BILINEAR GARCH TIME SERIES MODELS

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1 BILINEAR GARCH TIME SERIES MODELS Mahmoud Gabr, Mahmoud El-Hashash Dearme of Mahemacs, Faculy of Scece, Alexadra Uversy, Alexadra, Egy Dearme of Mahemacs ad Comuer Scece, Brdgewaer Sae Uversy, Brdgewaer, MA, USA Absrac I hs aer he class of BL-GARCH (Blear Geeral AuoregRessve Codoal Heeroskedascy) models s roduced. The roosed model s a modfcao o he BL-GARCH model roosed by Sor ad Vale (003). Saoary codos ad auocorrelao srucure for secal cases of hese ew models are derved. Maxmum lkelhood esmao of he model s also cosdered. Some smulao resuls are reseed o evaluae our algorhm. Keywords : Tme seres, ARCH models, GARCH models, Blear models, weak deedece,. 1. Iroduco A lo of me seres ecouered emrcal alcaos are olear ad o-saoary. Ther srucures such as meas ad varaces may vary over me. The roblem of olear me seres defcao ad modelg has araced cosderable aeo for he as 30 years dverse felds such as facal ecoomercs, bomercs, socoecoomcs, rasorao, elecrc ower sysems, ad aeroaucs whch exhb olear rocess. A good olear model should be able o caure some of he olear heomea he daa. Moreover, should also have some uve aeal. Therefore a umber of wde classes of olear me seres models have bee roosed, vesgaed ad suded. Oe of hese classes whch has receved a grea deal of aeo s ha of blear models. Blear me seres models ad s sascal ad robablsc roeres have bee exesvely suded by [7] Grager ad Aderse (1978), [14]Subba Rao (1981), [5] Gabr (199) ad comrehesvely surveyed by [15] Subba Rao ad Gabr (1984) ad [11] Pham (1993). A class of o-lear model, called a blear class, may be regarded as a lausble o-lear exeso of ARMA, raher ha of he AR model. Blear models cororae cross-roduc erms volvg lagged values of he me seres ad of he ovao rocess. The model may also cororae ordary AR ad MA erms. The geeral form of a blear me seres {X, 0, 1,,...} deoed by BL(, q, P, Q) s defed by q P Q X a X cj e j bj X e j e j 1 j 1 defe he model (1) as a blear me seres model BL (,r,m,k) ad he rocess {X } as a blear rocess. I ecoomercs, a vas leraure s devoed o he sudy of auoregressve codoally heeroskedasc (ARCH) models for facal daa. Oe of he bes-kow model s he GARCH model (Geeralzed Auoregressve Codoally Heeroskedasc) roduced by [3] Egle (198) ad [1] Bollerslev (1986). The classcal GARCH(,q) model s gve by he equaos (1) {e } s a se of deede radom varables. We ε =σ Z, h =σ h q q 1h 1 h q 0 j j j 1 h () 0 > 0, 0, j 0, q 0, 0 are model arameers ad {Z j, j=1,, 3, } are deede decally dsrbued (..d.) radom varables wh zero mea ad varace 1. The varables ε, σ, Z () are usually erreed as facal (log) reurs (ε ), her volales or codoal sadard devaos (σ ), ad so-called ovaos or shocks (Z ), resecvely; he ovaos are suosed o follow a cera fxed dsrbuo (e.g., sadard ormal). Laer, a umber of modfcaos of (4.1) were roosed, whch accou for asymmery, leverage effec, heavy als ad oher sylzed facs. Uder some addoal codos, smlarly as he case of ARMA models, he GARCH model ca be wre as ARCH( ) model.e., h ca be rereseed as a movg average of he as squared reurs s, s <, wh exoeally decayg coeffces (see [1] Bollerslev, 1986) ad absoluely summable exoeally decayg auocovarace fuco. For sace, he GARCH(, q) rocess of () ca be wre as

2 Z, h, 1 1 h 1 (1) 0 1 (B) (B) (B) 1B B ad B sads for he back-shf oeraor, B k X = X k. Ths leads o he ARCH( ) rereseao; Z, h 0 h b b (3) 1 wh b0 1 (1) 0 ad wh osve exoeally decayg weghs b, 1 defed by he geerag fuco (y) / 1 (y) b y. I s eresg o oe ha he o-egavy of he regresso coeffces α j, β j () s o ecessary for o-egavy of b j he corresodg ARCH( ) rereseao, see [10] Nelso ad Cao (199). Clearly, f E(Z / ε s, s < ) = 0, E( Z /ε s, s < ) =1 he ε has codoal mea zero ad a radom codoal varace,.e. E(ε / ε s, s < ) = 0, var( /ε s, s < ) = h The geeral framework leadg o he model () was roduced by [1] Robso (1991) he coex of esg for srog seral correlao ad has bee subsequely suded by [8] Kokoszka ad Leus (000) he chage-o roblem coex. The class of ARCH( ) models cludes he fe order ARCH ad GARCH models of [3] Egle (198) ad []Bollerslev (1986).. The Blear ARCH Models Formally, he classes AR, ARCH, LARCH (a leas, her fe memory couerars ARMA, GARCH, ARCH) all belog o he geeral class of blear model (1). [6] Gras ad Surgals (00) suded he heeroscedasc blear equao X Z 0 X 0 X (4) {Z, =1,, 3, } are..d. radom varables, wh zero mea ad varace 1, ad j, j, j 0 are real (o ecessary oegave) coeffces. Equao (4) aears aurally whe sudyg he class of rocesses wh he roery ha he codoal mea μ = E(X /X s, s < ) s a lear combao of X s, s <, ad he codoal varace h = Var(X /X s, s < ) s he square of a lear combaos of X s, s <, as s he case of (4):.e. E X / X s,s 0 X s 0 h var X / X,s X Clearly, he case j 0, j 1 gves he lear AR( ) equao, whle j 0 (j 0) resuls he Lear ARCH (LARCH) model, roduced by [1] Robso (1991), defed by he equao Z, h h c j j The ma advaage of LARCH s ha allows modelg of log memory as well as some characersc asymmeres (he leverage effec ). Boh hese roeres cao be modeled by he classcal ARCH( ) wh fe fourh mome. The coeffces c sasfy d 1 c j ~ k j for some 0 <d < ½, k> 0 whch mles he codo j c

3 Neher α or he c j are assumed osve ad, ulke (4.3), σ (o ), s a lear combao of he as values of ε, raher ha her squares. [4] Egle ad Ng (1993) roduced a olear asymmerc GARCH model whch caures asymmery by meas of eracos bewee as reurs ad volales I he smle (=1,q=1) case he codoal varace equao s gve by Z, h 0 a 1( j 1h 1) b1h 1 (5) wh he model becomg asymmerc whe he coeffce 1 s equal o zero. [13] Sar ad Vale (003) have geeralzed hs model o he followg BL-GARCH model Z, h a b h c h (6) 0 j j j j j j j j 1 j 1 j 1 a j, b j,c j j=1,,..., are cosas. Ths model has he advaage of beg characerzed by a more flexble aramerc srucure I hs model leverage effecs are exlaed by he eracos bewee as observaos ad volales To see he osvy of he codoal varace equao (6), we ca wre a j j bj h j cj jh j a j j bj h j c a b h j j j j j j=1,,, Hece for, α 0 > 0, a suffce codo for h > 0, (6), s gve by 4a j bj c j for j=1,,, (7) Z, h 0 j j jh j (8) From (6) ad (8), we ca see ha he wo models coa exacly he same umber of erms, alhough he umber of arameers requred by each model s dffere. I fac he umber of arameers (8) s less ha ha (6) by arameers. Moreover, we do o eed he codo of osvy of he arameers j, j of model (8). Theorem The Blear GARCH rocess (8) s saoary wde sese f ad oly f he roos u of he olyomal Proof (u) 1 u, le ousde he u crcle. The Blear GARCH rocess (8) ca be rewre as 0 j j j j j 0 j j j 1 j 1 h h Z h h (9) whch s a radom coeffce auoregressve rereseao for h j ( j Z j j) (10) Takg cosderao he roeres of {Z }, execao of (9) s gve by 0 j j 0 j j j 1 j 1 E h E h E E h he 0 j j E h j 0 j E h j j 1 j 1 Model (6) wh he codo (7) leads us o roduce a smler reduced arameer Blear GARCH model he form; Sce, E E E h E / 1

4 follows ha, defed by Therefore Y 0 j Y j (11) Y E. Leg B be he backward shf oeraor k B Y Y k, equao (11) ca be rewre as, Y 0 j 1 j B Y (11) coverges o a fe value f ad oly f all he roos u of he olyomal (u) 1 u le ousde he u crcle whch comlees he roof. The smles bu ofe very useful Blear GARCH rocess s ha of order 1 gve by / 1 ~ N(0,h ) (1) h h (13) wh 0 0. The ucodoal varace s 1 E(h ) E( 1) 1 E(h 1) E( ) E[E( / )] E( h h ) E( ) E( ) E( ) The sequece of varaces coverges o he cosa f suffces for wde sese saoary. Uder ormaly assumo E( ) E(E( / )) 3E(h ) From whch we oba, E h h 3E h h (1 1 1 ) E( ) 4 4 (15) (1 1 1 )( ) The ecessary ad suffce codo for he exsece of he fourh mome s (16) The coeffce of Kuross s E E( ) (.78) I fac s ycally foud ha he GARCH (1,1) model yelds a adequae descro of may facal me seres daa, see, for examle, [] Bollerslev,Chou, ad Kroer (199). A daa se whch requres a model of order greaer ha GARCH (1, ) or GARCH (, 1) s very rare. A seres of sze N=300 s geeraed from he smle BL- GARCH model Wh, whch mles ha The seres { } s a sequece of..d. N(0, 1). The al E( ) (14) values are chose as ad. The grah of he seres { } ad { } are reseed fgures (1) ad () resecvely

5 Fgure (1) Fgure () 3. MLE of BL-GARCH Parameers We ow cosder he maxmum lkelhood esmao of he arameers he BL-GARCH model (8). Le ( ) ad suose ha we x,...,x,x,...,x for he me have he observaos m seres {x }. Uder a reasoable assumo ha we have kow he σ-feld σ{ m 1,..., 1, 0}, we ca oba he jo codoal desy fuco of x 1,...,x gve he σ-feld {x m 1,...,x 0, m 1,..., 1, 0} as follows f(x 1,...,x / x 0,...,x m 1, 0,..., m 1) = f(x,...,x / x 1, x 0,...,x m 1, 0,..., m 1 ) f(x 1 / x 0,...,x m 1, 0,..., m 1).. = f (x / x 1,..., x m 1, 1,..., m 1 ) 1 1 = ex 1 h h Thus he MLE ˆ of he arameer vecor whch maxmzes he logarhm lkelhood fuco s he value of L( ) l f(x 1,...,x / x 0,...,x m 1, 0,..., m 1) = 1 l(h ) l( ) h = Q ( ) C 1 Usg he recursve Newo-Rahso erao algorhm, he MLE ˆ ca be obaed by he followg erao: (k) (k 1) (k) 1 (k) (k) H ( )G( ) s he se of esmaes obaed a he k h sage of erao. G( ) s he grade vecor of aral dervaves, G( ) L( )... L( ) 1 1 ad H( ) s he Hessa marx of secod order aral dervaves, H( ) L( ) j

6 The frs order aral dervaves are gve by are calculaed recursvely from he equaos: The secod order dervaves are gve by: Noe ha he esmaed he Hessa marx Ĥ( ) may be sgular ad some umercal roblems may arse. Oe commo way o deal wh hs roblem s he Leveberg- Marquard rocedure [9] (Marquard(1963)). 4. Moe Carlo Smulao The Newo-Rahso wh Marquord algorhm, descrbed he revous seco were red successfully o may ses of daa smulaed from several saoary BL-GARCH models. We shall cosder here he followg model

7 wh ad The seres { } s a sequece of..d. N(0, 1). The al values are chose as ad. The Newo-Rahso algorhm s aled a he above model wh samle sze N=300 ad relcae smulaos 100 mes. The resuls from he Moe- Carlo sudy shows, clearly, ha he mea of each arameer esmaes s close he rue value. The sadard devaos of he esmaes are small dcag ha he esmaors are cosse. Parameer esmaes N=100 True value Mea S.D Refereces [1] Bollerslev, T. (1986) Geeralzed auoregressve codoal heeroskedascy. J. Ecoomercs 31, [] Bollerslev,Chou, ad Kroer (199) Arch Modelg Face: A Revew of he Theory ad Emrcal Evdece. Joural of Ecoomercs, Arl-May 199, [3] Egle, R.F. (198) Auoregressve codoal heeroscedascy wh esmaes of he varace of Ued Kgdom flao. Ecoomerca 50, [4] Egle, R.F. ad V.Ng (1993) Measurg ad esg he Imac of News o Volaly. Joural of Face 48, [5] Gabr, M.M. (199) Recursve esmao of Blear Tme Seres Models. Commu. Sas.: Theory & Meh., 1(8), [6] Gras, L. ad Surgals, D. (00) ARCH-ye blear models wh double log memory. Soch. Process. Al. 100, [7] Grager, C.W.J. ad Aderse, A.P. (1978) A Iroduco o blear me seres Models. Goege: Vedehoek ad Rureche. [8] Kokoszka, P. ad Leus, R. (000) Chage-o esmao ARCH models. Beroull 6, [9] Maquard, D. (1963) A algorhm for leas squares esmao of olear arameers. J. Soc. Id. Al. Aah., [10] Nelso, D. B. ad Cao, C. Q. (199) Iequaly cosras he uvarae GARCH model. Joural of Busess & Ecoomc Sascs, 10, [11] Pham, D.T. (1993) Blear Tmes Seres Models. I Dmeso Esmao ad od-els (ed. H. Tog), Sgaore: World Sce.c Publshg Co. [1] Robso, P. M. (1991). Tesg for srog seral correlao ad dyamc codoal heeroskedascy mulle regressos. Joural of Ecoomercs, 47, [13] Sor ad Vale (003). BL-GARCH models ad asymmeres volaly. Sascal mehods & alcaos 1:19-39 [14] Subba Rao, T. (1981) O he heory of Blear Models. J. Roy. Sas. Soc. B, (43), [15] Subba Rao, T. ad Gabr, M. M. (1984) A Iroduco o Bsecral Aalyss ad Blear Tme Sere[14] Models. Lecure Noes Sascs, volume 4. Srger Verlag, New York.