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An asymptotic invariance property of the common trends under linear transformations of the data


  • Johansen, Søren
  • Juselius, Katarina


It is well known that if Xt is a nonstationary process and Yt is a linear function of Xt, then cointegration of Yt implies cointegration of Xt. We want to find an analogous result for common trends if Xt is generated by a finite order VAR with i.i.d. (0,Ωx) errors εxt. We first show that Yt has an infinite order VAR representation in terms of its white noise prediction errors, εyt, which are a linear process in εxt, the prediction error for Xt. We then apply this result to show that the limit of the common trends for Yt generated by εyt, are linear functions of the common trends for Xt, generated by εxt.

Suggested Citation

  • Johansen, Søren & Juselius, Katarina, 2014. "An asymptotic invariance property of the common trends under linear transformations of the data," Journal of Econometrics, Elsevier, vol. 178(P2), pages 310-315.
  • Handle: RePEc:eee:econom:v:178:y:2014:i:p2:p:310-315
    DOI: 10.1016/j.jeconom.2013.08.029

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    References listed on IDEAS

    1. Giese, Julia V., 2008. "Level, Slope, Curvature: Characterising the Yield Curve in a Cointegrated VAR Model," Economics - The Open-Access, Open-Assessment E-Journal, Kiel Institute for the World Economy (IfW), vol. 2, pages 1-20.
    2. 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.
    3. Saikkonen, Pentti, 1992. "Estimation and Testing of Cointegrated Systems by an Autoregressive Approximation," Econometric Theory, Cambridge University Press, vol. 8(01), pages 1-27, March.
    4. Saikkonen, Pentti & Lütkepohl, HELMUT, 1996. "Infinite-Order Cointegrated Vector Autoregressive Processes," Econometric Theory, Cambridge University Press, vol. 12(05), pages 814-844, December.
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    Cited by:

    1. Søren Johansen, 2018. "Cointegration and adjustment in the infinite order CVAR representation of some partially observed CVAR(1) models," Discussion Papers 18-05, University of Copenhagen. Department of Economics.
    2. Søren Johansen & Morten Nyboe Tabor, 0703. "Cointegration between trends and their estimators in state space models and CVAR models," CREATES Research Papers 2017-11, Department of Economics and Business Economics, Aarhus University.
    3. repec:eee:ecolet:v:165:y:2018:i:c:p:73-76 is not listed on IDEAS
    4. Katarina Juselius, 2018. "Searching for a theory that fits the data: A personal research odyssey," Discussion Papers 18-07, University of Copenhagen. Department of Economics.
    5. repec:gam:jecnmx:v:5:y:2017:i:3:p:36-:d:109242 is not listed on IDEAS

    More about this item


    Cointegration vectors; Common trends; Prediction errors;

    JEL classification:

    • 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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