Does the Fall in Crude Oil Prices Really Affect the Malaysian RinggitARIMA模型方法预测原油价格外文文献英文

Forecasting Crude Oil Prices Using Wavelet ARIMA Model Approach Nurull Qurraisya Nadiyya Md-Khair(st. The equation is described as follow, wu;st 1 ffi ffi s p w t ? u s ? :3 where w is the mother wavelet function, translated by the index of location u, which signify its position in the time domain, and dilated by the scale index s, that interprets the width of a daughter wavelet. Through wavelet transform, a time series is expressed in its time-frequency rep- resentation which result in wavelet coeffi cients. Three forms of wavelet transform can be recognized depending on the number of coeffi cients produced. They are continuous wavelet transform (CWT), discrete wavelet transform (DWT) and maximum overlap DWT (MODWT). In regard to DWT, only minimal number of coeffi cients required to reconstruct the original function xt are produced. To achieve this reduction, the parameters u and s are discretized so that u k2?jand s 2?jwhere j and k are integers. j defi nes the respective decomposition level where j 1;.;J and J is the number of decomposition levels produced. The process consists of a decomposition and reconstruction stage. In multiresolution analysis (MRA), time series is decomposed using wavelet transform into a smooth series AJ that contains smooth coeffi cients aJ;k, and a set of detail series Dj consisting of detail coeffi cients dj;k. MRA is a design method formulated based on the study of orthonormal, compactly supported wavelet bases. The smooth series which is described as a de-noised version of original time series is the main component of MRA according to Crowley 15, while the detail series captures fl uctuations around this smooth series. Finally, the original series can be reconstructed by summing the coeffi cients of the smooth series AJand the detail series Djusing the following equation: xt XJ j1 aJ;tdj;t:4 538N.Q.N. Md-Khair and R. Samsudin 2.4Wavelet ARIMA Combination Approach According to Schlter and Deuschle, highest forecast effi ciency can be accomplished if wavelets are utilized in a MRA that decomposes a time series into a smooth series 16. This individualsubseriesisthen forecastedusing time seriesmethodandfi nally summed back together to get the forecast of the original time series. Since the subseries decom- posed from wavelet transform tend to have a more stable variance and normally with no outliers, this approach can provide a more accurate forecasting result than the direct forecastingapproach13.Inwavelet-basedforecasting,DWTisobservedtobethemost used type of wavelet transform in most studies especially in the context of resource and economic therefore DWT will be applied in this proposed approach 6, 7, 13. As for the time series method, Dooley and Lenihan 17 used ARIMA and lagged forward price models in their forecasting approach and empirically proven that ARIMA model can provide a better performance in forecasting thus giving the reason why ARIMA model is suitable in this proposed approach. To allow the ARIMA models to be fi tted in a short time, the automatic ARIMA model fi tting algorithm introduced by Hyndman and Khandakar 18 is used. Using this automatic fi tting algorithm, a large number of fore- casts can be simulated within reasonable time. Figure 1 illustrates the framework for the proposed Wavelet ARIMA approach. 2.5Effectiveness Evaluation The prediction accuracy of each forecasting approach is calculated using by calculating the mean absolute error (MAE) and the root mean square error (RMSE) between the original price series and the forecasted price series. The calculation for MAE is as follow, MAE 1 n Xn t1 xt? ytjj:5 where n is the number of period forecast, xtis the actual price at time t and ytis the forecasted price at time t. The forecast result is also analyzed using RMSE to determine Fig. 1. Framework for forecasting with proposed approach. Forecasting Crude Oil Prices Using Wavelet ARIMA Model Approach539 the tendency of the proposed Wavelet ARIMA approach to large forecast errors using the equation, RMSE ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi 1 n Xn t1 xt? yt2: r 6 3Result and Discussion 3.1ARIMA Model Construction To preserve space, only explanation for Brent monthly is elaborated for the sake of clarifi cation. Nevertheless, the result obtained will be shown for the entire dataset. Figure 2 shows the graph for Brent monthly dataset where the x-axis shows the month count from year 2000 to 2016 and the y-axis shows the price. Figure 3 depicts the partial autocorrelation function (PACF) and autocorrelation function (ACF) of the in-sample data. The prices series is considered stationary if the ACF graph cut offfairly quickly, while it is considered nonstationary of the ACF graph dies down extremely slowly 5. From Fig. 3, it can be observed that the ACF graph dies down extremely slowly therefore the in-sample data is nonstationary. For this particular reason, ARIMA model will be utilized rather than ARMA model because ARIMA model deals with nonstationary data series by differencing it. In this particular study, the construction of the time series model is done using an automatic fi tting algorithm which includes the model identifi cation, estimation and validation as stated earlier. This process is done using a software environment named R which is utilized for statistical computing with the packages “forecast” and “tseries”. Fig. 2. Monthly dataset for Brent North Sea crude oil spot prices. 540N.Q.N. Md-Khair and R. Samsudin From Fig. 4, it can be observed that the ACF residuals die out from one lag on which conclude that the model fi ts the data well 3. Validation test which is proposed by Ljung and Box 19 is also illustrated in the result. From the generated result, the most fi tted model for the Brent monthly in-sample prices series is ARIMA (1, 1, 0). This model is used to forecast out-of-sample data to evaluate the effectiveness of the con- structed model. This whole ARIMA model construction steps will represent the direct application of ARIMA model approach and are repeated for the remaining data cate- gories stated earlier. Fig. 3. ACF (left) and PACF (right) of Brent monthly in-sample data. Fig. 4. Model validation generated from R software environment. Forecasting Crude Oil Prices Using Wavelet ARIMA Model Approach541 3.2Wavelet ARIMA Combination Implementation For the proposed approach, the prices series will be decomposed fi rst using discrete wavelet transform. For decomposition phase, decomposition level three is used because the prices series can be described in a meaningful and more thorough way 20. As a result, the original prices series is decomposed and into one smooth series and three detail series as shown in Fig. 5. This subseries will serve as the input for building ARIMA model for each of them. the model building steps will be the same as explained in the previous sub section. After the most fi tted ARIMA model for each subseries has been validated, the next phase which is the reconstruction of the decomposed subseries is done. The forecasted outputs for the subseries are converted back into the original prices series by applying inverse DWT using the equation stated in the wavelet transform section earlier. These decomposition and reconstruction phases are done using MATLAB software. Finally, the MAE and RMSE for both approaches are calculated using the forecast result with the original price series for the out-of-sample data to evaluate their forecasting accuracy. 3.3Effectiveness Evaluation Result The results of MAE and RMSE for the proposed approach and the direct application of ARIMA model used as benchmark model are shown in the Table 1 below respectively. The smaller the value of the MAE and RMSE, the more accurate the forecasting will be. It can be observed that all the results in the table of proposed work are smaller than the results in the table of direct ARIMA application. All results for all categories in Fig. 5. Decomposed wavelets of one smooth series & three detail series. 542N.Q.N. Md-Khair and R. Samsudin both datasets show that Wavelet-ARIMA approach is better than direct ARIMA approach. Hence, in can be concluded that the proposed work can provide a better and more accurate forecasting result than the direct usage of the ARIMA model as expected. 4Conclusion This research proposes a hybrid approach in forecasting the crude oil spot price. The proposed approach includes the combination of wavelet transform and ARIMA model to produce a better forecasting accuracy results. The wavelet transform changes the behaviour of the price series from containing unstable variance with outliers into a set of constitutive series with a more stable variance and without outliers. Each of the constitutive series are then forecasted using ARIMA model and fi nally shifted back into the original price series using inverse wavelet transform to obtain the prediction of the original series. The public dataset of crude oil spot prices market from WTI and Brent are used in this particular study. The experiment done comparing the proposed approach with the direct ARIMA model approach as a benchmark model verifi ed the effectiveness of the proposed approach by producing a more accurate forecasting than the direct ARIMA model approach. Acknowledgments. The authors would like to express their deepest gratitude to Research Management Center (RMC), Universiti Teknologi Malaysia (UTM), Ministry of Higher Edu- cation Malaysia (MOHE) and Ministry of Science, Technology and Innovation (MOSTI) for their fi nancial support under Grant Vot 4F875. References 1. Azevedo, V.G., Campos, L.M.S.: Combination of forecasts for the price of crude oil on the spot market. Int. J. Prod. Res. 54, 52195235 (2016) Table 1. MAE & RMSE result of out-of-sample data. Model Data type Out-of-sample (Testing) data Mean Absolute Error (MAE) Root Mean Square Error (RMSE) W-ARIMA ARIMAW-ARIMA ARIMA BrentDaily0.1491810.878396 0.2005491.161596 Weekly0.3791321.696445 0.4844162.200306 Monthly0.8639113.923116 1.0820175.108475 WTIDaily0.1482531.011699 0.3059751.318813 Weekly0.3488171.621453 0.4375952.10147 Monthly0.7838214.109613 0.9633885.076284 Forecasting Crude Oil Prices Using Wavelet ARIMA Model Approach543 2. Shabri, A., Samsudin, R.: Daily crude oil price forecasting using hybridizing wavelet and artifi cial neural network model. Math. Probl. Eng. (2014) 3. Yusof, N.M., Rashid, R.S.A., Mohamed, Z.: Malaysia crude oil production estimation: an application of ARIMA model. In: 2010 International Conference on Science and Social Research (CSSR 2010), pp. 12551259 (2010) 4. Wang, G., Wu, J.: Crude oil price forecasting based on the ARIMA and BP neural network combinatorial algorithm. 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