In. Journal of Mah. Analysis, Vol. 8, 2014, no. 28, 1377-1387 HIKARI Ld, www.m-hikari.com hp://dx.doi.org/10.12988/ijma.2014.45139 A Hybrid Model for Improving Malaysian Gold Forecas Accuracy Maizah Hura Ahmad 1, Pung Yean Ping 2, Sii Roslindar Yazir 3 and Nor Hamizah Miswan 4 1,2 Deparmen of Mahemaical Sciences, Faculy of Science Universii Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia 3 Faculy of Indusrial Sciences & Technology, Universii Malaysia Pahang, Malaysia 4 Fakuli Teknologi Kejurueraan, Universii Teknikal Malaysia Melaka, Malaysia Copyrigh 2014 Maizah Hura Ahmad e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac A hybrid model has been considered an effecive way o improve forecas accuracy. This paper proposes he hybrid model of he linear auoregressive moving average (ARIMA) and he non-linear generalized auoregressive condiional heeroscedasiciy (GARCH) in modeling and forecasing. Malaysian gold price is used o presen he developmen of he hybrid model. The goodness of fi of he model is measured using Akaike informaion crieria (AIC) while he forecasing performance is assessed using bias, variance proporion, covariance proporion and mean absolue percenage error (MAPE). Keywords: ARIMA-GARCH, hybrid model, heeroscedasiciy, volailiy clusering
1378 Maizah Hura Ahmad e al. 1 Inroducion A popular precious meal for invesmen is gold. In Malaysia, one of he highes gold invesmen demand is for is own gold bullion coins called Kijang Emas. The coins which come in hree differen sizes of 1 oz, ½ oz and ¼ oz are mined by he Royal Min of Malaysia. The daily selling and buying prices of hese coins are imporan o invesors in order o make an invesmen decision. Auoregressive inegraed moving average (ARIMA) models have been used for forecasing differen ypes of ime series o capure he long erm rend. In he case of financial ime series ha have been shown o have volailiy clusering where large changes in he daa end o cluser ogeher and resuling in persisence of he ampliudes of he changes, ARCH based models have been used. In he conex of Malaysian gold, he selling price of he 1 oz coins was modelled and forecas using ARIMA and GARCH models [1] [2]. While he models produced a good fi of he daa wih he GARCH being more superior, a hybrid of hose wo models is proposed o be able o improve forecasing accuracy [3]. In he curren sudy, a seleced series of Malaysian gold is modelled and forecas using he hybrid of ARIMA-GARCH. Akaike informaion crierion (AIC) is used o assess he goodness of fi. Bias, variance proporion, covariance proporion and mean absolue percenage error (MAPE) are used o evaluae he forecasing performances. All analyses are carried ou using a sofware called E-views. The paper is organized ino 4 secions. Secion 2 presens he mehodology of he sudy. Secion 3 presens he daa analysis. The sudy is concluded in Secion 4. 2 Mehodology Hybrid ARIMA-GARCH Models ARIMA models are he mos general class of models for forecasing a ime series, applied in cases where daa show evidence of non-saionariy [4]. Non-saionariy in mean can be removed by ransformaions such as differencing, while non-saionary in variance can be removed by a proper variance sabilizing ransformaion inroduced by Box and Cox [3]. The ARIMA(p,d,q) can be wrien as d ( B)(1 B) y ( B) p q where p p ( B) 1 1B... pb is he auoregressive operaor of order p;
Hybrid model 1379 q q ( B) 1 1 B... qb is he moving average operaor of order q; (1 B) d is he d h difference; B is backward shif operaor; and is he error erm a ime. The orders are idenified hrough he auocorrelaion funcion (ACF) and he parial auocorrelaion funcion (PACF) of he sample daa. The error erms are generally assumed o be independen idenically disribued random variables (i.i.d.) sampled from a normal disribuion wih zero mean, ha is ~ N(0,σ 2 ) where σ 2 is he variance. A his poin, he model can be used for forecasing. However, some ime series errors do no saisfy he assumpion of common variance. The variances are ime-varying and condiional. The auoregressive condiional heeroskedasiciy (ARCH) class of models pioneered by Engle in 1982 and generalized by Bollerslev in 1986 are popular class of economeric models for describing a series wih ime-varying condiional variance [5]. The generalized auoregressive condiional heeroskedasiciy (GARCH) family models were developed o capure volailiy clusering or he periods of flucuaions, and predic volailiies in he fuure [6]. Pas variances and pas variance forecass are used o forecas fuure variances. The GARCH (p, q) model is where u 2 h, ~ N(0,1) where,, for saionariy; p is he order of he GARCH erms 2, which is he las period forecas variance. q is he order of he ARCH erms 2, which is he informaion abou volailiy from he previous period measured as he lag of squared residual from he mean equaion. Augmened Dickey-Fuller (ADF) ADF is one of he widely used uni-roo ess o deermine saionariy. The esing procedure is applied o he model where y is he esed ime series, indicaes he firs difference, k is he lag order of he auoregressive process. Rejecion of he null hypohesis implies ha he series is saionary.
1380 Maizah Hura Ahmad e al. Breusch-Godfrey Lagrange Muliplier Tes (BG-LM) BG-LM is a es for auocorrelaion. The null hypohesis saes ha here is no serial correlaion of any order up o a cerain order lag. ARCH Lagrange Muliplier Tes (ARCH-LM) ARCH-LM is used o es he presence of heerocedasiciy. Le be he residual series. The squared series, is used o check he presence of ARCH effecs where i is defined as follows, where p is he lengh of ARCH lags and is he residual of he series. Tes saisic for LM es is he usual F saisics for he squared residuals regression. Rejecion of he null hypohesis implies ha ARCH effec exiss. Akaike Informaion Crierion (AIC) AIC is used o assess he goodness of fi of a model. I is defined as AIC = 2k 2 ln (L) where L is he maximized value of he likelihood funcion for he esimaed model and k is he number of free and independen parameers in he model. Mean Absolue Percenage Error (MAPE) MAPE measures he accuracy of forecas in erms of percenage. The formula is as follows: n ˆ MAPE = y y / 100% n 1 y where is he acual value; is he forecas value; n is he number of periods. 3 Daa Analysis and Resuls The daa used in he sudy are daily selling prices of he 1 oz Malaysian gold recorded from 18 h July 2001 unil 15 h April 2014 as ploed in Figure 1. A oal of 2875 observaions from 18 h July 2001 unil 25 h Sep 2012 which accoun for 90% of he daa are used for modelling. Ou-sample forecass are produced for observaions in he period from 26 h Sep 2012 unil 15 h April 2014.
Hybrid model 1381 7,000 SELLING 6,000 5,000 4,000 3,000 2,000 1,000 500 1000 1500 2000 2500 3000 Figure 1: Daily 1 oz Malaysian Gold Prices from 18 h Jul 2001 o 15 h Sep 2014 An upward rend exiss in he gold price daa. Le {y } be he ime series of he daily gold price. The reurn on he h day is defined as r = ln(y ) ln(y -1 ). Figure 2 shows he plo of he reurns which appears o be saionary. Mos of he daa are locaed around he mean of zero. DLN_KE.16.12.08.04.00 -.04 -.08 500 1000 1500 2000 2500 Figure 2: Plo of he Reurns The saionariy of he reurns is confirmed by he ADF uni-roo es as illusraed in Table 1. Based on he able, he null hypohesis ha he reurns are non-saionary is rejeced. Table 1: Uni Roo Tes of he Reurns Null Hypohesis: DLN has a uni roo Exogenous: None Lag Lengh: 0 (Auomaic based on SIC, MAXLAG=27) -Saisic Prob.* Augmened Dickey-Fuller es saisic -56.48802 0.0001 Tes criical values: 1% level -2.565767 5% level -1.940934 10% level -1.616625 *MacKinnon (1996) one-sided p-values.
1382 Maizah Hura Ahmad e al. Using ordinary leas squares mehod o esimae he parameers, he mos appropriae ARIMA model for his series is ARIMA(1,1,1) wih an AIC value of 10.08545 and MAPE value for in-sample forecas of 0.812356 [7]. The model is checked for serial correlaion using Breusch-Godfrey Serial Correlaion LM Tes. The resuls are shown in Table 2, which indicae ha wih significance level of 5%, he developed model does no suffer from serial correlaion up o lag 5. Table 2: Breusch-Godfrey Serial Correlaion LM Tes F-saisic 2.918448 Prob. F(5,2865) 0.0124 Obs*R-squared 14.55872 Prob. Chi-Square(5) 0.0124 The descripive saisics and plo of he residuals of he model are presened in Figure 3 and Figure 4 respecively. As presened in Figure 3, he residuals have excess kurosis and a mean which is very close o zero. From he Jarque-Bera saisic, he null hypohesis of residuals following he normal disribuion is rejeced. 1,200 1,000 800 600 400 200 0-200 -100 0 100 200 300 Series: Residuals Sample 3 2875 Observaions 2873 Mean -0.007668 Median -1.118173 Maximum 370.6464 Minimum -247.5455 Sd. Dev. 37.44663 Skewness 0.168772 Kurosis 15.00957 Jarque-Bera 17279.13 Probabiliy 0.000000 Figure 3: Descripive Saisics of he Residuals for ARIMA(1, 1, 1) 400 300 200 100 0-100 -200-300 500 1000 1500 2000 2500 Figure 4: Volailiy Cluserings D(SELLING) in he Residuals for ARIMA(1, 1, 1)
Hybrid model 1383 As ploed in Figure 4, here are volailiy clusering in he residuals. The residuals of he ARIMA are esed for ARCH effecs using he ARCH- LM es. The resuls are presened in Table 3. From he able, wih significance level of 5%, he null hypohesis of ARCH effecs do no exis is rejeced. Table 3: Heeroskedasiciy Tes for ARIMA(1, 1, 1) F-saisic 120.8169 Prob. F(1,2870) 0.0000 Obs*R-squared 116.0172 Prob. Chi-Square(1) 0.0000 Thus, alhough he hypohesis of no serial correlaion in he model is no rejeced, he presence of volailiy clusering in he residuals and he resuls of he ARCH LM es show ha he model is no a good fi. Hence, i is necessary o develop a beer model for Malaysian gold price. A GARCH model is proposed o handle heeroscedasiciy in he series. Table 4 presens he esimaion resuls for he hybrid ARIMA (1, 1, 1)-GARCH(2, 1) model as applied o he Malaysian gold price. Table 4: Esimaion Resul for ARIMA(1, 1, 1)-GARCH (2, 1) Variance Equaion C 1.744212 0.487440 3.578313 0.0003 RESID(-1)^2 0.104766 0.022469 4.662635 0.0000 RESID(-2)^2 0.048091 0.023100-2.081865 0.0374 GARCH(-1) 0.940387 0.006248 150.5098 0.0000 R-squared 0.003390 Mean dependen var 1.624782 Adjused R-squared 0.002695 S.D. dependen var 37.52772 S.E. of regression 37.47711 Akaike info crierion 9.285385 Sum squared resid 4031012. Schwarz crierion 9.299914 Log likelihood -13331.46 Hannan-Quinn crier. 9.290622 F-saisic 1.627001 Durbin-Wason sa 1.995463 Prob(F-saisic) 0.135481 In Table 4, wih significance level of 5%, boh he ARCH and GARCH effecs are significan. They are he inernal causes of volailiy in he residuals. The AIC value of he hybrid model is 9.299914 wih MAPE value for in-sample forecas of 0.808684. The residuals of he ARIMA-GARCH are esed for ARCH effecs using he ARCH- LM es. The resuls are presened in Table 5. Table 5: Heeroskedasiciy Tes for ARIMA (1, 1, 1)-GARCH (2, 1) F-saisic 0.005485 Prob. F(1,2870) 0.9410 Obs*R-squared 0.005488 Prob. Chi-Square(1) 0.9409
1384 Maizah Hura Ahmad e al. The resuls in Table 5 indicae ha a significance level of 5%, he null hypohesis of no ARCH effecs canno be rejeced. The hybrid model is hen esed for serial correlaion as presened in Table 6. Table 6: Ljung-Box Q-saisics on squared residuals for ARIMA(1,1,1)-GARCH(2,1) lags AC PAC Q-Sa Prob lags AC PAC Q-Sa Prob 1 0.001 0.001 0.0055 19-0.002-0.002 9.8829 0.908 2-0.019-0.019 1.0609 20 0.010 0.010 10.195 0.925 3 0.019 0.020 2.1524 0.142 21-0.007-0.007 10.349 0.944 4-0.007-0.008 2.3092 0.315 22 0.003 0.003 10.372 0.961 5 0.000 0.001 2.3099 0.511 23 0.011 0.010 10.733 0.968 6 0.026 0.026 4.3112 0.366 24 0.011 0.011 11.058 0.974 7-0.019-0.019 5.3550 0.374 25-0.029-0.030 13.484 0.941 8-0.009-0.008 5.6073 0.469 26 0.042 0.041 18.554 0.775 9 0.009 0.007 5.8215 0.561 27-0.010-0.012 18.868 0.803 10-0.017-0.016 6.6147 0.579 28-0.010-0.009 19.186 0.828 11-0.004-0.004 6.6683 0.672 29-0.018-0.021 20.089 0.827 12 0.003 0.001 6.6919 0.754 30-0.016-0.016 20.845 0.832 13-0.017-0.016 7.5387 0.754 31-0.017-0.017 21.711 0.832 14-0.009-0.009 7.7595 0.804 32-0.007-0.011 21.841 0.860 15-0.023-0.024 9.2705 0.752 33 0.008 0.009 22.030 0.882 16-0.006-0.004 9.3642 0.807 34-0.015-0.014 22.675 0.888 17-0.008-0.009 9.5398 0.848 35 0.003 0.003 22.707 0.911 18-0.011-0.011 9.8662 0.874 36-0.019-0.019 23.724 0.906 Based on he resuls in Table 6, he null hypohesis of no serial correlaion canno be rejeced. The descripive saisics of he residuals from he hybrid model are presened in Figure 5. 800 700 600 500 400 300 200 100 0-6 -4-2 0 2 4 6 8 Series: Sandardized Residuals Sample 3 2875 Observaions 2873 Mean 0.030987 Median -0.004199 Maximum 8.143539 Minimum -6.547355 Sd. Dev. 1.054654 Skewness 0.317749 Kurosis 7.479680 Jarque-Bera 2450.596 Probabiliy 0.000000 Figure 5: Descripive Saisics of he Residuals for ARIMA(1, 1, 1)-GARCH(2, 1)
Hybrid model 1385 From he Jarque-Bera saisic in Figure 5, he residuals are no normally disribued. However, he hybrid model is used for forecasing. The resuls of ou-sample forecasing are presened in Figure 6. 6,000 5,500 5,000 4,500 4,000 3,500 2900 2950 3000 3050 3100 3150 Forecas: SELLINGF Acual: SELLING Forecas sample: 2876 3194 Included observaions: 319 Roo Mean Squared Error 59.18209 Mean Absolue Error 39.06320 Mean Abs. Percen Error 0.827670 Theil Inequaliy Coefficien 0.006023 Bias Proporion 0.020026 Variance Proporion 0.000061 Covariance Proporion 0.979913 SELLINGF ± 2 S.E. 40,000 30,000 20,000 10,000 0 2900 2950 3000 3050 3100 3150 Forecas of Variance Figure 6: Forecasing Resuls of ARIMA (1, 1, 1)-GARCH (2, 1) For comparison purposes, he ou-sample forecass for ARIMA(1, 1, 1) are ploed in Figure 7. 6,000 5,600 5,200 4,800 4,400 Forecas: SELLINGF Acual: SELLING Forecas sample: 2876 3194 Included observaions: 319 Roo Mean Squared Error 59.59525 Mean Absolue Error 39.19383 Mean Abs. Percen Error 0.830665 Theil Inequaliy Coefficien 0.006063 Bias Proporion 0.034607 Variance Proporion 0.000627 Covariance Proporion 0.964766 4,000 3,600 2900 2950 3000 3050 3100 3150 SELLINGF ± 2 S.E. Figure 7: Ou-sample Forecass of ARIMA (1, 1, 1)
1386 Maizah Hura Ahmad e al. Conclusion Table 7 presens some resuls of modelling and forecasing of he daily prices of 1 oz Malaysian gold recorded from 18 h July 2001 unil 15 h April 2014. Two models were used, namely ARIMA and ARIMA-GARCH. Table 7: Modelling and Forecasing Resuls Models ARIMA ARIMA-GARCH AIC 10.08545 9.299914 MAPE of in-sample 0.812356 0.808684 MAPE of ou-sample 0.830665 0.827670 Bias Proporion 0.034607 0.020026 Variance Proporion 0.000627 0.000061 Covariance Proporion 0.964766 0.979913 Some informaion will always be los due o using one of he candidae models. Based on he AIC values, he model ha minimizes he esimaed informaion loss more is ARIMA-GARCH. ARIMA is 0.46 imes as probable as ARIMA-GARCH o minimize he informaion loss. The bias proporion, he variance proporion, and he covariance proporion sum up o 1. While he bias proporion measures how far he mean of he forecas is from he mean of he acual series, he variance proporion measures how far he variaion of he forecas is from he variaion of he acual series. The remaining unsysemaic forecasing errors are measured by he covariance proporion measures. Based on Table 7, he forecass produced by ARIMA-GARCH are beer since he bias and variance proporions are lower han hose produced by ARIMA. Furhermore, MAPE values for in-sample and ou-sample forecass for ARIMA-GARCH are lower han hose for ARIMA. I can be concluded ha in he case of he selling prices of 1 oz Malaysian gold, he hybrid model of ARIMA-GARCH can be an effecive way o improve forecasing accuracy achieved by using ARIMA only. Acknowledgemen This work was suppored by RUG Vo No: Q.J130000.2526.08H46. The auhors would like o hank Universii Teknologi Malaysia (UTM) for providing he funds and faciliies. References [1] Pung Yean Ping, Nor Hamizah Miswan and Maizah Hura Ahmad, Forecasing Malaysian Gold Using GARCH Model, Applied Mahemaical Sciences, 7 (58), 2013, 2879-2884.
Hybrid model 1387 [2] Maizah Hura Ahmad and Pung Yean Ping, Modelling Malaysian Gold Using Symmeric and Asymmeric GARCH Models, Applied Mahemaical Sciences, 8 (17), 2014, 817-822. [3] S.R. Yaziz, N.A. Azizan, R. Zakaria and M.H. Ahmad, The Performance of Hybrid ARIMA GARCH Modeling, In: 20h Inernaional Congress on Modelling & Simulaion 2013 (MODSIM2013), 1-6 December 2013, Adelaide, Ausralia. [4] G.E.P. Box, G.M. Jenkins, Time Series Analysis, Forecasing and Conrol, Holden-Day, San Francisco, 1970. [5] R. F. Engle, An Inroducion o he Use of ARCH/GARCH models in Applied Economerics, Journal of Business, New York (1982). [6] T. Bollerslev, Generalized Auorregressive Condiional Heeroskedasiciy, Journal of Economerics, 31 (1986), 307-327. [7] Nor Hamizah Miswan, Pung Yean Ping and Maizah Hura Ahmad, On Parameer Esimaion for Malaysian Gold Prices Modelling and Forecasing, Inernaional Journal of Mah Analysis, 7(22), 2013, 1059-1068. Received: May 15, 2014