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A Fast Fractional Difference Algorithm

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  • Andreas Noack Jensen
  • Morten Ørregaard Nielsen

Abstract

type="main" xml:id="jtsa12074-abs-0001"> 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-super-2, where T is the length of the time series. Our algorithm allows calculation speed of order TlogT. For moderate and large sample sizes, the difference in computation time is substantial. Copyright © 2014 John Wiley & Sons, Ltd.

Suggested Citation

  • Andreas Noack Jensen & Morten Ørregaard Nielsen, 2014. "A Fast Fractional Difference Algorithm," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(5), pages 428-436, August.
  • Handle: RePEc:bla:jtsera:v:35:y:2014:i:5:p:428-436
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    File URL: http://hdl.handle.net/10.1111/jtsa.12074
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    References listed on IDEAS

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    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. 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.
    4. 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.
    5. 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.
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    Citations

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    Cited by:

    1. Maggie E. C. Jones & Morten Ørregaard Nielsen & Micha Ksawery Popiel, 2014. "A fractionally cointegrated VAR analysis of economic voting and political support," Canadian Journal of Economics, Canadian Economics Association, vol. 47(4), pages 1078-1130, November.
    2. repec:eee:finlet:v:22:y:2017:i:c:p:274-279 is not listed on IDEAS
    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. Sepideh Dolatabadi & Paresh Kumar Narayan & Morten Ørregaard Nielsen & Ke Xu, 2017. "Economic significance of commodity return forecasts from the fractionally cointegrated VAR model," Working Papers 1337, Queen's University, Department of Economics.
    5. Søren Johansen & Morten Ørregaard Nielsen, 0111. "Testing the CVAR in the fractional CVAR model," CREATES Research Papers 2017-37, Department of Economics and Business Economics, Aarhus University.
    6. 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.
    7. Morten Ørregaard Nielsen & Michał Ksawery Popiel, 2018. "A Matlab program and user's guide for the fractionally cointegrated VAR model," Working Papers 1330, Queen's University, Department of Economics.
    8. 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.
    9. J. Eduardo Vera-Vald'es, 2018. "Nonfractional Memory: Filtering, Antipersistence, and Forecasting," Papers 1801.06677, arXiv.org.
    10. 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.
    11. repec:eee:econom:v:204:y:2018:i:1:p:54-65 is not listed on IDEAS
    12. repec:eee:econom:v:199:y:2017:i:1:p:1-11 is not listed on IDEAS
    13. 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.
    14. 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.
    15. Morten Ørregaard Nielsen & Sergei S. Shibaev, 2015. "Forecasting daily political opinion polls using the fractionally cointegrated VAR model," Working Papers 1340, Queen's University, Department of Economics.
    16. Håvard Hungnes, 2016. "Fractionality and co-fractionality between Government Bond yields," Discussion Papers 838, Statistics Norway, Research Department.
    17. 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

    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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