GARCH Modelling. Theoretical Survey, Model Implementation and

Transcription

1 Maser Thess GARCH Modellg Theorecal Survey, Model Imlemeao ad Robusess Aalyss Lars Karlsso

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3 Absrac I hs hess we survey GARCH modellg wh secal focus o he fg of GARCH models o facal reur seres The robusess of he esmao of he arameers he model s eamed wh hree dffere dsrbuoal assumos for he ovaos; Gaussa dsrbuo, Sude- dsrbuo ad GD Geeralsed rror Dsrbuo Boh he Sude - dsrbuo ad he GD have fa als The mamum-lkelhood aroach s used for he arameer esmao Usg backesg, he relaed resduals uder he hree dffere dsrbuoal assumos are eamed Furhermore, some fudameal coces of facal me seres aalyss wll be elaed ad some sylsed facs of real reurs wll be eamed 3

4 Ackowledgemes I would lke o hak Peer Alao ad Rober Thoré a Algorhmca Research AB for gvg me he ooruy of dog hs hess ad for her hel ad corbuos wh real-world facal kowledge I also would lke o hak Professor Boualem Djehche ad Seor Lecurer Ja ger, as well as Herk Hul, a he Dearme of Mahemacal Sascs a he Royal Isue of Techology KTH for her mora gudace ad valuable remarks Fally, I would lke o hak my famly ad grlfred for her suor durg he whole rojec secally, I would lke o eress my deees graude o my moher ad sser for her edless love, suor ad ecourageme 4

5 Coes Iroduco 7 Facal Tme Seres 8 mrcal Proeres 9 Kuross 9 Auocorrelao 3 Power Law Tals 4 Skewess 5 Holday ffec 3 3 Sylsed Facs Abou Facal Daa 4 4 Sochasc Volaly Models 6 5 The GARCH Famly 8 5 Symmerc GARCH Models 8 5 G A R C H 8 5 IGARCH 9 5 Asymmerc GARCH Models 5 GARCH 5 GJR-GARCH 53 APARCH 6 Parameer smao he GARCH Model 3 6 M a m u m- Lkelhood smao ML 3 6 S a o a r y 3 6 Gaussa Quas Mamum- Lkelhood smao 5 63 Fa -Taled Mamum - Lkelhood smao 6 6 Dsrbuos 7 6 Normal Dsrbuo 7 6 Sude- Dsrbuo 7 63 Geeralsed rror Dsrbuo GD 8 7 Robusess of smao 9 7 Resduals 9 7 Robusess of ML o Smulaed Daa 3 7 amles Wh Smulaed Daa 3 7 Cocluso of amles Robusess of ML o mrcal Daa amle Wh mrcal Daa Cocluso of mrcal amle M a m u m- Lkelhood Value Comarso Cocluso Mamum - Lkelhood Value Comarso 4 5 5

6 8 Varace Forecasg 46 8 GARCH, IGARCH,q Mulvarae GARCH Models 48 Aed A 49 S a o a r y 4 9 Proof of Corollary 4 9 Refereces 5 6

7 Iroduco The large crease he umber of raded asses has made he measureme of marke rsk, e, he rsk due o adverse marke movemes, o a rmary cocer he facal world May coveoal mehods for measurg rsk, assocaed wh asses, are doe hrough sudes of he varace volaly of he asse Ths measure of he ucodoal volaly does o ake o accou ha here mgh be a redcable aer h e sock marke volaly I he heory of facal reurs, a basc dealsao s ha reurs follow a saoary me seres model wh sochasc volaly srucure The resece of sochasc volaly mles ha reurs are o ecessarly deede over me I he year of 98, gle roosed a volaly rocess wh me varyg codoal varace; he AuoRegressve Codoal Heeroskedascy ARCH rocess However, emrcal evdece shows ha hgh ARCH order has o be seleced order o cach he dyamc of he codoal varace The hgh ARCH order mles ha may arameers have o be esmaed ad he calculaos ge burdesome Four years afer gel s roduco of he ARCH rocess, Bollerslev 986, roosed he Geeralsed ARCH GARCH model as a aural soluo o he roblem wh he hgh ARCH orders Ths model s based o a fe ARCH secfcao ad allows o dramacally reducg he umber of esmaed arameers from a fe umber o jus a few I Bollerslev s GARCH model he codoal varace s a lear fuco of as squared ovaos ad earler calculaed codoal varaces Wh hese models here are wo yes of reur dsrbuo o be cosdered; he codoal reur dsrbuo where he codog s he curre volaly ad he margal or saoary dsrbuo of he rocess Facal me seres ofe ehb some well- kow characerscs Frs, large chages ed o be followed by large chages ad small chages ed o be followed by small chages Secodly facal me seres ofe ehb leokuross, whch meas ha he dsrbuo of her reurs s fa-aled e relave hgh robably for ereme values The GARCH model successfully caures he frs roery descrbed above, bu somemes fals o caure he fa-al roery of facal daa Ths has lead o he use of o -ormal dsrbuos o beer model he fa - aled characersc ver sce Bollerslev roduced he GARCH model, ew GARCH models have bee roosed, eg oeal GARCH GARCH, wh dffere characerscs, advaages ad drawbacks 7

8 Facal Tme Seres Facal me seres aalyss s dreced o he udersadg of he mechasm ha drves a gve me seres of daa, or oher words: facal me seres aalyss focuses o he ruh behd he daa so ha oe ca fd hyscal models ha ela he emrcally observed feaures of real lfe daa Wh such models oe ca make dsrbuoal forecass for fuure values me seres Today here es may dffere yes of facal daa bu f oe focuses o share rces, sock dces ad foreg echage raes whch we deoe P,,,, where ca be mues, hours, days, ec, hey behave very smlar afer he rasformao: P log log P log P P The seres { s referred o as he log reur seres These seres has he advaage ha hey are free of u, ad ca herefore be comared wh each oher Oe ca ge a more uve udersadg of he log reur rasformao lookg a he relave reur seres: P P P T h e log reur seres s by a Taylor seres easo argume close o he relave reur seres ad hese descrbe he relave chages over me of he rce rocess The relave reurs are very small ad herefore very close o he log reur values Maybe he mos mora ssue of hs rasformao s ha we ca assume ha he me seres { ca be modelled by a saoary sochasc rocess, e a rocess whose characerscs do o chage wh me for a more formal defo, see aed A Naurally hs s o he case geeral for urasformed me seres see he grahs above for a llusrag eamle 8

9 However, oe should kee md ha for large samle szes large log- reur seres he assumo of saoary s secure Whe dealg wh log reur seres oe has o kee md ha qualave dffereces ca be eeced he me seres deedg o how small me us oe uses Wh relave small me us mues, secods, P wll vary very lle ad oe observes ha P does o chage over a relave log erod Comarg hs roery wh he behavour of P whe he me u s hours, days or weeks, oe realzes ha dffere models have o be aled deedg o how lo g he me u s I wha follows we hk of us of hours, days or weeks mrcal Proeres Kuross The observaos of he me seres { have a dsrbuo, whch ofe s assumed o be ormal Gaussa However, emrcal sudes of raccally ay facal me seres show ha hs s o que correc Oe way o quafy hs roery s o look a he k u r o s s of he dsrbuos Kuross s a measure of he ee o whch observed daa fall ea r he cere of a dsrbuo or he als: a laykurc dsrbuo has a kuross value less ha ha of a sadard, ormal dsrbuo Ths ye of dsrbuo has a fa mdrage o eher sde of he mea ad a low eak a leokurc dsrbuo has a kuross value greaer ha ha of a sadard, ormal dsrbuo whch gves he dsrbuo a hgh eak, a h mdrage, ad fa heavy als leokurc laykurc The laer characersc s commo observed rce, rae, ad reur me seres daa Ths mles ha here facal me seres s a hgher robably for ereme eves ha daa ha s ormally dsrbued Whe sudyg he kuross me seres, oe usually alks abou he Fsher kuross,,,: whch s defed as follows fo r he me seres { where µ 4 µ 4 γ 3 3, 4 µ k [ µ ] [ ] µ k, µ, 9

10 s he ceral mome of degree k Ths kuross measure ess oly f he fourh mome ess ad s f e Kuross s a ormalzed form of he fourh ceral mome of a dsrbuo I he Fsher kuross he dg hree s subraced o gve he ormal dsrbuo he kuross zero he dsrbuo s mesokurc Suosg ha ~ N,, he fourh ad secod ceral momes are gve by: µ [ µ ] [ ] e d 3 π µ [ ] Var µ whch shows ha he Fsher kuross for ormal dsrbuos s zero The ceral mome of degree k s esmaed wh: k µ k I he cases above we have he followg defos: laykurc dsrbuos: γ < leokurc dsrbuos: γ > ecess kuross mesokurc dsrbuos: γ Furher, a Quale-Quale lo QQ-lo oe ca check wheher or o a me seres comes from a cera dsrbuo Ths aalyss s a mora ool me seres a a l yss A QQ-lo s a lo of he emrcal quales agas he heorecal quales of a gve dsrbuo If he values come from hs gve ds rbuo, he lo wll be a aromaely sragh le y

11 Ths s a vsual mehod of aalyss where he aalyser ca ge a beer kowledge of he emrcal dsrbuo ad s devaos from a heorecal dsrbuo Auocorrelao Oe mora ool for assessg he degree of deedece observed daa s he samle auocorrelao fuco samle ACF of he daa Frs some defos: Le,, be observaos of a me seres The samle mea of,, s The samle auocovarace fuco s : where h h γ ˆ + h, < h <, γ C o v, h + h The samle auocorrelao fuco s: γˆ h ρ ˆ h, < h < γˆ A ycal feaure of facal me seres s he srucure of he auocorrelao fuco; he samle auocorrelaos are eglgble a almos all lags, e, he assumo of { beg whe ose feels que aural, bu he samle auocorrelaos of he squares or absolue values are dffere for a large umber of lags ad say almos cosa ad osve for large lags There are dffere oos of how hs roery s o be erreed ad he mos commo dea s o erre he slow decayg lags as a log-erm memory or log -rage deedece LRD Alhough some ooes clam ha hs mgh o be rgh due o o -esece of fourh momes, whch mles ha he ACF does o roduce ayhg meagful, ad secodly due o o -saoary effecs Defo: L o g- Rage Deedece s sad o ehb log- rage deedece f { h ρ h

12 Why boher lookg a he absolue ad squared log- reurs? Oe reaso s ha emrcal evdece shows ha he sequece of he sgs of a log- reur seres has smlar sascal roeres as a sequece of d deede, decally dsrbued symmerc Beroull radom varables Therefore umerous models are of he form sg, wher e he sequece { sg cosss of d symmerc Beroull radom varables, ad so oe s lef o model absolue reurs Varous oular models, for eamle GARCH models, are of he form see chaer 3, where { s a d symmerc sequece, he volaly rocess { s saoary ad o -egave, ad ad are deede for every fed Thus, he sequece { sg { sg s d Beroull, as desred Wha oe s eresed s he volaly of he log-reur, bu by s cosruco, cao be observed Therefore he observable quaes ad ofe are cosdered as surrogae values or esmaors of ad resecvely ad ha s why hey somemes are focus 3 Power Law Tals L o g- reur daa of facal me seres ofe ehbs h e a v y-aled dsrbuos Coveely, hese als ca be modelled wh dsrbuos wh ower law als, e, for large ad some osve umber he al de, P > u lm u P > u fa-aled dsrbuo ormal dsrbuo amles of dsrbuos sasfyg hs al codo are for eamle he Pareo dsrbuo wh a eac ower law ad he Sude- dsrbuo wh degrees of freedom I hs hess he Sude - dsrbuo s cosdered 4 Skewess Observaos of he emrcal dsrb uo of { ofe show ha he dsrbuo s leokurc Aoher roery ha devaes from he so ofe assumed Gaussa dsrbuo s ha he emrcal dsrbuo s o symmerc Skewess defes he degree of asymmery of a dsrbuo ad several yes of skewess are defed The Fsher skewess he mos commo ye of skewess, usually referred o smly as skewess s defed by: µ µ 3 γ, 3 µ 3 3 where µ 3 s he hrd ceral mome, ad µ / s he sadard devao A egave skewess value dcaes ha he daa has a dsrbuo skewed o he lef Ths meas ha he lef al s heaver ha he rgh al he dsrbuo Resecvely, a osve skew ess value dcaes a rgh skewed dsrbuo wh a rgh al heaver ha s lef al I hs hess he skewess of he me seres s egleced

13 5 Holday ffec Durg weekeds ad holdays formao accumulaes Ths could be refleced r c e s whe he markes reoe If he formao sream assumes o be cosa, he varace of he reurs over he erod from Frday close o he Moday close should be hree mes he varace from he Moday close o he Tuesday close However, he assumo of cosa formao sream s o accordace o he real lfe eerece The formao rae durg weekeds ad holdays s lower ha durg workg days, whch reduces he holday effec Ths roery s also egleced hs hess 3

14 3 Sylsed Facs Abou Facal Daa T h e l o g-reurs l o g P l o g P of share rces, sock dces ad foreg echage raes P,,,, ofe show he followg feaures: The frequecy of large ad small chages, relave o he rage of daa, s raher hgh whch leads us o beleve ha he daa do o come from a ormal, bu from a heavyaled leokurc dsrbuo relave hgh robably for ereme values Large ad small values a logreur samle ed o occur clusers Ths dcaes ha here s deedece he als Madelbro quoed 963: large chages ed o be followed by large chages -of eher sg - ad small chages by small chages Ths characersc s also called volaly cluserg Chages sock rces ed o be egavely correlaed wh chages volaly, e, volaly s hgher afer egave chocks ha afer osve chocks of same magude Ths roery s called he l e v e r a ge effec Log -rage deedece he daa Samle auocorrelaos of he daa are small whereas he samle auocorrelaos of he absolue ad squared values are sgfcaly dffere from zero eve for large lags Ths behavour suggess ha here s some kd of log- rage deedece he daa Aggregaoal Gaussay, e, he dsrbuo of log-reurs over larger erods of me such as a moh, half a year, a year s closer o he ormal dsrbuo ha for hourly or daly log- reur s Varous models have bee roosed order o descrbe hese feaures ad oe very commo model s of he ye: µ + Ζ, s a o -egave sochasc rocess such ha ad s called he volaly rocess sadard devao rocess I s ofe assumed ha he { s are sadard ormally dsrbued, ha s, { ~ d N, [ ] ad Var I wha follows, we wll always assume ha s symmerc ad has u varace The volaly rocess { ad he me seres { are assumed o be srcly saoary Moreover, we suose ha µ ca be esmaed from he daa ad herefore wll be covee o assume µ Here { s a sequece of d symmerc radom varables, ad { are deede for fed The rocess { 4

15 There are varous reasos for hs arcular choce of model: he dreco of he rce chages s modelled oly by he sg of, deedely of he order of magude of hs chage, whch s dreced by he volaly Ths s agreeme wh he emrcal observao ha s dffcul, or eve mossble, o redc he sg of rce chages sce ad are deede, a d s assumed o have mea zero ad varace, s he he codoal varace of gve Var [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 5

16 4 Sochasc Volaly Models As show he revous chaer, mos models for facal reurs are of he form: Ζ,, where { s a sequece of d symmerc radom varables, ad { s a o - egave sochasc rocess such ha ad are deede for fed There s srog emrcal suor for sochasc volaly facal me seres ad he resece of sochasc volaly mles ha reurs are o ecessarly deede over me The sadard assumo for he ose s ha { ~ d N, w h { deede of he sadard devao rocess { Volaly s a ceral ar of mos asse rcg models I hese models, oe ofe assumes ha he volaly s cosa over me However, s well kow ha facal me seres ehb me -varyg volaly I he year of 98, gle [6] roosed a : model for { + Ths model s called he AuoRegressve Codoal Heeroskedascy ARCH rocess where he auoregressve roery rcle meas ha old eve s leave waves behd a cera me afer he acual me of he aco The rocess deeds o s as The erms codoal heeroskedascy meas ha he varace codoal o he avalable formao vares ad deeds o old values of he rocess Oe ca resemble hs wh he rocess havg a shor-erm memory ad ha he behavour of he rocess s flueced by hs memory However, sce ca eeced ha s a me- chagg weghed average of as squared observaos, s que aural o defe, o oly as a weghed average of as s, bu also of as mrcal evdece shows ha hgh ARCH order has o be seleced order o c ach he dyamc of he codoal varace Ths leads o he Geeralsed ARCH model G A R C H roduced 986 by Bollerslev [] The volaly rocess s: + + β, where he s ad he β j s are o -egave arameers Ths model reduces he umber of esmaed arameers from fely may o oly jus a few Oe ca easly see ha he GARCH model s based o a fe ARCH secfcao See dervao chaer 5 q j j j 6

17 GARCH has gaed fas acceace ad oulary he facal world Ths ca be elaed by varous argumes: he GARCH rocess has a close relao o ARMA rocesses Ths suggess ha he heory behd he GARCH rocess mgh be closely relaed o h e h e o r y o f ARMA rocesses, whch s well suded ad wdely kow oe ca ge a reasoable good f o real lfe facal daa eve wh a GARCH, model wh oly hree arameers, rovded ha he samle s o oo log so ha he saoary assumo s urelable I he followg chaer a survey of some dffere GARCH models s doe 7

18 5 The GARCH Famly ver sce Bollerslev roduced he GARCH, q -model, ew models wh dffere characerscs have bee veed The esg models ca be dvded o wo ma caegores: symmerc ad asymmerc models I he symmerc models, he codoal varace oly deeds o he magude, ad o he sg, of he uderlyg asse Ths roery s seldom accordace wh e mrcal resuls where a leverage effec ofe s rese, e, volaly creases more afer egave reur shocks ha afer osve reur shocks of he same magude bad ews geeraes hgher volaly more ha good ews lowers he volal y However, he asymmerc models hese characerscs are more or less caured All he followg models buld o he mullcave form, Ζ f o r h e facal log-reurs as show earler The sad ard assumo for he ose s ha { s d symmerc radom varables wh zero mea ad u varace The volaly rocess { s a o-egave sochasc rocess such ha ad are deede for fed 5 Symmerc GARCH Models 5 GARCH Le deoe a real-valued dscree -me sochasc rocess The GARCH,q rocess roosed by Bollerslev s he gve by: ~ N, >, β j + >,, q + j,, q q j β j,,, or, usg he lag or backshf oeraor B defed as B, he GARCH,q model s: + B + β B, wh z z + z + + z q ad β z β z + β z + + β z q For q he rocess reduces o he ARCH rocess, ad for q s smly whe ose The GARCH, q rocess wh a d ose sequece { [ ] j such ha [ ], s srcly saoary wh fe varace f see chaer 6: ad 8

19 > ad + β j < The GARCH rocess has a close relao o ARMA rocesses By rearragg he GARCH, q model defg υ, follows ha: q j B + β B β B υ υ + +, whch defes a ARMAma,q, model for Ths relao o ARMA rocesses suggess ha he heory behd he GARCH rocesses mgh be closely relaed o ARMA rocess heory, whch s que easy ad wdely kow Alhough, oe has o be careful because he ose sequece { deeds o he s hemselves, so ha a comlcaed o -lear relaosh of he s s bul u Furhermore, s easy o see ha he GARCH model s based o a fe ARCH secfcao If all he roos of he olyomal β B le ousde he u crcle, we ge: β B + B, or equvalely: B, + + β β B β β λ q where he? s are suable cosas whch ogeher wh ~ N, may be see as a ARCH rocess 5 IGARCH Whe esmag he arameers he GARCH model oe ofe observes ha he sum of he arameers s close o oe For he arameer seg: q + β j j gle ad Bollerslev coed he ame Iegraed GARCH IGARCH Here, he egraed refers o he fac ha here mgh be a u roo roblem whch could lead o he o -esece of a saoary verso of { has fe varace However, hs s o he case for he IGARCH uder he codos of Theorem chaer 6 lus some mld addoal assumos see [] for furher formao Thus, he IGARCH has a srcly saoary soluo, bu wh fe varace To see hs we ake he eecaos of he codoal varace ad observe ha [ ] [ ] whch gves: [ ] + B [ ] + β B [ ] + + β [ ] 9

20 As ca be see hs s rue oly f he eecao s fy > s ecessary for saoary, hus, he IGARCH rocess has fe varace Ths roery s o ehbed real -l f e l o g- reurs I s foud ha he sum of he esmaed arameers he GARCH model ycally creases owards oe wh creasg samle sze Ths suors he hyohess ha he IGARCH effec s due o bad f of he GARCH model, ad ha he bad f of he model may be due o o-saoary, whch s more lkely fo r a larger samle sze I hs lo, we see ha he arameers sum u o values close o oe for large samle szes 5 Asymmerc GARCH Models To accommodae for he asymmery ha ess may facal me seres, umerous asymmerc GARCH m odels have bee derved May of hese models have large smlares wh each oher amles of some asymmerc models are: GARCH oeal GARCH GJR-GARCH Glose, Jagaaha ad Rukle GARCH APARCH Asymmerc Power ARCH The GARCH model wll be furher eamed he e chaer ad he oher asymmerc models wll be defed brefly 5 GARCH ve f he GARCH models successfully caure he hck al reurs, ad he volaly cluserg, hey are oor models f oe wshes o caure he leverage effec descrbed chaer 3 sce he codoal varace s a fuco oly of he magudes of he as

21 values ad o her sg The codoal varace of gve formao a me, obvously mus be o-egave wh robably oe I GARCH models hs roery s assured by makg a lear combao wh osve weghs of osve radom varables as he GARCH,q case Aoher way of ma kg o -egave s by makg l lear some fuco of me ad lagged s Ths formulao leads o he asymmerc GARCH model, oeal GARCH, of Nelso 99 [7]: q β j l j l g j The value of g deeds o several elemes Nelso oes, o accommodae he asymmerc relao bewee sock reurs ad volaly chages, he value of g mus be a fuco of boh he magude ad he sg of Ths leads o followg rereseao: [ [ ] { + g θ θ sg effec magude effec Wh hs cosruco, { g s a zero - mea, d radom sequece ach, comoe has mea zero Over he rage < <, g s lear wh sloe θ + θ, ad over he rage <, g s lear wh sloe θ θ T h u s g allows he codoal varace o resod asymmercally o rses ad falls sock rce To see ha he erm θ [ [ ]] rereses he magude effec oe frs assumes ha θ ad θ > Ths makes he ovao l + osve egave whe he magude of s larger smaller ha s eeced value Assumg ha θ < ad θ The ovao codoal varace s ow osve egave whe reurs ovaos are egave osve I coras o he GARCH models, he GARCH models do o have ay resrcos o he arameers he model The GARCH model always roduces a osve codoal varace deedely of he sgs of he esmaed arameers he model ad o resrcos are eeded Ths s referable whe he resrcos he GARCH model somemes creae roblems whe esmaed arameers volae he equaly cosras 5 GJR-GARCH GJR-GARCH Glose, Jagaaha ad Rukle GARCH: S + + ω S whe whe < + q j β j j

22 I hs model s suosed ha he effec of he o he codoal varace s dffere accordgly o he sg of ad hs s why he varable S s roduced Ths mles ha he model accommodaes he leverage effec 55 APARCH APARCH Asymmerc Power ARCH: δ q δ δ + γ + β j j j β j >, δ < γ,, j,, q <,, Mos of he GARCH models are o- esed hey ca o be wre as a resrced verso of a more geeral rocess, bu he APARCH model cludes seve oher ARCH secfcaos as secal cases: ARCH whe δ, γ,, ad β j j,, q GARCH whe δ ad γ,, Taylor 986 / Schwer 99 s GARCH whe δ, ad γ,, GJR-GARCH whe δ TARCH whe δ NARCH wh e γ,, ad β j j,, q L o g- ARCH by Geweke 986 ad Peula 986, whe δ

23 6 Parameer smao he GARCH Model To be able o redc he volaly for a me seres, oe frs has o f he GARCH -model o he me seres queso Ths s doe va esmao of he arameers he model The mos commo mehod of hs esmao s he mamum-l kelhood esmao ML 6 Mamum-Lkelhood smao ML The mamum- lkelhood esmao works as follows: The daa,, assumes o be radom observaos from a dsrbuo F ; θ ha deeds o he ukow arameers θ where θ [,,,, β,, β q ] he GARCH, q case wh he arameer sace Θ has he robably dsrbuo ; θ where ; θ deoes he robably ha, hus P S u o s g ha he robably fuco s kow ece from he ukow arameers s ossble o esmae he ukow θ s by ug u he lkelhood fuco he L fuco: L θ ; θ ; θ ; θ Obvously, L θ defes he robably ha eacly he values,, s observed as realsaos from he dsrbuo Now o he sohscaed dea behd he ML ; by leg he ukow θ assume all he values he arameer sace Θ, oe ca see for wha values of θ he Lθ has mamum value These values are deoed θ * Hece, he esmao of θ * s chose so ha he Lθ * -fuco s mamsed for observed,, 6 Saoary Whe dealg wh GARCH models he assumo of saoary of he me seres { s basc for he sascal aalyss of he daa Ths mles cosras o he esmaed arameers he mamum lkelhood - esmao Here follows wo heorems ha sae resrcos o he esmaed arameers he GARCH, q model for saoary he GARCH,q rocess Theorem: The GARCH, q r o c e s s, Ζ, wh he secfcao of he codoal varace secfed earler ad a d ose sequece { wh mea zero ad u varace, has a o-vashg srcly saoary causal verso f ad oly f > ad γ < Here, γ s he Lyaov eoe For furher formao of he eoe, see [] A suffce codo for γ < s gve by: q + β j j < 3

24 rovded ha [ ] ad [ ] I oher words, he GARCH,q rocess wh a d ose sequece { such ha [ ] ad [ ], { s srcly saoary wh fe varace f: q > ad + β j < Proof: See [] for formao Corollary: The GARCH,q rocess s weakly saoary wh: [ ] Var j ~ N, q + + β j q + β j j C o v, s for s f ad oly f + β j <, > q j Proof: See aed A j For dffere GARCH models here are dffere resrcos o he esmaed arameers 4

25 6 Gaussa Quas Mamum-Lkelhood smao he GARCH, q model of a gve order s d sadard ormal The s Gaussa N, gve he whole as,,, a d a codog argume yelds he desy fuco,, of,, hrough he codoal Gaussa deses of he s gve,, : Now, suose ha he ose {,,,, + +,,,, π + + e where s s a fuco of a, a,, a, ß,, ß q Codog o ad relacg + wh, he Gaussa log-lkelhood of,, s gve by: l,,,, β,, β q log + + log π For a geeral GARCH, q rocess he lkelhood fuco s mamsed as a fuco of he s ad β j s volved The resulg value he arameer sace s he Gaussa quas mamum-l k e l h o o d esmaor of he arameers of a GARCH,q rocess There are roblems assocaed wh hs esmao rocedure The assumo of Gaussa ose: I s assumed ha he ose { s Gaussa Alhough hs s o he mos realsc assumo; emrcal ess dcae ha he s are much beer modelled by a Sude - dsrbuo or a GD Theorecal work [7] ad Heyde 997 Quas- Lkelhood ad s Alcao: A Geeral Aroach o Omal Parameer s mao shows ha asymoc roeres such as -cossecy ad asymoc ormaly wh -rae of he Gaussa quas ML rema vald for large classes of ose dsrbuos Calculao of uobservable values: The formula of he lkelhood-fuco requres calculag he uobservable values,,,, from he observed samle,, Ths s obvously o ossble he geeral GARCH,q case Oe erao of he volaly rocess yelds ha oe has o kow all values,,,, for he calculao of,, Aleravely, oe eeds o kow fely may values of he uobservable values,, ad,, A commo echque for solvg hs roblem s o choose al values as he equlbrum values, e, for a GARCH, model Var ad Var The choce of he al values mles ha he calculaed,, cao be cosdered as a realsao of a saoary sequece Now, oe hoe ha he deedece of he al values dsaear for large values of a smlar way o a Markov cha wh arbrary al value whose dsrbuo becomes closer o he saoary 5

26 dsrbuo However, he Gaussa quas ML- fuco s a comlcaed fuco bul u o he s ad he s Therefore are he heorecal roeres of he Gaussa quas ML o easy o derve I hs hess, ML, s assumed ha he deedece of he al values dsaears wh reasoably large values of However, oe should bear md ha hs s a dffcul roblem 63 Fa-Taled Mamum-Lkelhood smao A alerave way of dealg wh o- Gaussa errors he frs roblem descrbed he chaer above s o assume a dsrbuo ha reflecs he feaures of he daa beer ha he ormal dsrbuo, ad esmae he arameers usg hs dsrbuo he lkelhood fuco sead of he Gaussa Thus, he roblem wh he calculao of uobservable values descrbed revous chaer s ye rese hs model Whe choosg a dsrbuo for he ovaos, QQ- los ca be very helful I hs hess wo dsrbuos, aar from he Gaussa, are cosdered; he Sude- Dsrbuo Dsrbuo ad he Geeralsed rror Dsrbuo GD The lkelhood fucos for wo dsrbuoal assumos are: h e l o g-lkelhood fuco for he Sude- dsrbuo: l ν + ν log Γ log Γ log π ν log ν + l o g + ν h e l o g-lkelhood -fuco for he GD: l ν log λ λ ν + ν log log Γ log ν where Γ s he gamma fuco, ad ν Γ ν λ 3 Γ ν T h e s e l o g-lkelhood fucos are mamsed wh resec o he ukow arameers he same rocedure as he Gaussa quas ML case 6

27 6 Dsrbuos As dscussed earler, observaos of he facal me seres { have a dsrbuo ha oe ofe assumes o be ormal Gaus sa bu, as show chaer, hey ofe ed o be leokurc fa aled QQ-los have bee show o be good ools whe decdg wha dsrbuo o use I hs hess he fa aled Sude- dsrbuo ad he GD are cosdered The GD ca be boh leokurc ad laykurc deedg o he chose degree of freedom Here follows some furher formao abou hese dsrbuos 6 Normal Dsrbuo The ormal or Gaussa dsrbuo s a symmerc dsrbuo wh desy fu co: f π e µ / where µ s he eecao value ad s he varace of he sochasc varable, hus ~N µ, The so- called sadard ormal dsrbuo s gve by akg µ ad The Fsher kuross s for he ormal dsrbuo er defo zero see chaer I he GARCH model, whe s assumed o be ormally dsrbued, he eecao he g fuco s gve by: 6 Sude- Dsrbuo [ ] π The Sude- dsrbuo, or dsrbuo, has followg desy fuco: [ ν + / ] f ; ν ν + / ν π Γ Γ [ ν / ] + / ν where ν s he degree of freedom ν > Lke he ormal dsrbuo, he dsrbuo s symmerc The mea, varace ad kuross of he dsrbuo are: µ for ν ν for ν 3 ν γ 6 ν 4 7

28 The Sude- dsrbuo wh u varace has he followg desy fuco: [ ν + / ] Γ f ; ν ν + / ν π Γ [ ν / ] + / ν I he GARCH model, whe s assumed o be Sude-ν dsrbued, he eecao he g fuco s gve by: [ ν + / ] Γ ν [ ] π ν Γ 63 Geeralsed rror Dsrbuo GD [ ν / ] The GD s a symmerc dsrbuo ha ca be boh leokurc ad laykurc deedg o he degree of freedom ν ν > The GD has he followg desy fuco: where f ν e ; ν ν + / λ λ Γ ν λ ν ν Γ[ ν ] [ 3 / ν ] Γ [ / ν ] The GD wh u varace has he followg desy fuco: ν λ ν e f ; ν ν + / ν λ Γ [ / ν ] ν ν For ν, he GD s a sadard ormal dsrbuo whereas he als are hcker ha he ormal case whe ν <, ad her whe ν > The GD becomes a uform dsrbuo o he erval [ 3, 3 ] whe ν I he GARCH model, whe s assumed o be GDν dsrbued, he eecao he g fuco s gve by: ν λ [ ] / Γ[ / ν ] Γ [ / ν ] 8

29 7 Robusess of smao As descrbed revous chaer, GARCH,q models are fed o he reur seres usg mamum-lkelhood esmao I he Gaussa quas ML mehod, hs esmao s doe uder he assumo ha he ovaos { have a Gaussa dsrbuo I he fa-aled ML, he ovaos are assumed o be leokurc Now, oe was o kow f he esmaos are robus, e: do he esmaos of he arameers,,,, β,, β q deed o he dsrbuoal assumo of he ovaos { do he resduals of he esmaed rocess have he same dsrbuo as he assumed dsrbuo of he ovaos 7 Resduals Whe he esmao of he ukow arameers θ s doe, esmaes of he sadard devao seres { ˆ,, ˆ ca be calculaed recursvely va he defo of he codoal varace for he GARCH, q rocess; + B + β B 9

30 I he wo grahs above, he log-reur rocess ad he esmaed codoal sadard devao rocess for a 5-day ecer from he S&P 5-de, are loed The esmaed codoal sadard devao rocess s derved from a GARCH, f Clearly, he esmaed codoal sadard devao rocess reflecs he behavoure of h e l o g- reur rocess By calculag resduals oe ca eame he adequacy of he GARCH modellg The resduals are calculaed as remember ha he lo g- reurs are modelled by : { z,, z ˆ,, ˆ The resduals should be d f he fed model s eable The ovaos { ~ mles ha he resduals also should be d d The auocorrelaos for he observaos, he squared observaos, he resduals ad he squared resduals for a 5 -day ecer from he S&P 5-de are here loed GARCH, model fed wh Gaussa quas ML Based o he los above, he GARCH model seems o be reasoable 3

31 Furher, wh a QQ-lo, oe ca eame he dsrbuo of he resduals Ths s a sor of verfyg he robusess he esmao To he rgh, he resduals of for a 5 - day ecer from he S&P 5 -de are loed Here he ML s doe uder he dsrbuoal assumo of Gaussa ovaos We ca see ha he resduals uder hs dsrbuoal assumo ge a heavy lef al If he QQ-lo shows ha he resduals are beer rereseed by aoher dsrbuo, eg a fa-aled dsrbuo, oe should cosder o do he ML uder he assumo of hs ew dsrbuo where he ovaos hoefully have a more arorae dsrbuo 7 Robusess of ML o Smulaed Daa To eame he robusess of mamum-lkelhood esmao models a corolled maer, oe ca do he GARCH f wh ML o smulaed rocesses wh kow dsrbuos o he ovaos The smulao ad robusess seco of a rocess are doe he followg maer: Assume good arameer values for a GARCH,q model If hs s dffcul, oe ca use esmaed arameer-values from a mamum-lkelhood esmao o real-lfe reurs Use he arameers o smulae a GARCH rocess uder a secfc dsrbuoal assumo of he ovaos I hs hess he Gaussa dsrbuo, he Sude- dsrbuo ad he GD are used 3 For he smulaed rocess, f a GARCH, q model o he daa wh mamumlkelhood esmaos uder he hree dffere dsrbuoal assumos of he ovaos 4 Comare he arameer values he hree cases 5 Calculae he resduals of he hree esmaed sadard dervao me seres 6 ame he dsrbuos of he hree resdual me seres qua l e- quale los Now follows hree eamles where GARCH, rocesses are smulaed uder hree dffere dsrbuoal assumos of he ovaos Mamum-lkelhood esmaos uder dffere dsrbuoal assumos of he ovaos are d oe ad he arameer values are dslayed ad he resduals are loed QQ-los 3

32 7 amles Wh Smulaed Daa amle : A GARCH, rocess s here smulaed wh Gaussa ovaos 4 daa Mamum-lkelhood esmao uder he assumo s o f Gaussa, Sude - ad GD ovaos yelds he arameers of he smulaed rocess The resduals of he rocess are calculaed ad loed a QQ -lo agas he Gaussa dsrbuo, he Sude- dsrbuo ad GD resecvely A GA R C H, f s d o e f o r h e smulaed rocess usg ML uder he assumo of Gaussa ovaos The resduals from he f o h e s m u l a e d d a a a r e c o m u e d a d l o e d a Q Q- l o agas he ormal dsrbuo A s w e c a s e e, l o o k s l k e h e resduals have a Gaussa dsrbuo o 77 β 554 N o w, h e G A R C H, f o h e s m u l a e d r o c e s s s d o e w h M L u d e r h e a s s u m o o f S u d e - dsrbued o v a o s T h e r e s d u a l s f r o m h e f a r e c o m u e d a d l o e d a Q Q- l o a g a s h e S u d e - dsrbuo wh a esmaed degree of freedom The f s good, bu he degree of f r e e d o m w a s e s m a e d o b e 3 5, w h c h d c a e s h a h e r o c e s s e h b s G a u s s a o v a o s h e Sude d s r b u o s c l o s e o h e o r m a l d s r b u o o 83 β 54 3

33 I h s l a s l o, h e G A R C H, f o h e s m u l a e d r o c e s s w h Gaus s a o v a o s s d o e w h M L u d e r h e a s s u m o o f G D o v a o s T h e r e s d u a l s f r o m h e f a r e c o m u e d a d l o e d a Q Q- l o a g a s h e G D w h a e s m a e d degree of freedom The f s good, bu he degree of f r e e d o m w a s e s m a e d o be 85, w h c h d c a e s h a h e r o c e s s e h b s G a u s s a o v a o s G a u s s a d s r b u o f ν o 78 β 556 smaed arameer values: Dsrbuoal assumo β Gaussa ovaos Sude- ovaos GD ovaos

34 amle : Now, a GARCH, rocess s smulaed wh GD3 ovaos 4 daa Mamum -lkelhood esmao uder he assumos of Gaussa, Sude - ad GD ovaos yelds he arameers of he smulaed rocess The resduals of he rocess are calculaed ad loed a QQ -lo agas he Gaussa dsrbuo, he Sude- dsrbuo ad GD resecvely A G A R C H, f s d o e f o r h e smulaed rocess wh GD3 ovaos usg ML uder he assumo of Gaussa ovaos The resduals from he f o h e smulaed daa are comued ad loed a QQ -lo agas he ormal dsrbuo The f s oor h e a l s a d h e l o s h o w s h a h e resduals are leokurc A more heavy- a l e d d s r b u o f o r h e o v a o s s r o b a b l y a b e e r a s s umo o 4 69 β 834 Now, he GARCH, f o he smulaed daa wh GD3 o v a o s s d o e w h M L u d e r he assumo of Sude - d s r b u e d o v a o s T h e r e s d u a l s f r o m h e f a r e c o m u e d a d l o e d a Q Q- l o a g a s h e S u d e - dsrbuo wh a e s m a e d d e g r e e o f f r e e d o m T h e f s o o r H e r e, h e r o b u s e s s o f h e m a m u m -lkelhood esmao uder he assumo of Sude - dsrbu e d o v a o s s quesoed o 795 β

35 I h s l a s l o, h e G A R C H, f o he smulaed rocess wh GD o v a o s s d o e w h M L u d e r he assumo of GD o v a o s T h e r e s d u a l s f r o m h e f a r e c o m u e d a d l o e d a Q Q- l o a g a s h e G D w h a e s m a e d degree of freedom The f s good ad he esmaed degree of freedom s close o he d e g r e e o f f r e e d o m h e s m u l a e d G A R C H, r o c e s s w h GD3 o v a o s o β 8589 smaed arameer values: Dsrbuoal assumo β Gaussa ovaos Sude- ovaos GD ovaos

36 amle 3: Fally a GARCH, rocess s smulaed wh Sude-4 dsrbued ovaos 4 daa Mamum -lkelhood esmao uder he assumos of Gaussa, Sude - ad GD ovaos yelds he arameers of he smulaed rocess The resduals of he rocess are calculaed ad loed a QQ-lo agas he Gaussa dsrbuo, he Sude- dsrbuo ad GD resecvely A G A R C H, f s d o e f o r h e smulaed rocess wh Sude - ovaos usg ML uder he assumo of Gaussa ovaos The resduals from he f o he smulaed daa are comued ad loed a QQ -lo agas he ormal dsrbuo The f good, bu he als dcae ha he resduals are leokurc A m o r e h e a v y- a l e d d s r b u o f o r h e o v a o s s r o b a b l y a b e e r assumo o 45 β 767 N o w, h e G A R C H, f o h e smulaed rocess wh Sude - o v a o s s d o e w h M L u d e r he assumo of Sude - d s r b u e d o v a o s T h e r e s d u a l s f r o m h e f a r e c o m u e d a d l o e d a Q Q- l o a g a s h e S ude - dsrbuo wh a esmaed degree of freedom T h e f s f a r l y g o o d, b u h e e s m a e d d e g r e e o f f r e e d o m d e v a e s f r o m h e s m u l a e d o 786 β

37 I h s l a s l o, h e G A R C H, f o h e s m u l a e d r o c e s s w h Sude - o v a o s s d o e w h M L u d e r h e a s s u m o o f G D o v a o s T h e r e s d u a l s f r o m h e f a r e c o m u e d a d l o e d a Q Q- l o a g a s h e G D w h a e s m a e d degree of freedom Th e f s g o o d, a d h e e s m a e d d e g r e e o f f r e e d o m d c a e s h a h e o v a o s c o m e f r o m a h e a v y- a l e d d s r b u o o 74 β 84 smaed arameer values: Dsrbuoal assumo β Gaussa ovaos Sude- ovaos GD ovaos

38 7 Cocluso of amles Frs, by lookg a he esmaed arameers for he dffere dsrbuoal assumos, oe ca coclude ha he models seem o be robus The values of he esmaed arameers do o dffer from each oher much uder he dffere dsrbuoal assumo s The los he eamles dcae ha he ML s robus uder he assumos of Gaussa ad GD dsrbued ovaos The overall erformace for he ML uder he assumo of Sude- dsrbued ovaos s ye o covcg The eamles suor he usefuless of QQ-los as ools for eamg he resduals Wh he ad of he QQ -los, a arorae model ca be chose If he ovaos come from a Gaussa dsrbuo, he mamum-lkelhood esmaos are good deede of he assumed dsrbuos The wo ML models ha assume fa- aled dsrbuos for he ovaos assg he degree of freedom so ha he fa-aled dsrbuo s close o he Gaussa O he oher had, f he ovaos come from a fa- a led dsrbuo, he bes f s acheved wh a ML ha assumes GD ovaos The mamum-lkelhood esmao uder he dsrbuoal assumo of Sude - dsrbued ovaos s quesoed The overall erformace for he esmao uder he assumo of Sude- dsrbued ovaos s o covcg Ths robusess eamao ca be used o choose he rgh esmao model for gve daa smae he arameers uder he assumo of Gaussa ovaos Plo he resdual s a Q Q-lo agas he ormal dsrbuo If he resduals are leokurc robably a beer dsrbuoal assumo s a fa -aled dsrbuo, eg GD ovaos 3 Now, esmae he arameers uder he assumo of leokurc ovaos, eg GD 4 Plo he resduals a QQ-lo agas he chose leokurc dsrbuo If he f s good, he dsrbuoal assumo mos robably s correc The smulaos hs eamle coss of oly 4 smulaed daa Larger smulaed samle szes should gve us more formao, bu large samles mly roblems wh he saoary assumo of { 3 8

39 73 Robusess of ML o mrcal Daa I followg eamle, he robusess of ML o emrcal daa s eamed 73 amle Wh mrcal Daa mrcal eamle: For he Volvo B log reurs, he arameers he GARCH, model are esmaed wh ML uder he assumo ha he ovaos are Gaussa dsrbued, Sude- dsrbued ad GD resecvely The resdu als are calculaed ad loed QQlos Frs, he mamum-lkelhood esmao s doe uder he assumo of Gaussa ovaos: o, 76 ad β 76 I hs frs l o, he GARCH, f o h e V o l v o r e u r s e r e s s d o e w h M L u d e r h e a s s u m o h a he ovaos comes from a Gaussa dsrbuo The resduals are calculaed ad loed agas he o r m a l d s r b u o T h e f s f a r l y g o o d, b u h e f h e als fal The resduals are leokurc A more heavy- a l e d d s r b u o o h e o v a o s h e ML would robably be more arorae 39

40 H e r e, h e r e s d u a l s c a l c u l a e d a b o v e are loed agas he Sude - d s r b u o w h h e c h o s e degree of f r e e d o m ν 5 T h e c h o s e d e g r e e o f f r e e d o m s h o w s ha he resduals have a dsrbuo c l o s e o h e G a u s s a, e c e f r o m h a h e y h a v e h e a v e r a l s Fally, he resduals are loed a g a s h e G D w h chose degree o f f r e e d o m ν 3 I hs case, he f s good, whch dcaes ha he resduals have a h e a v y- a l e d d s r b u o 4

41 Now, he same ML rocedure s doe, bu hs me uder he assumo ha he ovaos are Sude- dsrbued o, 4 ad β 78 I h s f r s l o, h e G A R C H, f o h e V o l v o r e u r s e r e s s d o e wh ML uder he assumo ha h e o v a os comes from a Sude - d s r b u o T h e r e s d u a l s are calculaed ad loed agas he o r m a l d s r b u o The f s farly good, bu he f h e a l s f a l T h e r e s d u a l s a r e leokurc A more heavy- a l e d d s r b u o o h e o v a o s he ML would robably be more arorae H e r e, h e r e s d u a l s a r e l o e d a g a s he Sude - dsrbuo wh he esmaed degree of freedom ν 6 T h e f s o o r a d h e r o b u s e s s o f h e M L u d e r h e a s s u m o o f Sude - d s r b u e d o v a o s s q u e s o e d 4

42 Fally, he resduals are loed agas he GD wh chose degree o f f r e e d o m ν 3 I h s c a s e, h e f s g o o d, w h c h d c a e s h a h e r e s d u a l s h a v e a heavy- a l e d d s r b u o Fally, he same ML rocedure s doe, bu hs me uder he assumo ha he ovaos are GD o, 966 ad β 735 N o w, h e G A R C H, f s d o e u d e r h e a s s u m o h a h e ovaos come from a GD The resduals are calculaed ad loed agas he ormal dsrbuo hs QQ - l o T h e f s o o r h e a l s T h e r e s d u a l s a r e l e o k u r c a d a m o r e h e a v y- a l e d d s r b u o would robably be more arorae 4

43 Secodly, he resduals are loed agas he Sude - d s r b u o w h c h o s e d e g r e e o f f r e e d o m ν T h e c h o s e d e g r e e o f f r e e d o m s h o w s ha he resduals have a dsrbuo close o he Gaussa, ece from he h e a v e r a l s Fally, he resduals are loed a g a s h e G D w h h e e s m a e d d e g r e e o f f r e e d o m ν 3 4 T h e f s g o o d smaed arameer values: Dsrbuoal assumo β Gaussa ovaos Sude- ovaos 4 78 GD ovaos

44 73 Cocluso of mrcal amle By lookg a he esmaed arameers for he dffere dsrbuoal assumos, oe sees ha he esmaed arameer values o dffer much from each oher Ths dcaes ha he models are robus The los he eamle dcae ha he ML s robus uder he assumos of Gaussa ad GD dsrbued ovaos The overall erformace for he ML uder he assumo of Sude- dsrbued ovaos s ye o covcg Aga, we ca see he usefuless of he QQ-los as ools for choosg he rgh dsrbuoal assumo for he ovaos Uder he assumo of Gaussa ovaos he ML, he QQ -lo of he resduals loed agas he ormal dsrbuo dslay ecess kuross The GD3 fs he resduals much beer, whch dcaes ha he resduals are leokurc Secodly, uder he assumo of Sude - ovaos he ML, he QQ-lo of he resduals loed agas he Sude - dsrbuo dslay a laykurc dsrbuo The erformace of he Sude - modellg s o covcg Aga, he GD3 seems o f he resduals much beer Fally, uder he assumo of GD ovaos he ML, he QQ-lo of he resduals loed agas he GD shows ha he model uder GD assumo s robus 4 4

45 74 Mamum-Lkelhood Value Comarso Aoher way of fdg he mos arorae model s o look a he egave log - lkelhood value a he mamum o for he dffere models A sgfca larger lkelhood value for a secfc dsrbuoal assumo he ML dcaes ha hs assumo mos lkely s he bes model beer ha he oher models Frs, we look a smulaed daa Dffere dsrbuoal assumos are used he esmaos ad he ovaos are assumed o have secfc dsrbuos Negave lθ * for smulaed reurs wh kow dsrbuo of he ovaos: Smulaed wh: Normal ML Sude- ν ML GD ν ML ormal ovaos ovaos ovaos GD3 ovaos Now, we look a emrcal daa Here follows a able of mamum egave loglkelhood-values for dffere yes of facal daa esmaed wh dffere dsrbuoal assumos for he ovaos Negave lθ * mamum- lkelhood values lθ * : Normal ML Sude- ν ML GD ν ML rcsso B 8,7 86,,3 8,8,7 Volvo B 99, 9, 6 9,4,3 OM -de 9, 93, 8 9,4,9 S&P 5-de 95,3 97,5 7, 97,,3 UR echage rae 789, 83, 7,5 799,6,4 GBP echage rae 89,8 9,5 7 95,,4 74 Cocluso Mamum-Lkelhood Value Comarso Geerally, s dffcul o draw coclusos by lookg a he mamum log-lkelhood values The l o g- lkelhood values do o dffer much, whch s a mlcao of ha he l o g-lkelhood surface s fla Larger samle szes could gve us more formao, bu large samles mly roblems wh he saoary assumo 45

46 8 Varace Forecasg Th e G A R C H-model cosdered hs hess assumes ha s a fuco of he as, e,,, ad,,, ad herefore s rcle kow a me For hs reaso, coeco wh he dsrbuo of ca be used o gve a dsrbuoal forecas of Assume for eamle ha s d N, The codoally o he as values,,,,, oday s reur has a N, dsrbuo [ ] [ ] [ ] [ ] Ths allows oe o gve a dsrbuoal forecas of he values For eamle, here s a 95% chace ha assumes values [- 9 6, 9 6 ] Hece s a easy maer o deerme he codoal VaR Value a Rsk of he sequece { 8 GARCH, Oce he arameers of he GARCH- models have bee esmaed oe s eresed he varace forecas for he uderlyg asse For he GARCH, model gve ha + β <, he eeced value of he oeerod varace a me s: + k where k + β k [ ] + + β + k + k + β + + β + Dervao of hs formula: β + [ + ] + [ + ] + β [ + ] [ ] [ ] { deede [ ] [ ] { ~ d N, [ ] + + β [ ] + β [ ] + [ ] + β [ ] + + β ec β + + β + + β β + + β Wh creasg value of k he varace forecas wll coverge o he ucodoal varace wh he rae + β : + β 4 6

47 47 Oe ca easly derve hs eresso by followg argumes for he geeral GARCH, q case: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] < s s s s s s s s,, Var, C o v hus, [ ] [ ] ad saoary gves: [ ] [ ] [ ] [ ] B B B B β β [ ] β I he grah above, a volaly forecas s doe for he OM-de As we ca see, he volaly forecas coverges o he ucodoal varace

48 8 IGARCH For he IGARCH,q models + β, he codoal eecao of he oe erod varace a me T s : T + k [ T k ] T k T + + Proof: [ + ] + [ + ] + β [ + ] [ + ] [ ] { deede [ ] [ ] { ~ dn, [ ] + + β [ + ] + + [ ] + [ ] + β [ ] + + β β + + β ec 9 Mulvarae GARCH Models Recall chaer 3 where dffere sylsed facs abou facal daa were cosdered I addo o hese, s worh meo aoher sylsed fac I facal daa he volales of dffere secures very ofe move ogeher, dcag ha here are lkages bewee markes ad ha some commo facors may ela he emoral varao codoal secod markes The aalyss of may ssues asse rcg ad orfolo allocao requre s a mulvarae framework 4 8

49 49 Aed A Saoary Defo: { s weakly saoary f [ ] o f s deede µ ad h each for o f s deede,, h C o v h + + γ Defo: { s a srcly saoary me seres f,,,, h h d + + for all egers h ad Here d s used o dcae ha he wo radom vecors have he same jo dsrbuo fuco Proof of Corollary: Src saoary mles weak sa oary See Theorem chaer 6 [ ] [ ] { ad are deede for every [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] < s s s s s s s s whe whe Var, C o v hus, [ ] [ ] lus saoary yelds: [ ] [ ] [ ] [ ] [ ] B B q j j Var B B β β β β

50 Refereces [] Bollerslev, T 986 Geeralzed Auoregressve Codoal Heeroskedascy Joural of coomercs 3, [] Bollerslev, T, Chou, RY ad KF Kroer 99 ARCH Modellg Face: A Revew of he Theory ad mrcal vdece Joural of coomercs 5, 5-59 [3] Bollerslev, T, gle RF ad DB Nelso 994 ARCH Models Hadbook of coomercs vol 4, [4] Brockwell, PJ ad RA Davs 99 Tme Seres: Theory ad Mehods Srger, New York [5] Brockwell, PJ ad RA Davs 996 Iroduco o Tme Seres ad Forecasg Srger, New York [6] gle, RF 98 Auoregressve Codoal Heeroskedascy wh smaes of he Varace of Ued Kgdom Iflao coomerca vol 5, [7] gle, RF 995 ARCH Seleced Readgs Oford Uversy Press [8] Gradell, J 998 Tme Seres Aalyss Lecure oes, KTH Sockholm [9] Hasso, B ad P Hördahl 998 Forecasg Varace Usg Sochasc Volaly ad GARCH Workg Paer Seres, Dearme of coomcs, School of coomcs ad Maageme, Uversy of Lud [] Mkosch, T Modellg Deedece ad Tals of Facal Tme Seres h://wwwmahkudk/~mkosch/semsa 5

51 [] Palm, FC 996 GARCH Models of Volaly Maddala, GS ad CR Rao eds, Hadbook of Sascs, Vol 4 Amserdam, lsever Scece BV, 9-4 [] Lueberger, DG 984 Lear ad No -lear Programmg Secod do Addso - Wesley, Readg, Massachuses [3] Sadard & Poor The S&P 5 Ide me seres h://wwwsadardadoorscom 5