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A general inversion theorem for cointegration

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  • Massimo Franchi
  • Paolo Paruolo

Abstract

A generalization of the Granger and the Johansen Representation Theorems valid for any (possibly fractional) order of integration is presented. This Representation Theorem is based on inversion results that characterize the order of the pole and the coefficients of the Laurent series representation of the inverse of a matrix function around a singular point. Explicit expressions of the matrix coefficients of the (polynomial) cointegrating relations, of the Common Trends and of the Triangular representations are provided, either starting from the Moving Average or the Auto Regressive form. This contribution unifies different approaches in the literature and extends them to an arbitrary order of integration. The role of deterministic terms is discussed in detail.

Suggested Citation

  • Massimo Franchi & Paolo Paruolo, 2019. "A general inversion theorem for cointegration," Econometric Reviews, Taylor & Francis Journals, vol. 38(10), pages 1176-1201, November.
    • Handle: RePEc:taf:emetrv:v:38:y:2019:i:10:p:1176-1201
    DOI: 10.1080/07474938.2018.1536100
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    Cited by:

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    2. James A. Duffy & Jerome R. Simons, 2020. "Cointegration without Unit Roots," Papers 2002.08092, arXiv.org, revised Apr 2023.
    3. Massimo Franchi & Paolo Paruolo, 2021. "Cointegration, Root Functions and Minimal Bases," Econometrics, MDPI, vol. 9(3), pages 1-27, August.
    4. Kheifets, Igor L. & Phillips, Peter C.B., 2023. "Fully modified least squares cointegrating parameter estimation in multicointegrated systems," Journal of Econometrics, Elsevier, vol. 232(2), pages 300-319.
    5. Berta, P. & Lovaglio, P.G. & Paruolo, P. & Verzillo, S., 2020. "Real Time Forecasting of Covid-19 Intensive Care Units demand," Health, Econometrics and Data Group (HEDG) Working Papers 20/16, HEDG, c/o Department of Economics, University of York.
    6. Mario Faliva & Maria Grazia Zoia, 2021. "Cointegrated Solutions of Unit-Root VARs: An Extended Representation Theorem," Papers 2102.10626, arXiv.org.
    7. 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.
    8. Franchi, Massimo & Paruolo, Paolo, 2020. "Cointegration In Functional Autoregressive Processes," Econometric Theory, Cambridge University Press, vol. 36(5), pages 803-839, October.
    9. Matteo Barigozzi & Marco Lippi & Matteo Luciani, 2020. "Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors," Econometrics, MDPI, vol. 8(1), pages 1-23, February.

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    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C33 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Models with Panel Data; Spatio-temporal Models
    • C55 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Large Data Sets: Modeling and Analysis

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