The Use of ARCH and GARCH Models for Estimating and Forecasting Volatility

Transcription

1 Kocael Ünverses Sosyal Blmler Ensüsü Dergs (4) 2007 / 2 : The Use of ARCH and GARCH Models for Esmang and Forecasng Volaly Bahadn Rüzgar İsme Kale Absrac: Ths paper presens he performance of ARCH-ype models each wh four dfferen dsrbuons combned wh ARMA specfcaons n condonal mean n esmang and forecasng he volaly of IMKB 00 sock ndces, usng daly daa over a 9 years perod. The resuls sugges ha fraconally negraed asymmerc models ouperform he non-fi versons and, usng skewed- and suden- dsrbuons provde beer f o he daa for almos every model n esmang volaly. In forecasng volaly a clear mprovemen s no observed by alerng a specfc model componen or dsrbuon. Keywords: GARCH; EGARCH; GJR; APARCH; IGARCH; FIGARCH; FIAPARCH; FIEGARCH; HYGARCH; ARMA; GED; Skewed-; Ox; G@RCH. Inroducon Unl he 80 s mos of he analycal research was on fndng he relaon beween facors and oucomes. For hs purpose was a mosly used assumpon for smplcy ha errors were random consans. The models ha bes represen relaons were hose ha produced mnmum errors. As a resul he errors were mnmzed n models, and remaned ou of he subjec of quanave predcon. However n some cases s exacly he quany of hose errors, he predcon of whch s mporan. The world of fnance s one of he frs who suppored he research as hey realzed ha hs error can be nerpreed as wha we may call he rsk. Generalzed AuoRegressv Condonal Heeroskedascy (GARCH) s a model of errors. I s mosly used n oher models o represen volaly. The models ha Yrd. Doç. Dr. Bahadn Rüzgar, Marmara Ünverses Bankacılık ve Sgoracılık Yüksekokulu Aküerya Bölümü n de öğrem üyesdr. İsme Kale, Marmara Ünverses Bankacılık ve Sgoracılık Yüksekokulu Bankacılık Bölümü nde Yüksek Lsans öğrencsdr.

2 The Use Of ARCH And GARCH Models. 79 make use of GARCH vary from predcng he spread of oxc gases n he amosphere o smulang neural acvy. Bu fnance s sll he leadng area and domnaes he research on GARCH. ARCH class models were frs nroduced by Nobel prce awarded Engle (982) wh he ARCH model. Snce hen, numerous exensons have been pu forward, all of hem modellng he condonal varance as a funcon of pas (squared) reurns and assocaed characerscs. In recen years, he remendous growh of radng acvy and radng losses of fnancal nsuons has led fnancal regulaors and supervsory commee of banks o favor quanave echnques whch apprase he possble loss ha hese nsuons can ncur. Value-a-Rsk (VaR) has become one of he mos sough-afer echnques. The compuaon of he VaR for a collecon of reurns requres he compuaon of he emprcal quanle a level α of he dsrbuon of he reurns of he porfolo. Because quanles are drec funcons of he varance n paramerc models, ARCH class models mmedaely ranslae no condonal VaR models. These condonal VaR models are mporan for characerzng shor erm rsk for nradaly or daly radng posons. In hs paper we nvesgae he esmang and forecasng capables of GARCH models when appled o daly IMKB 00 ndex daa. We furhermore am o undersand wheher IMKB daa exhbs he common caracerscs of fnancal me seres observed n developed counres. We hereby wsh o conrbue o he rsk managemen research n Turkey, he oucomes of whch wll be of crucal value afer he mplemenaon of Basel II regulaons n The res of he paper s organzed n he followng way. In Secon 2, we descrbe ARMA and GARCH processes as he buldng blocks of analysed varance models. These models are appled o daly sock ndex daa n Secon 3 where we assess her performances and conclude. 2. Formaon of varance models GARCH models are desgned o capure ceran characerscs ha are commonly assocaed wh fnancal me seres: fa als, volaly cluserng and leverage effecs. Probably dsrbuons for asse reurns ofen exhb faer als han he sandard normal, or Gaussan, dsrbuon. Tme seres ha exhb a fa al dsrbuon are ofen referred o as lepokurc. In addon, fnancal me seres usually exhb a characersc known as volaly cluserng, n whch large changes end o follow large changes, and small changes end o follow small changes. In eher case, he changes from one perod o he nex are ypcally of unpredcable

3 80 Bahadn Rüzgar, İsme Kale sgn. Large dsurbances, posve or negave, become par of he nformaon se used o consruc he varance forecas of he nex perod's dsurbance. In hs manner, large shocks of eher sgn are allowed o perss, and can nfluence he volaly forecass for several perods. Volaly cluserng, or perssence, suggess a me-seres model n whch successve dsurbances, alhough uncorrelaed, are noneheless serally dependen. Fnally, ceran classes of asymmerc GARCH models are also capable of capurng he so-called leverage effec, n whch asse reurns are ofen observed o be negavely correlaed wh changes n volaly. A sandard approach of me seres analyss s o ake a me seres ha exhbs complcaed behavor and ry o conver o a smpler form. Opmally, such smplfcaon would yeld me seres ha were so smple ha hey could reasonably be modeled as ndependen and dencally dsrbued (IID). In pracce, and especally n fnancal applcaons, hs s rarely possble. Saonary s a condon smlar o IID, bu no as srong. Two dfferen forms of saonary are defned: ) A process s sad o be srcly saonary f he uncondonal jon dsrbuon of any segmen (y, y +,..., y +r ) s dencal o he uncondonal jon dsrbuon of any oher segmen (y +s, y +s+,..., y +s+r ) of he same lengh. ) A process s sad o be covarance saonary f he uncondonal jon dsrbuon of any segmen (y, y +,..., y +r ) has means, sandard devaons and correlaons ha are dencal o he correspondng means, sandard devaons and correlaons of he uncondonal jon dsrbuon of any oher segmen (y +s, y +s+,..., y +s+r ) of equal lengh. Correlaons nclude auocorrelaons and cross correlaons. Src saonary s appealng because affords a form of homogeney across erms whou requrng ha hey be ndependen. Covarance saonary s he condon ha s more frequenly assumed n GARCH models. I does requre ha all frs and second momens exs whereas src saonary does no. In hs one respec, covarance saonary s a sronger condon (Holon, 996). In hs paper we are gong o consruc lnear models combnng mean and varance equaons holdng eher covarance or src saonary. We wll use ARMA for mean and GARCH for varance spesfcaon. 2. ARMA (R, S) And ARFIMA (R, D, S) Processes In The Condonal Mean Box and Jenkns nroduced a flexble famly of me seres models capable of expressng a varey of shor-range seral relaonshps n erms of lnear regresson, where he predcors are prevous observaons and prevous resdual errors.

4 The Use Of ARCH And GARCH Models. 8 One componen of he Box-Jenkns framework s he auoregressve (AR) equaon, where predcng varables are prevous observaons. An AR(r) model, where he predcors are he prevous r erms, s defned as y r = φ y + ε () = In equaon, he curren value y s parly based on he value a me - ( r), and parly based on a random varable ε, ypcally Gaussan nose. The nfluence of pror values s usually assumed o decay over me, such ha φ >φ 2 > >φ r. A second componen n he Box-Jenkns framework s he movng average (MA) process. In an MA process, he observaon y s dependen no on he prevous values of y, bu raher on he values of he nose random varable ε. A movng average model of order s, MA(s), s defned by s (2) y = θ ε + ε j j j j= where y depends on he prevous s errors ε -j (j s) and he curren error ε. Ooms and Doornk (999) presen he basc ARMA(r,s) model as φ φ ε θε θε y = y y and he general ARMA (r,s) s n he form r s y = + y + + r r s s φ φ ε φ ε 0 j j = j= where r s he order of he AR(r) par, φ s parameers, s he order of he MA(s) par, θ j s parameers and ε normally and dencally dsrbued nose or nnovaon process. The famly of ARMA models as defned by equaon 4 s flexble and able o concsely descrbe he seral dependences of seemngly complex me seres n erms of he number of parameers (.e., he order or hsory) of he AR and MA componens, and he values of hese parameers. In felds such as physcs and economcs, phenomena ha flucuae over me ofen dsplay long-range seral correlaons. In order o correcly denfy and parsmonously descrbe processes ha gve rse o perssen seral correlaons, radonal ARMA me seres models can be exended o allow for fraconal negraon o capure long-range correlaons. The resulng ARFIMA models, popular n economercs and hydrology, allow for smulaneous maxmum lkelhood esmaon of he parameers of boh shor-range and long-range processes. (3) (4)

5 82 Bahadn Rüzgar, İsme Kale Followng he descrpon of Lauren and Peers (2002), by usng lag polynomnals and nroducng a mean μ, he equaon 4 becomes φ( L)( y (5) μ ) = θ( L) ε where L s he lag operaor, μ s he uncondonal mean of y, r φ ( L ) = φ L and θ ( L ) = + θ L are he auoregressv and = he movng average operaors n he lag operaor. They are polnomnals of order r and s respecvely. Wh a fraconal negraon parameer d, he ARFIMA(r, d, s) model s wren as d φ( L)( L) ( y μ ) = θ( L) ε (6) The fraconal dfferencng operaor (-L) d s a noaon for he followng nfne polynomnal: d Γ( d) ( L) = L π ( ) (7) d L Γ ( + ) Γ( d) where π ( z) Γ( z)/ Γ ( + ) Γ( z) and Γ(.) s he Sandard gamma funcon. To ensure saonary and nverbly of he process y, d les beween -0.5 and 0.5. Gven daa seres y one can use condonal or exac lkelyhood mehod o specfy he order and parameers. The Ljung-Box sascs of resduals can check he f. Bhardwaj and Swanson (2004) found ha ARFIMA models perform beer for greaer forecas horzons and ha hey under ceran condons provde sgnfcanly beer ou-of-sample predcons han AR, MA, ARMA, GARCH, smple regme swchng, and relaed models. Throughou he paper ARMA spesfcaon wll only be used o model he mean of reurns. ARMA (0,0) mples a consan mean, ARMA(,0) s smply AR(). I s also possble o make he condonal mean a funcon of he condonal varance. In ha case he condonal varance derved from he GARCH model wll be a varable n he mean equaon. Ths hen wll be he so called ARCH-n-mean model, whch we denoe n hs paper wh (-m) n namng our models. s = = 0 = GARCH Processes o Model The Condonal Varance If he value of he sock marke ndex a me s marked P, he reurn of he ndex a me s gven by y = ln (P / P - ) where ln denoes naural logarhm. For he log reurn seres y, we assume s mean s ARMA modelled, hen le ε = y - μ be he mean correced log reurn. Sock marke ndex reurns can be modelled wh he help of he followng equaon: y = μ + ε, (8)

6 The Use Of ARCH And GARCH Models. 83 where µ s he mean value of he reurn, whch s expeced o be zero; s a random componen of he model, no auocorrelaed n me, wh a zero mean value. Sequence ε may be consdered a sochasc process, expressed as: ε = z σ (9) where z s a sequence of ndependenly and dencally dsrbued random varables, wh a dsrbuon E(z ) = 0 and Var(z ) =. By defnon ε s serally uncorrelaed wh a mean equal o zero, bu s condonal varance equals σ 2 and herefore may change over me, conrary o wha s assumed n he sandard regresson model. The condonal varance s he measure of our uncerany abou a varable gven a model and nformaon se. Followng Markowz' defnon of volaly as sandard devaon of he expeced reurn, σ s he volaly of log reurns a me, he changes of whch wll be modelled by means of he followng ARCH-ype models The ARCH Model Volaly cluserng, or perssence, suggess a me-seres model n whch successve dsurbances, alhough uncorrelaed, are noneheless serally dependen. Rob Engle had he grea nsgh o nroduce and sudy he class of auoregressve condonally heeroscedasc (ARCH) me seres models for modelng he me-varyng volaly cluserng phenomenon (Engle, 982). He used a weghed average of squared pas resduals over a long perod wh hgher weghs on he recen pas and small bu non-zero weghs on he dsan pas. The ARCH (q) model can be expressed as ε =z σ (0) z ~..d D(0,) σ 2 = σ 2 (ε -, ε -2,,, x, b) = σ 2 (σ -z -, σ -2z -2,,, x, b) where ε denoes he predcon error a me, x s a vecor of lagged exogenous varables, b s a vecor of parameers, D(.) s dsrbuon. The condonal varance of ε gven he nformaon a me - s σ 2. For he parameerzaon of hs varance many possbles are suggesed n he leraure. In s orgnal form ARCH can be wren as σ = α + α ε αqε q usng operaor equaon becomes q = σ α αε () = + (2) or by replacng ε = z σ, o more clearly noce he auoregresson, we ge

7 84 Bahadn Rüzgar, İsme Kale q z = σ α ασ = + (3) The ARCH model can descrbe volaly cluserng hrough he followng mechansm: f ε - was large n absolue value, σ 2 and hus ε s expeced o be large n absolue value as well. Even f he condonal varance of an ARCH model s me varyng, he uncondonal varance of ε s consan provded ha α 0 > 0 and α < q. Condonal varance σ 2 has o be posve for all. Suffcen condons are when α 0 > 0 and α 0. Evdence has shown ha a hgh ARCH order has o be seleced o cach he dynamcs of he condonal varance. Ths nvolves he esmaon of a large number of parameers. The generalzed ARCH (GARCH) model of Bollerslev (986) s based on an nfne ARCH spesfcaon and allows reducng he number of esmaed parameers by mposng nonlnear resrcons on hem. = The GARCH Model The GARCH model addonally assumes ha forecass of varance changng n me also depend on he lagged condonal varances of capal asses. An unexpeced ncrease or fall n he reurns of an asse a me wll generae an ncrease n he varably expeced n he perod o come. Inroduced by Engle (982) and Bollerslev (986) he mosly used GARCH (p,q) models make σ 2 a lnear funcon of lagged condonal varances and squared pas resdual σ p = α + α ε + + αqε q + β σ + + β σ p (4) usng operaor q p (5) σ = α + αε + βσ = j= where p s he degree of GARCH; q s he degree of he ARCH process, α 0 >0, α 0, β j 0. The covarance saonary condon s q p α + β j = j= <. Snce he equaon expresses he dependence of he varably of reurns n he curren perod on daa (.e. he values of he varables ε - 2 and σ -j 2 ) from prevous perods, we denoe hs varably as condonal. One can observe ha an mporan feaure of he GARCH (p,q) model s ha can be regarded as an ARMA (r,s), where r s he larger of p and q. Ths resul allows economercans o apply he analyss of ARMA process o he GARCH model. Usng he lag operaor, he GARCH (p,q) model can be rewren as: (ω=α 0 )

8 The Use Of ARCH And GARCH Models σ = ω + α(l) ε + β(l) σ (6) where L denoes he lag operaor and α(l) and β(l) denoe he AR and MA polynomnals respecvely, wh α(l) = α (L) + α 2 (L) α q (L) q and β(l) = β (L) + β 2 (L) β p (L) p. If all he roos of he polynomnal - β(l) =0 le ousde of he un crcle, we ge: σ 2 = ω - β(l) - + α(l) - β(l) - ε 2, (7) whch may be regarded as an ARCH ( ) process, snce he condonal varance lnearly depends on all prevous squared resduals. The uncondonal varance s gven by: 2 2 ω σ E ( ε ) = q p (8) α β = j= The basc and mos wdespread model s GARCH (, ), whch can be reduced o: σ = ω+ αε + βσ (9) As he varance s expeced o be posve, we expec ha he regresson coeffcens ω, α, β are always posve (α and β can also ake he value 0), whle he saonary of he varance s preserved, f he he sum of α and β s smaller han. Condonal varably of he reurns defned n equaon 9 s deermned by hree effecs:. The consan par, whch s gven by he coeffcen ω; 2. Par of varance expressed by he relaonshp αε 2 and desgnaed as ARCH componen; 3. Par gven by he predced varably from he prevous perod and expressed by he relaonshp βσ 2 -. The sum of regresson coeffcens (α+β) expresses he nfluence of he varably of varables from he prevous perod on he curren value of he varably. Ths value s usually close o, whch s a sgn of ncreased effecs of shocks on he varably of reurns on fnancal asses. Whle he basc GARCH model allows a ceran amoun of lepokurc behavour hs s ofen nsuffcen o explan real world daa. We herefore use 3 dsrbuons oher han normal n our analyss, namely Suden-, Skewed- (Lamber and Lauren, ) and Generalzed Error Dsrbuons whch help o allow for he fa als n he dsrbuon. The choce of he quadrac form for he condonal varance has he mporan consequence ha he mpac of he pas values of he nnovaon on he curren volaly s only a funcon of her magnude no of s sgn. The prncpal dsadvanage of he GARCH model s herefore s unsuably or modellng he frequenly observed asymmery ha occurs when a dfferen volaly s recorded sysemacally n he case of good and bad news. j

9 86 Bahadn Rüzgar, İsme Kale Falls and ncreases n he reurns can be nerpreed as good and bad news. If a fall n reurns s accompaned by an ncrease n volaly greaer han he volaly nduced by an ncrease n reurns, we may speak of a leverage effec. Followng classes of asymmerc GARCH models are capable of capurng hs effec The GJR and TARCH model The GJR model s an asymmerc model. I s proposed by Glosen, Jagannahan and Runkle (993). The generalzed verson may be wren as q q p = S + j j = = j= σ α αε γ ε β σ where S s a dummy varable wh S - =, f ε - < 0 and S - =0, f ε - 0. In hs model, s assumed ha he mpac of ε 2 on he condonal varance σ 2 s dfferen when ε s posve or negave. The TARCH model of Zakoan (994) s very smlar o he GJR, where he preferred o model he sandard devaon nsead of he condonal varance. Is basc varan s GJR (,), whch s expressed by: σ (2) = ω+ αε + γσ + βε S The model can be nerpreed ha unexpeced (unforeseen) changes n he reurns of he ndex y expressed n erms of ε, have dfferen effecs on he condonal varance of sock marke ndex reurns. An unforeseen ncrease s presened as good news and conrbues o he varance n he model hrough mulplcaor α. An unforeseen fall, whch s a bad news, generaes an ncrease n volaly hrough mulplcaors α and β. The asymmerc naure of he reurns s hen gven by he nonzero value of he coeffcen β, whle a posve value of β ndcaes a leverage effec. The covarance saonary condon s The EARCH Model q p 2 α( + γ ) + β j = j= <. The exponenal GARCH (EGARCH) model s nroduced by Nelson (99). In hs model, he condonal varance may be expressed as follows: q p 2 2 (22) ln σ = α + α sz ( ) + β ln( σ ) 0 j j = j= where z = ε / σ s he normalzed resduals seres. The funcon s(.) can be wren as s(z ) = δ z + δ 2 { z - E( z ) } (23) (20)

10 The Use Of ARCH And GARCH Models. 87 Therefore δ z adds he effec of he sgn of ε whereas δ 2 { z - E ( z ) adds s magnude effec. E ( z ) depends on he choce of he dsrbuon of reurn seres. For he normal dsrbuon E ( z ) = 2 π. Is basc varan s EGARCH (, ) wh normal dsrbuon s expressed by: 2 ε ε 2 2 ln σ = ω + α( δ + δ 2 ) + β ln σ (24) σ σ π The asymmerc naure of he reurns s hen gven by he nonzero value of he coeffcen δ, whle a posve value of δ ndcaes a leverage effec. The use of ln ransformaon ensures ha σ 2 s always posve and consequenly here are no resrcons on he sgn of he parameers. Moreover exernal unexpeced shocks wll have a sronger nfluence on he predced volaly han TARCH or GJR The APARCH Model In general, he ncluson of a power erm acs so as o emphasse he perods of relave ranqully and volaly by magnfyng he oulers n ha seres. Squared erms are herefore so ofen used n models. If a daa seres s normally dsrbued han we are able o compleely characerse s dsrbuon by s frs wo momens (McKenze and Mchell, 200). If we accep ha he daa may have a non-normal error dsrbuon, oher power ransformaons may be more approprae. Recognsng he possbly ha a squared power erm may no necessarly be opmal, Dng, Granger and Engle (993) nroduced a new class of ARCH model called he Power ARCH (PARCH) model. Raher han mposng a srucure on he daa, he Power ARCH class of models esmaes he opmal power erm. Dng, Granger and Engle (993) also specfed a generalsed asymmerc verson of he Power ARCH model (APARCH). The APARCH (p,q) model can be expressed as: q p δ δ δ (25) σ = α + α ( ε γ ε ) + β σ 0 j j = j= where α 0 > 0, δ 0, β j 0, α 0 and - < γ <. Ths model couples he flexbly of a varyng exponen δ wh he asymmery coeffcen γ o ake he leverage effec no accoun. Moreover, he APARCH ncludes ARCH, GARCH and GJR as specal cases: ARCH when δ = 2, γ = 0 ( =,...,p) and β j = 0 (j =,...,p), GARCH when δ = 2 and γ = 0 ( =,...,p) and

11 88 Bahadn Rüzgar, İsme Kale GJR when δ = 2 I also ncludes four oher ARCH exenons whch are no esed n hs paper TARCH when δ =, NARCH when γ = 0 ( =,...,p) and β j = 0 (j =,...,p), The Log-ARCH, when δ 0 and Taylor / Schwer GARCH when δ =, and γ = 0 ( =,..., p). A saonary soluon for APARCH model exss. See Dng, Granger and Engle (993) for deals The IGARCH Model In explanng he GARCH (p, q) model was menoned ha GARCH may be regarded as an ARCH ( ) process, snce he condonal varance lnearly depends on all prevous squared resduals. Moreover was saed ha a GARCH process s covarance saonary f and only f q p α + β j = j= <. Bu src saonary does no requre such a srngen resrcon ha he uncondonal varance does no depend on, n fac we ofen fnd n esmaon ha q p α + β j = j= s close o. Les denoe h as he melag beween he presen shock and fuure condonal varance. Then a shock o he condonal varance σ 2 has a decayng mpac on σ +h 2. When h ncreases hs mpac becomes neglecable ndcang a shor memory. However f q p α + β j, he effec on σ +h 2 does no de ou even for a very = j= hgh h. Ths propery s called perssence n he leraure. In many hgh frequency me seres applcaons, he condonal varance esmaed usng GARCH (p,q) process exhbs a srong perssence. I was also menoned ha he GARCH (p, q) process can be seen as an ARMA process. I s known ha such an ARMA process has a un roo when q p α + β j = j= =. When he sum of all AR coeffcens and MA coeffcens s equal o one, he ARMA process s negraed (ARIMA). Due o her smlary o ARMA models GARCH models are symerc and have shor memory.

12 The Use Of ARCH And GARCH Models. 89 A GARCH model ha sasfes q p α + β j = j= = (or equally rewren as α( L) + β ( L) = ) s known as an negraed GARCH (IGARCH) process (Engle and Bollerslev, 986), meanng ha curren nformaon remans of mporance when forecasng he volaly for all horzons. IGARCH (p, q) models are a knd of ARIMA models for volales. Recall ha GARCH model s: 2 2 defne υ ε σ σ = ω+ α( L) ε + β( L) σ(26) 2 2 ε (27) = ω+ ( α( L) + β( L)) ε + β( L) υ + υ Ths s an ARMA {max (p, q), p} model for he squared nnovaons. If α( L) + β ( L) = hen we have an Inegraed GARCH model (IGARCH). The IGARCH can be expressed as: 2 φ ( L)( L) ε = ω + β ( L) ν or (28) [ ] [ ]( ) φ ( L )( L ) ε = ω + β ( L ) ε σ, (29) where [ ]( ) θ( L) = α( L) β( L) L s a polynomnal of order {max (p,q)- } We can also express he condonal varance as a funcon of he squared resduals, and hen an IGARCH (p, q) becomes: 2 ω [ ] [ ] { 2 σ = + φ( L)( L) β ( L) } ε (30) β ( L ) Alhough a process y ha follows an IGARCH process s no covarance saonary, and s uncondonal varance s nfne, IGARCH process s sll mporan snce he uncondonal densy of y s he same for all, and hus he process can come from a srcly saonary process. However we may suspec ha IGARCH s more a produc of omed srucural breaks han he resul of rue IGARCH behavor. An negraed process wll be hereafer denoed as I(), a non-negrared process I(0). The assumpon of shor memory such as n GARCH models s usually no fulflled. Dng Granger and Engle (993) durng her research for he APARCH model have found ha he absolue reurns and her power ransformaons have a hghly sgnfcan long-erm memory propery as he reurns are hghly correlaed. For example, sgnfcan posve auocorrelaons were found a over 2,700 lags n 7,054 daly observaons of he S&P 500. Tha makes 2700 lags / 252 radng days = 0.7 years. On he oher hand he mplcaons of IGARCH models are oo srong whch leads o he consderaon of fraconally negraed models.

13 90 Bahadn Rüzgar, İsme Kale The FIGARCH Model As shown n Dng, Granger, and Engle (993) among ohers, he effecs of a shock can ake a consderable me o decay. Therefore, he dsncon beween I(0) and I() processes seems o be far oo resrcve. In an I(0) process he propagaon of shocks occurs a an exponenal rae of decay so ha only capures he shormemory, whle for an I() process he perssence of shocks s nfne. In he condonal mean, he ARFIMA specfcaon has been proposed o fll he gap beween shor and complee perssence, so ha he shor-run behavor of he me-seres s capured by he ARMA parameers, whle he fraconal dfferencng parameer allows for modellng he long-run dependence. The frs long memory GARCH model was he fraconally negraed GARCH (FIGARCH) nroduced by Balle, Bollerslev and Mkkelsen (996). The FIGARCH (p, d, q) model s a generalzaon of he IGARCH model by replacng he operaor (-L) of he IGARCH equaon by (-L) d, where d s he memory parameer. 2 φ( L) ( α) + α( L) d ε = ω+ [ β( L) ] ν (3) where [ ]( ) θ ( L ) = α ( L) β ( L) L s a polynomnal of order {max (p,q)- }(same as IGARCH). We can also express he condonal varance as a funcon of he squared resduals, hen a FIGARCH (p,d,q) becomes: 2 ω d [ ] [ ] { 2 σ = + φ( L)( L) β ( L) } ε (32) β ( L ) FIGARCH models exhb long memory. They nclude GARCH models (for d=) and IGARCH models (for d=). In conras o ARFIMA models, where he memory parameer d s -0.5 < d <+0.5, FIGARCH d s 0 < d <. FIGARCH-processes are non-saonary lke IGARCH-processes. Ths shows ha he concep of un roos can hardly be generalzed from lnear o nonlnear processes. Furhermore, he nerpreaon of he memory parameer d s dffcul n he FIGARCH se up The HYGARCH Model Davdson (200) exended he class of FIGARCH models o HYGARCH(p,α,d,q) models whch sands for hyperbolc GARCH. HYGARCH-models replace he operaor (-L) d n FIGARCH equaon by [(- α)+α(-l) d ]. The paramerzaon of HYGARCH-models s gven by

14 [ β ( L) ] The Use Of ARCH And GARCH Models. 9 { [ ] d ( L) ( L) { ( L) } } ω σ = + β φ + α ε 2 2 The parameers α and d are assumed o be non-negave. HYGARCH-models nes GARCH models (for α = 0), FIGARCH-processes (for α = ) and IGARCH-models (for α = d = ) FI process of Chung v.s. Balle, Bollerslev and Mkkelsen (BBM) Chung (999) underscores some lle drawbacks n he BBM model: here s a srucural problem n he BBM specfcaon snce he drec mplemenaon of he ARFIMA framework orgnally desgned for he condonal mean equaon s no perfec for he use n condonal varance equaon, leadng o dffcul nerpreaons of he esmaed parameers. Indeed he fraconal dfferencng operaor apples o he consan erm n he mean equaon (ARFIMA) whle does no n he varance equaon (FIGARCH). Chung (999) proposes a slghly dfferen process: 2 2 d 2 2 σ = σ + [ β ( L )] φ ( L )( L ) ( ε σ ) (34) or { } (33) σ = σ + λ ( L )( ε σ ) (35) λ (L) s an nfne summaon whch, n pracce, has o be runcaed. BBM propose o runcae λ (L) a 000 lags and nalze he unobserved ε 2 a her uncondonal momen. Conrary o BBM, Chung (999) proposes o runcae λ (L) a he sze of he nformaon se (-) and o nalze he unobserved (ε 2 - σ 2 ) a 0. In our analyss we hold he proposal of BBM and runcae a 000 lags The FIEGARCH and FIAPARCH Model The dea of fraconal negraon has been exended o oher GARCH ypes of models, ncludng he Fraconally Inegraed EGARCH (FIEGARCH) of Bollerslev and Mkkelsen (996) and he Fraconally Inegraed APARCH (FIAPARCH) of Tse (998). Smlarly o he GARCH (p, q) process, he EGARCH (p, q) can be exended o accoun for long memory by facorzng he auoregressve polynomal [ β ( L )] = φ ( L)( L) d where all he roos of φ (z) = 0 le ousde he un crcle. The FIEGARCH (p, d, q) s specfed as follows: [ ] 2 d ln( σ ) ω φ( L) ( L) α( L) s( z ) = + + (36)

15 92 Bahadn Rüzgar, İsme Kale And he FIAPARCH (p, d, q) model can be wren as: d { [ L ] L L } δ δ σ = ω + β( ) φ( )( ) ( ε γε ) (37) 3. Emprcal Applcaons In order o sudy he esmaon and forecasng performances of dfferen GARCH processes, models are appled o Isanbul Sock Exchange 00 Index log reurns. We used daly daa from o From a oal of 2200 radng days 2000 are used for esmaon and 200 are lef o es he forecass. The models appled are GARCH, EGARCH, GJR, APARCH, IGARCH, FIGARCH of BBM, FIGARCH of Chung, FIEGARCH, FIAPARCH of BBM, FIAPARCH of Chung and HYGARCH. The (p,q) = (,) varan of all models are sysemacally esed wh four dfferen dsrbuons, namely, Gaussan Normal, Suden-, Generalzed Error Dsrbuon (GED) and Skewed- dsrbuon. Tha makes 44 basc models. Moreover for maers of observaon 50 more models of hgher order combned wh ARMA are unsysemacally expermened o examne he effecs on esmaon and forecasng performances wh an emphass on Maxmum Lkelhood. We have wren an Ox code named IMKB_Esmae&Forecas.ox on he bass of he examples and objecs provded n he Ox 3.40 and s G@RCH 3.00 package and used for our analyss. G@RCH 3.0 of Lauren and Peers (2002) s a package dedcaed o GARCH models and many of s exensons. I s wren n he Ox programmng language (see Doornk, 999). G@RCH 3.0 can be downloaded free of charge for academc purposes a hp:// The program proved o be very flexble and fas. Currenly only sharewares provded for academc research are capable of analyzng such recen varey of models. Moreover mos sandard sofware do no allow for such a flexble combnaon of processes lke we appled n our analyss. Wh open-source code are able o add or modfy specfcaons, processes or graphcs n he fuure. Esmaon resuls are evaluaed on he bass of ML, Akake, Schwarz, Shbaa and Hannan-Qunn values, whereas forecasng resuls are ranked accordng o Mncer Zarnowz regresson R 2, Roo Mean Square Error, Mean Square Error and Mean Absolue Error value crera. 3. Esmaon Resuls I s apparen ha dsrbuons shall be preferred f one ams o oban a beer represenaon of he exsng daa. Among he frs 5 bes basc esmang models accordng o all fve crera all was eher suden- or skewed-. GED and Normal ds-

16 The Use Of ARCH And GARCH Models. 93 rbuons follow. Therefore n comparng he esmaon powers of models, we resrc our commens o suden and skewed- dsrbuons. Tes sascs of some models n esmang performances and forecasng performances accordng o dfferen crera are gven Table 3..5 and Table Table 3..: Frs 20 And Las Ten Models Wh Consan Mean İn Esmang Performances Accordng To Dfferen Crera 5 0 Akake Schwarz Shbaa Hannan-Qunn Log-L LogL k LogL kloglog [ ( n) ] LogL log ( k ) LogL n + 2k 2 + log n n n n n n n n FIAparchChSk FIAparchBBMS- FIgarchChS- FIAparchBBMS- FIgarchChS- FIAparchBBMSk FIAparchChSk FIgarchBBMS- FIAparchChSk FIgarchBBMS- FIAparchBBMS- FIAparchBBMSk HYGarchS- FIAparchBBMSk FIAparchBBMS- FIAparchChS- FIAparchChS- FIgarchChSk FIAparchChS- FIAparchChS- HYGarchSk FIgarchChS- FIGarchBBMSk FIgarchChS- HYGarchS- HYGarchS- FIgarchBBMS- FIAparchBBMS- FIgarchBBMS- FIgarchChSk FIgarchChSk HYGarchS- IgarchS- HYGarchS- FIGarchBBMSk FIGarchBBMSk FIgarchChSk FIAparchChS- FIgarchChSk FIAparchChSk FIgarchChS- FIGarchBBMSk GarchS- FIGarchBBMSk FIAparchBBMSk FIgarchBBMS- HYGarchSk GjrS- HYGarchSk HYGarchSk AparchSk GjrS- HYGarchSk GjrS- GjrS- GjrSk GjrSk FIAparchChSk GjrSk GjrSk AparchS- AparchS- FIAparchBBMSk AparchS- GarchS- GjrS- AparchSk IgarchSk AparchSk AparchS- FIAparchChGED GarchS- GarchSk GarchS- IgarchS- FIAparchBBMGED GarchSk GjrSk GarchSk GarchSk GarchSk FIAparchChGEDFIGarchChGED FIAparchChGEDAparchSk HYGarchGED FIAparchBBMGED AparchS- FIAparchBBMGED FIGarchChGED GarchS- FIGarchChGED FIGarchBBMGEDFIGarchChGED FIGarchBBMGED FIGarchChGED FIGarchBBMGEDAparchSk FIGarchBBMGEDIGarchSk FIgarchBBMN HYGarchN GarchN HYGarchN HYGarchN EGarchSk GjrN IgarchN GjrN GarchN AparchN GarchN GjrN GarchN GjrN GjrN AparchN AparchN AparchN AparchN GarchN EGarchSk EGarchSk EGarchSk IgarchN IgarchN IgarchN EGarchGED IgarchN EGarchSk EGarchGED EGarchGED FIEgarchN EGarchGED EGarchGED FIEgarchS- FIEgarchS- FIEgarchS- FIEgarchS- FIEgarchN FIEgarchN FIEgarchN EGarchN FIEgarchN FIEgarchS- EGarchN EGarchN FIAparchBBMGED EGarchN EGarchN Log-L = log lkelhood value, n = number of observaons, k = number of esmaed parameers

17 94 Bahadn Rüzgar, İsme Kale For opmzng maxmum lkelhood, skewed- performs beer han suden- for all models. On he oher hand f we evaluae accordng o he oher four crera, by whch more complcaed models are penalzed for he ncluson of addonal parameers, skewed- looses s apparen advanage, because requres an addonal skewness parameer. Especally Hannan-Qunn es seems o judge accordng o he dsrbuon raher han he model specfcaon and prefer suden-. We found ha he choce of models s a leas as mporan as he choce of dsrbuons, because bes performng models combned wh boh dsrbuons found place n he fron ranks, mosly successvely. For rankng models wh consan mean n esmang performances accordng o dfferen crera, es sascs, llusraed on Table 3..5, are used. Table 3..2: Mnmum Sum Of Rankngs Of Dfferen GARCH And FIAPARCH Specfcaons Log- L Akake Shbaa Hannan-Qunn Toal ARGarch22Sk Garch22Sk ARMA22Garch-m33Sk ARGarch-mSk ARGarchSk GarchS GarchSk GarchGED ARGarchN GarchN Log-L Akake Schwarz Shbaa Hannan-Qunn Toal ARFIAparchChSk ARMAFIAparchCh2Sk ARFIAparchCh2Sk ARFIAparchCh2S FIAparchChSk FIAparchChS FIAparchChGED ARFIAparchCh2GED ARFIAparchCh2N FIAparchChN Log Lkelhood resuls are conssen wh aggregae resuls. Normal dsrbuon esmaes worse for all models. Hgher orders alone mprove resuls

18 The Use Of ARCH And GARCH Models. 95 more han ARMA specfcaons alone. Togeher hey mprove more bu he margnal benef decreases. Among he basc models fraconally negraed ones, especally FIAPARCH of BBM and ha of Chung combned wh suden- and skewed- dsrbuons perform ousandng based on all crera. I s also o noe ha based on he esmaon power, he mehods of BBM and Chung repor only slgh dfferences. We can conclude ha among models wh he same dsrbuon and same mean specfcaon a FI () model s a beer esmaor han s FI (0) counerpar. Ths s a clear ndcaor ha IMKB 00 ndces shows srong perssence and he effec of shocks nfluence fuure reurns for long perods. Tha FI () performs beer hen I () show ha he perssence s no compleely permanen. Table 3..3: Comparson Of APARCH And FIAPARCH Models Model Srucure Model Mean Equaon Varance Equaon Dsrbuon Log-L ARMAAparch33Sk ARMA(,) Aparch (3,3) Skewed ARAparch22Sk ARMA(,0) Aparch (2,2) Skewed Aparch22Sk ARMA(0,0) Aparch (2,2) Skewed Aparch33Sk ARMA(0,0) Aparch (3,3) Skewed ARAparchSk ARMA(,0) Aparch (,) Skewed AparchSk ARMA(0,0) Aparch (,) Skewed AparchS- ARMA(0,0) Aparch (,) Suden ARMAAparch-mGED ARMA(,) Aparch (,) GED AparchGED ARMA(0,0) Aparch (,) GED AparchN ARMA(0,0) Aparch (,) Normal ARFIAparchBBM2Sk ARMA(,0) FIAparchBBM (2,d,) Skewed ARFIAparchBBMS- ARMA(,0) FIAparchBBM (,d,) Suden FIAparchBBMSk ARMA(0,0) FIAparchBBM (,d,) Skewed FIAparchBMMS- ARMA(0,0) FIAparchBBM (,d,) Suden ARFIAparchBBMGED ARMA(,0) FIAparchBBM (,d,) GED FIAparchBMMGED ARMA(0,0) FIAparchBBM (,d,) GED FIAparchBBMN ARMA(0,0) FIAparchBBM (,d,) Normal A rankng among he specfcaons would be he exsence of FI followed by he exsence of asymery. The coeffcens ndcang asymmery are mosly sgnfcan and showng ha here exs a leverage effec n IMKB 00. The smple GARCH (,) wh dsrbuons follow he more complex models wh dsrbuons bu s clearly beer han any model, even he FI() and asymmerc models, combned

19 96 Bahadn Rüzgar, İsme Kale wh GED or Normal dsrbuon. No EGARCH model could reach srong convergence durng numercal opmzaon and resuls are msleadng. Wh dsrbuons and mean specfcaons equal, he FI () verson ouperforms he FI (0) verson based on Log Lkelhood. Gven he same dsrbuon and same GARCH orders, usng an auoregresson AR() or auoregresson+movng average ARMA (,) or an ARCH-n-mean effec n he mean equaon mproves he performance of all models f he rankng s based on maxmum lkelhood. Agan he oher four crera ake he addonal parameers n accoun, however mos of he me he mprovemen n maxmum lkelhood s large enough o compensae for he esmaon burden of addonal parameers. The margnal mprovemen decreases as he mean equaon ges more complex. The sand-alone effec of ncreasng he orders s sgnfcanly more han sandalone effec of manpulang he mean equaon. Orders of (2, 2) or (3, 3) perform well, bu he chances of non-convergence and geng msleadng resuls also ncreases. Table 3..4: Bes 20 Esmang Models Based On Log Lkelhood Model Srucure Model Mean Equaon Varance Equaon Dsrbuon Log-L R 2 ARMAFIAparchCh2Sk ARMA(,) FIAparchBBM(2,d,)Skewed ARFIAparchCh2Sk ARMA (,0) FIAparchCh (2,d,) Skewed ARMAAparch33Sk ARMA (,) Aparch (3,3) Skewed ARMAGJR33Sk ARMA (,) GJR (3,3) Skewed ARFIAparchChSk ARMA (,0) FIAparchCh (,d,) Skewed ARFIAparchBBM2Sk ARMA (,0) FIAparchBBM(2,d,)Skewed ARFIAparchCh2S- ARMA (,0) FIAparchCh (2,d,) Suden ARGjr22Sk ARMA (,0) GJR (2,2) Skewed ARMA22Garch-m33Sk ARMA (2,2) ARCH-mGARCH (3,3) Skewed ARAparch22Sk ARMA (,0) Aparch (2,2) Skewed ARFIAparchBBMS- ARMA (,0) FIAparchBBM (,d,) Suden FIAparchChSk ARMA (0,0) FIAparchCh (,d,) Skewed FIAparchBBMSk ARMA (0,0) FIAparchBBM(,d,)Skewed Aparch22Sk ARMA (0,0) Aparch (2,2) Skewed FIAparchBBMS- ARMA (0,0) FIAparchBBM(,d,)Suden Gjr33Sk ARMA (0,0) GJR (3,3) Skewed Aparch33Sk ARMA (0,0) Aparch (3,3) Skewed FIAparchChS- ARMA (0,0) FIAparchCh (,d,) Suden ARHYGarch22Sk ARMA (,0) HYGarch (2,d,2) Skewed ARHYGarchSk ARMA (,0) HYGarch (,d,) Skewed

20 The Use Of ARCH And GARCH Models. 97 FIAPARCH models of dfferen specfcaons and dsrbuons domnae. Includng mean specfcaons clearly mprove Log-L resuls. Skewed- dsrbuon seems o be he soluon for fa als. The combnaon of more complex mean equaon wh hgher orders perform overall beer wh he cos of addonal parameers. As a resul we can conclude ha, based on he maxmum lkelhood, he more complex a model s he beer fs o he daa. Combned rankngs allow for he concluson ha maxmum lkelhood s a conssen evaluaon creron. Whle Akake, Shbaa, Schwarz and Hannan-Qunn may resul n complee dfferen rankngs, he aggregae resuls are conssen wh ha of he maxmum lkelhood. If skewed- or suden- dsrbuons are used, s always possble o ncrease he lkelhood by addng more alored processes, mplyng a beer f o he daa on he bass of numbers. However usng he graphs we can show ha even he smples GARCH (, ) s a sasfacory model n esmaon. The dfferences are no suble and usng a GARCH (, ) model would no lead o a dfferen decson han a decson based on a more complex ARMA (, ) FIAPARCH (2, ) model. GARCH models n general succeed n reproducng volaly cluserng, perssence, leverage effec and fa al behavor of real world daa. Graph 3..: ARMA(,)FIARCHCh(2,)Skewed-.

21 98 Bahadn Rüzgar, İsme Kale Bes esmaor n es, Log-L=424,82. Noe ha he seres and he resduals are almos dencal mplyng a good reproducon of characerscs, oulers are perfecly cached. Graph 3..2: GARCH(,)Normal. One of he worse esmaors n comparson, Log-L= Reproducon of he daa was able o cach he mporan oulers bu ends o say closer around mean. Resduals graph s hcker. Noe he hghes wo pons around daa 200 (November-December 2000). Condonal varance s hgher where medum sze resduals of around ± 0. cluser han where sngle bg resdual of around Ths s he oppose n above graph. GARCH (, ) Normal has slower responses. Table 3..5: Tes Sascs Of Each Model Informaon Crerum (mnmze) Model Log-L Akake Schwarz Shbaa Hannan-Qunn FIAparchChSk FIAparchBBMSk FIAparchBBMS FIAparchChS HYGarchSk HYGarchS FIgarchChSk FIGarchBBMSk FIgarchChS FIgarchBBMS AparchSk

22 Table 3..5 devamı The Use Of ARCH And GARCH Models. 99 GjrSk AparchS GjrS FIAparchChGED FIAparchBBMGED GarchSk HYGarchGED GarchS FIgarchChGED FIgarchBBMGED IgarchSk IgarchS GjrGED AparchGed GarchGED IgarchGED EGarchS FIEgarchSk FIEgarchGED FIAparchChN FIAparchBBMN FIgarchChN HYGarchN FIgarchBBMN EGarchSk AparchN GjrN GarchN IgarchN EGarchGED FIEgarchS FIEgarchN EGarchN Forecasng Resuls As we expeced, he bes models for esmaon are no necessarly he bes ones for forecasng. The same hng s also rue for he dsrbuons. The specfcaon of he model has a more clear and predcable effec on Mncer Zarnowz regresson R 2. As explaned by Lauren and Peers (2002) he Mncer-Zarnowz regresson has been largely used o evaluae forecass n he condonal mean. For he condonal varance, s compued by regressng he forecased varances on he acual varances. σ = α+ βσˆ + υ (34) 2 2

23 00 Bahadn Rüzgar, İsme Kale The oher crera are mnmzng errors and lead o dffcul nerpreaon and nconssen rankngs. Lke he maxmum lkelhood n esmaon, R 2 s n general more conssen wh he aggregaed rankng resuls. Table 3.2.: 20 Bes Forecasng Models Based On R 2 Model Srucure Model Mean Equaon Varance Equaon Dsrbuon Log-L R 2 IgarchSk ARMA (0,0) Igarch (,) Skewed IgarchN ARMA (0,0) Igarch (,) Normal IgarchGED ARMA (0,0) Igarch (,) GED Aparch33Sk ARMA (0,0) Aparch (3,3) Skewed IgarchS - ARMA (0,0) Igarch (,) Suden Gjr33Sk ARMA (0,0) GJR (3,3) Skewed ArIgarchSk ARMA (,0) Igarch (,) Skewed ARMAIgarchGED ARMA (,) Igarch (,) GED ARMAIgarchSk ARMA (,) Igarch (,) Skewed AparchSk ARMA (0,0) Aparch (,) Skewed GjrSk ARMA (0,0) GJR (,) Skewed ARMAGJR33Sk ARMA(,) GJR (3,3) Skewed AparchS- ARMA (0,0) Aparch (,) Suden GjrS- ARMA (0,0) GJR (,) Suden AparchN ARMA (0,0) Aparch (,) Normal GjrN ARMA (0,0) GJR (,) Normal GjrGED ARMA (0,0) GJR (,) GED AparchGed ARMA (0,0) Aparch (,) GED ARAparchSk ARMA (,0) Aparch (,) Skewed GarchSk ARMA (0,0) GARCH (,) Skewed IGARCH performance s worh nong. Rskmercs process of J.P. Morgan s also a knd of IGARCH. For deals of he model see Mna and Xao (200). Graph 3.2.: IGARCH(,) Sk s one of he negraed models ha proved o be a good forecas model based on R 2. MSE ells s he hrd worse.

24 The Use Of ARCH And GARCH Models. 0 Graph 3.2.2: EGARCH(,) GED converged only weakly afer 208 BFGS eraons n 52 seconds. However reached a record level R 2 and MSE. Graphcally seems o make a farly good condonal varance forecas of absolue reurns. A closer look predcs ha he peak pons of forecass follow acual daa wh a small delay. Based on MAE and RMSE hs model s he second worse. Graph 3.2.3: ARMA(,)FIAPARCH(2,d,) Sk was he bes esmaor accordng o Log-L, an average forecaser based on R 2. RMSE ranks o he boom. GED and skewed- dsrbuons performed well n predcons bu s no possble o favor any dsrbuon. Whle RMSE and MAE ranked he -dsrbuons beer, MSE favored GED and pus dsrbuons o he boom. R 2 made no mplcaons on he dsrbuon. Normal dsrbuons gve conssenly moderae resuls.

25 02 Bahadn Rüzgar, İsme Kale Table 3.2.2: Frs 20 And Las Ten Models Wh Consan Mean İn Forecasng Performances Accordng To Dfferen Crera R 2 RMSE R 2 Rank MSE R 2 Rank MAE R EGarchGED GarchSk 4 GjrGED 2 GarchGED 7 IgarchSk FIAparchBBMSk 9 AparchGed 3 FIEgarchSk 40 IgarchN FIAparchChSk 8 EGarchGED GarchSk 4 IgarchGED FIGarchBBMSk 25 FIEgarchN 38 FIAparchBBMSk 9 IgarchS- IgarchSk 2 AparchN 0 FIAparchChSk 8 AparchSk HYGarchSk 2 GjrN FIGarchBBMSk 25 GjrSk FIgarchChSk 22 FIAparchBBMG 20 IgarchSk 2 ED AparchS- AparchSk 6 FIAparchChGED 36 HYGarchSk 2 GjrS- GjrSk 7 FIEgarchGED 39 FIgarchChSk 22 AparchN HYGarchS- 32 GarchGED 7 FIAparchBBMS- 23 GjrN FIgarchChS- 33 FIgarchChGED 28 FIAparchChS- 37 GjrGED FIEgarchSk 40 FIAparchChN 35 IgarchS- 5 AparchGed IgarchN 3 IgarchGED 4 GarchS- 6 GarchSk FIAparchBBMS- 23 FIAparchBBMN 24 AparchSk 6 GarchN FIAparchChS- 37 HYGarchGED 26 GjrSk 7 GarchS- IgarchS- 5 FIgarchBBMGED 3 HYGarchS- 32 GarchGED GarchS- 6 AparchS- 8 FIgarchChS- 33 FIAparchChSk FIgarchBBMS- 34 GjrS- 9 FIgarchBBMS- 34 FIAparchBBMSk GjrGED 2 GarchN 5 FIgarchChGED 28 FIAparchBBMGED AparchGed 3 FIgarchBBMN 29 FIAparchChN 35. FIgarchBBMGE HYGarchGED 26 FIgarchChS- 33 FIAparchChGED 36 HYGarchS- FIgarchBBMGED 3 FIgarchBBMS- 34 FIEgarchGED 39 FIgarchChS- AparchS- 8 FIEgarchSk 40 IgarchGED 4 FIgarchBBMS- GjrS- 9 GarchSk 4 FIAparchBBMN 24 FIAparchChN GarchN 5 FIAparchBBMSk9 HYGarchGED 26 FIAparchChGED FIgarchBBMN 29 FIAparchChSk 8 FIgarchBBMGED 3 FIAparchChS- HYGarchN 30 FIGarchBBMSk 25 GjrGED 2 FIEgarchN FIgarchChN 27 IgarchSk 2 AparchGed 3 FIEgarchGED EGarchGED HYGarchSk 2 EGarchGED FIEgarchSk FIEgarchN 38 FIgarchChSk 22 FIEgarchN 38 Whle R 2 ranks accordng o model specfcaon, mnmum error crera seem o gve more mporance o dsrbuons. FI () models perform poor forecass based on R 2 whereas oher crera do no allow for a concluson

26 The Use Of ARCH And GARCH Models. 03 I s also hard o draw conclusons from he model specfcaon. Remarkable are he forecasng performances of GJR, IGARCH and GARCH. R 2 gave he wors performances wh FI () models whle he bes performers were I () and nonnegraed ones. Therefore we can conclude ha eher a complee negraon or no negraon s preferred o a fraconal negraon. In general one obans beer R 2 resuls he smpler a model s specfed. Increased parameers hrough modfcaons n mean or hgher orders provde poor R 2 resuls. Especally he order (2, 2) conssenly oupus very poor R 2. The order (3, 3) can be eher a good performer or a bad choce, bu s worh ryng. The resuls of FI processes of BMM and Chung are agan very smlar. Accordng o he evaluaon crera hey are eher among he frs or among he very las. Table 3.2.3: Mnmum Sum Of Rankngs For Dfferen IGARCH Specfcaons R 2 RMSE MSE MAE TOTAL IgarchN IgarchGED IgarchSk 4 5 IgarchS Igarch33Sk Igarch22Sk ArIgarchSk ARMAIgarchGED ARIgarch22Sk ARMAIgarchSk R 2 rankngs are conssen wh aggregae rankngs. In general smpler models wh less parameer perform beer forecass. Table 3.2.4: Mnmum Sum Of Rankngs For Dfferen GARCH Specfcaons R 2 RMSE MSE MAE TOTAL GarchGED 4 7 GarchN GarchS GarchSk ARGarchN ARGarchSk ARGarch-mSk Garch ARMA22Garch-m33Sk ARGarch22Sk

27 04 Bahadn Rüzgar, İsme Kale RMSE, MSE and MAE gve very dfferen rankngs n cross comparson. However n general mean specfcaons resrc he flexbly of all models and resul n a general rend and can no capure he oulers. Smple GARCH(,) performs generally well accordng o all crera. GARCH esmaon oupus he sum of all coeffcens very close o. Ths explans s forecasng success close o IGARCH. The sze of he sample s a crucal facor affecng he forecasng performance. Therefore we beleve ha mos models would behave dfferenly wh dfferen sample szes whch could be he opc of a separae research. Tes sascs for forecas evaluaon measures of some models are gven n he followng Table. Table 3.2.5: Tes Sascs Of Some Models İn Forecasng Performances Accordng To Dfferen Crera Forecas Evaluaon Measures Model R² MSE(M) MSE(V) MAE(M) MAE(V) RMSE (M) RMSE (V) ARMA22Garch-m33Sk 0, E ARGarch22Sk E Garch22Sk E ARGarch-mSk E ARGarchSk E GarchSk E GarchS E EGarchS E ARMAEgarch22Sk E AREgarchSk E EGarchSk E ARMAEgarch22GED E EGarchGED E EGarchN E ARMAGJR33Sk E ARGjr22Sk E Gjr33Sk E GjrSk E GjrS E ARGjrGED E GjrGED E Gjr33N E ARMAGJR22N E GjrN E ARMAAparch33Sk E ARAparch22Sk E Aparch22Sk E Aparch33Sk E ARAparchSk E