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Cointegration, Root Functions and Minimal Bases

Author

Listed:
  • Massimo Franchi

    (Department of Statistical Sciences, Sapienza University of Rome, P.le A. Moro 5, 00185 Rome, Italy)

  • Paolo Paruolo

    (European Commission Joint Research Centre, Via E. Fermi 2749, 21027 Ispra, Italy)

Abstract

This paper discusses the notion of cointegrating space for linear processes integrated of any order. It first shows that the notions of (polynomial) cointegrating vectors and of root functions coincide. Second, it discusses how the cointegrating space can be defined (i) as a vector space of polynomial vectors over complex scalars, (ii) as a free module of polynomial vectors over scalar polynomials, or finally (iii) as a vector space of rational vectors over rational scalars. Third, it shows that a canonical set of root functions can be used as a basis of the various notions of cointegrating space. Fourth, it reviews results on how to reduce polynomial bases to minimal order—i.e., minimal bases. The application of these results to Vector AutoRegressive processes integrated of order 2 is found to imply the separation of polynomial cointegrating vectors from non-polynomial ones.

Suggested Citation

  • Massimo Franchi & Paolo Paruolo, 2021. "Cointegration, Root Functions and Minimal Bases," Econometrics, MDPI, vol. 9(3), pages 1-27, August.
    • Handle: RePEc:gam:jecnmx:v:9:y:2021:i:3:p:31-:d:615763
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    More about this item

    Keywords

    VAR; cointegration; I(d); vector spaces;
    All these keywords.

    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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