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A Canonical Form for Unit Root Processes in the State Space Framework

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  • Dietmar Bauer
  • Martin Wagner

Abstract

In this paper we develop a canonical state space representation for rational stochastic processes containing unit roots with integer integration orders at arbitrary points on the unit circle. It is shown that the state space framework, which is -- in a certain sense made precise in the paper -- equivalent to the ARMA framework, is very suitable for the analysis of unit roots and cointegration issues. The advantages become especially prominent for systems with higher integration orders at the various roots on the unit circle. A unique state space representation is constructed that clearly reveals the integration and cointegration properties. The canonical form given in the paper can be used to construct a parameterization of the class of all rational processes with a given state space unit root structure, which is defined in the paper

Suggested Citation

  • Dietmar Bauer & Martin Wagner, 2003. "A Canonical Form for Unit Root Processes in the State Space Framework," Diskussionsschriften dp0312, Universitaet Bern, Departement Volkswirtschaft.
    • Handle: RePEc:ube:dpvwib:dp0312
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    7. Fragiskos Archontakis, 1998. "An alternative proof of Granger’s Representation Theorem forI(1) systems through Jordan matrices," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 7(2), pages 111-127, August.
    8. Hannes Leeb & Benedikt Poetscher, 1999. "The variance of an integrated process need not diverge to infinity," Econometrics 9907001, University Library of Munich, Germany.
    9. Dietmar Bauer & Martin Wagner, 2002. "Asymptotic Properties of Pseudo Maximum Likelihood Estimates for Multiple Frequency I(1) Processes," Diskussionsschriften dp0205, Universitaet Bern, Departement Volkswirtschaft.
    10. Aoki, Masanao & Havenner, Arthur, 1989. "A method for approximate representation of vector-valued time series and its relation to two alternatives," Journal of Econometrics, Elsevier, vol. 42(2), pages 181-199, October.
    11. Lee, Hahn Shik, 1992. "Maximum likelihood inference on cointegration and seasonal cointegration," Journal of Econometrics, Elsevier, vol. 54(1-3), pages 1-47.
    12. Dietmar Bauer & Martin Wagner, 2003. "On Polynomial Cointegration in the State Space Framework," Diskussionsschriften dp0313, Universitaet Bern, Departement Volkswirtschaft.
    13. repec:cup:etheor:v:8:y:1992:i:2:p:188-202 is not listed on IDEAS
    14. Gregoir, Stephane & Laroque, Guy, 1994. "Polynomial cointegration estimation and test," Journal of Econometrics, Elsevier, vol. 63(1), pages 183-214, July.
    15. Leeb, Hannes & Pötscher, Benedikt M., 2001. "The Variance Of An Integrated Process Need Not Diverge To Infinity, And Related Results On Partial Sums Of Stationary Processes," Econometric Theory, Cambridge University Press, vol. 17(4), pages 671-685, August.
    16. Bauer, Dietmar & Wagner, Martin, 2002. "Estimating cointegrated systems using subspace algorithms," Journal of Econometrics, Elsevier, vol. 111(1), pages 47-84, November.
    17. Johansen, Søren, 1992. "A Representation of Vector Autoregressive Processes Integrated of Order 2," Econometric Theory, Cambridge University Press, vol. 8(2), pages 188-202, June.
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    Cited by:

    1. Phillips, Peter C.B., 2005. "Automated Discovery In Econometrics," Econometric Theory, Cambridge University Press, vol. 21(1), pages 3-20, February.
    2. Bauer, Dietmar & Wagner, Martin, 2005. "Autoregressive Approximations of Multiple Frequency I(1) Processes," Economics Series 174, Institute for Advanced Studies.
    3. Bauer, Dietmar & Wagner, Martin, 2002. "Estimating cointegrated systems using subspace algorithms," Journal of Econometrics, Elsevier, vol. 111(1), pages 47-84, November.
    4. Paruolo Paolo, 2005. "Design of vector autoregressive processes for invariant statistics," Economics and Quantitative Methods qf0504, Department of Economics, University of Insubria.
    5. Dietmar Bauer & Martin Wagner, 2002. "Asymptotic Properties of Pseudo Maximum Likelihood Estimates for Multiple Frequency I(1) Processes," Diskussionsschriften dp0205, Universitaet Bern, Departement Volkswirtschaft.
    6. Martin Wagner, 2004. "A Comparison of Johansen's, Bierens’ and the Subspace Algorithm Method for Cointegration Analysis," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 66(3), pages 399-424, July.
    7. Dietmar Bauer & Martin Wagner, 2003. "The Performance of Subspace Algorithm Cointegration Analysis: A Simulation Study," Diskussionsschriften dp0308, Universitaet Bern, Departement Volkswirtschaft.
    8. Dietmar Bauer & Martin Wagner, 2003. "On Polynomial Cointegration in the State Space Framework," Diskussionsschriften dp0313, Universitaet Bern, Departement Volkswirtschaft.
    9. Bauer, Dietmar & Wagner, Martin, 2009. "Using subspace algorithm cointegration analysis: Simulation performance and application to the term structure," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 1954-1973, April.
    10. Søren Johansen, 2009. "Representation of Cointegrated Autoregressive Processes with Application to Fractional Processes," Econometric Reviews, Taylor & Francis Journals, vol. 28(1-3), pages 121-145.

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    More about this item

    Keywords

    canonical form; state space representation; unit roots; cointegration;
    All these keywords.

    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

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