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Testing Fractional Unit Roots with Non-linear Smooth Break Approximations using Fourier functions

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  • Gil-Alana, Luis A.
  • Yaya, OlaOluwa S

Abstract

In this paper we present a testing procedure for fractional orders of integration in the context of non-linear terms approximated by Fourier functions. The procedure is a natural extension of the linear method proposed in Robinson (1994) and similar to the one proposed in Cuestas and Gil-Alana (2016) based on Chebyshev polynomials in time. The test statistic has an asymptotic standard normal distribution and several Monte Carlo experiments conducted in the paper show that it performs well in finite samples. Various applications using real life time series, such as US unemployment rates, US GNP and Purchasing Power Parity (PPP) of G7 countries are presented at the end of the paper.

Suggested Citation

  • Gil-Alana, Luis A. & Yaya, OlaOluwa S, 2018. "Testing Fractional Unit Roots with Non-linear Smooth Break Approximations using Fourier functions," MPRA Paper 90516, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:90516
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    1. Kapetanios, George & Shin, Yongcheol & Snell, Andy, 2003. "Testing for a unit root in the nonlinear STAR framework," Journal of Econometrics, Elsevier, vol. 112(2), pages 359-379, February.
    2. Nelson, Charles R & Piger, Jeremy & Zivot, Eric, 2001. "Markov Regime Switching and Unit-Root Tests," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(4), pages 404-415, October.
    3. Bierens, Herman J., 2001. "Complex Unit Roots And Business Cycles: Are They Real?," Econometric Theory, Cambridge University Press, vol. 17(5), pages 962-983, October.
    4. Guglielmo Maria Caporale & Luis A. Gil‐Alana, 2007. "Nonlinearities and Fractional Integration in the US Unemployment Rate," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 69(4), pages 521-544, August.
    5. Gallant, A. Ronald & Souza, Geraldo, 1991. "On the asymptotic normality of Fourier flexible form estimates," Journal of Econometrics, Elsevier, vol. 50(3), pages 329-353, December.
    6. Elliott, Graham & Rothenberg, Thomas J & Stock, James H, 1996. "Efficient Tests for an Autoregressive Unit Root," Econometrica, Econometric Society, vol. 64(4), pages 813-836, July.
    7. L. A. Gil-Alana & P. M. Robinson, 2001. "Testing of seasonal fractional integration in UK and Japanese consumption and income," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(2), pages 95-114.
    8. DeJong, David N, et al, 1992. "Integration versus Trend Stationarity in Time Series," Econometrica, Econometric Society, vol. 60(2), pages 423-433, March.
    9. Gil-Alana, Luis A. & Moreno, Antonio, 2012. "Uncovering the US term premium: An alternative route," Journal of Banking & Finance, Elsevier, vol. 36(4), pages 1181-1193.
    10. Mirko Abbritti & Luis A. Gil-Alana & Yuliya Lovcha & Antonio Moreno, 2016. "Term Structure Persistence," Journal of Financial Econometrics, Oxford University Press, vol. 14(2), pages 331-352.
    11. Tanaka, Katsuto, 1999. "The Nonstationary Fractional Unit Root," Econometric Theory, Cambridge University Press, vol. 15(4), pages 549-582, August.
    12. Caporale, Guglielmo Maria & Gil-Alana, Luis A., 2008. "Modelling the US, UK and Japanese unemployment rates: Fractional integration and structural breaks," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4998-5013, July.
    13. Ferrara, Laurent & Guegan, Dominique, 2001. "Forecasting with k-Factor Gegenbauer Processes: Theory and Applications," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 20(8), pages 581-601, December.
    14. Lee, Dongin & Schmidt, Peter, 1996. "On the power of the KPSS test of stationarity against fractionally-integrated alternatives," Journal of Econometrics, Elsevier, vol. 73(1), pages 285-302, July.
    15. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
    16. Hylleberg, S. & Engle, R. F. & Granger, C. W. J. & Yoo, B. S., 1990. "Seasonal integration and cointegration," Journal of Econometrics, Elsevier, vol. 44(1-2), pages 215-238.
    17. L. A. Gil‐Alana, 2001. "Testing Stochastic Cycles in Macroeconomic Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 22(4), pages 411-430, July.
    18. Cuestas Juan Carlos & Gil-Alana Luis Alberiko, 2016. "Testing for long memory in the presence of non-linear deterministic trends with Chebyshev polynomials," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 20(1), pages 57-74, February.
    19. Jushan Bai & Pierre Perron, 2003. "Computation and analysis of multiple structural change models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 18(1), pages 1-22.
    20. Gil-Alana, Luis A. & Huijbens, Edward H., 2018. "Tourism in Iceland: Persistence and seasonality," Annals of Tourism Research, Elsevier, vol. 68(C), pages 20-29.
    21. Hassler, Uwe & Wolters, Jurgen, 1994. "On the power of unit root tests against fractional alternatives," Economics Letters, Elsevier, vol. 45(1), pages 1-5, May.
    22. Diebold, Francis X. & Rudebusch, Glenn D., 1991. "On the power of Dickey-Fuller tests against fractional alternatives," Economics Letters, Elsevier, vol. 35(2), pages 155-160, February.
    23. Gallant, A. Ronald, 1981. "On the bias in flexible functional forms and an essentially unbiased form : The fourier flexible form," Journal of Econometrics, Elsevier, vol. 15(2), pages 211-245, February.
    24. Tolga Omay & Rangan Gupta & Giovanni Bonaccolto, 2017. "The US real GNP is trend-stationary after all," Applied Economics Letters, Taylor & Francis Journals, vol. 24(8), pages 510-514, May.
    25. Robert M. Kunst, 1997. "Testing For Cyclical Non‐Stationarity In Autoregressive Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 18(2), pages 123-135, March.
    26. Gil-Alana, Luis A., 2002. "Seasonal long memory in the aggregate output," Economics Letters, Elsevier, vol. 74(3), pages 333-337, February.
    27. Juan J. Dolado & Jesus Gonzalo & Laura Mayoral, 2002. "A Fractional Dickey-Fuller Test for Unit Roots," Econometrica, Econometric Society, vol. 70(5), pages 1963-2006, September.
    28. Paulo M. M. Rodrigues & A. M. Robert Taylor, 2012. "The Flexible Fourier Form and Local Generalised Least Squares De-trended Unit Root Tests-super-," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 74(5), pages 736-759, October.
    29. Walter Enders & Junsoo Lee, 2012. "A Unit Root Test Using a Fourier Series to Approximate Smooth Breaks," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 74(4), pages 574-599, August.
    30. Enders, Walter & Lee, Junsoo, 2012. "The flexible Fourier form and Dickey–Fuller type unit root tests," Economics Letters, Elsevier, vol. 117(1), pages 196-199.
    31. Bierens, Herman J., 1997. "Testing the unit root with drift hypothesis against nonlinear trend stationarity, with an application to the US price level and interest rate," Journal of Econometrics, Elsevier, vol. 81(1), pages 29-64, November.
    32. Omay, Tolga, 2015. "Fractional Frequency Flexible Fourier Form to approximate smooth breaks in unit root testing," Economics Letters, Elsevier, vol. 134(C), pages 123-126.
    33. Johansen, Søren & Juselius, Katarina & Frydman, Roman & Goldberg, Michael, 2010. "Testing hypotheses in an I(2) model with piecewise linear trends. An analysis of the persistent long swings in the Dmk/$ rate," Journal of Econometrics, Elsevier, vol. 158(1), pages 117-129, September.
    34. Stephen Leybourne & Paul Newbold & Dimitrios Vougas, 1998. "Unit roots and smooth transitions," Journal of Time Series Analysis, Wiley Blackwell, vol. 19(1), pages 83-97, January.
    35. Fumitaka Furuoka, 2017. "A new approach to testing unemployment hysteresis," Empirical Economics, Springer, vol. 53(3), pages 1253-1280, November.
    36. Schmidt, Peter & Phillips, C B Peter, 1992. "LM Tests for a Unit Root in the Presence of Deterministic Trends," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 54(3), pages 257-287, August.
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    More about this item

    Keywords

    Fractional unit root; Chebyshev polynomial; Monte Carlo simulation; Nonlinearity; Smooth break; Fourier transform;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • 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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