Additive Outliers (AO) and Innovative Outliers (IO) in GARCH (1, 1) Processes
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1 Addve Oulers (AO) and Innovave Oulers (IO) n GARCH (, ) Processes MOHAMMAD SAID ZAINOL, SITI MERIAM ZAHARI, KAMARULZAMMAN IBRAHIM AZAMI ZAHARIM, K. SOPIAN Cener of Sudes for Decson Scences, FSKM, Unvers Teknolog MARA Shah Alam, Selangor MALAYSIA. Head Cenre for Engneerng Educaon Research, Faculy of Engneerng and Bul Envronmen, 6 UKM Bang, MALAYSIA Solar Energy Research Insue, Faculy of Engneerng and Bul Envronmen, 6 UKM Bang, MALAYSIA azam.zaharm@gmal.com, smbz_ma@yahoo.com, sad@msk.um.edu.my Absrac: - Ths sudy s abou ouler deecon n me seres daa. The man objecve s o derve and o es sascs for deecng addve ouler (AO) and nnovave ouler (IO) n GARCH(,) processes and subsequenly o develop a procedure for esng he presence of oulers usng he sascs. A es sasc has been derved for each ype of ouler. In he dervaon of he sascs, he mehod appled was o derve ouler deecon sascs for GARCH(,) by akng he analogy of GARCH(,) as beng equvalen o ARMA(,) for he ε ( ε beng he resdual seres). Because of he dffculy n deermnng he exac samplng dsrbuons of he ouler deecng sascs, crcal regons were esmaed hrough smulaons. The performance of he ouler deecon was evaluaed based on he ouler es crera and he ouler deecon procedure, usng smulaons. Resuls on he power of correcly deecng he ouler usng he ouler es crera and he power of correcly denfyng he ype of ouler, gven ha he locaon s correcly deeced were repored. Ths was done for each ype of ouler, ndvdually. In hs sudy he developed ouler deecon procedure was appled for esng he presence of he wo ouler ypes n he daly observaons of he Kuala Lumpur Compose Index (KLCI). An ouler was found o be presen n year 998 whch corresponded o he economc downurn of he perods. Keywords: Addve ouler, Innovave ouler; GARCH; smulaon; leas squares mehod Inroducon In he pas, researchers have focused a lo of works on he ssues of oulers and hey ofen reled on he assumpon of ndependenly negraed and dencally dsrbued (IID) o address her analyss. However, hs s no approprae when analysng he me seres daa snce he assumpon mples ha all he daa have he same dsrbuon whch s n fac, canno be relable. Based on ha assumpon, hose no well-mached observaons wll be consdered as oulers. Laer on, Fox (97) esablshed he sudy of ouler n dependen daa (me seres daa) and was he frs o defne wo ypes of ouler n an auoregressve model. Before hs arcle, pas researchers ofen reled on he assumpon of dencally and ndependenly dsrbued (IID) observaons for oulers. The assumpon made could lead o random sample procedures (Fox, 97). Fox ISSN: ISBN:
2 (97) proposed a mehod whch shows ha he power of he es performed beer han ha of he pas procedures regardless of wheher napproprae model was beng analysed. Aferward, researchers have formulaed an ARMA model o deec and modelled he presence of ouler n me seres (dependen) daa analyss. For nsance, Ljung (99) deeced he addve ouler n ARMA model. Laer, many sudes on deecng and modellng ouler have been exended o he oher classes of models ncludng ARIMA, ARCH, GARCH and oher models. Ths s due o he fac ha no all sascal me seres daa can be analysed appropraely usng he ARMA models for he purpose of ouler deecon. For nsance, n he case of usng fnancal daa, s more approprae f he class of ARCH or GARCH model s employed. Accordng o Goureroux (997), he ARMA f s poor n handlng fnancal problems. Laer, he sudy of ouler has been exended o non lnear me seres. Among he non lnear models are, blnear, EGARCH and LSTAR (logsc smooh ranson auoregresson) models. In non lnear me seres, van Djk e al. (999b) deeced he presence of oulers n LSTAR model. Chan and Cheung (99) examned he ouler occurrence n he hreshold auoregresson (TAR) model. Zaharm (996) deeced he presence of addve ouler, nnovave ouler, level change and emporary change n blnear model of order BIL(,,,). Ibrahm () exended he same ypes of oulers for BIL(,,,). For mullevel model, he ouler deecon has been suded by Langford and Lews (998). In he sudy, hey used he resduals analyss for ouler denfcaon. Laer, Sh and Chen (8) proposed a es o deec mulple oulers n mullevel daa. The es was consruced n such a way ha was able o overcome he problem of neglecng he effec of he esmaon of random parameer. Addonal revew on mullevel model can be referred n Goldsen (986). Franses and Ghjsels (999) modelled he addve ouler (AO) by adapng he analogy GARCH(,) model as equal o ha of he ARMA model for ε. They used he dea proposed by Chen and Lu (99) on he sudy of ouler n ARMA me seres. Laer, he work of Franses and Ghjsels (999) was exended by Charles and Darne () n order o deec he presence of he addve ouler (AO) and nnovave ouler (IO) n he GARCH models. There are several ypes of ouler ha have been nvesgaed n me seres daa. The addve ouler (AO) and nnovave ouler (IO) are he frs suded by prevous researchers. Laer, many prevous sudes showed ha here are oher ypes of oulers n me seres, namely emporary change (TC) or somemes called as ransen change and level change (LC) or be referred o as level shf (LS). These ypes of oulers were classfed by Tsay (988) who has also nroduced he ype of ouler, known as varance change (VC). Box and Tao (96) defned LC when here was a change n fscal polcy. Ths adjusmen mgh lead o a change n he level of some pons. However, he prevous four ypes of oulers (AO, IO, LC and TC) are he famous ypes, suded n many researches. Laer, Wu e al. (99) defned a new ype of ouler known as reallocaon ouler (RO). Dervaon of AO and IO effecs n GARCH(,) Processes A GARCH(,) processes s wren as: ε = z h, ε ~ N(, h ) (.) h ϕ ω α ε β (.) = + + h where h s he condonal sandard devaon derved from he condonal varance equaon of he GARCH model. The sequence of sandardzed resdual, { z } are dencally and ndependenly dsrbued (IID) random varable wh zeromean and consan varance. Alernavely, a GARCH(,) can be performed n an ARMA(,) for ε as shown n Bollerslev (986) : ε = ω+ ( α+ β) ε + v βv (.) Hence, he resdual s obaned by subsung he equaons (.) and (.): v = ε h (.) The equaon n (.) s recalled as an oulerfree model. Whereas ε s he ouler-free seres (rue seres) and v s he ouler-free resdual (rue resdual), respecvely and =,,... When ISSN: ISBN:
3 oulers are presen, he GARCH (,) represenaon s modelled as: * ε = ε + ξ ( L) ϖ I ( ) (.) * where, ε represen he conamnaed counerpar of ε, and () ϖ for =, denoes he ouler effec wh =, he ype of addve ouler (AO) and =, he ype of nnovave ouler (IO), () I ( ) s an ndcaor funcon for he occurrence of ouler effec and akes a value of a =, zero oherwse wh denoes he locaon of he occurrence of ouler. In general, he locaon s unknown and herefore, hs sudy assumes ha s unknown. () ξ (L) s a polynomal n L wh roos on he un crcle. I represens he dynamc paern of he ouler effec. For AO, ξ (L) = and for IO, ξ ( L) =.These has been dscussed n Chen and π ( L) Lu (99). Referrng o he work of Azam e al. (8), he effec of an AO on he ε and v of GARCH(,) processes s wren n equaons (.6) and (.7), respecvely. ε < * ε, AO = ε + ϖ = ε = + k, where k =,,..., n (.6) v < * v, AO = v + ϖ = k v + ( α β ) ϖ = + k, (.7) where k =,,..., n. In hs sudy, he effec of IO on he ε and v of GARCH(,) processes s wren n he followng manner: ε *, AO where ε = ε + ϖ ε + k [ α β ] ϖ k =,,..., n. < = = + k, (.8) The effec of an AO on he v of GARCH(,) processes s wren n he followng manner: v < * v, IO = v + ϖ = v = + k, where k =,,,..., n (.9) The leas squares esmae for AO and IO, can be smply wren n he followng manner: n ~ ϖ v x ~ ϖ (.) where + k, k, k= = n xk, k= k = xk, AO = k ( αβ ) k =,,,..., n and k = x = k, IO k =,,,..., n However, he occurrence of an ouler a any parcular me = s unknown. A sraegy for deecon s o compue he sandardzed sasc / ˆ ( ) ˆ ( ) = ˆ n ϖ xk a each me pon, σ v k = =,,...,n. The es sasc s = Max ˆ ( ), whch s he maxmum of he max =,,..., n absolue values of he compued ˆ ( ). The probably dsrbuon of he es sasc Max ˆ ( ) s dffcul o deermne and canno be =,,..., n specfed algebracally. The deermnaon of he dsrbuon s beyond he scope of hs sudy. Followng Chang e al. (988), he crcal values of he es sasc are deermned by smulaon. In deecng he presence of a sngle ouler, an approach ISSN: ISBN:
4 s o apply he lkelhood rao crera. The lkelhood rao creron s derved for esng he null hypohess agans he alernave hypohess for each ype of ouler. The deeced ype of ouler s based on he larges measure of he es sasc (n absolue value). The AO and IO Effec on GARCH(,) Processes The AO and IO effecs on he reurns seres and resduals of a smulaed GARCH(,) processes are ploed n Fgures. and.. The plo s mean o llusrae he behavour of addve ouler (AO) and nnovave ouler (IO) when hey are nfluencng he orgnal seres and resduals of a GARCH(,) varance model. The nfluence of an ouler s represened by red do lne and he orgnal seres and resduals are shown usng he sold lne. A sngle ouler s supposed o occur a = n sample sze of. The sze of each ouler s assumed o be ϖ =. A GARCH(,) process s smulaed usng a specfed coeffcen value α =., β =. 8 wh observaons as sample sze. The orgnal seres of a smulaed GARCH(,) model s ploed n Fgure.. The effec of an AO s ploed n Fgure.. As shown n Fgure., here s only an abrup jump a me pon =. The AO only affecs ha parcular observaon and lef he oher observaon afer =, unnerruped. The paern of AO effec s dffer o ha of IO n ha he IO affec he observaons from = and a few observaons aferward before des ou. The effec of an IO s shown n Fgure.. R e u rn Tme Fgure.: Plo of he orgnal reurn seres of a smulaed GARCH(,) model A O wh AO effec whou AO effec Tme Fgure.: The effec of AO on reurn seres ISSN: ISBN:
5 I O - - wh IO effec whou IO effec Tme Fgure.: The effec of IO on reurn seres In order o explan he behavour of hese ouler effecs on he smulaed GARCH(,) processes, he coeffcen values of α =. andβ =. are also used. The former model s referred as model A ( α =., β =. 8 ) and he laer as model B ( α =. andβ =. ). The effec of AO s llusraed usng boh model A and B and hey are shown n Fgures. and.6, respecvely. For AO case, s observed ha he effec of AO on he resduals s dffered from he effec of AO on he seres. From Fgures. and.6, an AO no only affecs he orgnal resduals a = bu also he subsequen observaons aferward. The magnude of changes of resduals ends o decrease and he number of conamnaed resduals also decreases as he coeffcen of α =. and 8 β =. changes o. and., respecvely. I s seen ha afer affecng a few of he observaons, he effec of AO has des ou. Fgures.7 and.8 llusraed he effec of IO on he resduals. From boh fgures, he IO only affecs he resduals a he parcular pon = wh sze of he ouler effec, ϖ =. and lef he subsequen pons unaffeced. For IO case, s noed ha he effec of IO on he resduals s no much dfferen from ha of AO when he ouler affec he reurn seres of a smulaed GARCH(,) models. Ths s due o he formulaon of he IO effec on he resduals whch akes no accoun for he effec a one parcular me pon (a = n hs example) and he effec urns o zero for he followng observaons afer ha me pon. Ths can be referred o equaon (.7) for AO and equaon (.9) for IO. 6 R e s d u a ls Tme Fgure.: Plo of resduals from smulaed model A ISSN: ISBN:
6 6 wh AO effec whou AO effec 6 wh IO effec whou IO effec A O I O Tme Fgure.: The effec of AO on resduals from smulaed model A Tme Fgure.7: The effec of IO on resduals from smulaed model A wh AO effec whou AO effec wh IO effec whou IO effec A O I O Tme Fgure.6: The effec of AO on resduals from smulaed model B Tme Fgure.8: The effec of IO on resduals from smulaed model B Tes Performance The focus of performance s based on ypes of ouler, sze of ouler effec, sample sze, crcal values and ypes of GARCH(,) varance model. ISSN: ISBN:
7 For each GARCH varance model, a sngle ouler was nroduced o he seres. Two dfferen szes of ouler, ϖ =. and 6. and wo sample szes, n= and n = were used. The selecon was made o examne he performance wh respec o he changes n he sze of he ouler and sample szes, respecvely. The sudy assumed ha ouler occurs n he mddle of he observaonal perod, specfcally a me pon = for n = and = for n =. These characerscs were employed o esmae he probably of correcly deecng he ouler-locaon a me pon = and = for sample sze and, respecvely. In he case of A, he frequency of correcly deecng he AO ype ncreases when sample sze ncreases, for all models. The suaon held when he sze of ouler effec, ϖ used s. and 6.. In addon, when he sze of ouler effec changes from. o 6., he frequency of deecng he AO ype ncreases for mos of he models, excep when crcal value was very low. In addon, as he crcal values ncreases from. o., he frequency ends o decrease. The same suaon occurs for each model and each ouler sze. When ϖ = 6., he percenages of correc AO deecon was 86.% when C =.. I decreases o.% when C =.. The performance of he es mproved when oulers were large. Mos models show beer performance when crcal values of. and. are examned. The resuls for IO shows ha he performance of correcly deecng n each model mproves when sample sze and ouler sze ncreases. I s also found ou ha he percenages of correcly deecng he IO es sasc s uny for every model, parcularly when crcal values of. s examned. The resul ends o be he same for he case of ϖ = 6.. I s observed ha he frequency of correcly deecng he AO s slgh hgher han ha of he IO, bu only n ceran models, dependng on he ouler sze and he crcal values used. In mos models here are no large dfferences beween he percenages of correcly deecng he ouler for AO and IO, regardless of whch sample sze and ouler sze are used. The performance of almos all models s beer when crcal values of. and.. In general, he paern of es performance for AO and IO shows an ncreasng funcon of ouler sze, ϖ parcularly, when crcal values of. and. are used. For AO and IO ypes, here are slgh changes n he frequency as ϖ ncreases. The sudy also found ha he frequency of correcly deecng he ouler for each ouler ype decreases wh respec o he ncrease n crcal values. Ths suaon happens for each sample sze and ouler sze examned n he sudy. Mos of he percenages for each ouler ype end o be uny for all models when C =. s consdered. A Sngle Ouler Deecon Procedure The deecon procedure consss of he followng seps:. A GARCH(,) model based on he orgnal daa s esmaed under he assumpon ha here s no ouler n he daa.. The esmaed resduals s obaned and hence, he ouler effec ϖ( ) and he resdual varance, σ v s compued. Usng he above nformaon, he es sascs ( ) s calculaed for all possble =,..., n where denoes ouler ype.. The maxmum of he absolue value of hese es sascs, = Max ˆ ( ) s max =,,..., n compued If he value of he es sasc exceeds he pre-specfed crcal value, C (sgnfcan) hen an ouler s deeced. Thus, he pon where max occurs s he pon deeced as havng he ouler. As suggesed by Fox (97), he ype of ouler can be dsngushed by comparng he larges es sascs for each ype of ouler. Those wh he maxmum absolue value of he es sascs wll be denfed as he ouler ype. Deal revew on how o dsngush he ype of ouler can be referred n Chang e al. (988) and Chen and Lu (99).. Ouler Deecon n a KLCI seres. Daly reurn seres of he KLCI, observed from December nd, 996 o December s, was ISSN: ISBN:
8 used. A fgure. shows he plo of he reurn seres of KLCI. R e u rn X: 6 Y:.8 Kuala Lumpur Compose Index/ Daly Reurns X: 6 Y:.6 X: 6 Y: Dec 996 Dec 998 Dec Dec Fgure.: Saonary me seres of KLCI reurns Ouler deecon procedure was appled o he reurn seres of KLCI. The presence of AO and IO were denfed hrough he whole daa ha consss of 87 daa. The resul of he es sascs s presened n Table. accordng o he locaon of he occurrence of ouler, dae and ype of oulers ha have been denfed. The larges value of es sasc (n absolue erm) s consdered as he ype of deeced ouler. I s observed ha ou of he whole daa, hree pons ndcaed he presence of suspeced oulers. They are a locaon 6, 6 and 6 (Fgure.). The deeced ouler s of AO ype a locaon 6 whch has he larges measure of es sascs, 97.7 as compared o he IO. The resul from Table. shows ha he presence of ouler durng hs perod, whch falls on Sepember 7, 998 could be due o he economc recesson n all ASEAN counres ncludng Malaysa. References: [] Azam Zaharm, S Meram Zahar, Mohammad Sad Zanol and K. Sopan. (8). Developng Crcal Regon n he Presence of Addve Ouler (AO) n GARCH(,) Processes Usng Smulaon Technque. Proc. European Journal of Scenfc Research. [] Box, G.E.P. & Tao, G.C. (96). A Change n Level of a Non-Saonary Tme Seres. Bomerka., 8-9. [] Chan, W.S. & Cheung, S.H. (99). On Robus Esmaon of Threshold Auoregressons. Journal of Forecasng., 7-9. [] Chang, I., Tao, G.C. & Chen, C. (988). Esmaon of Tme Seres Parameer n he Presence of Oulers. Technomercs., 9-. [] Charles, A. & Darne, O. (). Oulers and GARCH Models n Fnancal Daa. Economcs Leers. 86, 7. [6] Chen, C. & Lu, L.-M. (99a). Jon Esmaon of Model Parameers and Ouler Effecs n Tme Seres. Journal of he Amercan Sascal Assocaon. 88, [7] Fox, A.J. (97). Oulers n Tme Seres. Journal of he Royal Sascal Socey, Seres B., 6. [8] Franses, P.H. & Ghjsels, H. (999). Addve Oulers, GARCH and Forecasng Volaly. Inernaonal Journal of Forecasng., -9. [9] Goldsen, H. (986). Mullevel Mxed Lnear Model Analyss Usng Ierave Generalzed Leas Squares. Bomerka. 7, -6. [] Goureroux, C. (997). ARCH Models and Fnancal Applcaons. New York: Sprnger- Verlag. [] Ibrahm, M. (). Oulers n Blnear Tme Seres Model. PhD Thess, Unvers Teknolog MARA. [] Langford, I.H. & Lews, T. (998). Oulers n Mullevel Daa. Journal of he Royal Sascal Socey, Seres A. 6, -6. [] Ljung, G.M. (99). On Ouler Deecon n Tme Seres. Journal of he Royal Sascal Socey. Seres B. (Mehodologcal)., [] Sh, L. & Chen, G. (8). Deecon of Oulers n Mullevel Models. Journal of Sascal Plannng and Inference. 8, [] Tsay, R.S. (988). Oulers, Level Shfs and Varance Changes n Tme Seres. Journal of Forecasng. 7, -. [6] Van Djk, D., Franses, P.H. & Lucas, A. (999). Tesng for Smooh Transon Nonlneary n he Presence of Oulers. Journal of Busness and Economc Sascs. 7, 7. [7] Wu, L.S.-Y., Hoskng, J.R.M. & Ravshanker, N. (99). Reallocaon Oulers n Tme Seres. ISSN: ISBN:
9 Journal of he Royal Sascal Socey, Seres C (Appled Sascs).,. [8] Zaharm, A. (996). Oulers and Change Pons n Tme Seres Daa. PhD Thess. Unversy of Newcasle upon Tyne. [9] Zaharm, A. (996). Oulers and Change Pons n Tme Seres Daa. PhD Thess. Unversy of Newcasle upon Tyne. ISSN: ISBN:
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