The Standard Generalised Autoregressive Conditionally Heteroskedastic (sGARCH) model and the Functional Generalised Autoregressive Conditionally Heteroskedastic (fGARCH) model were applied to study the volatility of the Autoregressive Fractionally Integrated Moving Average (ARFIMA) model, which is the primary objective of this study. The other goal of this paper is to expand on the researchers' previous work by examining long memory and volatilities simultaneously, by using the ARFIMA-sGARCH hybrid model and comparing it against the ARFIMA-fGARCH hybrid model. Consequently, the hybrid models were configured with the monthly Brent crude oil price series for the period from January 1979 to July 2019. These datasets were considered as the global economy is currently facing significant challenges resulting from noticeable volatilities, especially in terms of the Brent crude prices, due to the outbreak of COVID-19. To achieve these goals, an R/S analysis was performed and the aggregated variance and the Higuchi methods were applied to test for the presence of long memory in the dataset. Furthermore, four breaks have been detected: in 1986, 1999, 2005, and 2013 using the Bayes information criterion. In the further section of the paper, the Hurst Exponent and Geweke-Porter-Hudak (GPH) methods were used to estimate the values of fractional differences. Thus, some ARFIMA models were identified using AIC (Akaike Information Criterion), BIC (Schwartz Bayesian Information Criterion), AICc (corrected AIC), and the RMSE (Root Mean Squared Error). In result, the following conclusions were reached: the ARFIMA(2,0.3589648,2)-sGARCH(1,1) model and the ARFIMA(2,0.3589648,2)-fGARCH(1,1) model under normal distribution proved to be the best models, demonstrating the smallest values for these criteria. The calculations conducted herein show that the two models are of the same accuracy level in terms of the RMSE value, which equals 0.08808882, and it is this result that distinguishes our study. In conclusion, these models can be used to predict oil prices more accurately than others.

ARFIMA, volatility, fGARCH, sGARCH, modelling and forecasting, hybrid model

AAMIR, M., SHABRI, A. B., (2015). Modelling and Forecasting Monthly Crude Oil Prices of Pakistan: A Comparative Study of ARIMA, GARCH and ARIMAGARCH Models. Sci.Int. (Lahore), 27(3), pp. 2365-2371.

AKRON, N., ISMAIL, Z., (2017). A hybrid GA-FEEMD for forecasting crude oil prices. Indian Journal of Science and Technology, 10(31), pp. 1-6.

AKTER, N., NOBI, A., (2018). Investigation of the Financial Stability of S&P 500 Using Realized Volatility and Stock Returns Distribution. Journal of Risk Financial Management, 11(22), pp. 1-10.

AL-GOUNMEEIN, R. S., ISMAIL, M. T., (2020). Forecasting the Exchange Rate of the Jordanian Dinar versus the US Dollar Using a Box-Jenkins Seasonal ARIMA Model. International Journal of Mathematics and Computer Science, 15(1), pp. 27-40.

AMBACH, D., AMBACH, O., (2018). Forecasting the oil price with a periodic regression ARFIMA-GARCH process. IEEE Second International Conference on Data Stream Mining & Processing, Lviv, Ukraine, pp. 212-217.

ALZGHOOL, R., (2017). Parameters estimation for GARCH (p,q) model: QL and AQL approaches. Electronic Journal of Applied Statistical Analysis, 10(1), pp.180-193.

AUE, A., HORVATH, L. and PELLATT, D. F., (2017). Functional generalized autoregressive conditional heteroskedasticity. Journal of Time Series Analysis, 38(1), pp. 3-21.

BAHAR, A., NOH, N. M. and ZAINUDDIN, Z. M., (2017). Forecasting model for crude oil price with structural break. Malaysian Journal of Fundamental and Applied Sciences, pp. 421-424.

BERAN, J., (1994). Statistics for Long Memory Processes, Chapman and Hall, p. 315.

BHARDWAJ, G., SWANSON, N. R., (2006). An empirical investigation of the usefulness of ARFIMA models for predicting macroeconomic and financial time series. Journal of Econometrics 131, pp. 539-578.

BOUTAHAR, M., MARIMOUTOU, V. and NOUIRA, L., (2007). Estimation Methods of the Long Memory Parameter: Monte Carlo Analysis and Application. Journal of Applied Statistics, 34(3), pp. 261-301.

BOX, G. E. P., JENKINS, G. M. and REINSEL, G. C., (2008). Time series analysis forecasting and control, Fourth Edition, Wiley & Sons, Inc, p. 746.

CRYER, J. D., CHAN, K., (2008). Time Series Analysis With Application in R, Second Edition, Springer, p. 491.

DIEBOLD, F. X., INOUE, A., (2001). Long Memory and Regime Switching. Journal of Econometrics, 105, pp. 131-159.

FAZELABDOLABADI, B., (2019). A hybrid Bayesian-network proposition for forecasting the crude oil price. Financial Innovation, 5(30), pp. 1-21.

FRANCQ, C., ZAKOIAN, J. M., (2019). GARCH Models: Structure, Statistical Inference and Financial Applications, Second Edition, John Wiley & Sons Ltd, p. 487.

GRANGER, C. W. J., HYUNG, N., (2004). Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. Journal of Empirical Finance, 11, pp. 399-421.

HE, X. J., (2018). Crude Oil Prices Forecasting: Time Series vs. SVR Models. Journal of International Technology and Information Management, 27(2), pp. 25-42.

HOSKING, J. R. M., (1981). Fractional differencing. Biometrika, 86(1), pp. 165-176.

IQELAN, B. M., (2015). Time Series Modeling of Monthly Temperature Data of Jerusalem / Palestine. MATEMATIKA, 31(2), pp. 159-176.

ISMAIL, M. T., AWAJAN, A. M., (2017). A new hybrid approach EMD-EXP for shortterm forecasting of daily stock market time series data. Electronic Journal of Applied Statistical Analysis, 10(2), pp. 307-327.

JIBRIN, S. A., MUSA, Y., ZUBAIR, U. A. and SAIDU, A. S., (2015). ARFIMA Modelling and Investigation of Structural Break(s) in West Texas Intermediate and Brent Series, CBN Journal of Applied Statistics, 6(2), pp. 59-79.

KANG, S. H., YOON, S., (2013). Modeling and forecasting the volatility of petroleum futures prices. Energy Economics, 36, pp. 354-362.

KARIA, A. A., BUJANG, I. and AHMAD, I., (2013). Fractionally integrated ARMA for crude palm oil prices prediction: case of potentially over difference. Journal of Applied Statistics, 40(12), pp. 2735-2748.

LEE, C. Y., HUH, S. Y., (2017). Forecasting Long-Term Crude Oil Prices Using a Bayesian Model with Informative Priors. Sustainability. 9, 190, DOI: 10.3390/su9020190.

LO, A. W., (1991). Long-term memory in stock market prices. Econometrica, 59(5), pp. 1279-1313.

MANDELBROT, B., (1972). Statistical Methodology for Nonperiodic Cycles: From the Covariance to R/S Analysis. Annals of Economic and Social Measurement, 1(3), pp. 259-290.

MANERA, M., MCALEER, M. and GRASSO, M., (2004). Modelling dynamic conditional correlations in the volatility of spot and forward oil price returns, 2nd International Congress on Environmental Modelling and Software - Osnabrück, Germany, 183, pp. 1-6.

MIAH, M., RAHMAN, A., (2016). Modelling Volatility of Daily Stock Returns: Is GARCH(1,1) Enough?. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS), 18(1), pp. 29-39.

MONTGOMERY, D. C., JENNINGS, C. L. and KULAHCI, M., (2015). Introduction To Time Series Analysis And Forecasting, Second Edition, Wiley & Sons, Inc, p. 643.

MOSTAFAEI, H., SAKHABAKHSH, L., (2012). Using SARFIMA Model to Study and Predict the Iran’s Oil Supply. International Journal of Energy Economics and Policy, 2(1), pp. 41-49.

NYANGARIKA, A., MIKHAYLOV, A. and RICHTER, U. H., (2019). Oil Price Factors: Forecasting on the Base of Modified Auto-regressive Integrated Moving Average Model. International Journal of Energy Economics and Policy, 9(1), pp. 149-159.

OHANISSIAN, A., RUSSELL, J. R. and TSAY, R. S., (2008). True or Spurious Long Memory? A New Test. Journal of Business & Economic Statistics, 26(2), pp. 161-175.

OLATAYO, T. O., ADEDOTUN, A. F., (2014). On the Test and Estimation of Fractional Parameter in ARFIMA Model: Bootstrap Approach. Applied Mathematical Sciences, 8(96), pp.4783-4792.

PALMA, W., (2007). Long-Memory Time Series: Theory and Methods, John Wiley & Sons, Inc, p. 285.

PRETIS, F., SCHNEIDER, L., SMERDON, J. E. and HENDRY, D. F., (2016). Detecting volcanic eruptions in temperature reconstructions by designed break-indicator saturation. Journal of Economic Surveys, 30(3), pp. 403-429.

RAMZAN, S., RAMZAN, S. and ZAHID, F. M., (2012). Modeling and Forecasting Exchange Rate Dynamics In Pakistan Using ARCH Family of Models. Electronic Journal of Applied Statistical Analysis, 5(1), pp. 15-29.

REISEN, V. A., (1994). Estimation of the Fractional Difference Parameter in the ARIMA(p,d,q) Model Using the Smoothed Periodogram. Journal of Time Series Analysis, 15(3), pp. 335-350.

SEHGAL, N., PANDEY, K. K., (2015). Artificial intelligence methods for oil price forecasting: a review and evaluation, Springer-Verlag Berlin Heidelberg, DOI: 10.1007/s12667-015-0151-y.

TELBANY, S., SOUS, M., (2016). Using ARFIMA Models in Forecasting Indicator of the Food and Agriculture Organization. IUGJEBS, 24(1), pp. 168-187.

TENDAI, M., CHIKOBVU, D., (2017). Modelling international tourist arrivals and volatility to the Victoria Falls Rainforest, Zimbabwe: Application of the GARCH family of models. African Journal of Hospitality, Tourism and Leisure, 6(4), pp. 1-16.

YIN, X., PENG, J. and TANG, T., (2018). Improving the Forecasting Accuracy of Crude Oil Prices. Sustainability. 10, 454, DOI: 10.3390/su10020454.

YU, L., WANG, S. and LAI, K. K., (2008). Forecasting crude oil price with an EMDbased neural network ensemble learning paradigm. Energy Economics, 30, pp. 2623-2635.

YU, L., ZHANG, X. and WANG, S., (2017). Assessing Potentiality of Support Vector Machine Method in Crude Oil Price Forecasting. EURASIA Journal of Mathematics, Science and Technology Education, 13(12), pp. 7893-7904.

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