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Local Whittle estimation of multivariate fractionally integrated processes

Author

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  • Frank S. Nielsen

    (Aarhus University and CREATES)

Abstract

This paper derives a semiparametric estimator of multivariate fractionally integrated processes covering both stationary and non-stationary values of d. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the multivariate local Whittle estimator of Shimotsu (2007) to cover non-stationary values of d. We show consistency and asymptotic normality for d between -1/2 and infinity. A simulation study illustrates the performance of the proposed estimator for relevant sample sizes. Empirical justification of the proposed estimator is shown through an empirical analysis of log spot exchange rates. We find that the log spot exchange rates of Germany, United Kingdom, Japan, Canada, France, Italy, and Switzerland against the US Dollar for the period January 1974 until December 2001 are well decribed as I (1) processes.

Suggested Citation

  • Frank S. Nielsen, 2009. "Local Whittle estimation of multivariate fractionally integrated processes," CREATES Research Papers 2009-38, Department of Economics and Business Economics, Aarhus University.
    • Handle: RePEc:aah:create:2009-38
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    File URL: https://repec.econ.au.dk/repec/creates/rp/09/rp09_38.pdf
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    References listed on IDEAS

    as
    1. Nielsen M.O., 2004. "Optimal Residual-Based Tests for Fractional Cointegration and Exchange Rate Dynamics," Journal of Business & Economic Statistics, American Statistical Association, vol. 22, pages 331-345, July.
    2. Frederiksen, Per & Nielsen, Frank S. & Nielsen, Morten Ørregaard, 2012. "Local polynomial Whittle estimation of perturbed fractional processes," Journal of Econometrics, Elsevier, vol. 167(2), pages 426-447.
    3. Shimotsu, Katsumi, 2007. "Gaussian semiparametric estimation of multivariate fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 137(2), pages 277-310, April.
    4. Lobato, Ignacio N & Velasco, Carlos, 2000. "Long Memory in Stock-Market Trading Volume," Journal of Business & Economic Statistics, American Statistical Association, vol. 18(4), pages 410-427, October.
    5. Donald W. K. Andrews & Yixiao Sun, 2004. "Adaptive Local Polynomial Whittle Estimation of Long-range Dependence," Econometrica, Econometric Society, vol. 72(2), pages 569-614, March.
    6. Violetta Dalla & Liudas Giraitis & Javier Hidalgo, 2006. "Consistent estimation of the memory parameter for nonlinear time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(2), pages 211-251, March.
    7. Dalla, Violetta & Giraitis, Liudas & Hidalgo, Javier, 2006. "Consistent estimation of the memory parameter for nonlinear time series," LSE Research Online Documents on Economics 6813, London School of Economics and Political Science, LSE Library.
    8. Nielsen, Morten Orregaard & Shimotsu, Katsumi, 2007. "Determining the cointegrating rank in nonstationary fractional systems by the exact local Whittle approach," Journal of Econometrics, Elsevier, vol. 141(2), pages 574-596, December.
    9. Lobato, Ignacio N., 1999. "A semiparametric two-step estimator in a multivariate long memory model," Journal of Econometrics, Elsevier, vol. 90(1), pages 129-153, May.
    10. Abadir, Karim M. & Distaso, Walter & Giraitis, Liudas, 2007. "Nonstationarity-extended local Whittle estimation," Journal of Econometrics, Elsevier, vol. 141(2), pages 1353-1384, December.
    11. 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.
    12. Katsumi Shimotsu & Peter C.B. Phillips, 2000. "Local Whittle Estimation in Nonstationary and Unit Root Cases," Cowles Foundation Discussion Papers 1266, Cowles Foundation for Research in Economics, Yale University, revised Sep 2003.
    13. Violetta Dalla & Liudas Giraitis & Javier Hidalgo, 2006. "Consistent estimation of the memory parameterfor nonlinear time series," STICERD - Econometrics Paper Series 497, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    14. Carlos Velasco, 2003. "Gaussian Semi‐parametric Estimation of Fractional Cointegration," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(3), pages 345-378, May.
    15. Frank S. Nielsen, 2008. "Local polynomial Whittle estimation covering non-stationary fractional processes," CREATES Research Papers 2008-28, Department of Economics and Business Economics, Aarhus University.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. de Truchis, Gilles, 2013. "Approximate Whittle analysis of fractional cointegration and the stock market synchronization issue," Economic Modelling, Elsevier, vol. 34(C), pages 98-105.
    2. Kristoufek, Ladislav, 2013. "Mixed-correlated ARFIMA processes for power-law cross-correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(24), pages 6484-6493.
    3. Marina Balboa & Paulo M. M. Rodrigues & Antonio Rubia & A. M. Robert Taylor, 2021. "Multivariate fractional integration tests allowing for conditional heteroskedasticity with an application to return volatility and trading volume," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 36(5), pages 544-565, August.
    4. 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.
    5. Abry, Patrice & Didier, Gustavo, 2018. "Wavelet eigenvalue regression for n-variate operator fractional Brownian motion," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 75-104.
    6. Kristoufek, Ladislav, 2015. "On the interplay between short and long term memory in the power-law cross-correlations setting," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 421(C), pages 218-222.
    7. Zouhaier Dhifaoui, 2021. "Robustness of Detrended Cross-correlation Analysis Method Under Outliers Observations," Working Papers hal-03411380, HAL.

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    More about this item

    Keywords

    fractional integration; local Whittle; long memory; multivariate semiparametric estimation; exchange rates;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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