Forecasting Malaysian Gold Using. a Hybrid of ARIMA and GJR-GARCH Models

Transcription

1 Applied Mahemaical Sciences, Vol. 9, 15, no. 3, HIKARI Ld, hp://dx.doi.org/1.1988/ams Forecasing Malaysian Gold Using a Hybrid of ARIMA and GJR-GARCH Models Maizah Hura Ahmad 1, Pung Yean Ping, Sii Roslindar Yaziz 3 and Nor Hamizah Miswan 4 1, Deparmen of Mahemaical Sciences, Faculy of Science Universii Teknologi Malaysia, 8131 UTM Skudai, Johor, Malaysia 3 Faculy of Indusrial Sciences & Technology, Universii Malaysia Pahang, Malaysia 4 Fakuli Teknologi Kejurueraan, Universii Teknikal Malaysia Melaka, Malaysia Copyrigh 15 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 An effecive way o improve forecas accuracy is o use a hybrid model. This paper proposes a hybrid model of linear auoregressive moving average (ARIMA) and non-linear GJR-GARCH model also known as TARCH in modeling and forecasing Malaysian gold. The goodness of fi of he model is measured using Akaike informaion crieria (AIC) while he forecasing performance is assessed using mean absolue percenage error (MAPE), bias proporion, variance proporion and covariance proporion. Keywords: ARIMA-GJR, TARCH, hybrid model, heeroscedasiciy, volailiy clusering 1 Inroducion Malaysian gold bullion coins called Kijang Emas are legal ender coins whose marke price depends on heir gold conen. The price depends on he prevailing inernaional gold price. They are invesmen coins where he daily selling and buying prices of hese coins are imporan o invesors in order o make an invesmen decision.

2 149 Maizah Hura Ahmad e al. For forecasing purposes, Auoregressive Inegraed Moving Average (ARIMA) models have been widely used o capure he long erm rend in a ime series. In ime series where volailiy clusering, he siuaion when large changes in he daa end o cluser ogeher and resuling in persisence of he ampliudes of he changes are prevalen, ARCH based models have been used. In he case of Malaysian gold prices, a hybrid model was considered an effecive way o improve forecas accuracy [1]. ARIMA-GARCH model was developed and i ouperformed ARIMA model. However, in he sudy of symmeric and asymmeric Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) models for forecasing Malaysian gold prices, a varian of GARCH, called TGARCH was shown o ouperform GARCH, GARCH-M and EGARCH models []. This paper proposes a hybrid of linear auoregressive moving average (ARIMA) and a varian of non-linear generalized auoregressive condiional heeroscedasiciy (GARCH) called GJR-GARCH in modeling and forecasing Malaysian gold price. In his sudy, he goodness of fi of he model is measured using Akaike informaion crieria (AIC) while he forecasing performance is assessed using mean absolue percenage error (MAPE), bias proporion, variance proporion and covariance proporion. All analyses are carried ou using a sofware called E-views. In he nex secion, he mehodology of he sudy is presened. This is followed by daa analysis in Secion 3. The sudy is concluded in Secion 4. Mehodology Hybrid ARIMA-GJR Models Box and Jenkins developed a general class of models called ARIMA for forecasing non-saionary ime series [3]. Non-saionariy exiss in mean and/or in variance. To remove non-saionariy in mean, ransformaions such as differencing can be applied. Non-saionary in variance on he oher hand, can be removed by a proper variance sabilizing ransformaion inroduced by Box and Cox [4]. The ARIMA (p,d,q) can be wrien as d p ( B)(1 B) y q ( B) where p p ( B) 11B... pb is he auoregressive operaor of order p; q q ( B) 11 B... qb is he moving average operaor of order q; (1B) d is he d h difference; B is backward shif operaor; and is he error erm a ime. ) Using a sample daa, he orders are idenified hrough he auocorrelaion funcion (ACF) and he parial auocorrelaion funcion (PACF). The error erms are generally

3 Forecasing Malaysian gold using a hybrid 1493 assumed o be independen idenically disribued random variables (i.i.d.) sampled from a normal disribuion wih zero mean, ~ N(,σ ) where σ is he variance. A his poin, he model can be used for forecasing. No all ime series errors saisfy he assumpion of common variance. Someimes, he variances are ime-varying and condiional. Engle in 198 developed auoregressive condiional heeroskedasiciy (ARCH) class of models o describe a series wih ime-varying condiional variance. These models were generalized by Bollerslev in 1986 and are known as GARCH models [5]. The GARCH models are able o capure volailiy clusering or he periods of flucuaions, and predic volailiies in he fuure [6]. In he GARCH model, pas variances and pas variance forecass are used o forecas fuure variances. The sandard GARCH model is symmeric in response o pas volailiy and variance. The GARCH (p, q) model is y where y = ime series daa; u h, ~ N(,1) h h 1 h.. q p i 1 i j i1 j 1 where 1 1, h, for saionariy, i, j The GARCH erm is, where he las period forecas variance is of order p, The ARCH erm is, which is he informaion abou volailiy from he previous period measured as he lag of squared residual from he mean equaion. I is of order q. Good news and bad news have differen effecs on volailiy [7]. Beween good and bad, bad news is said o have more effec on fuure volailiy of reurns. When his happens, symmeric GARCH models are unable o capure he asymmery of volailiy response. A characerisic of asymmeric volailiy is leverage effec. Leverage effec is asymmery in volailiy induced by big posiive and negaive asse reurns. Asymmeric GARCH models are able o explain he leverage effecs by enabling condiional variance o respond asymmerically o rises and falls in volailiy reurns. A model ha reas posiive and negaive news symmerically as proposed by Glosen, Jagannahan and Runkle is Glosen-Jagannahan-Runkle GARCH (GJR-GARCH) which is also known as TARCH [8]. Wih posiive or good news, -i < and wih negaive or bad news, -i >. TARCH can capure he phenomenon of posiive news hiing on he financial marke wih he marke being in a calm period; and he negaive news hiing on he financial marke wih he marke enering ino a flucuaing period and high volailiy. The model is as follows: where h h q p i 1 1d1 jh i 1 j 1, is he leverage erm and i, j j and are consan

4 1494 Maizah Hura Ahmad e al. parameers. d is an indicaor imiaion variable where d 1, (bad news), here is a leverage effec, (good news), vice versa The GJR (p,q) model has p GARCH coefficiens associaed wih lagged variances, q ARCH coefficiens associaed wih lagged squared innovaions, and q leverage coefficiens associaed wih he square of negaive lagged innovaions. Augmened Dickey-Fuller (ADF) A uni-roo es called ADF can be used o deermine saionariy of a ime series. The null hypohesis saes ha he series is non-saionary. The esing procedure is applied o he model y y k y 1 i 1 i 1 where y = he esed ime series, = he firs difference, k = he lag order of he auoregressive process and y are he series residual. Akaike Informaion Crierion (AIC) The goodness of fi of a model can be assessed using AIC = k ln (L), where L = he maximized value of he likelihood funcion for he esimaed model and k = he number of free and independen parameers in he model. Breusch-Godfrey Lagrange Muliplier Tes (BG-LM) Auocorrelaion is esed using BG-LM es. Rejecion of he null hypohesis sae ha here exiss serial correlaion of any order up o a cerain order lag. ARCH Lagrange Muliplier Tes (ARCH-LM) The presence of heerocedasiciy is deermined by using ARCH-LM es. The squared series, defined as p p is used o check he presence of ARCH effecs 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. The null hypohesis saes ha ARCH effecs do no exis. Jarque-Bera Tes The Jarque Bera es is a es of wheher sample daa have he skewness and kurosis maching a normal disribuion. The null hypohesis saes ha he sample daa follows a normal disribuion. The es saisic is defined as n JB S K 3 where n = he number of observaions, S = he sample skewness and K = he sample kurosis.

5 Forecasing Malaysian gold using a hybrid 1495 Mean Absolue Percenage Error (MAPE) The accuracy of forecass (measured in erms of percenage) is measured using MAPE wih he following formula: n ˆ MAPE = y y / 1% n 1 y where y = he acual value, ŷ = he forecas value and n = he number of periods. 3 Daa Analysis and Resuls The daily selling prices of 1 oz Malaysian gold recorded from 3 January 11 unil January 15 were used. The daa are ploed in Figure 1. 6,4 Selling Price 6, 5,6 5, 4,8 4,4 4, Figure 1: Daily 1 oz Malaysian Gold Prices from 3 Jan 11 o Jan 15 Reurns were used since a downward rend exiss in he daa. The reurn on he h day is defined as r = ln(y)ln(y-1). The saionariy of he reurns was confirmed by using ADF uni-roo es. Niney percen of he observaions, ha is from 3 January 11 unil Augus 14 which accoun for 9% of he daa were used for modeling o obain an ARIMA model. Using ordinary leas squares mehod o esimae he parameers, an appropriae ARIMA model for his series is ARIMA (, 1, ) wih an AIC value of When he model was used for forecasing, he MAPE value for in-sample forecas is Ou-sample forecass were produced for observaions in he period from 1 Augus 14 unil January 15 wih MAPE value of Breusch-Godfrey Serial Correlaion LM Tes was performed on ARIMA (, 1, ) and he model was confirmed o no suffer from serial correlaion as illusraed in Table 1.

6 1496 Maizah Hura Ahmad e al. Table 1: Breusch-Godfrey Serial Correlaion LM Tes F-saisic.335 Prob. F(, 887).8169 Obs*R-squared Prob. Chi-Square().8156 Figure presens he descripive saisics of he residuals where he mean of he residuals is close o zero and he residuals have excess kurosis. Based on he Jarque-Bera saisic, he null hypohesis of residuals following he normal disribuion is rejeced Series: Residuals Sample Observaions 894 Mean.19 Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy. Figure : Descripive Saisics of he Residuals for ARIMA(, 1, ) Figure 3 presens he plo of he residuals where here exiss clear volailiy clusering in he residuals D(SELLING_PRICE) Residuals Figure 3: Volailiy Cluserings in he Residuals for ARIMA(, 1, ) Using ARCH-LM es, ARIMA(, 1, ) residuals were esed for ARCH effecs. The resuls as presened in Table indicae ha a 5% significance level, he null hypohesis of ARCH effecs do no exis is rejeced. Table : Heeroskedasiciy Tes for ARIMA(, 1, ) F-saisic Prob. F(1,891). Obs*R-squared Prob. Chi-Square(1).

7 Forecasing Malaysian gold using a hybrid 1497 Based on he presence of volailiy clusering in he residuals and he ARCH- LM es resul, i can be concluded ha he model was no a good fi. A beer model for forecasing Malaysian gold was deemed necessary. A hybrid model was considered an effecive way o improve forecas accuracy [1]. In he sudy of symmeric and asymmeric Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) models for forecasing Malaysian gold prices, a varian of GARCH, called TGARCH was shown o ouperform GARCH, GARCH-M and EGARCH models []. The TGARCH model is a GARCH varian ha includes leverage erms for modeling asymmeric volailiy clusering. Hence, he curren sudy proposes using ARIMA-GRJ model o analyze he series undersudied. Table 3 presens he esimaion resuls for variance equaion of he hybrid ARIMA (, 1, )-GJR (1, 1) model as applied o Malaysian gold. Table 3: Esimaion Resuls for Variance Equaion of ARIMA (, 1, )-GJR (1, 1) Variance Equaion C RESID(-1)^ RESID(-1)^*(RESID(-1)<) GARCH(-1) R-squared.1391 Mean dependen var Adjused R-squared.5938 S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood Hannan-Quinn crier F-saisic Durbin-Wason sa Prob(F-saisic) In Table 3, since he coefficien of RESID (-1)^*(RESID(-1)<) is posiive and significan, we can conclude ha he model has leverage effecs. This means ha bad news can have more impac on he condiional variance han good news. The AIC value of he model is The residuals of he model are esed for ARCH effecs using ARCH-LM es, wih he resuls presened in Table 4. Table 4: Heeroskedasiciy Tes for ARIMA (, 1, )-GJR (1, 1) F-saisic.919 Prob. F(1,891).767 Obs*R-squared.9145 Prob. Chi-Square(1).764 Based on Table 4, 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 5. From he resuls in Table 5, he null hypohesis of no serial correlaion canno be rejeced.

8 1498 Maizah Hura Ahmad e al. Table 5: Ljung-Box Q-saisics on squared residuals for ARIMA(,1,)-GJR(1,1) lags AC PAC Q-Sa Prob lags AC PAC Q-Sa Prob The descripive saisics of he residuals from ARIMA-GJR model are presened in Figure Series: Sandardized Residuals Sample Observaions 894 Mean Median -.9 Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy. Figure 4: Descripive Saisics of he Residuals for ARIMA (, 1, )-GJR (1, 1) The residuals are no normally disribued as implied by Jarque-Bera saisic in Figure 4. However, he hybrid model is used for forecasing. The resuls of in-sample and ou-sample forecasing are presened in Figure 5 and Figure 6 respecively.

9 Forecasing Malaysian gold using a hybrid ,5 6, 5,5 5, 4,5 4, 3, Forecas: SELLING_PRF Acual: SELLING_PRICE Forecas sample: Adjused sample: Included observaions: 894 Roo Mean Squared Error Mean Absolue Error Mean Abs. Percen Error Theil Inequaliy Coefficien.556 Bias Proporion.37 Variance Proporion.8 Covariance Proporion SELLING_PRF ± S.E. 8, 6, 4,, Forecas of Variance Figure 5: In-Sample Forecasing Resuls of ARIMA (, 1, )-GJR (1, 1) 5, 4,8 4,6 4,4 4, 4, 3, Forecas: SELLING_PRF Acual: SELLING_PRICE Forecas sample: Included observaions: 1 Roo Mean Squared Error Mean Absolue Error Mean Abs. Percen Error Theil Inequaliy Coefficien.51 Bias Proporion.1179 Variance Proporion.3719 Covariance Proporion SELLING_PRF ± S.E. 1, 8, 6, 4,, Forecas of Variance 4 Conclusion Figure 6: Ou-Sample Forecasing Resuls of ARIMA (, 1, )-GJR (1, 1) The resuls of modelling and forecasing of 1 oz Malaysian gold daily prices recorded from 3 January 11 unil January 15 using ARIMA-GJR are abulaed

10 15 Maizah Hura Ahmad e al. in Table 5. The resuls are compared wih he resuls obained by using ARIMA model. Table 5: Modelling and Forecasing Resuls Models ARIMA ARIMA-GJR AIC MAPE of in-sample MAPE of ou-sample Bias Proporion of in-sample..37 Variance Proporion of in-sample.13.8 Covariance Proporion of in-sample Bias Proporion of ou-sample Variance Proporion of ou-sample Covariance Proporion of ou-sample Based on AIC values, ARIMA-GJR is a beer model. In erms of forecasing, MAPE of boh in-sample and ou-sample using ARIMA-GJR are lower han using ARIMA only. There are no much differences in bias proporion which measures how far he mean of he forecas is from he mean of he acual series and in he variance proporion which measures how far he variaion of he forecas is from he variaion of he acual series. There is also no much difference in he remaining unsysemaic forecasing errors as measured by covariance proporion. However, i can be concluded ha a hybrid model of ARIMA-GJR is a beer forecasing model since even hough he residuals do no follow a normal disribuion, he model does no suffer from serial correlaion and here are no ARCH effecs. Acknowledgemens. This work was suppored by RUG Vo No: Q.J H46. The auhors would like o hank Universii Teknologi Malaysia (UTM) for providing he funds and faciliies. References [1] Maizah Hura Ahmad, Pung Yean Ping, Sii Roslindar Yaziz and Nor Hamizah Miswan, A Hybrid Model for Improving Malaysian Gold Forecas Accuracy, Inernaional Journal of Mahemaical Analysis, 8 (8), 14, hp://dx.doi.org/1.1988/ijma [] Maizah Hura Ahmad and Pung Yean Ping, Modelling Malaysian Gold Using Symmeric and Asymmeric GARCH Models, Applied Mahemaical Sciences, 8 (17), 14, hp://dx.doi.org/1.1988/ams [3] G.E.P. Box, G.M. Jenkins, Time Series Analysis, Forecasing and Conrol,

11 Forecasing Malaysian gold using a hybrid 151 Holden-Day, San Francisco, 197. hp://dx.doi.org/1.37/3855 [4] S.R. Yaziz, N.A. Azizan, R. Zakaria and M.H. Ahmad. The Performance of Hybrid ARIMA GARCH Modeling, In: h Inernaional Congress on Modelling & Simulaion 13 (MODSIM13), 1-6 December 13, Adelaide, Ausralia. [5] R. F. Engle, An Inroducion o he Use of ARCH/GARCH models in Applied Economerics, Journal of Business, New York (198). [6] T. Bollerslev, Generalized Auorregressive Condiional Heeroskedasiciy, Journal of Economerics, 31 (1986), hp://dx.doi.org/1.116/34-476(86)963-1 [7] F. Black, Sudies of Sock Price Volailiy Changes, Proceedings of he Business and Economics Secion of he American Saisical Associaion, 1976, [8] L. R. Glosen, R. Jagannahan and D. E. Runkle, On he Relaion beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks, The Journal of Finance, 48 (5), 1993, hp://dx.doi.org/1.37/3967 Received: January 1, 15; Published: February 5, 15