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A State Space Canonical Form For Unit Root Processes


  • Bauer, Dietmar
  • Wagner, Martin


In this paper we develop a canonical state space representation of autoregressive moving average (ARMA) processes with unit roots with integer integration orders at arbitrary unit root frequencies. The developed representation utilizes a state process with a particularly simple dynamic structure, which in turn renders this representation highly suitable for unit root, cointegration, and polynomial cointegration analysis. We also propose a new definition of polynomial cointegration that overcomes limitations of existing definitions and extends the definition of multicointegration for I(2) processes of Granger and Lee ( 1989a , Journal of Applied Econometrics 4, 145–159). A major purpose of the canonical representation for statistical analysis is the development of parameterizations of the sets of all state space systems of a given system order with specified unit root frequencies and integration orders. This is, e.g., useful for pseudo maximum likelihood estimation. In this respect an advantage of the state space representation, compared to ARMA representations, is that it easily allows one to put in place restrictions on the (co)integration properties. The results of the paper are exemplified for the cases of largest interest in applied work.

Suggested Citation

  • Bauer, Dietmar & Wagner, Martin, 2012. "A State Space Canonical Form For Unit Root Processes," Econometric Theory, Cambridge University Press, vol. 28(06), pages 1313-1349, December.
  • Handle: RePEc:cup:etheor:v:28:y:2012:i:06:p:1313-1349_00

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    Cited by:

    1. repec:eee:ecolet:v:155:y:2017:i:c:p:39-42 is not listed on IDEAS
    2. Massimo Franchi, 2017. "On the structure of state space systems with unit roots," DSS Empirical Economics and Econometrics Working Papers Series 2017/4, Centre for Empirical Economics and Econometrics, Department of Statistics, "Sapienza" University of Rome.
    3. repec:eee:ecolet:v:165:y:2018:i:c:p:73-76 is not listed on IDEAS
    4. Massimo Franchi & Paolo Paruolo, 2017. "A general inversion theorem for cointegration," DSS Empirical Economics and Econometrics Working Papers Series 2017/3, Centre for Empirical Economics and Econometrics, Department of Statistics, "Sapienza" University of Rome.

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