Procdngs of th IConSSE FS SWCU (), pp SC ISB: 7-6-7--7 SC Two-lvl paramtr stmats GSTARX-GS modl Andra Prma Dtago and Shartono Statstcs Dpartmnt, Splh ovmbr of Insttt Tchnology, Srabaya 6, Indonsa Abstract GSTAR s a spcal form of th VAR modl and s on of th commonly sd modls for modlng and forcastng tm srs data and locaton At GSTAR modlng, stmaton mthod sd s OS, th mthod s consdrd to hav a waknss, whch wll rslt n an nffcnt stmator Ths, on approprat mthod s GS In ths stdy, condctd modlng GSTARX two lvls by addng a prdctor of calndar varaton modl Paramtr stmaton of th frst lvl modls mad of prdctors wth a lnar rgrsson modl, whl th scond lvl modls sng rror modls whch s don on frst lvl wth GSTAR modl Calndar varaton modl dscssd s th mpact of Ramadhan ffct Rslts of th smlaton stdy showd that GSTAR-GS modls prodcs a mor ffcnt stmator than GSTAR-OS, sn from th obtand standard rror smallr Kywords calndar varatons, ffcnt, GS, GSTARX, Ramadhan, two-lvl Introdcton On approach that can b sd to handl th data spac-tm s a Gnralzd Spac Tm Atorgrssv modls (GSTAR) Th modl provds a mor flxbl and s an xtnson of th modl STAR (Borovkova t al, ) Dffrnt from STAR modls, GSTAR dos not rqr that th vals of th sam paramtrs for all locatons Thrfor GSTAR mor ralstc, bcas n ralty s mor fond modls wth dffrnt paramtrs for dffrnt locatons Thortcal stds rlatng to th natr of th paramtrs GSTAR asymptotcally and wghtng btwn locatons gvn (ophaa & Borovkova, ) Implmntaton of GSTAR has bn don on th prodcton of ol and Gross Domstc Prodct (GDP) n 6 Wstrn Eropan contrs (ran, ) In addton, rsarch on comparson of btwn VARA wth GSTAR modls (Shartono, ) showd that th forcastng s mor accrat GSTAR modl Howvr, n th modl bldng procss n trms of thortcal and appld by th statstcal program packag was fond that th modl s mor flxbl and prfct VARA In addton, stds rlatd to GSTAR spcally for paramtr stmaton s stll lmtd to sng Ordnary ast Sqar (OS) (Borovkova t al, ) and axmm klhood mthod (Trz, ) Paramtr stmaton sng OS n th mltvarat modl wth rsdal corrlatd assssd as havng a waknss, whch wll rslt n nffcnt stmators Howvr, ovr th yars, dvlopd Gnralzd ast Sqar (GS) stmaton mthod GS mthods commonly appld to th modl Smngly Unrlatd Rgrsson (SUR), bcas on of th approachs that can b sd to stmat th paramtrs n th modl SUR s th GS mthod (Baltag, ) SUR s a systm of qatons that consst of mltpl rgrsson qatons, whr ach qaton has a dffrnt rspons and possbl prdctors hav dffrnt also Advantags of systm of qatons SUR s abl to accommodat th corrlaton btwn th rror qaton wth th othr qatons SUR modls wr frst
SC AP Dtago, Shartono appld n th cas of gross nvstmnt dmand n th two compans (Zllnr, 6) Th rslts obtand ar th stmatd paramtrs by GS for th ovrall modl s mor ffcnt than th OS paramtr stmats for ach modl In addton, SUR modls ar also appld to th spato-tmporal domans (Wang & Kocklman, 7) SUR modls wr appld to stmat th paramtrs GSTAR provd assranc that th rror of th modl s a mltvarat wht nos (Wtsqa & Shartono, ) Along wth ts dvlopmnt, GSTAR modl can b xpandd to GSTARX In ths cas X s a notaton for a prdctor or npt Prdctors can b a mtrc and or non-mtrc scal form For ths form mtrc, prdctor wll b condctd by transfr fncton modl, whras for th form of non-mtrc condctd by dmmy varabls In th cas of non-mtrc, th varabl may b th ffct of th ntrvnton, otlrs and calndar varatons Ths rsarch wll b sd prdctor of th calndar varaton modl to captr of Ramadhan ffcts Implmntaton of th GSTARX modl n ths stdy wll b dscssd throgh a smlaton stdy wth th am to gt th rght modl bldng procdr accordng wth th condtons of ral data atrals and mthods Estmaton procss n GSTAR can b don wth two mthods of stmaton,, OS and GS For xampl, appld n GSTAR( ) for locatons, t can b wrttn Z X β, whr β whr ( φ, φ,, φ, φ)' In th matrx form, can b wrttn z( ) z( ) ( ) ( ) z z z ( ) ( ) T z T ( ) z z ( ) z ( T) j V ( ) V ( ) V ( T ) O O z z ( ) ( ) z ( T ) ( ) ( ) φ ( ) φ T, V ( ) ( ) φ V ( ) ( ) φ V ( T ) ( T ) V ( t ) w Z ( t ) Paramtr stmaton of β condctd sng OS mthod by mans j j of mnmzng Z Xβ, so th stmator for β φ, φ,, φ, φ )' s obtand ˆ β ( X' X) X' Z ( Whras stmats for GS s obtand by mnmzng gnralzd sm of sqar ε' Ω whr ε ( Y Xβ), so that th qaton ε' Ω ε ( Y Xβ)' Ω ( Y Xβ) thn do dcras to th paramtrs, so that wold b obtand stmator whr Ω Σ I, ˆ ε, β ( X'Ω X) X'Ω Y, ()
Two-lvl paramtr stmats GSTARX-GS modl SC σ σ σ σ σ σ Σ dan I, O O σ σ σ n whch Σ s a rror varanc-covaranc matrx sz ( x ) and I s a dntty matrx of sz (T x T) In ths stdy, paramtr stmaton of th GSTARX modl wll b don n two-lvl odls for frst lvl,, th rgrsson modls wth calndar varaton: Y β f D, D ), () whr, t ( g, t g, t, t ( s th total calndar varaton ffcts, D g,t and f Dg, t, Dg, t ) αg Dg,t γg Dg,t g g D g,t rspctvly rprsnts dmmy varabl for th drng month Ed and on month bfor Ed, g ndcat th nmbr of days pror to th dat of Ed,, t s th rror componnt, and s a notaton for nmbr of locaton odls for scond lvl,, GSTAR modls: ( φ ( t ) φ w w ( t ) ( ( ) ( ) ( ) ( ) t φ t φ w w t t () ( t ) φ ( t ) φ w w ( t ) ( t ) From Eq (), th modls for scond lvl can b wrttn n th form ( φ φ φ ( t ) ( ( ) ( ) ( ) t φ φ φ t t, ( t ) φ φ φ ( t ) ( t ) whr φ φ, for,, and φ j w j φ, for, j,, whr j Stags of smlaton stdy carrd ot n two-lvl GSTARX modls ar as follows Stp : Dtrmn th ffcts of calndar varaton drng th spcfd prod Tabl Ed clbraton for th prod to Yar Dat Yar Dat Yar Dat 6 7 7- Aprl 6-7 Aprl - Aprl -6 arch - arch - arch - Fbrary - Fbrary - ary - ary - ary - Dcmbr 7- Dcmbr 6-7 Dcmbr -6 ovmbr - ovmbr 6 7 - ovmbr - Octobr - Octobr - Octobr - Sptmbr - Sptmbr Stp : Dtrmn coffcnt paramtrs of th vctor AR() modls:,,, Φ,,,,,, Stp : Gnrat rsdal at thr locatons mltvarat normal dstrbton wth a man of zro and varanc covaranc matrx Ω Stp : Dtrmnng two smlaton scnaros
SC AP Dtago, Shartono - for th frst cas, th rsdal s not corrlatd n thr locatons:,,, Ω,,,,,,, - for th scond cas, th rsdals ar corrlatd n thr locatons:,,, Ω,,,,,, Stp : Dtrmnng th dmmy varabl for th prod of calndar varatons (s Tabl ) Stp 6: Prform paramtr stmaton modl for frst lvl sng th OS mthod, sch as Eq () Stp 7: Dtrmnng th spatal wghts (W ) ar sd Stp : Prform paramtr stmaton modl for scond lvl sng th OS and GS mthod, sch as Eq () Stp : Calclat th ffcncy (%) of GS mthod, wth form SE OS ( ˆ) β SE GS ( ˆ) β x SE OS ( ˆ) β Stp : Ths phas s don by addng p th val of ot-sampl forcastng of th frst and scond lvl modl Rslts and dscsson Th frst stp s to dntfy th ffcts of calndar varaton from plot tm srs for a spcfd prod accordng to Tabl Plot tm srs of data smlaton th ffcts of calndar varaton shown n Fgr okas Apr ' ar ' ar ' Fb '7 ' Dc ' Dc ' ov ' Oct '6 Oct ' Sp ' Apr ' Apr ' ar ' Fb '6 ' ' Dc ' ov ' ov ' Oct '7 Sp ' okas Apr ' ar ' ar ' Fb '7 ' Dc ' Dc ' ov ' Oct '6 Oct ' Sp ' Apr ' Apr ' ar ' Fb '6 ' ' Dc ' ov ' ov ' Oct '7 Sp ' 6 onth Yar 6 6 onth Yar 6 6 okas 7 6 Apr ' ar ' ar ' Fb '7 ' Dc ' Dc ' ov ' Oct '6 Oct ' Sp ' Apr ' Apr ' ar ' Fb '6 ' ' Dc ' ov ' ov ' Oct '7 Sp ' onth Yar 6 6 Fgr Tm srs plot of smlaton data wth th vctor AR() modl
Two-lvl paramtr stmats GSTARX-GS modl SC6 Fgr shows of th tm srs plot of th vctor AR() modl wth th data contanng th ffcts of calndar varatons A vrtcal (dottd ln) was ncldd n ths plot to mphasz th months of Ed that occrrd drng ths prod Stag on n modlng GSTARX s to stmat of paramtrs th frst lvl Sch as th followng rslts - odl for locaton Y, t,d, t 6,77D 6,D 7,7D 6,D 6,6D 7,D,D - odl for locaton Y, t,7d 6,77D 7,D,D 67,D,D,D, t 7,D,7D 6, t 6, t, t, t, t 76,D, t, t - odl for locaton Y,D, t 6,D 6,6D,7D,D,D, t 6, t, t 6,D,D,D,7D,6D 6,7D 7,D, t, t,7d 6, t, t, t, t, t, t 6, t 6, t, t, t, t, t, t,d 7,7D,6D,6D 6,D,D, t, t, t,6d 7,6D 6,D, t 7, t, t, t,77d,d, t, t, t,d,d 6,D 7,D, t 7,76D 6,D,7D, t 7,D,777D, t,76d 7,D, t,7d 7, t, t, t, t, t,6d,7d,d 7,76D,D, t 7,6D, t 7, t,d, t,d, t, t, t,6d 6,D,7D, t 7,7D, t (),766D,6D 7, t, t, t 6,7D, t, t,6d,d 67,7D 7, t,6d 67,D, t, t, t 6,D,6D, t, t, t,7d 76,D, t,7d,d, t,6d,7d,d, t 6, t 6,6D, t 7, t 6, t,d, t,67d,6d, t, t 7,D, t, t, t,d, t, t,7d,d,d 7,D,D, t, t 7, t,6d, t () 7,D 6, t 6, t 7,D,D 7, t, t,d,77d, t, t,6d, t 7, t 7,D, t 6,D, t (6),6D,6D 7, t, t, t, t, t, t, t 6, t 6, t 7, t, t, t, t,d 6,D, t 7, t From th stmaton paramtrs sch as th frst lvl of Eq () to Eq (6) obtand rsdal modls (,, ) Frthrmor, th rsdal modls sd for stmaton at scond t lvl,, GSTAR modls wth paramtr btwn locatons (spatal) Charactrstcs of GSTAR modls s wghtd wth th locaton Spatal wghtng mthod sd n ths stdy s lmtd only by nfrnc Partal Corrlaton ormalzd Cross (IPKS) Ths mthod s basd on th hgh or low val of th partal cross corrlaton btwn locatons Statstcal nfrnc procss s don by sng a % confdnc ntrval
SC7 AP Dtago, Shartono Tabl Estmats normalzaton of cross corrlaton nfrnc partal data smlaton cas on Paramtr Estmats % confdnc ntrval owr Uppr Conclson P() 6 7 Vald and concrrnt P() 7 Vald and concrrnt P() 6 Vald and concrrnt P() 7 Vald and concrrnt P() 7 Vald and concrrnt P() 67 Vald and concrrnt Basd on th calclaton of th amont of th cross-corrlaton btwn th locaton at th tm to lag, th procss of nfrnc statstcs n Tabl shows that th confdnc ntrval gvs th sam amont (th rlatonshp) Ths, th dcson obtand ar vald and comparabl, t showd no dffrnc n wghtng btwn locatons Ths, th approprat wghtng mthod n ths cas s nform W (7) For th scond cas, th wghtng mthod s th sam as Eq (7) By sng th wght of locatons th rslts of paramtr stmaton GSTAR( ) modl shown n Tabl Tabl Comparson of paramtr stmats from OS and GS mthod for th frst and scond cas OS GS Efsns (%) Cas Paramtr Estmas SE Estmas SE GS φ 6 6 φ 7 φ 7 6 6 6 φ 7 7 φ 6 6 φ 6 φ 7 67 6 66 φ 7 φ 6 6 76 6 6 φ 77 7 76 6 φ 7 6 φ 7 7 7 Tabl shows th rslts of stmaton GSTAR modls thr s a dffrnc n th standard rror of th stmaton OS wth GS mthod Th standard rror of th GS mthod s smallr than th OS, th dffrnc s largly occrs on all paramtrs For th scond cas can b statd that th paramtr stmaton sng GS bttr than OS Ths can b sn n
Two-lvl paramtr stmats GSTARX-GS modl SC almost all GS ffcncy coffcnt s worth abov fv prcnt In addton, comparson of th ffcncy of th standard rror of ach paramtr GSTAR modl can also b shown throgh th crv probablty dstrbton fncton (pdf) n Fgr 7 6 OS GS OS GS Dnsty Dnsty ps - - ps 7 6 OS GS 6 OS GS Dnsty Dnsty ps ps 7 6 OS GS OS GS Dnsty Dnsty ps ps 6 Fgr Paramtr dstrbton plot for φ (lft) dan mthod (a) locaton, (b) locaton, dan (c) locaton φ (rght) wth OS and GS Effcncy of ach paramtr stmaton by sng GS mthod look mor ffcnt than OS mthod, t s markd on th shap of th crv pdf bl color a mor narrow In addton, th vrtcal a dottd ln shows th actal coffcnt vals of ach paramtr Vsally, th coffcnt of paramtr stmaton approach wth a tr val Frthrmor, for th frst cas GSTAR( )-OS modl can b wrttn (,,, ( t ) ( ( t ),,7, ( t ) ( t ) ( ),,, ( ) ( ) t t t and GSTAR( )-GS can b wrttn (,,, ( t ) ( ( ),,6, ( ) ( ) t t t ( t ),,, ( t ) ( t )
SC AP Dtago, Shartono As for th scond cas GSTAR( )-OS modl can b wrttn (,7,7,7 ( t ) ( ( t ),7,6,7 ( t ) ( t ) ( ),,, ( ) ( ) t t t and GSTAR( )-GS can b wrttn (,,66,66 ( t ) ( ( ),,76, ( ) ( ) t t t ( t ),,, ( t ) ( t ) Conclson and rmarks Th rslts of rsarch, partclarly n th smlaton stdy showd that GSTAR nvolvng th ffcts of calndar varaton stmaton, f don drctly (smltanosly) thr ar obstacls, that rsltd n th form of non-lnar modls So n ths stdy, proposd a procdr for bldng a modl,, two-lvl GSTARX, X basd on rgrsson modls wth calndar varaton Th frst lvl modl s sd to stmat of paramtrs th ffcts of calndar varatons, whras at th scond lvl modls for th spatal of paramtrs (basd on rror modls of th frst lvl) Two-lvl modl bldng procdr rfrs to W (6) Th smlaton rslts show that f th rsdals ar corrlatd btwn all locatons, or only a fw locatons only, thn th modl GSTAR-GS wll rslt n a mor ffcnt paramtr stmaton than GSTAR-OS modls Ths s shown on th standard rror vals gnratd by th modl GSTAR-GS s smallr than GSTAR-OS anwhl, f th rsdal s not corrlatd btwn all th qatons, th standard rror vals gnratd by th modl GSTAR-OS and GSTAR-GS ar th sam For frthr rsarch s ncssary to stmat a paramtr wth mlltvarat rgrsson modl for th frst lvl modl It s also ncssary to stdy a smlaton nvolvng a combnaton data typ (mtrc and non-mtrc) Rfrncs Baltag, B () On smngly nrlatd rgrssons wth rror componnts Economtrca,, 7 Borovkova, SA, ophaa, HP, & Rchjana, B () Consstncy and asymptotc normalty of last sqar stmators n Gnralzd STAR modls Jornal Complaton Statstca rlandca, 6(), ophaa, HP, & Borovkova, S () Asymptotc proprts of last sqars stmators n gnralzd STAR modls Tchncal Rport, Dlft Unvrsty of Tchnology ran, B () Pmodlan krva prodks mnyak bm mnggnakan modl gnralsas S-TAR Jrnal Form Statstka dan Komptas, IPB, Bogor Pffr, PE, & Dtsch, SJ () A thr stag tratv procdr for spac-tm modlng Tchnomtrcs, (), 7 Shartono () Prbandngan antara modl GSTAR dan VARIA ntk pramalan data drt wakt dan lokas Prosdng Smnar asonal Statstka, ITS, Srabaya Shartono, & Sbanar (6) Th optmal dtrmnaton of spac wght n GSTAR modl by sng crosscorrlaton nfrnc Jornal of Qanttatv thod, Jornal Dvotd to th athmatcal and Statstcal Aplcaton n Varos Fld, (), Trz, S () axmm lklhood stmaton of a gnralzd STAR (p; p) modl Jornal of th Italan Statstcal Socty, (), 77
Two-lvl paramtr stmats GSTARX-GS modl SC Wang, X, & Kocklman, K (7) Spcfcaton and stmaton of a spatally and tmporally atocorrlatd smngly nrlatd rgrsson modl: Applcaton to crash rats n Chna Jornal of Transportaton,, W, WWS (6) Tm srs analyss nvarat and mltvarat mthods Canada: Addson- Wsly Pblshng Company, Inc Wtsqa, DU, & Shartono () Sasonal mltvarat tm srs forcastng on torsm data by sng VAR-GSTAR modl Jrnal IU DASAR, (), Zllnr, A (6) An ffcnt mthod of stmatng smngly nrlatd rgrsson qatons and tsts for aggrgaton bas Jornal of th Amrcan Statstcal Assocaton, 7, 6