الملخص: 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.

الملخص: The aim of this article is to study the chromatic polynomial of cycle graph, and to describe some algebraic properties about the chromatic polynomial’s coefficients and roots to the same graph.

الملخص: This research compares the symmetric and asymmetric effects of generalized autoregressive conditional heteroscedasticity (GARCH)-type models to investigate the volatility of the autoregressive fractionally integrated moving average (ARFIMA) model using the monthly Brent crude oil price series for the period of January 1979–July 2019. The best model of volatility is determined by comparing 13 hybrid models of GARCH (sGARCH, fGARCH, EGARCH, TGARCH, IGARCH, AVGARCH, NGARCH, NAGARCH, APARCH, apARCH, GJRGARCH, gjrGARCH, and csGARCH) in terms of symmetric and asymmetric effects at the level of (1,1). R/S analysis is used to achieve this target. The aggregated variance method, the Higuchi method, and the structural break test are performed to determine the presence of long memory in the dataset. Furthermore, the Hurst exponent method and the Geweke and Porter–Hudak method are used to estimate the fractional difference values. The ARFIMA(2,0.3589648,2)–IGARCH(1,1) model under normal distribution is selected as the best model based on the Akaike information criterion, Schwartz Bayesian information criterion, and by the smallest value for root-mean-squared error, in which this model can be used to predict more accurately than other models

الملخص: The autoregressive fractional integrated moving average (ARFIMA) has become one of the popular linear models in time series modeling and forecasting in the past decades. Recent research in modeling and forecasting with artificial neural networks (ANN) suggests that these networks are a promising alternative to the traditional linear and nonlinear methods. ARFIMA models and ANNs are often compared with mixed conclusions in terms of superiority in forecasting performance. In this research, a hybrid methodology that combines both ARFIMA and multilayer perceptron (MLP) models is proposed to take advantage of the unique strength of the ARFIMA and MLP models in linear and nonlinear modeling, which is the primary objective of this study. This research uses the monthly Brent crude oil price series for the period of January 1979-July 2019. As for our other goal, the researchers’ previous works are also extended by examining the linear and nonlinear methods for the dataset simultaneously and comparing individual models with the hybrid models. The best model is determined by comparing 19 individual and hybrid models in terms of forecasting accuracy based on the root mean squared error and Ljung–Box test. Empirical results with real datasets indicate that the ARFIMA (1,0.3589648,0)–MLP (1,2,1) hybrid model outperforms the separately used models and the other hybrid models, and the Akaike information criterion value is not the smallest for this model.

الملخص: Seasonal Autoregressive Integrated Moving Average (SARIMA) model was fitted for the time series data either to better understand the data or to predict the future points in the series (forecasting). Using the forecasting for the exchange rate is very important at the national, regional and international levels. It can help investors minimize financial risk as well maximize earnings in the volatility of the global economy. The aim of this study was to use the time series model to forecast the exchange rate of Jordan dinar based on the monthly data collected for Jordanian dinar vs US dollar. Exchange rate prediction is performed using two methods; Autoregressive Integrated Moving Average (ARIMA) and Seasonal Autoregressive Integrated Moving Average (SARIMA) time series. After comparing the forecasting method using ARIMA (1, 0, 1) and SARIMA (1, 0, 1)(1, 0, 0)12 models, we find that the second model shows a smaller mean absolute percentage error (MAPE), root mean square error (RMSE), mean absolute error (MAE) and mean absolute square error (MASE) as compared to the first; That is, SARIMA (1, 0, 1)(1, 0, 0)12 model is the most appropriate method to forecast the exchange rate of the Jordanian dinar vs US dollar.

الملخص: Verifying the existence of long-memory feature is a crucial activity performed during the development process of the autoregressive integrated moving average (ARIMA) model. The verifying step will determine whether a researcher needs to use the ARIMA model or the autoregressive fractionally integrated moving average (ARFIMA) model, which depends on the estimated value of the fractional difference (d). This study focuses on analytical techniques for verifying the long-memory feature (graphs and statistical tests); determines estimation methods/functions for approximating long-memory parameters (ie, d), limitations, extensions, comparisons, applications, and performs an in-depth review of the recent literature on the ARFIMA model. The discussion will also include the hybrid method for forecasting in different fields. Although the validation of the existence of the long-memory feature and its estimation, limitations, extensions, comparisons, and applications has been extensively investigated, specific criteria should be considered to avoid obtaining invalid or wrong ARFIMA models remain unclear. We examine the literature to validate these issues and identify effective methods and tests to avoid these errors. Thus, the results of this study can provide an initial classification of the literature on long memory and the ARFIMA model that can be used as a basis for future work when the value of d is a non-integer number.