The simulation studies for Generalized Space Time Autoregressive-X (GSTARX) model

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1 Procdings of h IConSSE FSM SWCU 5 pp. SC. 7 ISBN: SC. Th simulaion sudis for Gnralid Spac Tim Auorgrssiv- GSTAR modl Juna Di Kurnia a Siaan b Sani Puri Rahayu c a b c Dparmn of Saisics Insiu Tknologi Spuluh Nopmbr Surabaya Kampus ITS Sukolilo Surabaya 6 Indonsia Absrac Gnralid Spac Tim Auorgrssiv- GSTAR is a modl ha involv h prdicor variabl inroducd by Pfifr dan Dusch. Gnralid Spac Tim Auorgrssiv GSTAR is on of mulivaria im sris modls ha combin lmns of im and locaion or spaial daa or im sris. Variabl in GSTAR is a symbol ha has a mric and non-mric scal. For h cas of univaria im sris using h prdicor ih mric scal calld h Transfr Funcion Modl hil for non-mric scal calld h Inrvnion Modl and Calndar Variaions. Th liraur sudis shod ha sudis rgarding h approach of mulivaria im sris by using GSTAR- is sill limid o modls involving variabl ih non-mric scal so ha in his rsarch rsricd us a variabl ih a mric scal. GSTAR- simaion mhod for using h Gnralid Las Squar GLS as ll as h simaion mhod on h modl Smingly Unrlad Rgrssion SUR ha inroducd by llnr. Th purpos of his rsarch is o obain a paramr simaion from GSTAR- modl ih simulaion sudy. Rsuls of h simulaion sudy shod ha if h rsidual of simulaion ar corrlad i ill gnra a rror sandard of paramrs sima valus ar small in GSTAR-SUR modl han GSTAR-OLS so i can b said ha h paramr simaion using GSTAR-SUR is mor fficin han GSATR-OLS. Kyords GSTAR-SUR GSTAR-OLS mric prdicor. Inroducion GSTAR is on of mulivaria im sris modls ha involv mor han on rspons and corrlad. GSTAR is h dvlopmn of modls Spac Tim Auorgrssiv STAR inroducd by Pfifr & Dusch 98. This modl is a modl ha combins lmns ih h lmns of h spaial dpndncy of im or locaion. STAR modl islf is a dvlopmn of h Modl Vcor Auorgrssiv Ingrad Moving Avrag VARIMA bu h VARIMA modl has no bn paying anion im ih spaial dpndncis. Thrfor dvlopd a mhod ha combins lmns of im and locaion dpndncis mulivaria ih spaially hrognous lmns hich as hn calld h mhod GSTAR Ruchjana GSTAR mhod involving h prdicor variabls calld GSTAR. Variabl in GSTAR is a symbol ha has a mric and non-mric scal. For h cas of univaria im sris using h prdicor ih mric scal calld h Transfr Funcion Modl hil for non-mric scal calld h Inrvnion Modl and Variaions Calndar. Rsarch on Transfr Funcion Modl can b sn in Wu & Tsay on h rol of s saisics on a limid sampl cas hrough simulaions using Transfr Funcion Modl. For rsarch on inrvnion modls hav bn idly applid on by Suharono 7 on h ffc of h firs Bali bombing agains a fiv-sar hol occupancy. Whil on sudy on Variaion Modl Calndar conducd by L al. for h sals daa mal Muslim clohing by adding h ffc of Ramadhan.

2 SC. J.D. Kurnia Siaan S.P. Rahayu Whil rsarch has bn conducd by GSTAR Suharono al. 5 concrning GSTAR modl for forcasing h daa spaio mporal in h cas of inflaion of four ciis in Eas Java ih -scal non-mric i Eid vns and facors ris in ful prics as ll as rsarch by Okanindya 4 rgarding h inrvnion modl GSTAR and a sp puls is applid o h cas of forign ouriss forcasing. Sudis of mulivaria im sris approach using GSTAR is sill limid o modls involving variabl ih non-mric scal. GSTAR simaion mhod using h GLS as ll as h simaion mhod on h modl quaions SUR. Ordinary Las Squar mhod OLS can no b usd for mulivaria modl consising of mulipl quaions ha ar corrlad bcaus i ill produc a simaor is lss fficin in h sns ha h rsuling varianc ould b vry larg. Basd on h dscripion ha has bn dscribd abov in his sudy ill b conducd furhr sudis on mulivaria im sris modl ih variabl mric using GLS simaion. Th aim of his sudy is o obain simas of h modl paramrs GSTAR hrough simulaion sudis.. Marials and mhods. Mulivaria im sris Tim sris analysis usd in daa ha hav dpndncis im hr hr is a rlaionship bn h occurrnc of a priod ih h prvious priod. A h im sris analysis has h priod or h sam obsrvaion inrval Wi 6. Tim sris analysis involvs only a singl vn or a phnomnon calld h univaria im sris analysis hil involving som vn or phnomnon hich occurs corrlaion or rlaionship bn h incidnc of on anohr calld mulivaria im sris analysis. Similarly in h analysis of univaria im sris mulivaria im sris analysis o also pay anion o saionary hich can b sn on h plo Marix Cross Corrlaion Funcion MCCF and plo Marix Parial Cross Corrlaion Funcion MPCCF. On modl is a mulivaria im sris modl VARMA ha can gnrally b rin ino h form of h folloing quaion. Φ B Θ B a Whr p is a vcor ih mulivaria im sris Φ p B auorgrssiv ordr p marix and Θ B is a polynomial moving avrag ordr q. q. GSTAR modls GSTAR a gnraliaion of STAR modls. Diffrnc bn STAR modls ih GSTAR is auorgrssion paramr in h modl STAR assumd o b qual o any locaion hil h auorgrssion paramr of GSTAR b diffrn for ach locaion and h diffrnc bn h locaion shon in h form of ighing marix Borovkova al. 8. GSTAR in h form of a marix is givn by p λ s k s k Φ s Φ sk W s GSTAR modl ih on ordr of im and spaial ordr for hr diffrn locaions is givn by [ Φ ΦW ] ha can b prsnd in h form of a marix: q

3 Th simulaion sudis for Gnralid Spac Tim Auorgrssiv- GSTAR modl SC. To drmin h ordr of im in h modl can b usd AIC criria hras for h spaial ordr is gnrally limid o only ordr on cours bcaus of h highr ordr ill b difficul o inrpr Wusqa al.. Wighing on GSTAR hr ar four namly uniform igh invrs disanc normalid cross corrlaion and infrnc parial normaliaion of cross corrlaion Suharono & Aok 6.. Paramr simas Th OLS simaors β is ar as follos: ˆ - β Y. Whras h form of paramr sima from GLS simaor is Park 967: βˆ 'Ω 'Ω Y hr Ω Σ I so h abov quaion ill b: βˆ ' Σ I ' Σ IY..4 Mhods GSTAR-OLS and GSTAR- SUR ighd cross corrlaion normalid parial corrlaion. Sps for simulaion sudy ar as follos. a Gnraing h daa x and y for locaions ih n mulivaria normal disribuion ih a man of ro and varianc covarianc marixω. b Drmining h valu of cofficin paramrs usd in h modl GSTAR ih a saionary condiion. c Applying sps a and b in six simulaions i.. Simulaion for rsidual bn locaions is no corrlad ih h sam varianc. Simulaion o rsidual bn locaions dos no corrla ih diffrn variancs. Simulaion for rsidual bn locaions all corrlad ih h sam varianc. 4 Simulaion 4 for rsidual bn locaions is no all corrlad ih h sam varianc. 5 Simulaion 5 for rsidual bn locaions all corrlad ih diffrn variancs. 6 Simulaion 6 o rsidual bn locaions all corrlad ih diffrn variancs. d Evaluaing ordr ARIMA rsiduals. Ging sris yi and xi o locaions. f Incorporaing ordr ransfr funcion for ach simulaion Cas sudy using h ordr of b s r ino h quaion y i ω ω Β. i Cas sudy using h ordr of b s r ino h quaion y i ω ωβ ω B. i g GSTAR-OLS modl building and GLS. h Ging h modl paramr simaion GSTAR-OLS and GSTAR-SUR.

4 J.D. Kurnia Siaan S.P. Rahayu SC. i Comparing h rsuls of modl paramr simaion GSTAR-ols and GSTAR-SUR.. Rsuls and discussion Sudy of simulaion in his sudy using h VAR modl hich is hn usd o build h modl GSTAR ih h paramrs in h folloing quaion cofficin marix Φ As dscribd in h prvious chapr ha sag simulaion sudis conducd hrough six ays ih ach simulaion consisd of o cas sudis. Th firs cas sudy using h ordr of h ransfr funcion b s r and b s r. For h simulaion sudy usd a marix of parial normaliaion of cross corrlaion ighing. Rsuls of h simulaion sudy cas sudy ih a rsidual valu of bn locaions ar no muually corrlad h valu of h parial normaliaion of cross corrlaion ighing orh valid and comparabl on all paramrs hich mans a parial amoun of h cross-corrlaion bn h scond and hird locaion o h firs locaion is qually gra in h lag- and h valu of h parial cross-corrlaion bn h firs and hird locaion o h scond locaion is qually gra in h lag- as ll as h valu of h parial cross-corrlaion bn h firs and h scond locaion o a hird locaion is qually gra in h lag-. I is hrfor appropria ighing o simula on scond cas sudy is uniform ighing. Th ighing valu usd o form h rsidual bcom GSTAR modl paramr simaion in ordr o obain rsuls using h mhod of OLS and SUR in h folloing quaions For h firs simulaion cas sudy also producs a ighd valu of h parial normalid cross corrlaion is valid and comparabl hrfor b usd o obain a uniform ighd rsidual valu and h rsuling valu of h paramr simas in h folloing quaions

5 Th simulaion sudis for Gnralid Spac Tim Auorgrssiv- GSTAR modl SC Esimaion of paramrs in simulaion cas sudy and by using h Esimaion Mhod OLS and SUR gnraing paramr valus simad by OLS can b said o b no much diffrn or produc narly all of h sam valu by using h GLS simaion mhod as ll as h rsuling sandard rrors OLS and SUR. This mans GSTAR-OLS modl is as good as GSTAR-SUR in cass hr rsidual daa bn locaions ar no muually corrlad. Th sam hing is shon in simulaion cas sudis and hr h rsidual bn locaions is no corrlad o produc sandard rror simaion paramrs ih h sam valu hich mans GSTAR-OLS modl is as good as GSTAR-SUR. Th comparison of sandard rrors bn GLS and OLS in simulaion is prsnd in Tabl. Tabl. Comparison sandard rror of OLS and GLS in simulaion. Paramr OLS GLS simasi SE simasi SE psi psi psi psi psi Cas sudy psi Paramr OLS GLS simasi SE simasi SE psi psi psi psi psi psi Cas sudy As for h simulaion of 4 5 and 6 for h rsidual bn locaions corrlad produc GSTAR-SUR modl is mor fficin han h GSTAR-OLS bcaus i producs a

6 J.D. Kurnia Siaan S.P. Rahayu SC.5 sandard rror of sima paramr valus ar smallr. In h hird simulaion cas sudy gnrad valu ighd parial normaliaion of cross corrlaion ar valid and comparabl and hrfor a uniform ighing may b applid o his cas rsuling paramr simas OLS and SUR in h folloing quaions Simulaion in h cas sudy also producs a ighd valu of h parial normalid cross corrlaion is valid and comparabl hrfor b usd o g a uniform ighd rsidual valu and h rsuling valu of h paramr simas in h folloing quaions In h hird simulaion cas sudis and hr h rsidual daa bn locaions ar corrlad o produc sandard rror of sima paramr valus ha ar smallr in GSTAR-SUR Modl compard ih GSTAR-OLS. This mans ha h modl is mor fficin GSTAR-SUR applid o h cas hr corrlad rsiduals bn sis. Valu Modl GSTAR-SUR fficincy can b sn in Tabl. Simulaion of 4 5 and 6 o h sam conclusion as in h simulaion. 4. Conclusion Th rsuls of simulaion sudis ih a ransfr funcion GSTAR Modl ih is a mric variabl can b concludd ha if h rsiduals from h simulad daa ar no muually corrlad bn locaions h modl GSTAR-OLS and GSTAR-SUR ill gnra a sandard rror of paramr sima valu ar h sam. Hovr if h rsidual

7 Th simulaion sudis for Gnralid Spac Tim Auorgrssiv- GSTAR modl SC.6 of simulaion daa ar corrlad i ill gnra a sandard rror of paramr sima valus ar small in GSTAR-SUR Modl compard ih GSTAR-OLS Modl. So i can b said ha h paramr simaion using GSTAR-SUR Modl mor fficin smallr sandard rrors compard ih GSTAR-OLS for rsidual cass of simulaion daa ar corrlad. Cas sudy Cas sudy Tabl. Efficincy valu of GSTAR-SUR modl in simulaion. Paramr OLS GLS Efisinsi GLS simasi SE simasi SE % psi psi psi psi psi psi Paramr OLS GLS Efisinsi GLS simasi SE simasi SE % psi psi psi psi psi psi Rfrncs Borovkova S.A. Lopuhaa H.P. & Ruchjana B.N. 8. Consisncy and asympoic normaliy of las squar simaors in gnralid STAR modls. Saisica Nrlandica L M.H. Suharono & Hamah N.A.. Calndr variaion modl basd on ARIMA for forcasing sals daa ih Ramadhan ffc. Procdings of h Rgional Confrnc on Saisical Scincs Okanidya K.S. 4 Pmodlan GSTAR dngan inrvnsi puls dan sp unuk pramalan isaaan mancangara Unpublishd masr s hsis. Surabaya : ITS. Parks R.W Efficin simaion of sysm of rgrssion quaions hn disurbancs ar boh srially and conmporanously corrlad. Journal of h Amrican Saisical Associaion

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