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Summability of stochastic processes—A generalization of integration 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, 2014. "Summability of stochastic processes—A generalization of integration for non-linear processes," Journal of Econometrics, Elsevier, vol. 178(P2), pages 331-341.
    • Handle: RePEc:eee:econom:v:178:y:2014:i:p2:p:331-341
    DOI: 10.1016/j.jeconom.2013.08.031
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    1. Pötscher, Benedikt M., 2004. "Nonlinear Functions And Convergence To Brownian Motion: Beyond The Continuous Mapping Theorem," Econometric Theory, Cambridge University Press, vol. 20(1), pages 1-22, February.
    2. Berkes, István & Horváth, Lajos, 2006. "Convergence Of Integral Functionals Of Stochastic Processes," Econometric Theory, Cambridge University Press, vol. 22(2), pages 304-322, April.
    3. Arteche, J. & Orbe, J., 2005. "Bootstrapping the log-periodogram regression," Economics Letters, Elsevier, vol. 86(1), pages 79-85, January.
    4. Phillips, P.C.B., 1986. "Understanding spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 33(3), pages 311-340, December.
    5. Franses,Philip Hans & Dijk,Dick van, 2000. "Non-Linear Time Series Models in Empirical Finance," Cambridge Books, Cambridge University Press, number 9780521770415.
    6. Terasvirta, Timo & Tjostheim, Dag & Granger, Clive W. J., 2010. "Modelling Nonlinear Economic Time Series," OUP Catalogue, Oxford University Press, number 9780199587155, Decembrie.
    7. Park, Joon Y & Phillips, Peter C B, 2001. "Nonlinear Regressions with Integrated Time Series," Econometrica, Econometric Society, vol. 69(1), pages 117-161, January.
    8. Johansen, Soren, 1995. "Likelihood-Based Inference in Cointegrated Vector Autoregressive Models," OUP Catalogue, Oxford University Press, number 9780198774501, Decembrie.
    9. 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.
    10. Stock, James H., 1991. "Confidence intervals for the largest autoregressive root in U.S. macroeconomic time series," Journal of Monetary Economics, Elsevier, vol. 28(3), pages 435-459, December.
    11. Engle, Robert & Granger, Clive, 2015. "Co-integration and error correction: Representation, estimation, and testing," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 39(3), pages 106-135.
    12. Qiying Wang & Peter C. B. Phillips, 2009. "Structural Nonparametric Cointegrating Regression," Econometrica, Econometric Society, vol. 77(6), pages 1901-1948, November.
    13. 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.
    14. 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.
    15. Liu, Ming, 1998. "Asymptotics Of Nonstationary Fractional Integrated Series," Econometric Theory, Cambridge University Press, vol. 14(5), pages 641-662, October.
    16. Granger, Clive W. J. & Terasvirta, Timo, 1993. "Modelling Non-Linear Economic Relationships," OUP Catalogue, Oxford University Press, number 9780198773207, Decembrie.
    17. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-1580, November.
    18. 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.
    19. Nelson, Charles R. & Plosser, Charles I., 1982. "Trends and random walks in macroeconmic time series : Some evidence and implications," Journal of Monetary Economics, Elsevier, vol. 10(2), pages 139-162.
    20. 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.
    21. Granger, Clive W J, 1995. "Modelling Nonlinear Relationships between Extended-Memory Variables," Econometrica, Econometric Society, vol. 63(2), pages 265-279, March.
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    2. Davide Delle Monache & Stefano Grassi & Paolo Santucci de Magistris, 2015. "Testing for Level Shifts in Fractionally Integrated Processes: a State Space Approach," CREATES Research Papers 2015-30, Department of Economics and Business Economics, Aarhus University.
    3. Ghoshray, A., 2018. "The Dynamic Properties of Natural Resource Prices," 2018 Conference, July 28-August 2, 2018, Vancouver, British Columbia 277210, International Association of Agricultural Economists.
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    5. Niels Haldrup & Robinson Kruse, 2014. "Discriminating between fractional integration and spurious long memory," CREATES Research Papers 2014-19, Department of Economics and Business Economics, Aarhus University.
    6. 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.
    7. Shinhye Chang & Matthew W. Clance & Giray Gozgor & Rangan Gupta, 2019. "A Reconsideration of Kuznets Curve across Countries: Evidence from the Co-summability Approach," Working Papers 201970, University of Pretoria, Department of Economics.
    8. Ben Nasr, Adnen & Gupta, Rangan & Sato, João Ricardo, 2015. "Is there an Environmental Kuznets Curve for South Africa? A co-summability approach using a century of data," Energy Economics, Elsevier, vol. 52(PA), pages 136-141.
    9. Kasparis, Ioannis & Andreou, Elena & Phillips, Peter C.B., 2015. "Nonparametric predictive regression," Journal of Econometrics, Elsevier, vol. 185(2), pages 468-494.
    10. Apergis, Nicholas & Christou, Christina & Gupta, Rangan, 2017. "Are there Environmental Kuznets Curves for US state-level CO2 emissions?," Renewable and Sustainable Energy Reviews, Elsevier, vol. 69(C), pages 551-558.
    11. Adnen Ben Nasr & Mehmet Balcilar & Seyi Saint Akadiri & Rangan Gupta, 2019. "Kuznets Curve for the US: A Reconsideration Using Cosummability," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 142(2), pages 827-843, April.
    12. María Dolores Gadea Rivas & Jesús Gonzalo, 2022. "A tale of three cities: climate heterogeneity," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 13(1), pages 475-511, May.
    13. Ding, Shusheng & Zhang, Yongmin, 2020. "Cross market predictions for commodity prices," Economic Modelling, Elsevier, vol. 91(C), pages 455-462.
    14. Davide Delle Monache & Stefano Grassi & Paolo Santucci de Magistris, 2017. "Does the ARFIMA really shift?," CREATES Research Papers 2017-16, Department of Economics and Business Economics, Aarhus University.
    15. Seref Bozoklu & A. Oguz Demir & Sinan Ataer, 2020. "Reassessing the environmental Kuznets curve: a summability approach for emerging market economies," Eurasian Economic Review, Springer;Eurasia Business and Economics Society, vol. 10(3), pages 513-531, September.
    16. Hassler, Uwe & Hosseinkouchack, Mehdi, 2020. "Estimating the mean under strong persistence," Economics Letters, Elsevier, vol. 188(C).
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    18. Chevillon, Guillaume & Mavroeidis, Sophocles, 2018. "Perpetual learning and apparent long memory," Journal of Economic Dynamics and Control, Elsevier, vol. 90(C), pages 343-365.

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    More about this item

    Keywords

    Co-integration; Co-summability; Integrated processes; Non-linear balanced relationships; Non-linear processes; Summability;
    All these keywords.

    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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