARFIMA Models, with an Application to Price Dynamics in Commodity Spot and Futures Markets," CREATES Research Papers 2014-22, School of Economics and Management, University of Aarhus.
  9. Tschernig, Rolf & Weber, Enzo & Weigand, Roland, 2013. "Fractionally Integrated VAR Models with a Fractional Lag Operator and Deterministic Trends: Finite Sample Identification and Two-step Estimation," University of Regensburg Working Papers in Business, Economics and Management Information Systems 471, University of Regensburg, Department of Economics.
  10. Morten Ørregaard Nielsen, 2008. "Nonparametric Cointegration Analysis of Fractional Systems With Unknown Integration Orders," Working Papers 1174, Queen's University, Department of Economics.
  11. Niels Haldrup & Robinson Kruse, 2014. "Discriminating between fractional integration and spurious long memory," CREATES Research Papers 2014-19, School of Economics and Management, University of Aarhus.

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