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Long Memory, Fractional Integration, and Cross-Sectional Aggregation

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  • Niels Haldrup

    (Aarhus University and CREATES)

  • J. Eduardo Vera-Valdés

    (Aarhus University and CREATES)

Abstract

It is commonly argued that observed long memory in time series variables can result from cross-sectional aggregation of dynamic heterogeneous micro units. For instance, Granger (1980) demonstrated that aggregation of AR(1) processes with a Beta distributed AR coefficient can exhibit long memory under certain conditions and that the aggregated series will have an autocorrelation function that exhibits hyperbolic decay. In this paper, we further analyze this phenomenon. We demonstrate that the aggregation argument leading to long memory is consistent with a wide range of definitions of long memory. In a simulation study we seek to quantify Granger's result and find that indeed both the time series and cross-sectional dimensions have to be rather significant to reflect the theoretical asymptotic results. Long memory can result even for moderate T,N dimensions but can vary considerably from the theoretical degree of memory. Also, Granger's result is most precise in samples with a relatively high degree of memory. Finally, we show that even though the aggregated process will behave as generalized fractional process and thus converge to a fractional Brownian motion asymptotically, the fractionally differenced series does not behave according to an ARMA process. In particular, despite the autocorrelation function is summable and hence the fractionally differenced process satisfy the conditions for being I(0), it still exhibits hyperbolic decay. This may have consequences for the validity of ARFIMA time series modeling of long memory processes when the source of memory is due to aggregation.

Suggested Citation

  • Niels Haldrup & J. Eduardo Vera-Valdés, 2015. "Long Memory, Fractional Integration, and Cross-Sectional Aggregation," CREATES Research Papers 2015-59, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2015-59
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    References listed on IDEAS

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    Cited by:

    1. J. Eduardo Vera-Valdés, 2021. "Nonfractional Long-Range Dependence: Long Memory, Antipersistence, and Aggregation," Econometrics, MDPI, vol. 9(4), pages 1-18, October.
    2. Mawuli Segnon & Manuel Stapper, 2019. "Long Memory Conditional Heteroscedasticity in Count Data," CQE Working Papers 8219, Center for Quantitative Economics (CQE), University of Muenster.
    3. Javier Haulde & Morten Ørregaard Nielsen, 2022. "Fractional integration and cointegration," CREATES Research Papers 2022-02, Department of Economics and Business Economics, Aarhus University.
    4. Gianluca Cubadda & Alain Hecq & Antonio Riccardo, 2018. "Forecasting Realized Volatility Measures with Multivariate and Univariate Models: The Case of The US Banking Sector," CEIS Research Paper 445, Tor Vergata University, CEIS, revised 30 Oct 2018.
    5. J. Eduardo Vera‐Valdés, 2020. "On long memory origins and forecast horizons," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 39(5), pages 811-826, August.
    6. Kunal Saha & Vinodh Madhavan & Chandrashekhar G. R. & David McMillan, 2020. "Pitfalls in long memory research," Cogent Economics & Finance, Taylor & Francis Journals, vol. 8(1), pages 1733280-173, January.
    7. Chevillon, Guillaume & Hecq, Alain & Laurent, Sébastien, 2018. "Generating univariate fractional integration within a large VAR(1)," Journal of Econometrics, Elsevier, vol. 204(1), pages 54-65.
    8. Prados de la Escosura, Leandro & Rodríguez-Caballero, C. Vladimir, 2022. "War, pandemics, and modern economic growth in Europe," Explorations in Economic History, Elsevier, vol. 86(C).
    9. J. Eduardo Vera-Valdés, 2021. "Temperature Anomalies, Long Memory, and Aggregation," Econometrics, MDPI, vol. 9(1), pages 1-22, March.
    10. Proietti, Tommaso & Maddanu, Federico, 2024. "Modelling cycles in climate series: The fractional sinusoidal waveform process," Journal of Econometrics, Elsevier, vol. 239(1).
    11. J. Eduardo Vera-Vald'es, 2018. "Nonfractional Memory: Filtering, Antipersistence, and Forecasting," Papers 1801.06677, arXiv.org.
    12. Leschinski, Christian & Sibbertsen, Philipp, 2018. "The Periodogram of Spurious Long-Memory Processes," Hannover Economic Papers (HEP) dp-632, Leibniz Universität Hannover, Wirtschaftswissenschaftliche Fakultät.

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    More about this item

    Keywords

    Long memory; Fractional Integration; Aggregation;
    All these keywords.

    JEL classification:

    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
    • 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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