IDEAS home Printed from https://ideas.repec.org/p/qed/wpaper/1244.html
   My bibliography  Save this paper

A Necessary Moment Condition For The Fractional Functional Central Limit Theorem

Author

Listed:
  • Morten Ø. Nielsen

    (Queen's University and CREATES)

  • S Johansen

    (University of Copenhagen and CREATES)

Abstract

We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x_{t} = Delta^{-d} u_{t}, where d in (-1/2,1/2) is the fractional integration parameter and u_{t} is weakly dependent. The classical condition is existence of q≥2 and q>1/(d+1/2) moments of the innovation sequence. When d is close to -1/2 this moment condition is very strong. Our main result is to show that when d in (-1/2,0) and under some relatively weak conditions on u_{t}, the existence of q≥1/(d+1/2) moments is in fact necessary for the FCLT for fractionally integrated processes, and that q>1/(d+1/2) moments are necessary for more general fractional processes. Davidson and de Jong (2000) presented a fractional FCLT where only q>2 finite moments are assumed. As a corollary to our main theorem we show that their moment condition is not sufficient, and hence that their result is incorrect.

Suggested Citation

  • Morten Ø. Nielsen & S Johansen, 2010. "A Necessary Moment Condition For The Fractional Functional Central Limit Theorem," Working Paper 1244, Economics Department, Queen's University.
    • Handle: RePEc:qed:wpaper:1244
    as

    Download full text from publisher

    File URL: https://www.econ.queensu.ca/sites/econ.queensu.ca/files/wpaper/qed_wp_1244.pdf
    File Function: First version 2010
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. de Jong, Robert M. & Davidson, James, 2000. "The Functional Central Limit Theorem And Weak Convergence To Stochastic Integrals I," Econometric Theory, Cambridge University Press, vol. 16(5), pages 621-642, October.
    2. Marinucci, D. & Robinson, P. M., 2000. "Weak convergence of multivariate fractional processes," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 103-120, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Søren Johansen & Morten Ørregaard Nielsen, 2018. "Testing the CVAR in the Fractional CVAR Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(6), pages 836-849, November.
    2. Søren Johansen & Morten Ørregaard Nielsen, 2019. "Nonstationary Cointegration in the Fractionally Cointegrated VAR Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 40(4), pages 519-543, July.
    3. Javier Hualde & Morten {O}rregaard Nielsen, 2022. "Fractional integration and cointegration," Papers 2211.10235, arXiv.org.
    4. Mustafa R. K{i}l{i}nc{c} & Michael Massmann, 2024. "The modified conditional sum-of-squares estimator for fractionally integrated models," Papers 2404.12882, arXiv.org.
    5. Morten Ørregaard Nielsen, 2015. "Asymptotics for the Conditional-Sum-of-Squares Estimator in Multivariate Fractional Time-Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(2), pages 154-188, March.
    6. Man Wang & Ngai Hang Chan, 2016. "Testing for the Equality of Integration Orders of Multiple Series," Econometrics, MDPI, vol. 4(4), pages 1-10, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Nielsen, Morten, 2008. "A Powerful Tuning Parameter Free Test of the Autoregressive Unit Root Hypothesis," Working Papers 08-05, Cornell University, Center for Analytic Economics.
    2. Nielsen, Morten Ørregaard, 2009. "A Powerful Test Of The Autoregressive Unit Root Hypothesis Based On A Tuning Parameter Free Statistic," Econometric Theory, Cambridge University Press, vol. 25(6), pages 1515-1544, December.
    3. Hongshuai Dai, 2013. "Convergence in Law to Operator Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 26(3), pages 676-696, September.
    4. Lavancier, Frédéric & Philippe, Anne & Surgailis, Donatas, 2009. "Covariance function of vector self-similar processes," Statistics & Probability Letters, Elsevier, vol. 79(23), pages 2415-2421, December.
    5. Sibbertsen, Philipp & Wenger, Kai & Wingert, Simon, 2020. "Testing for Multiple Structural Breaks in Multivariate Long Memory Time Series," Hannover Economic Papers (HEP) dp-676, Leibniz Universität Hannover, Wirtschaftswissenschaftliche Fakultät.
    6. Uwe Hassler & Jan Scheithauer, 2011. "Detecting changes from short to long memory," Statistical Papers, Springer, vol. 52(4), pages 847-870, November.
    7. Davidson, James & Hashimzade, Nigar, 2009. "Type I and type II fractional Brownian motions: A reconsideration," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2089-2106, April.
    8. Yuzo Hosoya, 2005. "Fractional Invariance Principle," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(3), pages 463-486, May.
    9. Wensheng Wang, 2024. "The Moduli of Continuity for Operator Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2097-2120, September.
    10. Javier Hualde & Peter M Robinson, 2006. "Semiparametric Estimation of Fractional Cointegration," STICERD - Econometrics Paper Series 502, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    11. Fabian Knorre & Martin Wagner & Maximilian Grupe, 2021. "Monitoring Cointegrating Polynomial Regressions: Theory and Application to the Environmental Kuznets Curves for Carbon and Sulfur Dioxide Emissions," Econometrics, MDPI, vol. 9(1), pages 1-35, March.
    12. Hassler, U. & Marmol, F. & Velasco, C., 2006. "Residual log-periodogram inference for long-run relationships," Journal of Econometrics, Elsevier, vol. 130(1), pages 165-207, January.
    13. Victor Chernozhukov & Iván Fernández‐Val & Blaise Melly, 2013. "Inference on Counterfactual Distributions," Econometrica, Econometric Society, vol. 81(6), pages 2205-2268, November.
    14. Richard T. Baillie & George Kapetanios, 2006. "Nonlinear Models with Strongly Dependent Processes and Applications to Forward Premia and Real Exchange Rates," Working Papers 570, Queen Mary University of London, School of Economics and Finance.
    15. Chi-Young Choi & Ling Hu & Masao Ogaki, 2005. "Structural Spurious Regressions and A Hausman-type Cointegration Test," RCER Working Papers 517, University of Rochester - Center for Economic Research (RCER).
    16. Chang Sik Kim & Peter C.B. Phillips, 2006. "Log Periodogram Regression: The Nonstationary Case," Cowles Foundation Discussion Papers 1587, Cowles Foundation for Research in Economics, Yale University.
    17. Robinson, P.M. & Iacone, F., 2005. "Cointegration in fractional systems with deterministic trends," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 263-298.
    18. Kapetanios, G. & Pesaran, M. Hashem & Yamagata, T., 2011. "Panels with non-stationary multifactor error structures," Journal of Econometrics, Elsevier, vol. 160(2), pages 326-348, February.
    19. Philipp Sibbertsen, 2004. "Long memory versus structural breaks: An overview," Statistical Papers, Springer, vol. 45(4), pages 465-515, October.
    20. Politis, Dimitris, 2016. "HEGY test under seasonal heterogeneity," University of California at San Diego, Economics Working Paper Series qt2q4054kf, Department of Economics, UC San Diego.

    More about this item

    Keywords

    Fractional integration; functional central limit theorem; long memory; moment condition; necessary condition;
    All these keywords.

    JEL classification:

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

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:qed:wpaper:1244. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Mark Babcock (email available below). General contact details of provider: https://edirc.repec.org/data/qedquca.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.