In this paper, we introduce the class of autoregressive fractionally integrated moving average - generalized autoregressive conditional heteroskedasticity (ARFIMA-GARCH) models with level shift type intervention that are capable of capturing three key features of time series: long range dependence, volatility and level shift. The main concern is on detection of mean and volatility level shift in a fractionally integrated time series with volatility. We will denote such a time series as level shift autoregressive fractionally integrated moving aver age (LS-ARFIMA) and level shift generalized autoregressive conditional heterosk edasticity (LS-GARCH). Test statistics that are useful to examine if mean and volatility level shifts are present in an autoregressive fractionally in tegrated moving average - generalized autoregressive conditional heteroskedas ticity (ARFIMA-GARCH) model are derived. Quasi maximum likelihood esti mation of the model is also considered.
Granger, C.W.J. and Joyeux, R. (1980) An Introduction to Long-Memory Time Series Models and Fractional Differencing. Journal of Time Series Analysis, 1, 15-29. https://doi.org/10.1111/j.1467-9892.1980.tb00297.x
Robinson, P.M. and Zaffaroni, P. (1998) Nonlinear Time Series with Long Memory: A Model for Stochastic Volatility. Journal of Statistical Planning and Inference, 68, 359-371. https://doi.org/10.1016/S0378-3758(97)00149-3
Bollerslev, T. (1986) Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31, 307-327. https://doi.org/10.1016/0304-4076(86)90063-1
Bollerslev, T., Engle, R.F. and Wooldridge, J.M. (1988) A Capital Asset Pricing Model with Time-Varying Covariances. Journal of Political Economy, 96, 116-131. https://doi.org/10.1086/261527
Weiss, A.A. (1984) ARMA Models with ARCH Errors. Journal of Time Series Analysis, 5, 129-143. https://doi.org/10.1111/j.1467-9892.1984.tb00382.x
Ling, S.Q. and Li, W.K. (1997) On Fractionally Integrated Autoregressive Moving-Average Time Series Models with Conditional Heteroscedasticity. Journal of the American Statistical Association, 92, 1184-1194. https://doi.org/10.1080/01621459.1997.10474076
Reisen, V.A., Sarnaglia, A.J.Q., Reis Jr., N.C., Levy-Leduc, C. and Santos, J.M. (2014) Modeling and Forecasting Daily Average PM10 Concentrations by a Seasonal Long-Memory Model with Volatility. Environmental Modelling & Software, 51, 286-295. https://doi.org/10.1016/j.envsoft.2013.09.027
Tong, H. (2011) Nonlinear Time Series Analysis. In: Lovric, M., Ed., International Encyclopedia of Statistical Science, Springer, Berlin, Heidelberg, 955-958. https://doi.org/10.1007/978-3-642-04898-2_411
Narayan, P.K., Liu, R.P. and Westerlund, J. (2016) A GARCH Model for Testing Market Efficiency. Journal of International Financial Markets, Institutions and Money, 41, 121-138. https://doi.org/10.1016/j.intfin.2015.12.008
Box, G.E.P., Jenkins, G.M., Reinsel, G.C. and Ljung, G.M. (2015) Time Series Analysis: Forecasting and Control. John Wiley & Sons, Hoboken.
Adenstedt, R.K. (1974) On Large-Sample Estimation for the Mean of a Stationary Random Sequence. The Annals of Statistics, 2, 1095-1107. https://doi.org/10.1214/aos/1176342867
Beran, J. (2017) Statistics for Long-Memory Processes. Routledge, Abingdon-on-Thames. https://doi.org/10.1201/9780203738481
Baillie, R.T., Bollerslev, T. and Mikkelsen, H.O. (1996) Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 74, 3-30. https://doi.org/10.1016/S0304-4076(95)01749-6
Robinson, P.M. (2003) Long Memory Time Series. In: Robinson, P.M., Ed., Time Series with Long Memory, Oxford University Press, Oxford, 4-32.
Engle, R.F. (1982) Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica: Journal of the Econometric Society, 50, 987-1007. https://doi.org/10.2307/1912773
Bollerslev, T., Engle, R.F. and Nelson, D.B. (1994) Arch Models. In: Engle, R.F. and McFadden, D.L., Eds., Handbook of Econometrics, Vol. 4, Elsevier, Amsterdam, 2959-3038. https://doi.org/10.1016/S1573-4412(05)80018-2
Baillie, R.T., Chung, C.-F. and Tieslau, M.A. (1996) Analysing Inflation by the Fractionally Integrated ARFIMA-GARCH Model. Journal of Applied Econometrics, 11, 23-40. https://doi.org/10.1002/(SICI)1099-1255(199601)11:1 3.0.CO;2-M
Montanari, A., Rosso, R. and Taqqu, M.S. (2000) A Seasonal Fractional ARIMA Model Applied to the Nile River Monthly Flows at Aswan. Water Resources Research, 36, 1249-1259. https://doi.org/10.1029/2000WR900012
Giraitis, L. and Leipus, R. (1995) A Generalized Fractionally Differencing Approach in Long-Memory Modeling. Lithuanian Mathematical Journal, 35, 53-65. https://doi.org/10.1007/BF02337754
Yang, Q. and Wang, Y.S. (2019) Application of the Improved Generalized Autoregressive Conditional Heteroskedast Model Based on the Autoregressive Integrated Moving Average Model in Data Analysis. Open Journal of Statistics, 9, 543-554. https://doi.org/10.4236/ojs.2019.95036
Bisognin, C. and Lopes, S.R.C. (2009) Properties of Seasonal Long Memory Processes. Mathematical and Computer Modelling, 49, 1837-1851. https://doi.org/10.1016/j.mcm.2008.12.003
Ishida, I. and Engle, R.F. (2002) Modeling Variance of Variance: The Square Root, the Affine, and the CEV GARCH Models. Working Papers, Dept. Finances, New York.
Belkhouja, M. and Mootamri, I. (2016) Long Memory and Structural Change in the g7 Inflation Dynamics. Economic Modelling, 54, 450-462. https://doi.org/10.1016/j.econmod.2016.01.021
Chareka, P., Matarise, F. and Turner, R. (2006) A Test for Additive Outliers Applicable to Long Memory Time Series. Journal of Economic Dynamics and Control, 30, 595-621. https://doi.org/10.1016/j.jedc.2005.01.003
Chang, I., Tiao, G.C. and Chen, C. (1988) Estimation of Time Series Parameters in the Presence of Outliers. Technometrics, 30, 193-204. https://doi.org/10.1080/00401706.1988.10488367
Leadbetter, M.R., Lindgren, G. and Rootzen, H. (1983) Extreme and Related Properties of Random Sequence and Processes. Springer-Verlag, New York.
Embrechts, P., Klppelberg, C. and Mikosch, T. (1997) Modeling Extremal Events for Insurance and Finance. Springer-Verlag, Berlin. https://doi.org/10.1007/978-3-642-33483-2