Testing Goodness-of-Fit in Autoregressive Fractionally Integrated Moving- Average Models with Conditional Hetroscedastic Errors of Unknown form

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1 Rsarch Joural of Rc Scics ISSN 77-5 Vol. (5, 9-4, May ( Rs.J.Rc Sci. Tsig Goodss-of-Fi i Auorgrssiv Fracioally Igrad Movig- Avrag Modls wih Codiioal roscdasic Errors of Uow form Absrac Ali Amad, Salahuddi ad Alamgir Darm of Saisics, Islamia Collg Uivrsiy Pshawar, PAKISTAN Darm of Saisics, Uivrsiy of Pshawar, Pshawar, PAKISTAN Availabl oli a: Rcivd 8 h Dcmbr, rvisd 8 h Jauary, accd h Fbruary This ar cosidrs sig goodss-of-fi i Auorgrssiv fracioally igrad movig-avrag modls wih codiioal hroscdasiciy. W xd h alicabiliy of og s ad owr rasformd og s s saisics as goodss-of-fi ss i ARFIMA-GARC modls, whr h srucural form of GARC modl is uow. Simulaio sudy is rformd o assss h siz ad owr rformac of boh ss. Ky Words: Codiioal hroscdasiciy, ARFIMA, GARC, Goodss-of-Fi Tss. Iroducio I is a orivial as o fid a aroria or a arsimoious modl i rgrssio ad im sris daa aalysis. Rsiduals aalysis is commoly usd as modl diagosics i im sris modl buildig. Th adquacy of h fid im sris modl is commoly sd by chcig h assumio of whi ois rsiduals. If h aroria modl has b chos, hr will b zro auocorrlaio i h rsiduals sris. L b h sris of h rsiduals from h fid modl, h i hyohsis sig sigs w ca sa our ull ad alraiv hyohsis as : ρ ( = for all vrsus : ρ ( for som. I frqucy domai aroach h abov hyohsis ca b sad as : f ( v = / π, vrsus : f ( v / π for som v ( π, π, whr f ( θ = (π ρ ( z iθ is h ormalizd scral dsiy fucio of. Rcig h abov ull hyohsis imlis h iadquacy of h fid modl. Svral ss hav b dvlod o s h hyohsis of zro auocorrlaio. Box ad Pirc hav dvlod a ormaau s o s h adquacy of h fid im sris modl. Th s saisic is giv as: = ˆ m ˆρ ( ( h whr ρ ( h is h auocorrlaio of a lag h ad m is assumd o b fixd. Thy showd ha for larg (saml siz, h saisic has chi-squar disribuio wih m dgrs of frdom assumig ha sris is iddly ad idically disribud. If ar h rsiduals from a fid im sris modl, h is disribud as χ wih m dgrs of frdom, whr is h umbr of aramrs i h modl. Davis al. showd ha h disribuio of ca dvia from chi-squar ad h ru sigificac lvl is lily o b lowr ha h rdicd sigificac lvl. A modifid vrsio of Box ad irc s saisic was roosd by Lug ad Box, which has h followig form: m = ( + ˆ ρ ( h / ( Thy rformd a comaraiv sudy of hir s wih h s of Box ad Pirc ad showd ha hir s has subsaially imrovd aroximaio o chi-squar disribuio. For various choics of m, Lug 4 xamid h roris of Box ad Pirc s saisic. Thy suggsd a modifid vrsio of Box ad Pirc s saisic ha allowd h us of various valus of m. Thir simulaio sudis showd ha h modifid s is mor owrful udr various iovaios disribuios. og 5 iroducd hr classs of cosis o sidd ss for sig srial corrlaio of h rsiduals of h liar dyamic modl ha iclud boh laggd dd ad idd variabls. Udr h ull hyohsis of zro auocorrlaio, hy showd ha h sadardizd form of all hs s saisics is asymoically N (,. To imrov asymoic ormaliy of og s ss, Ch ad Do 6 iroducd owr rasformd og s s. Thy xamid h rformac of og s ad owr rasformd og s s saisics as goodss-of-fi ss Iraioal Scic Cogrss Associaio 9

2 Rsarch Joural of Rc Scics ISSN 77-5 Vol. (5, 9-4, May ( Rs. J. Rc Sci. for diffr im sris modls wih idically idd rrors. I h curr sudy, w cosidr modl diagosic chcig of ARFIMA modls wh is iovaios ar codiioally hroscdasic of uow form. Th Modl Log mmory rocsss hav b widly usd i h aalysis of im sris daa. Nil rivr daa is a ousadig xaml which xhibis log mmory bhaviour 7. Ohr xamls ar h Ehr raffic im sris sudid by Llad.al. 8 ad forig xchag ra rurs sudid by Goodhar ad ar 9. Th commo faur of hs im sris is ha h dcay of h auocorrlaio fucio is li a owr fucio rahr ha xoial as i h cas of shor mmory im sris. Th scral dsiy of such rocsss bhavs us li a owr fucio ad divrgs as h frqucy gos o zro. Auorgrssiv fracioally igrad movig avrag rocss (ARFIMA(, d, q is a wll ow class of log mmory im sris. Ths modls a io accou h hyrbolic dcay of auocorrlaio fucio. ARFIMA(, d, q wr iddly iroducd by Gragr ad Joyux ad osig. This modl is a gralizaio of h ARIMA(, d, q modl, whr d is a o b a igr. I is dfid as φ B X B d B ( ( = θ ( (, whr = i φ ( B = φi B ad θ ( = θ B, = i= q B θ, φ ( ad B is h bacward shif oraor, ar h auorgrssiv ad movig-avrag oraors rscivly; φ(b ad θ (B hav o commo roos, ( B d is fracioally diffrcig oraor dfid by h biomial xasio d Γ( + d ( B = B, =,,,..., Γ( + for d <.5, d, -, -,... ad is a whi ois squc wih fii variac. If d >, h sris xhibi log mmory. ARFIMA modls hav rov usful ools i h aalysis of log rag ddc rocsss. Auorgrssiv fracioally igrad movig avrag (ARFIMA modls wih GARC rrors hav b widly usd i im sris daa aalysis. Bailli al. usd ARFIMA-GARC modls o aalyz h iflaio of diffr couris. To modl daily daa o h Swiss - moh Euromar irs ra durig h riod , ausr ad Kus usd fracioally igrad modls wih ARC rrors. Ohr alicaios of fracioally igrad modls wih codiioally hroscdasic rrors ca b foud i (4 ausr ad Kus, Li ad Ts 4, Elc ad Marus 5 ad Kooma al 6. A wo sag modl buildig sragy is grally usd o fi a ARFIMA-GARC modl. I h firs s a ARFIMA modl is fid o h giv sris ad h a GARC modl o h rsiduals of h ARFIMA modl. So, i is imora o slc a corrc ARFIMA modl i h firs sag. Th misscificaio of ARFIMA modl i h firs sag will lad o misscificaio of h GARC modl i h scod sag 7. Th ss dvlod by Ch ad Do 8, Dlgado al. 9, Dlgado ad Vlasco ad idalgo ad Kriss all wor for log mmory im sris modls. owvr, hy assumd Gaussia or liar rocsss wih codiioally homoscdasic ois rocsss. Lig ad Li ad Li ad Li hav sudid BP y ss for modl diagosics of ARFIMA-GARC modls bu assumig ha h aramric form of GARC modl is ow. I h rs wor, w cosidrd modl diagosis of ARFIMA modls wih GARC rrors of uow form. W ivsiga h rformac of og;s saisic as a goodss of fi s for ARFIMA-GARC modls hrough simulaio sudy. W also xami h rformac of owr rasformd og s saisic of Ch ad Do 6 i h abov sigs. Th Ts Saisics I a smial ar og 5 iroducd svral s saisics ha ar gralizaio of h Box ad irc s saisic. Ths ss ar basd o h disac bw h rl basd scral dsiy simaor ad h scral dsiy of h ois udr h ull hyohsis. Th sadardizd form of h og s s saisics wih quadraic disac is giv by ( / ˆ ( C ( /(D ( / = ρ (5 whr C ( = ( / ( /, 4 D ( = ( / ( ( + / ( /, (. is h rl fucio which is o-gaiv ad symmric ad is h badwidh ha dds o h saml siz. Udr h assumio of i.i.d rrors of h modl, wh = o( ad, og 5 showd ha h asymoic ull disribuio of is sadard ormal. og ad L 4 xdd h abov rsul rlaxig h assumio of i.i.d rrors ad sablishd h rsuls assumig h codiioal hroscdasic rrors of uow form. Simulaio rsuls of Ch ad Do 6 foud ha for small samls h disribuio of og s s is righ swd, which rsuls o h siz disorio of h s. To dal wih his roblm, Ch ad Do 6 iroducd a owr rasformd vrsio of og s s saisics. Th ida bhid his rasformaio is o iduc ormaliy. Iraioal Scic Cogrss Associaio 4

3 Rsarch Joural of Rc Scics ISSN 77-5 Vol. (5, 9-4, May ( Rs. J. Rc Sci. Thy showd ha h aroria owr β o b usd such ha β bcom aroximaly ormal is giv by = ( ( 4 6 β. (6 Mo Carlo sudy of Ch ad Do 8 showd ha for h abov choic of β, h disribuio of could b wll aroximad by ormal disribuio. I our Mo Carlo simulaios h abov valu of β is usd. Mo Carlo Evidc I his scio, w ivsiga, hrough simulaios, h fii saml rformac of h og s ad owr rasformd og s s saisics as goodss-of-fi ss for ARFIMA (, d, q modls wih dd rrors. W us wo saml sizs = ad =. Th rror disribuio is a o b sadard ormal. W us h followig four rls for boh ss o xami h ffc of diffr rls. Dail (: ( w = si( w / w, w (, Parz(PAR: 6( π w / πw / 6 ( w = ( πw / 6 : ( w = (9 / 5w Barl(: π β w / π / π w 6 / π ohrwis { si( 5 / πw / 5 / πw cos( 5 / πw } w (, z ( w = z ohrwis To ivsiga h ffc of. = [ l( ], [ ], w us hr diffr ras:. = ad [ ] =. For = hs ras dlivr = 5, 8, ad for = hs ras ma = 8,, 7. To xami h siz rformac of og s ad owr rasformd og s s saisics w cosidr h followig modls. M: ARFIMA (,.4, - GARC((.5,.,.85 M: ARFIMA(.5,.4, - GARC((.5,.,.85 For owr rformac of boh ss h followig modls ar usd. M: ARFIMA(,.,.4, - GARC((.5,.,.85 alraiv fiig modl as ARFIMA(, d, M4: ARFIMA(,.4,. - GARC((.5,.,.85 alraiv fiig modl as ARFIMA(, d, M5: ARFIMA(.5,.4,. - GARC((.5,.,.85 alraiv fiig modl as ARFIMA(, d,. Th rsuls for M M5 hav b show i abl o 5. Ths rsuls ror h rcag rcio ras a omial lvls of 5% ad % basd o 5 rlicaios. For small saml siz = siz disorios occur for boh ss bu com clos o h omial siz for =. Th owr rasformd s is mor udrsizd as comard o og s s saisics. Th siz is br for M comard o M. Thr is o sigifica ffc of diffr rls o h siz of boh ss. Th siz bcoms br as w icras h badwidh. This is ru for boh saml sizs, ss ad diffr rls. Boh ss hav good owr rformac for diffr saml sizs bu h owr icrass as w icras h saml siz from o. Diffr rls hav o sigifica ffc o h owr of boh ss. Tabl- Rcio ra udr h ARFIMA(.,.4,-GARC((.5,.,.85 alraiv, fiig modl ARFIMA(,d, = = =5 = = = Iraioal Scic Cogrss Associaio 4

4 Rsarch Joural of Rc Scics ISSN 77-5 Vol. (5, 9-4, May ( Rs. J. Rc Sci. = Tabl- Rcio ra udr h ARFIMA(,.4,-GARC((.5,.,.85 = = = = = Tabl- Rcio ra udr h ARFIMA (,.4,.-GARC((.5,.,.85 alraiv fiig modl ARFIMA(,d, = = =5 = = = Tabl-4 Rcio ra udr h ARFIMA (.5,.4,-GARC((.5,.,.85 alraiv fiig modl ARFIMA(,d, = = =5 = = = Coclusio W alid h og s ad owr rasformd og s of Ch ad Do 6 for goodss-of-fi of auorgrssiv fracioally igrad movig avrag modls wih codiioally hroscdasic rrors of uow form. Our simulaio sudy rvals ha for larg saml siz ( = boh h ss hav good siz ad owr rformac, wh alid o diffr log mmory modls wih codiioally hroscdasic rrors bu for small saml ( = boh ss ar udrsizd. Th owr rasformd s is mor udrsizd comard o og s s. This siz disorio occurs du o h fac ha h ma ad variac of hs s saisics ar basd o h asymoic hory ad could b misladig i small samls as rord by Ch ad Do 6. Th abov rsuls show ha som siz corrcio dvics ar dd i h abov s saisics for ARFIMA modls wih dd rrors of uow form. Iraioal Scic Cogrss Associaio 4

5 Rsarch Joural of Rc Scics ISSN 77-5 Vol. (5, 9-4, May ( Rs. J. Rc Sci. Tabl-5 Rcio ra udr h ARFIMA(.5,.4,.-GARC((.5,.,.85 alraiv fiig modl ARFIMA(,d, = = =5 = = = Rfrcs. Box G.E.P ad Pirc D.A., Disribuio of h rsiduals auocorrlaios i auorgrssiv-igrad movig avrag im sris modls, JASA, 65(, 59-5 (97. Davis N., Triggs C.M. ad Nwbold P. Sigificac lvls of h Box-Pirc ormaau saisics i fii samls, Biomria, 64(, 57- (977. Lug G.M. ad Box GEP, O a masur of lac of fi i im sris modls, Biomrica, 65(, 97- ( Lug G.M., Diagosic sig of uivaria im sris modls, Biomrica, 7(, 75- ( og Y., Cosis sig for srial corrlaio of uow form, Ecoomrica, ( Ch W. ad Do R.S., Powr rasformaio o iduc ormaliy ad hir alicaios, Joural of Royal Saisical Sociy Sris B Saisical Mhodology, 66, 7- (4b 7. urs., Log-rm sorag caaciy of rsrvoirs, Trasacios of h Amrica Sociy of Civil Egirs, 6, (95 8. Llad W., Taqqu M., Willigr W. ad Wilso D., O h slfsimilar aur of Ehr rafic (xdd vrsio, IEEE/ACM Tras. Nwor, (, -5 ( Goodhar C.A.E., ad M. O'ara., igh Frqucy Daa i Fiacial Mars: Issus ad Alicaios, Joural of Emirical Fiac, 4, 7-4 (997. Gragr C.W.J. ad Joyux R., A Iroducio o Log-Mmory Tim Sris Modls ad Fracioal Diffrcig, Joural of Tim Sris Aalysis,, 5-9 (98. osig J.R.M., Fracioal diffrcig, Biomria, 68, (98. Bailli R.T., Chug C.F. ad Tislau M.A., Aalysig iflaio by h fracioally igrad ARFIMA-GARC modl, Joural of Alid Ecoomrics,, -4 (996. ausr M.A. ad Kus R.M., Fracioally igrad modls wih ARC rrors: Wih a alicaio o h Swiss -moh uromar irs ra, Rviw of uaiaiv Fiac ad Accouig,, 95 ( Li D. ad Ts Y.K., Forcasig h Nii so idx wih fracioal coigraio, Joural of Forcasig, 8, 59 7 ( El P. ad Marus L., A log rag dd modl wih oliar iovaios for simulaig daily rivr flows, Naural azards ad Earh Sysm Scics, 4, 77 8 (4 6. Kooma S.J., Oohs M. ad Carro M.A., Priodic sasoal Rg-ARFIMA-GARC modls for daily lcriciy so rics, Joural of h Amrica Saisical Associaio,, 6 7 (7 7. Lumsdai R.L. ad Ng S., Tsig for ARC i h rsc of a ossibly misscifid codiioal ma, Joural of Ecoomrics, 9, ( Ch W. ad Do R.S., A gralizd ormaau goodss of fi s for im sris modls, Ecoomric Thory,, (4a 9. Dlgado ad Vlasco, Disribuio-fr ss for im sris modl scificaio, Joural of Ecoomrics, 55, 8-7 (7. Dlgado M.A., idalgo J. ad Vlasco C., Disribuio fr goodss-of-fi ss for liar rocsss, Aals of Saisics,, (5. idalgo J. ad Kriss J.P., Boosra scificaio ss for liar covariac saioary rocsss, Joural of Ecoomrics,, (6. Lig S. ad Li W.K., O fracioally igrad auorgrssiv movig-avrag im sris modls wih codiioal hroscdasiciy, Joural of h Amrica Saisical Associaio, 9, (997. Li G. ad Li W.K., Las absolu dviaio simaio for fracioally igrad auorgrssiv movig avrag im sris modls wih codiioal hroscdasiciy, Biomria, 95, (8 4. og Y. ad L Y.J., Cosis Tsig for Srial Corrlaio of Uow Form Udr Gral Codiioal rosdasiciy, Tchical Ror, Darms of Ecoomics ad Saisical Scics, Corll Uivrsiy, ( Iraioal Scic Cogrss Associaio 4

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