FORECASTS GENERATING FOR ARCH-GARCH PROCESSES USING THE MATLAB PROCEDURES

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1 FORECASS GENERAING FOR ARCH-GARCH PROCESSES USING HE MALAB PROCEDURES Dušan Marček, Insiue of Comuer Science, Faculy of Philosohy and Science, he Silesian Universiy Oava he Faculy of Managemen Science and Informaics, Universiy of Žilina Absrac he urose of he aer is o demonsrae he overall forecasing roblems by develoing and assessing models of ARCH rocesses forecas. Procedures were develoed o deermine aroriae forecass for variance and values of he SAX index ime series. hese rocedures were based on exension of recen develomens in ime series analysis and alied o forecasing sysems. he comarison of classical forecasing model versus Brown s quadraic exonenial smoohing is also resened. Keywords: Variance funcion, ARCH/GARCH rocesses, whie noise, Brown s exonenial smoohing. Inroducion he auoregressive condiional heeroscedasic (ARCH) rocesses described by Engle [3] are a owerful class of ime series models for modelling a wide variey of financial rocesses. he general ARCH rocess in erms of ψ (he informaion se available a ime ) can be wrien as y ψ ~ Ν ( xb, h) h = h( ε, ε,..., ε, x, x,..., x, a ) () ε = y xb where h is he variance funcion, h is a convenien funcion secificaion usually linear in he arameers. x is a vecor of lagged endogenous and lagged and curren exogenous variables, ε is whie noise. hese x s and ε also ener he informaion se. hen he h in () becomes simle form h = h( ψ, a ), where a is a vecor of unknown arameers of variance funcion, b is a vecor of unknown arameers of regression model and he suerscri denoing he marix or vecor ransose. Esimaing he a and b arameers can be execued by he maximum likelihood mehod. Inclusion of he regression y = xb u and he variance funcion h = h( ψ, a ) leads o leas number of arameers in he siri of he arsimony (ha is in simles form). he variance funcion h in () may be used o obain easily variance forecas. In secion, we will consider he exension of hese idea and discuss he echniques for forecas variance modelling in ARCH-GARCH rocesses hrough he use of variance funcion exressed in erms of he {ε } and { y }, where. In [] several assumions above arameers esimaion of he ARCH-GARCH models were discussed. Esimaion of he model arameers requires ha he successive observaions are reresened by a linear combinaion of indeenden errors (whie noise rocess). es of hyoheses and inerval esimaion assume ha he errors are normally disribued. he analys should always consider he validiy of hese assumions. In Secion we give some mehodological remarks o he consrucion of oin forecas for variance. In secion 3 we concenrae on he modelling and forecasing ARCH-GARCH using MALAB oolboxes: GARCH, Saisics and Oimizaion. o comare he forecas accuracy of he ARCH-GARCH mehodology, in Secion 4 his mehod is comared agains Brown s quadraic exonenial smoohing mehod. We illusrae an examle of hese mehodologies for he non-linear deendence of he sock rice index SAX ime series. Finally concluding remarks are resened.

2 Poin forecas for variance Once he variance funcion of he ARCH rocess has been seleced, i can be used o generae forecass for fuure ime eriods ha are oimal in a minimum mean square error sense. he variance funcion exressed in form () is called he general form of he variance funcion. As we menioned above, i can include error comonens {ε }, lagged deenden { y }, exogenous variables, and can also combine all hese variables. Denoe he curren eriod by, τ eriods ino he fuure. Le h $ τ ( ) reresen he variance oin forecas for eriod τ made a origin. he forecas is generaed by aking execaion a origin of he variance funcion a ime τ. Generally, he forecas for eriod τ mus be buil u successively from he forecass for eriods,,..., τ - [4]. We suose ha in his rocedure he h ha have no occurred a ime are relaced by he forecass h $ ( ), he errors comonen ε ha have no occurred a ime are relaced by zero, and he ε ha have occurred are relaced by residuals e( ) = h h$ ( ). In saring he forecasing rocess i will be necessary o assume ha ε = 0 for - 0. As an illusraion, consider forecasing he variance funcion in (). A τ he funcion is h = a a y a y ε τ 0 τ τ τ he oin esimae of his funcion for τ = and aking execaion a ime on boh sides of he equaion, we obain Eh [ ] h$ ( ) a$ ay $ ay $ = 0 e ( ) (3) When we now consider he variance forecass using he funcion exressed only in erm of heε variables (random shock of he model [4]), we may wrie h τ = aε τ aε τ... aτ ε (4) a ε a ε... ε τ τ τ hus he forecasing funcion corresonding o Eq. (4) by aking execaion as before is h$ τ( ) = aτe( ) aτ e( )... (5) he Eq. (5) was obained formally from (4) in which a ime > he corresonding ε was relaced by zero and a ime by relacing ε by e (). he las equaion rovides a simle algorihm for udaing he forecass a end of each ime eriod. his algorihm is described in [4]. In addiion, Eq. (4) and (5) admi a simle algorihm for obaining redicion limis on he τ -se ahead variance forecas. For more deails on his algorihm see [4]. 3 Daa and idenificaion of ARCH-GARCH o illusrae he ARCH-GARCH mehodology, consider he sock rice SAX index ime readings. We would like o develo a ime series model for his rocess so ha a redicor for he rocess ouu can be develoed. he daa was colleced for he eriod July, 995 o Ocober 3, 005 which rovided a oal of 50 observaions (see Fig. ). Afer some exerimenaion, using Malab oolboxes we have idenified resuling GARCH(,) model ()

3 Figure. : he daa for SAX index (July 995-Ocober 005) y h = ε = h ε he samle auocorrelaion funcion [] of his model for normalised residuals gives evidence ha he residuals are hose of a whie noise rocess. Acual and esimaed values for SAX index are grahically deiced in Fig.. Nex, he model for SAX ime series has been develoed by using only las 500 observaions. o build a forecas model he samle eriod for analysis y 0č9,..., y 450 was defined, i.e. he eriod over which he forecasing model was develoed and he ex os forecas eriod (validaion daa se), y 45,..., y 50 as he ime eriod from he firs observaion afer he end of he samle eriod o he mos recen observaion. By using only he acual and forecas values wihin he ex os forecasing eriod only, he accuracy of he model can be calculaed. he aroriae model was idenified as ARMA(,)/GARCH(,) as follows y h =.0957e y = h 0.898ε 0.867ε (6) (7) Figure : Acual (full line) versus esimaed values (dashed line) for SAX index (GARCH(, ) model

4 he daa for SAX sock rice index and he ex os forecas values of he ARMA(,)/GARCH(,) esimaed by model (7) and ex ane forecas for en eriod are grahically deiced in Fig. 3. Figure: 3 Ex os forecass and ex ane forecas for en eriods of SAX index (dashed line) and acual daa 4 Brown s quadraic exonenial smoohing o comare he redicive ower of anoher ime series models using daily daa of SAX, in his secion we focus on he Brown s quadraic exonenial smoohing aroach. Finally we comare he forecasing accuracy of our ime series models. he daa for SAX sock rice index and he ex os forecas values of he Brown s quadraic exonenial smoohing aroach are grahically deiced in Fig. 4. Figure 4 Brown s quadraic exonenial smoohing aroach - ex os forecas and ex ane forecas for en eriods of SAX index (dashed line) and acual daa As is sandard in he economic lieraure, we hen comued he Roo Mean Squared Error (RMSE). he accuracy of our en day forecas are resened in able From his able can be seen ha he ARMA(,)/GARCH(,) model is he bes.

5 5 Conclusion able : RMSE FOR EN EX ANE FORECASS* MODEL GARCH(, ) ARMA(,)/ GARCH(,) Brown s ex. smoohing RMSE his aer has focused on he roblems associaed wih forecasing economic variables, which disurbances follow an ARCH rocess. he suggesed mehods are alicable o rocess models, where he underlying forecas variance may change over ime. We aemed o use he Brown s quadraic exonenial smoohing aroach for forecasing an ARCH rocess. he use of he Brown s quadraic exonenial smoohing aroach for forecasing an ARCH model of he SAX index ime series had limied success. Because he resuls were based on chosen SAX index values and daa se, hey were difficul o generalise o oher siuaions. Ye, he resuls cerainly rovide a raional way for imrovemen of forecasing abiliy in chaoic economic sysems. Acknowledgemen: his work was suored by Slovak gran foundaion under he gran No. /68/05 and from he Gran Agency of he Czech Reublic under he gran No. 40/05/768. I hank M. Lieskovská for daa ses and comuaional suor. References: [] BOLLERSLEV,.: Generalized Auoregressive Condiional Heeroskedasiciy. Journal of Economerics 3, 986, [] BOX, G.,E., JENKINS, G., M.: ime Series Analysis, Forecasing and Conrol. Holden-Day, San Francisco, CA 976. [3] ENGLE, R.,F.: Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kindom Inflaion. Economerica, Vol. 50, No.4, (July, 98), [4] MONGOMERY, D.C., JOHNSON, L.A., GARDINER, J.S.: Forecasing and ime Series Analysis. McGraw-Hill, Inc., 990 Dušan Marček: marcek@fria.uc.sk