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Representation of Cointegrated Autoregressive Processes with Application to Fractional Processes

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  • Søren Johansen

Abstract

We analyze vector autoregressive processes using the matrix valued characteristic polynomial. The purpose of this article is to give a survey of the mathematical results on inversion of a matrix polynomial in case there are unstable roots, to study integrated and cointegrated processes. The new results are in the I(2) representation, which contains explicit formulas for the first two terms and a useful property of the third. We define a new error correction model for fractional processes and derive a representation of the solution.

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  • 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.
    • Handle: RePEc:taf:emetrv:v:28:y:2009:i:1-3:p:121-145
    DOI: 10.1080/07474930802387977
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    References listed on IDEAS

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    1. Engle, R. F. & Granger, C. W. J. (ed.), 1991. "Long-Run Economic Relationships: Readings in Cointegration," OUP Catalogue, Oxford University Press, number 9780198283393.
    2. Mario Faliva & Maria Grazia Zoia, 2006. "Topics in Dynamic Model Analysis," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-29239-5, December.
    3. Dietmar Bauer & Martin Wagner, 2002. "A Canonical Form for Unit Root Processes in the State Space Framework," Diskussionsschriften dp0204, Universitaet Bern, Departement Volkswirtschaft.
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    Cited by:

    1. Federico Carlini & Katarzyna (K.A.) Lasak, 2018. "Likelihood based inference for an Identifiable Fractional Vector Error Correction Model," Tinbergen Institute Discussion Papers 18-085/III, Tinbergen Institute.
    2. Anders Rygh Swensen, 2022. "On causal and non‐causal cointegrated vector autoregressive time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(2), pages 178-196, March.
    3. Beare, Brendan K. & Seo, Won-Ki, 2020. "Representation Of I(1) And I(2) Autoregressive Hilbertian Processes," Econometric Theory, Cambridge University Press, vol. 36(5), pages 773-802, October.
    4. Massimo Franchi & Paolo Paruolo, 2012. "On ABCs (and Ds) of VAR representations of DSGE models," Working Paper series 56_12, Rimini Centre for Economic Analysis, revised Aug 2012.
    5. Massimo Franchi & Paolo Paruolo, 2015. "Minimality of State Space Solutions of DSGE Models and Existence Conditions for Their VAR Representation," Computational Economics, Springer;Society for Computational Economics, vol. 46(4), pages 613-626, December.
    6. Massimo Franchi & Paolo Paruolo, 2019. "A general inversion theorem for cointegration," Econometric Reviews, Taylor & Francis Journals, vol. 38(10), pages 1176-1201, November.
    7. Tim Bollerslev & Daniela Osterrieder & Natalia Sizova & George Tauchen, 2011. "Risk and Return: Long-Run Relationships, Fractional Cointegration, and Return Predictability," CREATES Research Papers 2011-51, Department of Economics and Business Economics, Aarhus University.
    8. Berta, Paolo & Lovaglio, Pietro Giorgio & Paruolo, Paolo & Verzillo, Stefano, 2020. "Real Time Forecasting of Covid-19 Intensive Care Units demand," JRC Working Papers in Economics and Finance 2020-08, Joint Research Centre, European Commission.
    9. Katarzyna Łasak & Carlos Velasco, 2015. "Fractional Cointegration Rank Estimation," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(2), pages 241-254, April.
    10. de Truchis, Gilles, 2013. "Approximate Whittle analysis of fractional cointegration and the stock market synchronization issue," Economic Modelling, Elsevier, vol. 34(C), pages 98-105.
    11. Lasak, Katarzyna, 2010. "Likelihood based testing for no fractional cointegration," Journal of Econometrics, Elsevier, vol. 158(1), pages 67-77, September.
    12. 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.
    13. repec:hum:wpaper:sfb649dp2016-001 is not listed on IDEAS
    14. Chen, Cathy Yi-Hsuan & Chiang, Thomas C. & Härdle, Wolfgang Karl, 2016. "Downside risk and stock returns: An empirical analysis of the long-run and short-run dynamics from the G-7 Countries," SFB 649 Discussion Papers 2016-001, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    15. Franchi, Massimo & Paruolo, Paolo, 2011. "A characterization of vector autoregressive processes with common cyclical features," Journal of Econometrics, Elsevier, vol. 163(1), pages 105-117, July.
    16. Katarzyna Łasak & Johannes Lont, 2020. "Observation Driven Long Run Equilibria," Computational Economics, Springer;Society for Computational Economics, vol. 55(2), pages 551-575, February.
    17. Daniela Osterrieder & Daniel Ventosa-Santaulària & J. Eduardo Vera-Valdés, 2015. "Unbalanced Regressions and the Predictive Equation," CREATES Research Papers 2015-09, Department of Economics and Business Economics, Aarhus University.
    18. Brendan K. Beare & Juwon Seo & Won-Ki Seo, 2017. "Cointegrated Linear Processes in Hilbert Space," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(6), pages 1010-1027, November.
    19. Federico Carlini & Katarzyna Lasak, 2014. "On an Estimation Method for an Alternative Fractionally Cointegrated Model," Tinbergen Institute Discussion Papers 14-052/III, Tinbergen Institute.
    20. Søren Johansen & Katarina Juselius, 2010. "An invariance property of the common trends under linear transformations of the data," CREATES Research Papers 2010-72, Department of Economics and Business Economics, Aarhus University.
    21. Chen, Xiaoyu & Chiang, Thomas C., 2016. "Stock returns and economic forces—An empirical investigation of Chinese markets," Global Finance Journal, Elsevier, vol. 30(C), pages 45-65.
    22. H. Peter Boswijk & Paolo Paruolo, 2017. "Likelihood Ratio Tests of Restrictions on Common Trends Loading Matrices in I(2) VAR Systems," Econometrics, MDPI, vol. 5(3), pages 1-17, June.

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