IDEAS home Printed from https://ideas.repec.org/p/aah/create/2008-53.html
   My bibliography  Save this paper

Maximum likelihood estimation of fractionally cointegrated systems

Author

Listed:
  • Katarzyna Lasak

    () (School of Economics and Management, University of Aarhus, Denmark and CREATES)

Abstract

In this paper we consider a fractionally cointegrated error correction model and investigate asymptotic properties of the maximum likelihood (ML) estimators of the matrix of the cointegration relations, the degree of fractional cointegration, the matrix of the speed of adjustment to the equilibrium parameters and the variance-covariance matrix of the error term. We show that using ML principles to estimate jointly all parameters of the fractionally cointegrated system we obtain consistent estimates and provide their asymptotic distributions. The cointegration matrix is asymptotically mixed normal distributed, while the degree of fracional cointegration and the speed of adjustment to the equilibrium matrix have joint normal distribution, which proves the intuition that the memory of the cointegrating residuals affects the speed of convergence to the long-run equilibrium, but does not have any influence on the long-run relationship. The rate of convergence of the estimators of the long-run relationships depends on the cointegration degree but it is optimal for the strong cointegration case considered. We also prove that misspecification of the degree of fractional cointegation does not affect the consistency of the estimators of the cointegration relationships, although usual inference rules are not valid. We illustrate our results in finite samples by Monte Carlo analysis.

Suggested Citation

  • Katarzyna Lasak, 2008. "Maximum likelihood estimation of fractionally cointegrated systems," CREATES Research Papers 2008-53, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2008-53
    as

    Download full text from publisher

    File URL: ftp://ftp.econ.au.dk/creates/rp/08/rp08_53.pdf
    Download Restriction: no

    References listed on IDEAS

    as
    1. Hualde, J. & Robinson, P.M., 2007. "Root-n-consistent estimation of weak fractional cointegration," Journal of Econometrics, Elsevier, vol. 140(2), pages 450-484, October.
    2. Robinson, P.M. & Iacone, F., 2005. "Cointegration in fractional systems with deterministic trends," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 263-298.
    3. Engle, Robert & Granger, Clive, 2015. "Co-integration and error correction: Representation, estimation, and testing," Applied Econometrics, Publishing House "SINERGIA PRESS", vol. 39(3), pages 106-135.
    4. Breitung, Jorg & Hassler, Uwe, 2002. "Inference on the cointegration rank in fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 110(2), pages 167-185, October.
    5. Willa Chen & Clifford Hurvich, 2004. "Semiparametric Estimation of Fractional Cointegrating Subspaces," Econometrics 0412007, EconWPA.
    6. Marinucci, D & Robinson, Peter M., 2001. "Semiparametric fractional cointegration analysis," LSE Research Online Documents on Economics 2269, London School of Economics and Political Science, LSE Library.
    7. P. M. Robinson & J. Hualde, 2003. "Cointegration in Fractional Systems with Unknown Integration Orders," Econometrica, Econometric Society, vol. 71(6), pages 1727-1766, November.
    8. Robinson, Peter M. & Yajima, Yoshihiro, 2002. "Determination of cointegrating rank in fractional systems," Journal of Econometrics, Elsevier, vol. 106(2), pages 217-241, February.
    9. Nielsen, Morten Orregaard, 2007. "Local Whittle Analysis of Stationary Fractional Cointegration and the ImpliedRealized Volatility Relation," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 427-446, October.
    10. Michael Dueker & Richard Startz, 1998. "Maximum-Likelihood Estimation Of Fractional Cointegration With An Application To U.S. And Canadian Bond Rates," The Review of Economics and Statistics, MIT Press, vol. 80(3), pages 420-426, August.
    11. Ingolf Dittmann, 2004. "Error Correction Models for Fractionally Cointegrated Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(1), pages 27-32, January.
    12. Lasak, Katarzyna, 2010. "Likelihood based testing for no fractional cointegration," Journal of Econometrics, Elsevier, vol. 158(1), pages 67-77, September.
    13. Hassler, Uwe & Marmol, Francesc, 1998. "Fractional cointegrating regressions in the presence of linear time trends," DES - Working Papers. Statistics and Econometrics. WS 9794, Universidad Carlos III de Madrid. Departamento de Estadística.
    14. Marinucci, D. & Robinson, P. M., 2001. "Semiparametric fractional cointegration analysis," Journal of Econometrics, Elsevier, vol. 105(1), pages 225-247, November.
    15. D Marinucci & Peter M Robinson, 2001. "Semiparametric Fractional Cointegration Analysis," STICERD - Econometrics Paper Series 420, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    16. Andersson, Michael K. & Gredenhoff, Mikael P., 1999. "On the maximum likelihood cointegration procedure under a fractional equilibrium error," Economics Letters, Elsevier, vol. 65(2), pages 143-147, November.
    17. 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, March.
    18. 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.
    19. Luis A. Gil-Alana, 2004. "A joint test of fractional integration and structural breaks at a known period of time," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(5), pages 691-700, September.
    20. Chen, Willa W. & Hurvich, Clifford M., 2003. "Semiparametric Estimation of Multivariate Fractional Cointegration," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 629-642, January.
    21. 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.
    22. Morten Ørregaard Nielsen & Per Frederiksen, 2008. "Fully Modified Narrow-Band Least Squares Estimation of Stationary Fractional Cointegration," Working Papers 1171, Queen's University, Department of Economics.
    23. Granger, Clive W J, 1986. "Developments in the Study of Cointegrated Economic Variables," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 48(3), pages 213-228, August.
    24. Javier Hualde & Peter Robinson, 2006. "Semiparametric Estimation of Fractional Cointegration," Faculty Working Papers 07/06, School of Economics and Business Administration, University of Navarra.
    25. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-1580, November.
    26. Juan J. Dolado & Francesc Marmol, 2004. "Asymptotic inference results for multivariate long-memory processes," Econometrics Journal, Royal Economic Society, vol. 7(1), pages 168-190, June.
    27. Chen, Willa W. & Hurvich, Clifford M., 2003. "Estimating fractional cointegration in the presence of polynomial trends," Journal of Econometrics, Elsevier, vol. 117(1), pages 95-121, November.
    28. Lyhagen, Johan, 1998. "Maximum likelihood estimation of the multivariate fractional cointegrating model," SSE/EFI Working Paper Series in Economics and Finance 233, Stockholm School of Economics.
    29. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
    30. Christensen, Bent Jesper & Nielsen, Morten Orregaard, 2006. "Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting," Journal of Econometrics, Elsevier, vol. 133(1), pages 343-371, July.
    31. Ignacio N. Lobato & Carlos Velasco, 2006. "Optimal Fractional Dickey-Fuller tests," Econometrics Journal, Royal Economic Society, vol. 9(3), pages 492-510, November.
    32. Davidson, James, 2002. "A model of fractional cointegration, and tests for cointegration using the bootstrap," Journal of Econometrics, Elsevier, vol. 110(2), pages 187-212, October.
    33. Francesc Marmol & Carlos Velasco, 2004. "Consistent Testing of Cointegrating Relationships," Econometrica, Econometric Society, vol. 72(6), pages 1809-1844, November.
    34. Peter C.B. Phillips & Bruce E. Hansen, 1988. "Statistical Inference in Instrumental Variables," Cowles Foundation Discussion Papers 869R, Cowles Foundation for Research in Economics, Yale University, revised Apr 1989.
    35. Sims, Christopher A & Stock, James H & Watson, Mark W, 1990. "Inference in Linear Time Series Models with Some Unit Roots," Econometrica, Econometric Society, vol. 58(1), pages 113-144, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Federico Carlini & Katarzyna Lasak, 2014. "On an Estimation Method for an Alternative Fractionally Cointegrated Model," Tinbergen Institute Discussion Papers 14-052/III, Tinbergen Institute.
    2. Søren Johansen, 2010. "An Extension of Cointegration to Fractional Autoregressive Processes," Discussion Papers 10-28, University of Copenhagen. Department of Economics.
    3. Katarzyna Łasak & Carlos Velasco, 2015. "Fractional Cointegration Rank Estimation," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(2), pages 241-254, April.
    4. Marcel Aloy & Gilles De Truchis, 2012. "Estimation and Testing for Fractional Cointegration," Working Papers halshs-00793206, HAL.
    5. Lasak, Katarzyna, 2010. "Likelihood based testing for no fractional cointegration," Journal of Econometrics, Elsevier, vol. 158(1), pages 67-77, September.
    6. Caporin, Massimiliano & Ranaldo, Angelo & Santucci de Magistris, Paolo, 2013. "On the predictability of stock prices: A case for high and low prices," Journal of Banking & Finance, Elsevier, vol. 37(12), pages 5132-5146.
    7. Eduardo Rossi & Paolo Santucci de Magistris, 2009. "A No Arbitrage Fractional Cointegration Analysis Of The Range Based Volatility," CREATES Research Papers 2009-31, Department of Economics and Business Economics, Aarhus University.
    8. Eduardo Rossi & Paolo Santucci de Magistris, 2013. "A No‐Arbitrage Fractional Cointegration Model for Futures and Spot Daily Ranges," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 33(1), pages 77-102, January.

    More about this item

    Keywords

    Error correction model; Gaussian VAR model; Maximum likelihood estimation; Fractional cointegration;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:aah:create:2008-53. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (). General contact details of provider: http://www.econ.au.dk/afn/ .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.