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Granger's representation theorem: A closed-form expression for I(1) processes

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  • Peter Reinhard Hansen

Abstract

The Granger representation theorem states that a cointegrated vector autoregressive process can be decomposed into four components: a random walk, a stationary process, a deterministic part, and a term that depends on the initial values. In this paper, we present a new proof of the theorem. This proof enables us to derive closed-form expressions of all terms of the representation and allows a unified treatment of models with different deterministic specifications. The applicability of our results is illustrated by examples. For example, the closed-form expressions are useful for impulse response analyses and facilitate the analysis of cointegration models with structural changes. Copyright 2005 Royal Economic Society

Suggested Citation

  • Peter Reinhard Hansen, 2005. "Granger's representation theorem: A closed-form expression for I(1) processes," Econometrics Journal, Royal Economic Society, vol. 8(1), pages 23-38, March.
  • Handle: RePEc:ect:emjrnl:v:8:y:2005:i:1:p:23-38
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    Cited by:

    1. Hansen, Peter Reinhard, 2015. "A martingale decomposition of discrete Markov chains," Economics Letters, Elsevier, vol. 133(C), pages 14-18.
    2. Adolfson, Malin & Laséen, Stefan & Lindé, Jesper & Villani, Mattias, 2008. "Evaluating an estimated new Keynesian small open economy model," Journal of Economic Dynamics and Control, Elsevier, vol. 32(8), pages 2690-2721, August.
    3. Paruolo Paolo, 2005. "Design of vector autoregressive processes for invariant statistics," Economics and Quantitative Methods qf0504, Department of Economics, University of Insubria.
    4. Håvard Hungnes, 2010. "Identifying Structural Breaks in Cointegrated Vector Autoregressive Models," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 72(4), pages 551-565, August.
    5. Mikael Juselius & Moshe Kim & Staffan Ringbom, 2015. "Do markup dynamics reflect fundamentals or changes in conduct?," Empirical Economics, Springer, vol. 48(3), pages 1119-1147, May.
    6. Alexander Chudik & Vanessa Smith, 2013. "The GVAR approach and the dominance of the U.S. economy," Globalization and Monetary Policy Institute Working Paper 136, Federal Reserve Bank of Dallas.
    7. 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.
    8. Òscar Jordà, 2005. "Estimation and Inference of Impulse Responses by Local Projections," American Economic Review, American Economic Association, vol. 95(1), pages 161-182, March.
    9. Hallin, Marc & van den Akker, Ramon & Werker, Bas J.M., 2016. "Semiparametric error-correction models for cointegration with trends: Pseudo-Gaussian and optimal rank-based tests of the cointegration rank," Journal of Econometrics, Elsevier, vol. 190(1), pages 46-61.
    10. Swensen, Anders Rygh, 2011. "A bootstrap algorithm for testing cointegration rank in VAR models in the presence of stationary variables," Journal of Econometrics, Elsevier, vol. 165(2), pages 152-162.
    11. Hansen, Peter R. & Lunde, Asger, 2006. "Realized Variance and Market Microstructure Noise," Journal of Business & Economic Statistics, American Statistical Association, vol. 24, pages 127-161, April.
    12. Niklas Ahlgren & Mikael Juselius, 2012. "Tests for cointegration rank and the initial condition," Empirical Economics, Springer, vol. 42(3), pages 667-691, June.
    13. Maria Grazia Zoia, 2006. "A New Algebra ic Approach to Representation Theorems for (Co)integrated Processes up to the Second Order," Research Papers by the Institute of Economics and Econometrics, Geneva School of Economics and Management, University of Geneva 2006.06, Institut d'Economie et Econométrie, Université de Genève.
    14. Hallin, M. & Werker, B.J.M. & van den Akker, R., 2015. "Optimal Pseudo-Gaussian and Rank-based Tests of the Cointegration Rank in Semiparametric Error-correction Models," Discussion Paper 2015-001, Tilburg University, Center for Economic Research.
    15. repec:bis:bisifc:46-10 is not listed on IDEAS
    16. McCauley, Joseph L. & Bassler, Kevin E. & Gunaratne, Gemunu H., 2009. "Is integration I(d) applicable to observed economics and finance time series?," International Review of Financial Analysis, Elsevier, vol. 18(3), pages 101-108, June.

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