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On Polynomial Cointegration in the State Space Framework

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  • Dietmar Bauer
  • Martin Wagner

Abstract

This paper deals with polynomial cointegration, i.e. with the phenomenon that linear combinations of a vector valued rational unit root process and lags of the process are of lower integration order than the process itself (for definitions see Section 2). The analysis is performed in the state space representation of rational unit root processes derived in Bauer and Wagner (2003). The state space framework is an equivalent alternative to the ARMA framework. Unit roots are allowed to occur at any point on the unit circle with arbitrary integer integration order. In the paper simple criteria for the existence of non-trivial polynomial cointegrating relationships are given. Trivial cointegrating relationships lead to the reduction of the integration order simply by appropriate differencing. The set of all polynomial cointegrating relationships is determined from simple orthogonality conditions derived directly from the state space representation. These results are important for analyzing the structure of unit root processes and their polynomial cointegrating relationships and also for the parameterization for system sets with given cointegration properties.

Suggested Citation

  • Dietmar Bauer & Martin Wagner, 2003. "On Polynomial Cointegration in the State Space Framework," Diskussionsschriften dp0313, Universitaet Bern, Departement Volkswirtschaft.
    • Handle: RePEc:ube:dpvwib:dp0313
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    References listed on IDEAS

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    1. Johansen, Soren & Schaumburg, Ernst, 1998. "Likelihood analysis of seasonal cointegration," Journal of Econometrics, Elsevier, vol. 88(2), pages 301-339, November.
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    3. Stock, James H & Watson, Mark W, 1993. "A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems," Econometrica, Econometric Society, vol. 61(4), pages 783-820, July.
    4. Gregoir, Stephane & Laroque, Guy, 1994. "Polynomial cointegration estimation and test," Journal of Econometrics, Elsevier, vol. 63(1), pages 183-214, July.
    5. 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.
    6. Gregoir, Stéphane, 1999. "Multivariate Time Series With Various Hidden Unit Roots, Part Ii," Econometric Theory, Cambridge University Press, vol. 15(4), pages 469-518, August.
    7. Hannes Leeb & Benedikt Poetscher, 1999. "The variance of an integrated process need not diverge to infinity," Econometrics 9907001, University Library of Munich, Germany.
    8. Dietmar Bauer & Martin Wagner, 2002. "A Canonical Form for Unit Root Processes in the State Space Framework," Diskussionsschriften dp0204, Universitaet Bern, Departement Volkswirtschaft.
    9. Haldrup, Niels & Salmon, Mark, 1998. "Representations of I(2) cointegrated systems using the Smith-McMillan form," Journal of Econometrics, Elsevier, vol. 84(2), pages 303-325, June.
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    Cited by:

    1. Dietmar Bauer & Martin Wagner, 2002. "A Canonical Form for Unit Root Processes in the State Space Framework," Diskussionsschriften dp0204, Universitaet Bern, Departement Volkswirtschaft.
    2. Dietmar Bauer & Martin Wagner, 2002. "Asymptotic Properties of Pseudo Maximum Likelihood Estimates for Multiple Frequency I(1) Processes," Diskussionsschriften dp0205, Universitaet Bern, Departement Volkswirtschaft.
    3. Dietmar Bauer & Lukas Matuschek & Patrick de Matos Ribeiro & Martin Wagner, 2020. "A Parameterization of Models for Unit Root Processes: Structure Theory and Hypothesis Testing," Econometrics, MDPI, vol. 8(4), pages 1-54, November.

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    More about this item

    Keywords

    Unit roots; polynomial cointegration; state space representation;
    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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