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A Fast Fractional Difference Algorithm


  • Andreas Noack Jensen

    (University of Copenhagen)

  • Morten Ø. Nielsen

    () (Queen's University and CREATES)


We provide a fast algorithm for calculating the fractional difference of a time series. In standard implementations, the calculation speed (number of arithmetic operations) is of order T^2, where T is the length of the time series. Our algorithm allows calculation speed of order T log(T). For moderate and large sample sizes, the difference in computation time is substantial.

Suggested Citation

  • Andreas Noack Jensen & Morten Ø. Nielsen, 2013. "A Fast Fractional Difference Algorithm," Working Paper 1307, Economics Department, Queen's University.
  • Handle: RePEc:qed:wpaper:1307

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    References listed on IDEAS

    1. Sowell, Fallaw, 1992. "Maximum likelihood estimation of stationary univariate fractionally integrated time series models," Journal of Econometrics, Elsevier, vol. 53(1-3), pages 165-188.
    2. Søren Johansen & Morten Ørregaard Nielsen, 2012. "Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model," Econometrica, Econometric Society, vol. 80(6), pages 2667-2732, November.
    3. Chen, Willa W. & Hurvich, Clifford M. & Lu, Yi, 2006. "On the Correlation Matrix of the Discrete Fourier Transform and the Fast Solution of Large Toeplitz Systems for Long-Memory Time Series," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 812-822, June.
    4. Bollerslev, Tim & Osterrieder, Daniela & Sizova, Natalia & Tauchen, George, 2013. "Risk and return: Long-run relations, fractional cointegration, and return predictability," Journal of Financial Economics, Elsevier, vol. 108(2), pages 409-424.
    5. Doornik, Jurgen A. & Ooms, Marius, 2003. "Computational aspects of maximum likelihood estimation of autoregressive fractionally integrated moving average models," Computational Statistics & Data Analysis, Elsevier, vol. 42(3), pages 333-348, March.
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    Cited by:

    1. Maggie E. C. Jones & Morten Ørregaard Nielsen & Michał Ksawery Popiel, 2014. "A fractionally cointegrated VAR analysis of economic voting and political support," Canadian Journal of Economics/Revue canadienne d'économique, John Wiley & Sons, vol. 47(4), pages 1078-1130, November.
    2. Klein, Tony & Walther, Thomas, 2017. "Fast fractional differencing in modeling long memory of conditional variance for high-frequency data," Finance Research Letters, Elsevier, vol. 22(C), pages 274-279.
    3. Lunina, Veronika, 2016. "Joint Modelling of Power Price, Temperature, and Hydrological Balance with a View towards Scenario Analysis," Working Papers 2016:30, Lund University, Department of Economics.
    4. 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.
    5. 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.
    6. Sepideh Dolatabadi & Paresh Kumar Narayan & Morten Ørregaard Nielsen & Ke Xu, 2018. "Economic significance of commodity return forecasts from the fractionally cointegrated VAR model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 38(2), pages 219-242, February.
    7. Dolatabadi, Sepideh & Nielsen, Morten Ørregaard & Xu, Ke, 2016. "A fractionally cointegrated VAR model with deterministic trends and application to commodity futures markets," Journal of Empirical Finance, Elsevier, vol. 38(PB), pages 623-639.
    8. Sepideh Dolatabadi & Morten Ørregaard Nielsen & Ke Xu, 2015. "A Fractionally Cointegrated VAR Analysis of Price Discovery in Commodity Futures Markets," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 35(4), pages 339-356, April.
    9. Dark, Jonathan, 2015. "Futures hedging with Markov switching vector error correction FIEGARCH and FIAPARCH," Journal of Banking & Finance, Elsevier, vol. 61(S2), pages 269-285.
    10. Morten Ørregaard Nielsen & Sergei S. Shibaev, 2015. "Forecasting daily political opinion polls using the fractionally cointegrated VAR model," Working Paper 1340, Economics Department, Queen's University.
    11. 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.
    12. Håvard Hungnes, 2016. "Fractionality and co-fractionality between Government Bond yields," Discussion Papers 838, Statistics Norway, Research Department.
    13. Giuseppe Brandi & T. Di Matteo, 2020. "A new multilayer network construction via Tensor learning," Papers 2004.05367,
    14. Gao, Guangyuan & Ho, Kin-Yip & Shi, Yanlin, 2020. "Long memory or regime switching in volatility? Evidence from high-frequency returns on the U.S. stock indices," Pacific-Basin Finance Journal, Elsevier, vol. 61(C).
    15. Shi, Yanlin & Ho, Kin-Yip, 2015. "Long memory and regime switching: A simulation study on the Markov regime-switching ARFIMA model," Journal of Banking & Finance, Elsevier, vol. 61(S2), pages 189-204.
    16. Morten Ø. Nielsen & Michal Ksawery Popiel, 2018. "A Matlab Program And User's Guide For The Fractionally Cointegrated Var Model," Working Paper 1330, Economics Department, Queen's University.
    17. J. Eduardo Vera-Vald'es, 2018. "Nonfractional Memory: Filtering, Antipersistence, and Forecasting," Papers 1801.06677,
    18. Morten Ørregaard Nielsen & Antoine L. Noël, 2020. "To infinity and beyond: Efficient computation of ARCH(\infty) models," Working Paper 1425, Economics Department, Queen's University.
    19. 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.
    20. Haldrup, Niels & Vera Valdés, J. Eduardo, 2017. "Long memory, fractional integration, and cross-sectional aggregation," Journal of Econometrics, Elsevier, vol. 199(1), pages 1-11.
    21. Cheung, Ying Lun, 2020. "Nonstationarity-extended Whittle estimation with discontinuity: A correction," Economics Letters, Elsevier, vol. 187(C).
    22. Jochen Heberle & Cristina Sattarhoff, 2017. "A Fast Algorithm for the Computation of HAC Covariance Matrix Estimators," Econometrics, MDPI, Open Access Journal, vol. 5(1), pages 1-16, January.

    More about this item


    Circular convolution theorem; fast Fourier transform; fractional difference;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C87 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Econometric Software


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