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Comparative study of Predication evolution of ARMA, AR GARCH and ARMA GARCH Sunspot Cycles as a case study

. Asma Zaffar, ovais Siraj Department of Mathematics, Sir Syed University of Engineering and Technology, Karachi, Pakistan


This study focuses on the appropriateness of forecasting evolution of various Autoregressive and moving average models and generalized autoregressive conditional heteroskedasticity (GARCH) models in terms of their performance for delivering volatility forecasts for Sunspot cycles. It is verified that the sunspot cycles have stationary nature with second difference and Autocorrelation (AC), Partial Autocorrelation (PAC) and Ljung-Box Q-statistics test are used to check the presence of white noise (strongly correlated) in the time series data. Under model Identification and Estimation, diagnostic checking and Forecasting the ARMA, AR (p)-GARCH and ARMA-GARCH models are found to be most appropriate. To detect the appropriateness of Autoregressive Conditional Heteroscedastic (ARCH) effect on sunspot cycles data, Lagrange Multiplier test is used. The selection of the model is based on residual diagnostic checking’s such as ARCH LM, normality test and correlogram squared residuals. Forecasting evolutions are verified by Root Mean Squared Error (RMSE), Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE) and Theil’s U-Statistics test (U test). Least square Estimation is used to investigate the ARMA process and the Gaussian quasi maximum likelihood estimation (QMLE) is used to estimate GARCH (1, 1) process for the specification of AR (p) and ARMA (p, q) models. The selection of models is based on the least value of Durbin-Watson state. The novelty of this study to forecasts for the evolution of sunspot cycles obtained by ARMA, AR-GARCH and ARMA-GARCH models are compared. RMSE, MAE and U test is utilized to check the appropriateness of various ARMA models. Only the MAPE exhibited the appropriateness of ARMA-GARCH model apart from cycles 1st, 9th, 10th and 17th follows ARMA models and cycles 2nd, 3rd, 4th, 5th, 6th and 20th follows AR-GARCH model.


Keywords: ARMA, AR-GARCH, ARMA-GARCH, Durbin-Watson, Theil’s U-Statistics


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