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Summability of stochastic processes: a generalization of integration and co-integration valid for non-linear processes

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  • Berenguer Rico, Vanessa
  • Gonzalo, Jesús

Abstract

The order of integration is valid to characterize linear processes; but it is not appropriate for non-linear worlds. We propose the concept of summability (a re-scaled partial sum of the process being Op(1)) to handle non-linearities. The paper shows that this new concept, S (δ): (i) generalizes I (δ); (ii) measures the degree of persistence as well as of the evolution of the variance; (iii) controls the balancedness of non-linear relationships; (iv) opens the door to the concept of co-summability which represents a generalization of co-integration for non-linear processes. To make this concept empirically applicable, an estimator for δ and its asymptotic properties are provided. The finite sample performance of subsampling confidence intervals is analyzed via a Monte Carlo experiment. The paper finishes with the estimation of the degree of summability of the macroeconomic variables in an extended version of the Nelson-Plosser database.

Suggested Citation

  • Berenguer Rico, Vanessa & Gonzalo, Jesús, 2011. "Summability of stochastic processes: a generalization of integration and co-integration valid for non-linear processes," UC3M Working papers. Economics we1115, Universidad Carlos III de Madrid. Departamento de Economía.
  • Handle: RePEc:cte:werepe:we1115
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    References listed on IDEAS

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    1. Leybourne, S J & McCabe, B P M & Tremayne, A R, 1996. "Can Economic Time Series Be Differenced to Stationarity?," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(4), pages 435-446, October.
    2. Park, Joon Y. & Phillips, Peter C.B., 1999. "Asymptotics For Nonlinear Transformations Of Integrated Time Series," Econometric Theory, Cambridge University Press, vol. 15(3), pages 269-298, June.
    3. Park, Joon Y. & Phillips, Peter C.B., 1988. "Statistical Inference in Regressions with Integrated Processes: Part 1," Econometric Theory, Cambridge University Press, vol. 4(3), pages 468-497, December.
    4. Liu, Ming, 1998. "Asymptotics Of Nonstationary Fractional Integrated Series," Econometric Theory, Cambridge University Press, vol. 14(5), pages 641-662, October.
    5. Arteche, J. & Orbe, J., 2005. "Bootstrapping the log-periodogram regression," Economics Letters, Elsevier, vol. 86(1), pages 79-85, January.
    6. Phillips, P.C.B., 1986. "Understanding spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 33(3), pages 311-340, December.
    7. de Jong, Robert & Wang, Chien-Ho, 2005. "Further Results On The Asymptotics For Nonlinear Transformations Of Integrated Time Series," Econometric Theory, Cambridge University Press, vol. 21(2), pages 413-430, April.
    8. Johansen, Soren, 1995. "Likelihood-Based Inference in Cointegrated Vector Autoregressive Models," OUP Catalogue, Oxford University Press, number 9780198774501.
    9. MÜller, Ulrich K., 2008. "The Impossibility Of Consistent Discrimination Between I(0) And I(1) Processes," Econometric Theory, Cambridge University Press, vol. 24(3), pages 616-630, June.
    10. Berenguer Rico, Vanessa & Gonzalo, Jesús, 2013. "Co-summability from linear to non-linear cointegration," UC3M Working papers. Economics we1312, Universidad Carlos III de Madrid. Departamento de Economía.
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    Citations

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    Cited by:

    1. Kasparis, Ioannis & Andreou, Elena & Phillips, Peter C.B., 2015. "Nonparametric predictive regression," Journal of Econometrics, Elsevier, vol. 185(2), pages 468-494.
    2. P.A.V.B. Swamy & I-Lok Chang & Jatinder S. Mehta & William H. Greene & Stephen G. Hall & George S. Tavlas, 2016. "Removing Specification Errors from the Usual Formulation of Binary Choice Models," Econometrics, MDPI, Open Access Journal, vol. 4(2), pages 1-1, June.
    3. Gadea Rivas, María Dolores & Gonzalo, Jesús, 2020. "Trends in distributional characteristics: Existence of global warming," Journal of Econometrics, Elsevier, vol. 214(1), pages 153-174.
    4. Markus Eberhardt, 2013. "Nonlinearities in the Relationship between Debt and Growth: Evidence from Co-Summability Testing," Discussion Papers 2013/06, University of Nottingham, Centre for Finance, Credit and Macroeconomics (CFCM).
    5. Hong, Seung Hyun & Wagner, Martin, 2011. "Cointegrating Polynomial Regressions," Economics Series 264, Institute for Advanced Studies.
    6. Banerjee Anurag & Pitarakis Jean-Yves, 2014. "Functional cointegration: definition and nonparametric estimation," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 18(5), pages 1-14, December.
    7. Markus Eberhardt & Andrea Filippo Presbitero, 2013. "This Time They're Different: Heterogeneity;and Nonlinearity in the Relationship;between Debt and Growth," Mo.Fi.R. Working Papers 92, Money and Finance Research group (Mo.Fi.R.) - Univ. Politecnica Marche - Dept. Economic and Social Sciences.
    8. Markus Eberhardt & Andrea F. Presbitero, 2013. "This Time They’re Different: Heterogeneity and Nonlinearity in the Relationship between Debt and Growth," Discussion Papers 2013/10, University of Nottingham, Centre for Finance, Credit and Macroeconomics (CFCM).

    More about this item

    Keywords

    Co-integration;

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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