Statistical Science

Fractionally Differenced Gegenbauer Processes with Long Memory: A Review

G. S. Dissanayake, M. S. Peiris, and T. Proietti

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The main objective of this paper is to review and promote the usefulness of generalized fractionally differenced Gegenbauer processes in time series and econometric research endeavours. In particular, theoretical and computational aspects centered around fractionally differenced Gegenbauer processes with long memory together with a number of interesting and elegant extensions will be discussed. In-depth conceptual developments and large scale simulation study results are presented for clarity and completeness. This survey highlights a number of gaps in the existing literature of this subject area and becomes a valuable reference source for time series practitioners.

Article information

Statist. Sci., Volume 33, Number 3 (2018), 413-426.

First available in Project Euclid: 13 August 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Gegenbauer process long memory heteroskedasticity fractional difference volatility spectral density stationarity invertibility


Dissanayake, G. S.; Peiris, M. S.; Proietti, T. Fractionally Differenced Gegenbauer Processes with Long Memory: A Review. Statist. Sci. 33 (2018), no. 3, 413--426. doi:10.1214/18-STS649.

Export citation


  • Abrahams, M. and Dempster, A. (1979). Research on seasonal analysis. Progress report on the asa/census project on seasonal adjustment. Technical report. Dept. Statistics, Harvard Univ., Boston, MA.
  • Anděl, J. (1986). Long memory time series models. Kybernetika (Prague) 22 105–123.
  • Anderson, B. D. O. and Moore, J. B. (1979). Optimal Filtering. Prentice-Hall, New York.
  • Anh, V. V., Angulo, J. M. and Ruiz-Medina, M. D. (1999). Possible long-range dependence in fractional random fields. J. Statist. Plann. Inference 80 95–110.
  • Anh, V. V., Lunney, K. and Peiris, S. (1997). Stochastic models for characterisation and prediction of time series with long-range dependence and fractality. Environ. Model. Softw. 12 67–73.
  • Aoki, M. (1990). State Space Modeling of Time Series, 2nd ed. Springer, Berlin.
  • Arteche, J. (2007). The analysis of seasonal long memory: The case of Spanish inflation. Oxf. Bull. Econ. Stat. 69 749–772.
  • Arteche, J. (2012). Standard and seasonal long memory in volatility: An application to Spanish inflation. Empir. Econ. 42 693–712.
  • Arteche, J. and Robinson, P. M. (2000). Semiparametric inference in seasonal and cyclical long memory processes. J. Time Series Anal. 21 1–25.
  • Baillie, R. T. (1996). Long memory processes and fractional integration in econometrics. J. Econometrics 73 5–59.
  • Baillie, R. T., Bollerslev, T. and Mikkelsen, H. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. J. Econometrics 74 3–30.
  • Beaumont, P. and Ramachandran, R. (2001). Robust estimation of GARMA model parameters with an application to cointegration among interest rates of industrialized countries. Comput. Econ. 17 179–201.
  • Beran, J. (1992). Statistical methods for data with long-range dependence. Statist. Sci. 7 404–416.
  • Beran, J. (1993). Fitting long-memory models by generalized linear regression. Biometrika 80 817–822.
  • Beran, J. (1994). Statistics for Long-Memory Processes. Monographs on Statistics and Applied Probability 61. Chapman & Hall, New York.
  • Beran, J., Feng, Y., Ghosh, S. and Kulik, R. (2013). Long-Memory Processes: Probabilistic Properties and Statistical Methods. Springer, Heidelberg.
  • Bisognin, C. and Lopes, S. R. C. (2009). Properties of seasonal long memory processes. Math. Comput. Modelling 49 1837–1851.
  • Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307–327.
  • Box, G. E. P. and Jenkins, G. M. (1970). Times Series Analysis. Forecasting and Control. Holden-Day, San Francisco, CA.
  • Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • Brockwell, P. J. and Davis, R. A. (1996). Introduction to Time Series and Forecasting. Springer, New York.
  • Chan, N. H. and Palma, W. (1998). State space modeling of long-memory processes. Ann. Statist. 26 719–740.
  • Chan, N. H. and Palma, W. (2006). Estimation of long-memory time series models: A survey of different likelihood-based methods. Adv. Econom. 20 89–121.
  • Chung, C.-F. (1996). A generalized fractionally integrated autoregressive moving-average process. J. Time Series Anal. 17 111–140.
  • Corsi, F. (2009). A simple approximate long-memory model of realized volatility. J. Financ. Econom. 7 174–196.
  • Dissanayake, G. S. and Peiris, M. S. (2011). Generalized fractional processes with conditional heteroskedasticity. Sri Lankan J. Appl. Stat. 12 1–12.
  • Dissanayake, G. S., Peiris, M. S. and Proietti, T. (2014). Estimation of generalized fractionally differenced processes with conditionally heteroskedastic errors. In International Work Conference on Time Series, Proceedings ITISE 2014 (I. R. Ruiz and G. R. Garcia, eds.). Copicentro Granada S L 871–890.
  • Dissanayake, G. S., Peiris, M. S. and Proietti, T. (2015). State space modeling of seasonal Gegenbauer processes with long memory. Working Paper, School of Mathematics and Statistics, Univ. Sydney, Australia.
  • Dissanayake, G. S., Peiris, M. S. and Proietti, T. (2016). State space modeling of Gegenbauer processes with long memory. Comput. Statist. Data Anal. 100 115–130.
  • Dissanayake, G. S., Peiris, M. S., Proietti, T. and Wang, Q. (2015). Nearly efficient testing and asymptotics of a long memory $\operatorname{GARMA}(0,\delta,0)$ process, Working Paper, School of Mathematics and Statistics, Univ. Sydney, Australia.
  • Dolado, J. J., Gonzalo, J. and Mayoral, L. (2002). A fractional Dickey–Fuller test for unit roots. Econometrica 70 1963–2006.
  • Durbin, J. and Koopman, S. J. (2001). Time Series Analysis by State Space Methods. Oxford Statistical Science Series 24. Oxford Univ. Press, Oxford.
  • Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987–1007.
  • Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1953). Higher Transcendental Functions, Vol. II, Bateman Manuscript Project. McGraw-Hill. New York.
  • Ferrara, L. and Guegan, D. (2001). Forecasting with k-factor Gegenbauer processes: Theory and applications. J. Forecast. 20 581–601.
  • Ferrara, L., Guegan, D. and Lu, Z. (2010). Testing fractional order of long memory processes: A Monte Carlo study. Comm. Statist. Simulation Comput. 39 795–806.
  • Giraitis, L., Hidalgo, J. and Robinson, P. M. (2001). Gaussian estimation of parametric spectral density with unknown pole. Ann. Statist. 29 987–1023.
  • Giraitis, L., Koul, H. L. and Surgailis, D. (2012). Large Sample Inference for Long Memory Processes. Imperial College Press, London.
  • Giraitis, L. and Leipus, R. (1995). A generalized fractionally differencing approach in long-memory modeling. Liet. Mat. Rink. 35 65–81.
  • Gonçalves, E. (1987). Une généralisation des processus ARMA. Ann. Écon. Stat. 5 109–145.
  • Gould, H. W. (1974). Coefficient identities for powers of Taylor and Dirichlet series. Amer. Math. Monthly 81 3–14.
  • Gradshteyn, I. S. and Ryzhik, I. M. (1980). Tables of Integrals Series and Products. Academic Press, New York.
  • Granger, C. W. J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Series Anal. 1 15–29.
  • Grassi, S. and Santucci de Magistris, P. (2014). When long memory meets the Kalman filter: A comparative study. Comput. Statist. Data Anal. 76 301–319.
  • Gray, H. L., Zhang, N.-F. and Woodward, W. A. (1989). On generalized fractional processes. J. Time Series Anal. 10 233–257.
  • Gray, H. L., Zhang, N.-F. and Woodward, W. A. (1994). A correction: “On generalized fractional processes” [J. Time Ser. Anal. 10 (1989), no. 3, 233–257; MR1028940 (90m:62208)]. J. Time Series Anal. 15 561–562.
  • Guegan, D. (2000). A new model: The k-factor GIGARCH process. J. Signal Process. 4 265–271.
  • Guégan, D. (2005). How can we define the concept of long memory? An econometric survey. Econometric Rev. 24 113–149.
  • Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge Univ. Press, Cambridge.
  • Harvey, A. C. and Proietti, T., eds. (2005). Readings in Unobserved Components Models. Oxford Univ. Press, Oxford.
  • Hassler, U. (1994). (Mis)specification of long memory in seasonal time series. J. Time Series Anal. 15 19–30.
  • Hassler, U. and Wolters, J. (1994). On the power of unit root tests against fractional alternatives. Econom. Lett. 45 1–5.
  • Hassler, U. and Wolters, J. (1995). Long memory in inflation rates: International evidence. J. Bus. Econom. Statist. 13 37–45.
  • Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68 165–176.
  • Hsu, N.-J. and Tsai, H. (2009). Semiparametric estimation for seasonal long-memory time series using generalized exponential models. J. Statist. Plann. Inference 139 1992–2009.
  • Jansson, M. and Nielsen, M. Ø. (2012). Nearly efficient likelihood ratio tests of the unit root hypothesis. Econometrica 80 2321–2332.
  • Kalman, R. E. (1961). A new approach to linear filtering and prediction problems. Trans. Am. Soc. Mech. Eng. 83D 35–45.
  • Kalman, R. E. and Bucy, R. S. (1961). New results in linear filtering and prediction theory. Trans. Am. Soc. Mech. Eng. 83 95–108.
  • Koopman, S. J., Ooms, M. and Carnero, M. A. (2007). Periodic seasonal Reg-ARFIMA-GARCH models for daily electricity spot prices. J. Amer. Statist. Assoc. 102 16–27.
  • Lieberman, O. and Phillips, P. C. B. (2008). Refined inference on long memory in realized volatility. Econometric Rev. 27 254–267.
  • Ling, S. and Li, W. K. (1997). On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. J. Amer. Statist. Assoc. 92 1184–1194.
  • Lobato, I. N. and Savin, N. E. (1998). Real and spurious long-memory properties of stock-market data. J. Bus. Econom. Statist. 16 261–283.
  • Lu, Z. and Guegan, D. (2011). Estimation of time-varying long memory parameter using wavelet method. Comm. Statist. Simulation Comput. 40 596–613.
  • McAleer, M. and Medeiros, M. C. (2008). A multiple regime smooth transition heterogeneous autoregressive model for long memory and asymmetries. J. Econometrics 147 104–119.
  • Montanari, A., Rosso, R. and Taqqu, M. S. (2000). A seasonal fractional ARIMA modelapplied to Nile river monthly flows at Aswan. Water Resour. Res. 36 1249–1259.
  • Ohanissian, A., Russell, J. R. and Tsay, R. S. (2008). True or spurious long memory? A new test. J. Bus. Econom. Statist. 26 161–175.
  • Ooms, M. (1995). Flexible seasonal long memory and economic time series. Technical Report EI-9515/A. Econometric Institute, Erasmus Univ., Rotterdam.
  • Oppenheim, G. and Viano, M.-C. (2004). Aggregation of random parameters Ornstein–Uhlenbeck or AR processes: Some convergence results. J. Time Series Anal. 25 335–350.
  • Palma, W. (2007). Long-Memory Time Series: Theory and Methods. Wiley-Interscience, Hoboken, NJ.
  • Palma, W. and Chan, N. H. (2005). Efficient estimation of seasonal long-range-dependent processes. J. Time Series Anal. 26 863–892.
  • Pearlman, J. G. (1980). An algorithm for the exact likelihood of a high-order autoregressive–moving average process. Biometrika 67 232–233.
  • Peiris, M. S. (2003). Improving the quality of forecasting using generalized AR models: An application to statistical quality control. Stat. Methods 5 156–171.
  • Peiris, S., Allen, D. and Peiris, U. (2005). Generalized autoregressive models with conditional heteroscedasticity: An application to financial time series modeling. In Proceedings of the Workshop on Research Methods: Statistics and Finance 75–83.
  • Peiris, S. and Asai, M. (2016). Generalized fractional processes with long memory and time dependent volatility revisited. Econometrics 4 37.
  • Peiris, S. and Thavaneswaran, A. (2007). An introduction to volatility models with indices. Appl. Math. Lett. 20 177–182.
  • Phillips, P. C. B. and Xiao, Z. (1998). A primer on unit root testing. J. Econ. Surv. 12 423–470.
  • Porter-Hudak, S. (1990). An application of the seasonal fractionally differenced model to the monetary aggregates. J. Amer. Statist. Assoc. 85 338–344.
  • Rainville, E. D. (1960). Special Functions. The Macmillan Co., New York.
  • Ray, B. K. (1993). Modeling long-memory processes for optimal long-range prediction. J. Time Series Anal. 14 511–525.
  • Reisen, V. A., Rodrigues, A. L. and Palma, W. (2006). Estimation of seasonal fractionally integrated processes. Comput. Statist. Data Anal. 50 568–582.
  • Robinson, P. M. (1991). Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. J. Econometrics 47 67–84.
  • Shephard, N. (1996). Statistical aspects of ARCH and stochastic volatility. In Time Series Models: In Econometrics, Finance and Other Fields (D. R. Cox, D. B. Hinkley, and O. E. Barndorff-Nielsen, eds.), Chapman & Hall, London.
  • Shitan, M. and Peiris, S. (2008). Generalized autoregressive (GAR) model: A comparison of maximum likelihood and Whittle estimation procedures using a simulation study. Comm. Statist. Simulation Comput. 37 560–570.
  • Shitan, M. and Peiris, S. (2009). On properties of the second order generalized autoregressive $\operatorname{GAR}(2)$ model with index. Math. Comput. Simulation 80 367–377.
  • Shitan, M. and Peiris, S. (2013). Approximate asymptotic variance–covariance matrix for the Whittle estimators of $\operatorname{GAR}(1)$ parameters. Comm. Statist. Theory Methods 42 756–770.
  • Slutsky, E. (1927). The summation of random causes as the source of cyclic processes. Econometrica 5 105–146.
  • Spolia, S. K., Chandler, S. and O’Connor, K. M. (1980). An autocorrelation approach for parameter estimation of fractional order equal-root autoregressive models using hypergeometric functions. J. Hydrol. 47 1–18.
  • Taylor, A. M. R. (2005). Fluctuation tests for a change in persistence. Oxf. Bull. Econ. Stat. 67 207–230.
  • Velasco, C. and Robinson, P. M. (2000). Whittle pseudo-maximum likelihood estimation for nonstationary time series. J. Amer. Statist. Assoc. 95 1229–1243.
  • Wang, Q., Lin, Y.-X. and Gulati, C. M. (2003). Asymptotics for general fractionally integrated processes with applications to unit root tests. Econometric Theory 19 143–164.
  • Woodward, W. A., Cheng, Q. C. and Gray, H. L. (1998). A $k$-factor GARMA long-memory model. J. Time Series Anal. 19 485–504.
  • Yule, G. U. (1926). Why do we sometimes get nonsense-correlations between time-series?—A study in sampling and the nature of time-series. J. Roy. Statist. Soc. 89 1–63.