Detection of Level Change (LC) Outlier in GARCH (1, 1) Processes

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1 Proceedigs of he 8h WSEAS I. Cof. o NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS Deecio of Level Chage () Oulier i GARCH (, ) Processes AZAMI ZAHARIM, SITI MERIAM ZAHID, MOHAMMAD SAID ZAINOL AND K. SOPIAN 3 Head Cere for Egieerig Educaio Research 3 Direcor of Solar Eergy Research Isiue, Faculy of Egieerig ad Buil Evirome, 436 UKM Bagi, Selagor Deparme of Quaiaive Scieces, FTMSK, Uiversii Teologi MARA 44 Shah Alam, Selagor MALAYSIA azamielli@gmail.com, smbz_ma@yahoo.com, saidzaiol@gmail.com Absrac: - A oulier is a exreme observaio ha may have a severe effec o daa aalysis. Their occurreces migh cause problems such as bias or disorio of parameer esimaio. Whe a ime series wih ouliers is modelled ad forecased wihou aig io accou heir presece or wihou removig heir effecs, he es saisics of esimaed parameers i he model will be disored. A GARCH (,) process is o excepioal from beig affeced by ouliers. I his sudy we cosruc es saisics for ideifyig ouliers i GARCH (,) processes wih special focus o he emporary chage () ype. The saisic was developed by he leas squares mehod, ad cosequely a simulaio was carried ou for he purpose of fidig he criical regio. This sudy is a exesio of Che ad Liu s (993) wor o oulier deecio.. The simulaio was doe for eigh sample sizes;,,,,,, ad. The 9 h, 9 h ad 99 h perceiles were compued for esimaig he disribuio of he es saisic. The criical value C. was seleced based o he simulaio sudy ad was used i deecig he presece of level chage () i he reur series of Idusrial Produc Idex (IPI). For he period of aalysis, he resuls idicae ha oulier occurred i 998. Keywords: - oulier; GARCH; simulaio; leas squares mehod Iroducio I he simples sese, a oulier ca be described as a exreme value or exraordiary value i a daa se as compared o oher values. The exremiy of he value may be idicaed wih respec o he variace of he daa se. A srage observaio relaive o oher observaios i a daa se migh also be a poeial cadidae for a oulier. Early sudies i peraiig o ouliers iclude, Bare (978), Hawis (98), Becma ad Coo (983) ad Bare ad Lewis (994). The presece of ouliers has bee argued o have a bad effec o daa aalysis. Their occurrece migh cause problems such as bias or disorio of esimaes, disorio of p -values due o he presece of a few or eve oe uusual daa. The mos serious effec is whe a ime series wih ouliers is modelled ad forecased wihou aig io accou he presece of ouliers or wihou removig he effec of he ouliers. The esimaed parameers i he model will be disored. Sice ime series daa is widely used i ecoomics ad fiace, i is very impora o acowledge he presece of ouliers i he daa by deecig hem ad removig heir effecs. Differe approaches o chec for oulier presece i ime series have bee followed by previous researchers. Oulier presece i GARCH processes has bee sudied by Saaa ad Whie (998) usig a robus esimaio approach. Hoa ad Tsay (998) ad Doori ad Ooms () have examied he presece of addiive level oulier (ALO) ad addiive volailiy oulier (AVO) i GARCH(,) processes. Doori ad Ooms () used he lielihood raio ess isead of Lagrage muliplier ess ha were used i Hoa ad Tsay (998). Frases ad Ghijsels (999) sudied he presece of addiive oulier (AO) based upo he deecio procedure of Che ad Liu (993). A sudy by Charles ad Dare () ivolved he effecs of addiive oulier (AO) ad iovaive oulier (IO) i GARCH(,) processes. The Naure of Level Chage () Effec Ouliers ISSN: ISBN:

2 Proceedigs of he 8h WSEAS I. Cof. o NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS The level () oulier was defied by Box ad Tiao (96). I is called as level chage sice is behavior is o chage he level or mea of he observed series afer he firs effec ad his oulier creaes a sudde ad permae chage i he observed series. I a simple equaio, he chages ca be deoed as a φ( L) φ( L) chage from Y e o Y e + ϖ. θ ( L) θ ( L) ϖ L ca eiher be i a posiive or egaive form. The dyamic paer ξ (L) for level chage oulier is equal o. Thus, as show i Che ad Liu ( L) (993), a model ca be wrie as * Y Y + ϖi ( τ ) ( L) The GARCH(,) processes was used i pracice, sice i is ad equivale o ARMA(,) for ε, as show i Bollerslev (988). I his sudy, he mehod applied by Che ad Liu (993) o derive oulier deecio saisics for ARMA processes is exeded o GARCH(,) by aig he aalogy of GARCH(,) as beig equivale o ARMA(,) for he ε ( ε beig he residual series). This approach has bee moivaed by he sudy of Frases ad Ghijsels (999) whe deecig he presece of addiive oulier (AO) i GARCH(,) processes. Frases ad Ghijsels (999) showed ha coamiaed residuals * v v + ϖx where x for < τ, x for τ ad x π for > τ. I his sudy, he derivaio of emporary chage effec o he series of a GARCH(,) variace equaio is summarized as below: v < τ * v, v + ϖ τ + i v αβ ϖ τ +,,,... i () The leas squares esimae for he effec of a sigle level chage () ca be expressed as: ~ ϖ τ vτ +, x, τ x, () where i x, α β,,... i 3 Oulier Tes Saisic The saisic for esig he presece of oulier is, ϖˆ w here σ x τ The disribuio of he es saisic is difficul o deermie. This has bee poied i, ier alia, Fox (97), Chag e al. (988), Che & Liu (993), Frases ad va Dij (). Hece a simulaio sudy has o be carried ou i order o esimae he criical regio for he es of he presece of ouliers. The ull hypoheses for he es is H : ϖ (o oulier) agais H : ϖ (oulier prese). The saisic for level chage () is, ϖˆ where The compoes of he σ es saisic are: x, τ. Oulier esimae: ϖˆ τ. Variace of oulier esimaes: var( ϖ ˆ ) σ τ v x τ 3. Residuals of GARCH(,) variace equaio: v x, if < τ if τ i α β if > τ, i v x x,,... ISSN: ISBN:

3 Proceedigs of he 8h WSEAS I. Cof. o NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS 4. Locaio of he occurrece of oulier: τ 4 Simulaio A simulaio was carried ou i order o approximae he disribuio of he esimaed saisics for deecig he presece of oulier. The seps of his process are as follows: i. Geerae series for each of sample sizes,,,,,, ad from he GARCH(,) model ii. Geerae he series usig various combiaios of GARCH(,) coefficies deoes as α ad β whereas ω ( α + β) iii. Compue he oulier deecio saisic max for each series. I his way a approximaio of he disribuio of he max uder he ull hypohesis of o oulier is obaied Oulier Deecio The presece of oulier is deeced usig he followig seps: (i) Esimae a iiial GARCH(,) model (ii) Obai he esimae of he codiioal variace, ĥ ad hece, vˆ ε hˆ (i ii) Compue he esimae of oulier effec ϖ (τ ) for all possible τ,,..., (iv) If he value of he es saisic is less ha he criical value, here is o oulier a he observaio for which he -saisic of ϖ (τ ) is a maximum (i absolue value). A oulier is prese a he observaio for which he -saisic of ϖ (τ ) is maximum (i absolue value) if he value of he es saisic is greaer ha he criical value. 6 Resuls ad Discussio The resuls of he esimaed perceiles of he es saisics of, T were graphically show i Figure. The disribuios of he esimaed es saisics were ploed io 9 h, 9 h ad 99 h perceiles. I is observed ha, he es saisic shows o clear paer whe sample size icreases from o. This is observed for each model ad each perceile. The excepioal case occurs oly for model 7 i which, he value has icreased as sample size icreases from o, whe he 9 h perceile is examied. For he esimaed 99 h perceile, he smalles value of he es saisics is ideified o be.49 for model wih sample size of ad he larges value is foud o be 3.7 whe sample size of is used for model 7. For es saisics, he values of he esimaed 9 h perceile rage from 8.7 o.4 as sample size icreases from o, ad he criical value of 9 is he highes frequecy for es saisics. A 9 h perceile, he value rage from 6.8 o.8 as icreases from o, wih he value of 8. is he highes frequecy. While a 99 h perceile, he value rages from.9 o 3.7, wih 3. is he highes frequecy for es saisics. Hece, he rage for he criical values is bewee 6.8 ad 3.7. Therefore, his sudy had decided o es for he criical values of.,.,.,. ad. for he ex simulaio sudy. The simulaio experime was mea o chec for he probabili y of correcly deecig he oulier, a ime poi τ for sample size of, usig five criical values. Resuls of he performace sudy were repored i Table. The resuls show ha here are large differeces bewee he frequecy of correcly deecig he if sample size of ad are compared for all models. For isace, he frequecy chages from o 998 whe sample size icreases from o, for model wih oulier size, ϖ 6.. As he criical values icreases from. o., he frequecy of TC eds o dramaically decreases. I mos of he models, he frequecy is whe criical values. ad. are examied. I is observed ha model is beer as compared o oher models i erm of he highes frequecy of correcly deecig he. The criical value of. is he highes frequecy of deecig he oulier as compared o he oher criical values. Usig he oulier deecio procedure, he daa which cosiss of a oal of 33 weely reurs of Idusrial Produc Idex (IPI) was sudied. The daa was colleced November s, 993 o Augus 6 h,. The resul obaied was show i Table 3. The resul idicaes ha, oulier occurs i year 998. The is deeced a locaio 7 wih es saisic, 7.7. The es saisic is also high a locaio 44 wih he values of.33. I is believed ha he ecoomic recessio durig ha year has coribued o he occurrece of ouliers i ha paricular period. ISSN: ISBN:

4 Proceedigs of he 8h WSEAS I. Cof. o NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS Level Chage () h Perceile Model Level Chage () h Perceile Model h Perceile Level Chage () Model Figure : Plo of criical values of level chage () ISSN: ISBN:

5 Proceedigs of he 8h WSEAS I. Cof. o NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS Table : Frequecy of correc deecio of oulier for usig crierio Criical Value ϖ ϖ () 999 (4) 7 () () (68) 68 () () 997 (94) 6 () 998 () 4 () 983 (78) 63 (3) () 987 (96) 68 (4) (99) 9 (3) 8 () () (997) 778 (48) 39 () () () 9 (7) 8 (9) () () 933 (74) 6 (9) () (99) 77 (9) 6 (9) 3 (7) 6 () () 8 (89) 6 () 3 (7) 6 () (96) 4 (9) 8 (4) 9 () () (979) 439 (67) 86 (4) 9 () () The frequecy of correc deecio of ouliers was based o a simulaio sudy wih replicaios ad usig 7 models of GARCH(,) processes. This performace sudy was examied for sample size of ad. (The parehesis idicaed he performace sudy for sample size of ). Table : The values of Level Chage () es saisic Dae Locaio Tes Saisic of // /4/ Refereces: [] Bare, V. (978). The Sudy of Ouliers: Purpose ad Model. Applied Saisics. 7, 4-. [] Bare V. & Lewis T Ouliers i Saisical Daa. 3 rd. Ed. Chicheser: Wiley. [3] Becma, R.J. & Coo, R.D. (983). Oulier..s. Techomerics., [4] Bollerslev, T. (986). Geeralized Auoregressive Codiioal Heerosedasiciy. Joural of Ecoomerics. 3, [] Box, G.E.P. & Tiao, G.C. (96). A Chage i Level of a No-Saioary Time Series. Biomeria., 8-9. [6] Chag I., Tiao G.C. & Che C Esimaio of Time Series Parameer i he Presece of Ouliers. Techomerics. 3, [7] Charles, A. & Dare, O. (). Ouliers ad GARCH Models i Fiacial Daa. Ecoomics Leers. 86, [8] Che C. & Liu L.M Joi Esimaio of Model Parameers ad Oulier Effecs i Time Series. Joural of he America Saisical Associaio. 88, [9] Doori, J.A. & Ooms, M. (). Oulier Deecio i GARCH Models. Worig Paper. Nuffield College, Uiversiy of Oxford. [] Fox A. J. 97. Ouliers i Time Series. Joural of he Royal Saisical Sociey B. 34, [] Frases, P.H. & Ghijsels, H. (999). Addiive Ouliers, GARCH ad Forecasig Volailiy. Ieraioal Joural of Forecasig., -9. [] Frases P.H. & va Dij D.. Noliear Time Series Models i Empirical Fiace. Cambridge: Cambridge Uiversiy Press. [3] Frases, P.H., Va Dij, D., & Lucas, A. (4). Shor Paches of Ouliers, ARCH ad Volailiy Modelig. Applied Fiacial Ecoomics. 4, -3. ISSN: ISBN:

6 Proceedigs of he 8h WSEAS I. Cof. o NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS [4] Hawis D.M. 98. Ideificaio of Ouliers. [6] Saaa, S. & Whie, H. (998). High Lodo: Chapma ad Hall. Breadow Poi Codiioal Dispersio [] Hoa, L.K. & Tsay, R.S. (998). Ouliers i GARCH Processes. Mimeo. IMECC, Brazil Esimaio wih Applicaio o S&P Daily Re urs Volailiy. Ecoomerica. 66, ad Uiversiy of Chicago. ISSN: ISBN: