Modelling and Forecasting Volatility of Gold Price with Other Precious Metals Prices by Univariate GARCH Models

Transcription

1 Deparmen of Saisics Maser's Thesis Modelling and Forecasing Volailiy of Gold Price wih Oher Precious Meals Prices by Univariae GARCH Models Yuchen Du 1 Supervisor: Lars Forsberg 1 Yuchen.Du.84@suden.uu.se

2 Modelling and Forecasing Volailiy of Gold Price wih Oher Precious Meals Prices by Univeriae GARCH Models Yuchen Du, June, 1 Absrac This paper aims o model and forecas he volailiy of gold price wih he help of oher precious meals. The daa applied for applicaion par in he aricle involves hree financial ime series which are gold, silver and plainum daily spo prices. The volailiy is modeled by univariae Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) models including GARCH and EGARCH wih differen disribuions such as normal disribuion and suden- disribuion. A he same ime, comparisons of esimaion and forecasing he volailiy beween GARCH family models have been done. Keywords: gold price, volailiy, precious meal, GARCH, EGARCH, volailiy forecasing

3 CONTENT 1.Inroducion Background Brief Inroducion o he mehodology....mehodology Sandard GARCH Model Exponenial GARCH (EGARCH) Model Univariae GARCH Models wih error erms Disribuions of Error Terms Roo Mean Square Error (RMSE) Daa Reurns Plos Daa Analysis Esimaion GARCH Models EGARCH Models Forecas Conclusion Reference... Appendix... 1

4 1.Inroducion 1.1 Background During he end of he World War II, in 1944, fory-four counries around he world signed he well-known Breon Woods Agreemen o esablish a new currency sysem based on gold. They specified he value of he US Dollar is conneced o gold and oher currencies are conneced o he US Dollar. This sysem indicaed where gold sanding, no only jus as an inves good. This meal has been he only commodiy ha served as currency in inernaional financial business for cenuries. Gold is so ideal ha a number of currencies were based on he value of i. For residens, i is an excellen invesmen good since i could proec hem from asses devaluaion. For a counry, i is an imporan reserve asse; he siuaion is somewha he same as he oher precious meals. The bes-known precious meals are gold and silver. While boh have indusrial uses, hey are beer known for heir usage in ar, jewellery as well. Oher precious meals include he plainum groupmeals: palladium, osmium, iridium, and plainum which is he mos widely raded one in his group. Hisorically, precious meals have commanded much higher prices han common indusrial meals, like copper and iron. During he pas few years, especially afer he chaos of he US dollar sysem caused by he financial crisis, invesors who have been able o maser he reasonable invesmen of he precious meals markes have goen remarkable reurns. These meals could play an essenial role in a porfolio because of heir abiliy o ac as a sore of value and heir sabiliy agains inflaion as well. Therefore, he modeling and forecasing of he volailiy of gold price is indispensable. ''Precious Meal", hp:// See more in <hp://en.wikipedia.org/wiki/precious_meal.> 1

5 Wih he common sense ha higher profi comes wih higher risk, modeling volailiy in asse reurns is concerned by financial invesors. Basically, volailiy is widely-used for measuring asses risk; invesors wan guaranees for invesing in risky asses for heir invesmen decision. The demand of assising invesors o handle he volailiy of commodiies prices need o be supplied by some creaive and relaive financial insrumens. 1. Brief Inroducion o he mehodology Alhough i is an urgen need of modelling volailiy, here were sill some limiaions and difficulies. As common financial ime series, he usual commodiy spo price daa always have properies like volailiy clusering, fa-ailness, excess kurosis and skewness. The widely-used classical linear regression model usually considers he residuals have homoscedasiciy which means random disurbances have equal variance. Tha is he exac difficuly using radiional linear ools. "Models of Auoregressive Condiional Heeroskedasiciy (ARCH) form he mos popular way of parameerizing his dependence", Teräsvira (6), p.. The auoregressive condiional heeroskedasiciy (ARCH) which is he firs model of condiional heeroskedasiciy was proposed by Engle (198). Then he generalized auoregressive condiional heeroskedasiciy (GARCH) model was proposed by Bollerslev and Taylor (1986) separaely. Since i was brough ino use, GARCH model shows beer capabiliy of financial ime series analysis in some empirical sudies. Thereafer, many GARCH family models came up, such as exponenial GARCH which was brough up by Nelson (1991), Asymmeric Power ARCH which was carried ou by Ding, Z., Engle, R.F. and Granger, C.W.J., (1993). Among he worldwide inroduced GARCH family models, Teräsvira (6) hough ha he overwhelmingly mos popular GARCH model in applicaions has been he GARCH (1,1) model.

6 . Mehodology Since Engle (198) inroduced Auoregressive Condiional Heeroskedasiciy which allows he condiional variance o change over ime as a funcion of pas errors and Bollerslev (1986) generalized i which allows for a much more flexible lag srucure. Nelson (1991) brough ou a new model wih a new form of heeroskedasiciy, called Exponenial GARCH (EGARCH)..1 Sandard GARCH Model Le {r } denoe a financial ime series, presening reurns in his paper. r E ( r 1 ) Here, 1 is an informaion se conained all he informaion hrough ime period 1, following Bollerslev (1986). We can see here he condiional mean and variance boh change wih 1. The funcion of he condiional mean ( ), E( r 1 ) ( ) r 1 1 pr p 1 1 q q which can also be considered as an AuoRegressive MovingAverage (ARMA)(p,q) process. Then an AuoRegressive Condiional Heeroskedasiciy - ARCH model proposed by Engle (198) can be regarded as he variance funcion, Var(r 1) E( 1) h ( ) Here, here is an assumpion ha he condiional disribuion of is supposed 3

7 o be normal disribuion wih zero mean and variance equals o h. The ARCH(q) process funcion form is like his: h 1 ~ N(, h ) q i1 i i The Generalized ARCH (GARCH) process inroduced by Bollerslev (1986) is given below wih he same assumpion. h 1 ~ N(, h ) q i1 i i p j1 h j j where p, q,, i, i 1,..., q, i, j 1,..., p p We can call h j j par as GARCH erms. For p= which means here is no j1 GARCH erms, he process becomes an ARCH(q) process; for p=q=, is simply whie noise. As Teräsvira said, usually he simples GARCH(1,1) model works very well: -1 ~ N(,h ) h 1 h 1 wih,,... Exponenial GARCH (EGARCH) Model The Exponenial GARCH(EGARCH) Model was inroduced by Nelson (1991) in order o model asymmeric variance effecs. I has some special properies which 4

8 makes i especially efficien in an asse pricing conex. The basic frame for he mean and variance is similar o sandard GARCH model, he condiional variance funcion par is very unique. Given he EGARCH(p,q) process, wih similariy of he concepion in sandard GARCH, r E ( r 1 ) 1 ~ N(, h ) ln h p q jg j( Z j) j j1 j1 ln h j where g( Z j) Z j ( Z j E( Z j )), j 1,..., q and Z could be a sandard normal variable. The par relaive o Z allows he sign and he magniude effecs affec he volailiy separaely which is he very propery making EGARCH may be more suiable for price reurn daa. By a more specific way, an EGARCH (p,q) process can also be presened like his, 1 ~ N(, h ) ln( h ) p j1 ln( h j j ) q i1 i / h E( / h ) ( / h ) i i i i i i.3 Univariae GARCH Models wih exra error erms In his paper, he aim is o model and forecas volailiy of gold price wih he help of oher precious meals, i is proper o rea he silver and plainum error erms as regressors. The unique process is similar o generalized linear model in his paper. Since we consider hese hree kind of precious meals, price daa are correlaed wih 5

9 each oher, we could add he erms we choose of he oher wo meals o he original GARCH (1,1) Model based on is populariy and applicabiliy. The new univariae GARCH Model wih exra error erms behaves like his, r E ( r 1 ) 1 ~ N(, h ) h h gold, gold, 1 gold, 1 1 silver, 1 plainum, 1 Here, 1 sands for a new informaion se which conains no only gold informaion during ime period -1, bu also boh silver and plainum informaion during ha ime. The firs equaion for he reurns and disribuion of he residuals are proper o all hree series. 1 and here sand for he special parameers of exra erms. For a cerain paern of he univariae sandard GARCH model case above, he firs-order model is he mos popular EGARCH model in pracice as well. Then i would closely resemble he he siuaion of sandard GARCH(1,1) model, he error erms of silver and plainum reurn prices are added up o original EGARCH(1,1) model respecively..4 Disribuions of Error Terms.4.1 Normal Disribuion Normal disribuion is one of he mos imporan and common disribuion in saisical analysis which is also known as Gaussian disribuion. 6

10 The probabiliy densiy funcion of normal disribuion is presened as, f ( ) 1 h e ( ) h where sands for he mean and h sands for he variance. Normal disribuion is a coninuous and symmeric disribuion wih a bell-shape curve. When 1, h, he disribuion is considered as sandard normal disribuion, inerpreed as N(,1)..4. Suden- Disribuion Suden- disribuion, also known as disribuion is proposed by Gosse (198). The lile difference o normal disribuion is found by him based on a large number of empirical applicaions. disribuion is also a symmeric, coninuous and bell-shaped disribuion, all like normal disribuion only wih a faer ail which is perfecly corresponded o he financial ime series. Esimaion GARCH models wih suden- disribuion were firs proposed by Bollerslev (1987). Le he condiional disribuion of r, 1,..., T, be disribued wih variance h 1 and degrees of freedom. r E ( r 1 ) 1 ~ f ( 1) f ( 1) denoes he condiional densiy funcion for 7

11 f ( 1 1 ( ) ) ( ) 1 ( ) h 1 (1 ( )) h ( ) ( 1) / Where is considered as a number of degree of freedom, (z) is gamma funcion. When 1, disribuion urns ino sandard Cauchy disribuion, having no mean. When is large, disribuion is could approximae sandard normal disribuion. 3.5 Roo Mean Square Error (RMSE) In his paper, we prefer a crierion called RMSE for goodness of forecasing. The Roo Mean Square Error is also known as Roo Mean Square Deviaion (RMSD). I is used o measure he differences beween he forecas values calculaed from a model or an esimaor and he values acually observed generally. Here is he funcion form of RMSE, RMSE N 1 N E where E Y F, Y is he acual value a period, and F is he forecasing value for period. N here is he number of ou-of-sample observaions. Finally, he square roo of he average is aken. The crierion of RMSE requires ha a smaller value which reflecs a beer forecasing he model compleed. Bu for more accuracy, we canno definiely judge which model is beer wih only comparing he quaniies of some crierions. Since here hardly is such significance concluded from 3 More deails abou disribuion could be seen in Saisical Inference ex book nd Ediion (Casella, Berger, ) 8

12 only crierion comparison. 3. Daa The price of gold is deermined hrough rading in he gold and is derivaives markes, however a procedure known as he Gold Fixing in London which sprang from Sepember 1919, provides a daily benchmark price o he indusry. The afernoon fixing was inroduced in 1968 o provide a price when US markes are open 4 which is caused by ime differences. The oher precious meals use he similar pricing mehod. The daa in his paper is downloaded from Kico via wikiposi.org. 5 All of hem are he afernoon fixing prices. 3.1 Reurns Le { y } be he financial ime series of he daily price of some financial asses. The reurn on he h day is defined as r log( y ) log( y 1) where r is defined as reurn of he h variable during ime period. Someimes he reurns are hen muliplied by 1 so ha hey can be reaed as percenage changes in he price making some sense in economics. Furhermore, his procedure may decrease numerical errors as he raw reurns could be very small numbers and lead o large rounding errors in some calculaions, following Cryer and Chan (8). 4 "Gold fixing".hp:// More deails here <hp://en.wikipedia.org/wiki/gold_fixing> 5 The daa is downloaded from <hp://wikiposi.org/uid?kitco> 9

13 3. Plos Figure 1: Plo of original daily daa Figure 1 shows he plo of he original daabase used in his paper. I is a plo of daily spo prices including 3 kinds of precious meals, gold, silver and plainum during he ime period Oc. 7 h o Mar. 11 h. 11. We can see ha gold price appears o be increasing in general rend, meanwhile silver and plainum prices have more severe up and down han gold price. Figure : Plo of reurn daa Figure shows he plo of he resuls of log-difference. The analysis, consis of esimaion and forecas, we will do needs reurn daa insead of he raw price daa hrough he way ha using he firs-order log-difference resul muliplied by one hundred of each daily afernoon fixing prices. 1

14 3.3 Daa Analysis Table 1 presens some summary saisics of he reurn daa. All hese hree ime series have he same amoun of observaions. The mean of each is very close o zero, as well as he median and a no very large sandard deviaion. Among hem, only gold reurn daa does have a no-negaive skewness and each of hem has excess kurosis. The ARCH-LM es here is a mehodology o es for he lag lengh of ARCH errors using he Lagrange muliplier es which was proposed by Engle (198) 6. This es should be done before we apply GARCH models o he daa. The p-values for his es are all very small ha he null hypohesis, say he daase has no ARCH effecs should be rejeced. Based on his es, i is secured and proper ha we can fi his daa o a GARCH model. Table 1: Summary Saisics Gold Reurns Silver Reurns Plainum Reurns Min Max Median Mean Sd.Dev Skewness Kurosis ARCH-LM Tes p-value <.e-16 p-value <.e-16 p-value <.e-16 Noe: The daa is colleced since o and each series has 364 observaions in oal. 6 More deails abou ARCH-LM Tes could be seen in Appendix. 11

15 The Figure 3 demonsraes he auocorrelaion funcion resuls. As a financial ime series, having he auocorrelaion means i can be predicable because fuure values depend on curren and pas value and his dependence could las for long erm. Teräsvira (6) said ha observaions in reurn series of financial asses observed a weekly and higher frequencies are no independen. While observaions in hese series are uncorrelaed or nearly uncorrelaed, he series conain higher-order dependence. This can explain he auocorrelaion funcion resuls below precisely. Figure 3: Auocorrelaion Funcion of Reurns, Reurns Square and absolue value of Reurn daa We can see ha gold reurn daa is no correlaed; silver and plainum eiher. However, reurn square daa shows ha boh gold and silver daa 1

16 are highly correlaed and also have a long-erm memory. The reurn square daa shows auocorrelaion since i is formed as square which is exacly corresponded o he form of condiional variance. The absolue values of reurn daa urns ou higher correlaion han boh wo before. 4. Esimaion We apply univariae GARCH models and EGARCH models o he daa for esimaion disinguishingly wih differen error erm disribuions. 4.1 GARCH Models Table below presens he resuls of esimaion of univariae GARCH model, only from log-likelihood and AIC values of he models we can see ha when we boh add silver and plainum error erms, he model has a smaller AIC value and larger log-likelihood. From he poin of view, differen disribuions make a difference on esimaion indeed. Wih a suden- disribuion, he models all provide smaller AIC and larger log-likelihood. Besides, he parameer of silver erm ofen urns ou o be very small, someimes no significan which suggess ha we could model he volailiy wihou silver erm. 13

17 Table : Esimaion of Sandard GARCH Models wih differen disribuions sandard +Pla +Silver +Pla sandard +Pla +Silver +Pla +Silver +Silver Normal disribuion disribuion (.) (.) (.) (.) (.3) (.3) (.3) (.3) (.5) (.5) (.5) (.5) (.9) (.9) (.9) (.9) (.6) (.7) (.7) (.7) (.9) (.1) (.1) (.9) (.6) Log-likeli hood AIC Noes: 1 in he model equals o zero suggess ha here is no plainum error erm in i, sands for he parameer of silver error erm. Boh of hem equal o zero means sandard GARCH process. 14

18 4. EGARCH Models Table 3: Esimaion of Univariae EGARCH models sandard +pla +Silver +Pla sandard +plai +Silver +Pla +silver +silver Normal Disribuion disribuion (.7) (.8) (.8) (.8) (.1) (.13) (.1) (.1) (.1) (.1) (.1) (.1) (.16) (.17) (.16) ( (.6) (.6) (.6) (.6) (.1) (.11) (.11) (.11) (.) (.4) (.3) (.3) (.3) (.5) (.4) (.4) (.1) Log-likeli hood AIC Noes: here is he parameer in he g ) par, he same as he in he ( Z j g( Z j) Z j ( Z j E( Z j )), j 1,..., q. Following he same paern of secion 4.1, we can ge he esimaion resuls shown in 15

19 Table 3. Based on AIC and log-likelihood, we can ge a similar resul as univariae sandard GARCH models. When adding boh plainum and silver reurn's error erms since i has a smaller AIC value and larger log-likelihood. Only under he assumpion wih disribuion, adding silver error erm makes he model change a lile bi; i has a smaller AIC value. 4.3 Comparison Compared o he univariae GARCH models, he average of all he six univariae EGARCH models gives a beer performance in boh log-likelihood and AIC value. Wih differen disribuion assumpion, alhough he sample size is large enough ha we could approximae suden- disribuion o sandard normal disribuion, he esimaion sill is influenced by disribuion difference. Among all hese models, he model based on EGARCH(1,1) model wih silver erms under he suden- disribuion has boh he smalles AIC and larges log-likelihood value. 5. Forecas Besides he esimaion comparison of models, we can ell he goodness of fiing by using ha model forecasing and hen compared o some exising daa, such as an 1-ou-of-sample forecasing. Through ha way, as menioned before, RMSE is a quie useful ool and has more exacness. In his paper, we complee forecasing following his mehod: Firs of all, here are 364 observaions in oal in he daase. Now we reserve he las 1 observaions in order o compare o he forecasing value. Then we should regard he fron 354 values as a window which has a cerain lengh and 354 observaions. Secondly, we esimae he model wih his window, afer ha we can do a one-sep 16

20 forecas of volailiy. Take he univariae sandard GARCH as an example of one-sep forecas, h h gold, gold, 1 gold, 1 1 silver, 1 plainum, 1 h h gold, 1 gold, gold, 1 silver, plainum, Nex, we repea he firs sep excep for creaing a new window from 354 observaions o 355 observaions. The window ges wider and wider. We can keep moving forward by repeaing his wo seps one-hundred imes and hen we can ge 1 forecas values. Evenually, we can calculae RMSEs of all he values from all he models and compare hem. Firs, we compare RMSEs of GARCH models. Table 4: The resuls of RMSE of GARCH models Sandard +pla +silver +pla Sandard +pla +silver +pla +silver +silver Normal Disribuion Suden- Disribuion RMSE * Noe: RMSE is shor for Roo Mean Square Error. From Table 4, adding plainum and silver error erms a he same ime seems has a much beer resul han every model else, meanwhile he group wih disribuion makes beer performance han he oher. The mos imporan hing is ha exra error erms acually can help wih forecasing gold price volailiy since mos of hem urn ou a beer resul han using sandard GARCH models wihou all he regressors no maer wha disribuion we assume. 17

21 Nex, we compare he resuls of EGARCH models. The resuls in Table 5 do no have a specacular beer number among all. Bu in boh normal disribuions and disribuion, adding only silver erm shows is efficiency in helping forecasing gold volailiy. Similarly he group wih disribuions performs beer. Also same as GARCH models, mos models wih exra erms have smaller RMSE. Compared he resuls from he view of he kinds of model, by average, EGARCH models provide a beer final resul in forecasing han GARCH models no maer wih or wihou regressors, especially excellen wih disribuion. Table 5: The resuls of RMSE of EGARCH models Sandard +pla +silver +pla Sandard +pla +silver +pla +silver +silver Normal Disribuion Suden- Disribuion RMSE * Noe: RMSE is shor for Roo Mean Square Error. 6. Conclusion In summary, his paper aims o forecas gold prices under he help of oher precious meals such as silver and plainum. Among all he resuls we ge, for models, EGARCH models seem o be more efficien in forecasing volailiy; for disribuions, unique properies of disribuion makes i more appropriae han normal disribuion in forcasing. The resul acually confirms a hypohesis we did no menion much, wheher he precious meals could help wih forecasing volailiy of gold price or no. Now we can know i for sure ha adding silver and plainum error erms makes GARCH 18

22 models beer. The exra regressors increase he accuracy of he models in forecasing. Bu here are sill exising limiaions. Firs we assume all he esimaion and forecasing are processed under he assumpion ha he error erm denoes a symmeric disribuion such as normal disribuion which is no ha enough for real siuaion. Second, we consider all he daa do no conain any breaks, however, afer he world-wide financial crisis, here may be some srucural breaks during ha ime. This needs many more seps of analysis before we applied he daa o he models. There have been a lo empirical applicaions of analyzing financial ime series, hey seem o be he same kind of financial daa and all applied o GARCH family models, however he combinaion of differen daa and differen poin of views leads o oally differen resuls. In his paper, i is oo complicaed o explain ha how and why he silver and plainum could help wih modeling and forecasing his way. We can dig i deeper and deeper and find more deails o improve he whole esimaion and forecas sysem of GARCH family models. 19

23 Reference Bollerslev, T., (1986), Generalized auoregressive condiional heeroskedasiciy. Journal of Economerics, 31(3), pp Bollerslev, T., (1987), A Condiionally Heeroskedasic Time Series Model for Speculaive Prices and Raes of Reurn, The Review of Economics and Saisics, Vol. 69, No. 3, pp Cryer J.D, Chan, (8), Times Series Analysis wih Applicaions in R, nd ed., Springer Science Business Media, LLC Casella, Berger, (1) Saisical Inference ex book nd Ediion, Duxbury Press Ding, Z., Engle, R.F., & Granger, C.W.J., (1993). A long memory propery of sock marke reurns and a new model. Journal of Empirical Finance, 1, pp Engle R.F. (198). Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion. Economerica.Vol. 5(4), pp Gosse W.S., (198) The Probable Error Of A Mean, Biomerika. 6 (1): 1 5. Nelson, D. B. (1991). "Condiional heeroskedasiciy in asse reurns: A new approach", Economerica 59: Taylor, S., (1986), Modelling Financial Time Series, Wiley, Chicheser. Teräsvira,(6), An Inroducion o Univariae GARCH Models. SSE/EFI Working Papers, Economics and Finance, No. 646

24 Appendix ARCH - Lagrange muliplier (LM) es The ARCH - Lagrange muliplier (LM) es is a mehodology o es for he lag lengh of ARCH errors, in he oher word, The ARCH-LM Tes is for esing wheher he series has ARCH effecs a all (Engle,198). In he ARCH-LM es, here is a regression, p ( i1 ) i p e In his regression, is he null hypohesis, meanwhile, he 1 q alernaive hypohesis H or or means ha no all i 1 : 1 being no significan a he same ime. Besides, he es saisics follows disribuion wih p degrees of freedom. 1