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A Non-Linear Approach with Long Range Dependence Based on Chebyshev Polynomials

Author

Listed:
  • Juan Carlos Cuestas

    () (Department of Economics, The University of Sheffield)

  • Luis A. Gil-Alana

    () (Department of Economics, Universidad de Navarra)

Abstract

This paper examines the interaction between non-linear deterministic trends and long run dependence by means of employing Chebyshev time polynomials and assuming that the detrended series displays long memory with the pole or singularity in the spectrum occurring at one or more possibly non-zero frequencies. The combination of the non-linear structure with the long memory framework produces a model which is linear in parameters and therefore it permits the estimation of the deterministic terms by standard OLS-GLS methods. Moreover, we present a procedure that permits us to test (possibly fractional) orders of integration at various frequencies in the presence of the Chebyshev trends with no effect on the standard limit distribution of the method. Several Monte Carlo experiments are conducted and an empirical application, using data of real exchange rates, is also carried out at the end of the article.

Suggested Citation

  • Juan Carlos Cuestas & Luis A. Gil-Alana, 2012. "A Non-Linear Approach with Long Range Dependence Based on Chebyshev Polynomials," Working Papers 2012013, The University of Sheffield, Department of Economics.
  • Handle: RePEc:shf:wpaper:2012013
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    File URL: http://www.shef.ac.uk/economics/research/serps/articles/2012_013.html
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    References listed on IDEAS

    as
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    3. Violetta Dalla & Javier Hidalgo, 2005. "A Parametric Bootstrap Test for Cycles," STICERD - Econometrics Paper Series 486, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    4. Henry L. Gray & Nien‐Fan Zhang & Wayne A. Woodward, 1994. "On Generalized Fractional Processes – A Correction," Journal of Time Series Analysis, Wiley Blackwell, vol. 15(5), pages 561-562, September.
    5. Granger, C. W. J., 1980. "Long memory relationships and the aggregation of dynamic models," Journal of Econometrics, Elsevier, vol. 14(2), pages 227-238, October.
    6. Granger, Clive W. J. & Hyung, Namwon, 2004. "Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns," Journal of Empirical Finance, Elsevier, vol. 11(3), pages 399-421, June.
    7. Dalla, Violetta & Hidalgo, Javier, 2005. "A parametric bootstrap test for cycles," LSE Research Online Documents on Economics 6829, London School of Economics and Political Science, LSE Library.
    8. Zivot, Eric & Andrews, Donald W K, 2002. "Further Evidence on the Great Crash, the Oil-Price Shock, and the Unit-Root Hypothesis," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 25-44, January.
    9. Ching‐Fan Chung, 1996. "A Generalized Fractionally Integrated Autoregressive Moving‐Average Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 17(2), pages 111-140, March.
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    11. Lothian, James R. & Taylor, Mark P., 2000. "Purchasing power parity over two centuries: strengthening the case for real exchange rate stability: A reply to Cuddington and Liang," Journal of International Money and Finance, Elsevier, vol. 19(5), pages 759-764, October.
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    15. 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.
    16. Papell, David H. & Prodan, Ruxandra, 2006. "Additional Evidence of Long-Run Purchasing Power Parity with Restricted Structural Change," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 38(5), pages 1329-1349, August.
    17. Junsoo Lee & Mark C. Strazicich, 2003. "Minimum Lagrange Multiplier Unit Root Test with Two Structural Breaks," The Review of Economics and Statistics, MIT Press, vol. 85(4), pages 1082-1089, November.
    18. Dalla, Violetta & Hidalgo, Javier, 2005. "A parametric bootstrap test for cycles," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 219-261.
    19. Ray, Bonnie K., 1993. "Long-range forecasting of IBM product revenues using a seasonal fractionally differenced ARMA model," International Journal of Forecasting, Elsevier, vol. 9(2), pages 255-269, August.
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    23. Henry L. Gray & Nien‐Fan Zhang & Wayne A. Woodward, 1989. "On Generalized Fractional Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 10(3), pages 233-257, May.
    24. Gil-Alaña, L. A. & Robinson, Peter M., 2001. "Testing of seasonal fractional integration in UK and Japanese consumption and income," LSE Research Online Documents on Economics 298, London School of Economics and Political Science, LSE Library.
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    Citations

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    Cited by:

    1. Gil-Alana, Luis A. & Gupta, Rangan, 2014. "Persistence and cycles in historical oil price data," Energy Economics, Elsevier, vol. 45(C), pages 511-516.
    2. Gil-Alana, Luis A. & Yaya, OlaOluwa S, 2018. "How do Stocks in BRICS co-move with REITs?," MPRA Paper 88753, University Library of Munich, Germany.
    3. Guglielmo Caporale & Luis Gil-Alana, 2016. "Persistence and cyclical dependence in the monthly euribor rate," Journal of Economics and Finance, Springer;Academy of Economics and Finance, vol. 40(1), pages 157-171, January.
    4. Yaya, OlaOluwa A & Gil-Alana, Luis A., 2018. "Modelling Long Range Dependence and Non-linearity in the Infant Mortality Rates of Africa Countries," MPRA Paper 88752, University Library of Munich, Germany.
    5. Cuestas, Juan Carlos & Regis, Paulo José, 2013. "Purchasing power parity in OECD countries: Nonlinear unit root tests revisited," Economic Modelling, Elsevier, vol. 32(C), pages 343-346.
    6. Luis A. Gil-Alana & Rangan Gupta & Fernando Perez de Gracia, 2016. "Persistence, mean reversion and non-linearities in the US housing prices over 1830--2013," Applied Economics, Taylor & Francis Journals, vol. 48(34), pages 3244-3252, July.
    7. Guglielmo Maria Caporale & Luis A. Gil‐Alana & James C. Orlando, 2016. "Linkages Between the US and European Stock Markets: A Fractional Cointegration Approach," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 21(2), pages 143-153, April.
    8. Luis Alberiko Gil-Alana & Zeynel Abidin Ozdemir & Aysit Tansel, 2019. "Long Memory in Turkish Unemployment Rates," Emerging Markets Finance and Trade, Taylor & Francis Journals, vol. 55(1), pages 201-217, January.
    9. Caporale, Guglielmo Maria & Carcel, Hector & Gil-Alana, Luis, 2018. "The EMBI in Latin America: Fractional integration, non-linearities and breaks," Finance Research Letters, Elsevier, vol. 24(C), pages 34-41.
    10. Giorgio Canarella & Luis A. Gil-Alana & Rangan Gupta & Stephen M. Miller, 2016. "Modeling U.S. Historical Time-Series Prices and Inflation Using Various Linear and Nonlinear Long-Memory Approaches," Working Papers 201683, University of Pretoria, Department of Economics.
    11. Guglielmo Maria Caporale & Luis A. Gil-Alana & Manuel Monge, 2019. "Energy Consumption in the GCC Countries: Evidence on Persistence," CESifo Working Paper Series 7470, CESifo.
    12. Luis Alberiko Gil-Alana & Zeynel Abidin Ozdemir & Aysit Tansel, 2019. "Long Memory in Turkish Unemployment Rates," Emerging Markets Finance and Trade, Taylor & Francis Journals, vol. 55(1), pages 201-217, January.
    13. Guglielmo Maria Caporale & Hector Carcel & Luis A. Gil-Alana, 2015. "Modelling African inflation rates: nonlinear deterministic terms and long-range dependence," Applied Economics Letters, Taylor & Francis Journals, vol. 22(5), pages 421-424, March.
    14. Guglielmo Maria Caporale & Luis A. Gil-Alana, 2017. "Trends and Cycles in Macro Series: The Case of US Real GDP," CESifo Working Paper Series 6728, CESifo.
    15. Guglielmo Caporale & Luis Gil-Alana, 2016. "Persistence and cyclical dependence in the monthly euribor rate," Journal of Economics and Finance, Springer;Academy of Economics and Finance, vol. 40(1), pages 157-171, January.
    16. Luis A. Gil-Alana & Rangan Gupta & Fernando Perez de Gracia, 2016. "Persistence, mean reversion and non-linearities in the US housing prices over 1830--2013," Applied Economics, Taylor & Francis Journals, vol. 48(34), pages 3244-3252, July.

    More about this item

    Keywords

    Chebyshev polynomials; long run dependence; fractional integration;

    JEL classification:

    • 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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