Automobile Sales Modeling using Granger-Causality Graph with PROC VARMAX in SAS 9.3

Transcription

1 uomobie Saes Modeig usig Grager-Causai Grah wih PROC VRMX i SS 9.3 BSTRCT kkarao Sa-gasoogsog ad Saish T.S. Bukkaaam Okahoma Sae Uiversi, Siwaer, OK, US Predicio of saes has aed a crucia roe for roducio aig i auomoive idusr where ica coce-o-reease ime is -60 mohs og. However, due o osaioar characerisic ad deedecies wih diverse macroecoomic variabes, he use of hisorica daa of auomobie saes aoe is o sufficie o derive accurae redicio of he fuure of auomobie saes. This aer reses a modeig of auomobie saes usig Grager-Causai GC grah wih PROC VRMX i SS 9.3. The GC grah is a rereseaio of ideified causa reaioshis of auomobie saes ad ecoomic idicaors i which he verices rerese he comoes of ime series, icudig auomobie saes ad ecoomic idicaors, ad he edges rerese he causa reaios amog comoes i he grah. I his aer, we aso address he issues eraiig o he redicio of auomobie saes wih he aid of GC Grah, icudig seasoai ad osaioari of variabes. Based o ideified GC Grah, Vecor uoregressive VR e modes are fia esimaed usig PROC VRMX i SS 9.3. Comariso of redicio accurac wih cassica ime series echiques demosraes saisfacor erformace of auomobie saes modeig wih GC grah mode. INTRODUCTION Saes redicio is esseia ar of a busiess acivi. Esecia for roducio aig aciviies, reiabe forecass make a imora coribuio o efficie roducio aig. s i he case of auomobie roducs, where ica coce-o-reease imes are og -60 mohs [], saes redicio serves as iu o ma busiess decisios, icudig roducio ad oeraio aig, iveor coro, maeria maageme, ec. These decisios usua affec rofiabii of he orgaizaio. Hece, saes redicio is criica comoe of successfu roducio aig aciviies. Saes is kow o be ifueced b ma facors, such as, adverisig, saes romoios, reai rice ad echoogica sohisicaio []. I auomobie marke, adverisig ad saes romoios ed o have subsaia effecs, however, hese effecs o saes are rare ersise [3-6]. Due o osaioar characerisic, he use of hisorica daa of auomobie saes aoe is o sufficie o derive accurae redicio of he fuure of auomobie saes. Some ecoomic idicaors, such as, housig sars, cosumer rice idex, gas rices, ec., have bee suggesed o have ersise effecs o saes. However, ideificaio of a srucura reaioshi of hese idicaors wih auomobie saes is robemaic due o ukow uderig rocesses of he ssem. I he coex of ecoomeric ime series research, here is a deveome i muivariae ime series echiques ha ca be used o deermiig a redicabii of oe variabe from he ohers. This echique is so caed Grager Causai [7]. Grager Causai has bee broad regarded as a owerfu heordrive mehod, ad wide used i ecoomeric ime series research sice is mome of emergece. Rece, he oio of Grager Causai has bee merged wih grah heor o iroduce a ew grahica aroach for modeig, ideificaio ad visuaizaio of he causa reaioshis bewee he comoes of a muivariae ime series. This ew grahica aroach is caed Grager-Causai GC grah [8-0] for ime series i which verices rerese he comoes of ssem coeced b direced edges accordig o he Grager-Causai reaioshis bewee variabes. The advaage of GC grah is ha i ca easi visuaized ad hus eads o a ideificaio of causa reaioshis of variabes i he ssem. Moreover, he coce of GC grah ca be exeded furher o buid a Vecor uoregressive VR rocess [-5]. This aer reses a mehodoog o ideif causa reaioshis, ad aso o deveo VR modes of auomobie saes ad ecoomic idicaors usig GC grah wih PROC VRMX i SS 9.3. I his aer, we aso address he issues eraiig o he redicio of auomobie saes wih he aid of GC Grah, icudig seasoai ad osaioari of variabes. The orgaizaio of his aer is as foows:

2 uomobie Saes Modeig usig Grager-Causai Grah wih PROC VRMX i SS 9.3, coiued he daa secio reses descriio ad characerisic of each variabe i deais. Issues reaed o redicio mode, icudig seasoai ad osaioari of variabes are reseed i he daa rerocessig secio. GC grah ad VR mode are reseed i he mehodoog secio. Emirica resus are reored i imemeaio deais ad resus secio. Summar ad cocusios are reseed i he as secio of his aer. DT I his aer, he mai ime series is he oa auomobie saes i heav & medium rucks segme durig a eriod of from idusr source. The daase cosiss of saes ad five hohesized variabes see Fig., icudig Housig Sars HS, Cosumer Price Idex CPI, Gas Prices GP, Emome Rae EMP ad verage Overa Exediure er Vehice VG. These variabes are hohesized o have causa reaioshis wih saes. Due o roriear reasos, auomobie saes daa are coded io a rage of rea umber from -8 o 8. Deais ad Descriio of each variabe are summarized as show i Tabe. Variabe Name [Usage] Housig Sars [HS] Cosumer Price Idex Iems [CPI] GS Prices [GP] Emome [EMP] verage overa Exediure er Car [VG] uomobie Saes Tabe : Daa Summar Daa Characerisics Descriio Series Tie: Privae Owed Housig Sars: -Ui Srucures Uis: Thousads of Uis djusme: Seasoa djused ua Rae SR Source: U.S. Dearme of Commerce Cesus Bureau Series Tie: Cosumer Price Idex Urba Cosumers rea: U.S. Ci verage Seasoa djused Source: U.S. Dearme of Labor Bureau of Labor Saisics Series Tie: Moh U.S. Reguar Formuaios Reai Gasoie Prices Uis: Ces er Gao Seasoa djused Source: U.S. Eerg Iformaio dmiisraio Series Tie: Toa o farmig emome Uis: Thousads Seasoa djused Source: U.S. Dearme of Labor Bureau of Labor Saisics Series Tie: verage overa Exediure er car Uis: Doars Seasoa djused Source: U.S. Dearme of Commerce Bureau of Ecoomic asis Series Tie: Toa Vehice Saes Icudig Heav & Medium Trucks Uis: Coded djusme: ukow Source: U.S. Dearme of Commerce Bureau of Ecoomic asis & Idusr The auomobie saes Fig. a seems o have ccica aer from Jauar 975 or before ui he midde of ear 993. This ma be exaied b ecoomic u ad dow ur durig hese eriods. Nex, a icreasig aer ca be see i he begiig of ear 000. Saes he seem o sabiize wih wo oca eaks Ocober 00, Ju 005 ui he begiig of ear 007. The fiacia ad ecoomic crisis riggered b U.S. bakig ssem ad arge fiacia isiuios has show is effec o auomobie saes from he ed of 007. HS Fig. b shows ver simiar aer wih saes daa. The red aer i Saes ad HS, as show i Fig. a, is ossib ifueced he same grou of uderig facors. However, HS eds o receive he effecs of hese facors faser ha saes, as show b he urig ois oca exremes i he saes daa usua foow hose of HS. GP Fig. c shows sow icreasig aer from Jauar 975 ui he begiig of ear 000. GP he icreases wih a much faser rae bewee Oe ca aso see ha GP sars icreasig drasica as saes i droig i some oca ime eriods See Fig. b, Regios, ad 4. so, saes

3 uomobie Saes Modeig usig Grager-Causai Grah wih PROC VRMX i SS 9.3, coiued seems o icrease if GP remais sabe for a og ime See Fig. b, Regio 3. EMP, CPI ad VG Fig. d-f have simiar icreasig red aer, ad are hohesized o be exogeous variabes ha ca be used o he redic saes. a b c d e f Figure : a uomobie Saes b Housig Sars c Gas Prices d Emome rae e Cosumer Price Idex f verage Overa Exediure Per Car a b Figure : a Co-evouio of Saes ad Housig Sars b Saes vs. Gas Prices DT PRE-PROCESSING I his secio, issues eraiig o redicio of auomobie saes wih GC grah are reseed. The firs issue is seasoai. Seasoai is defied as reeiive comoe of ime series ha recurs ever caedar ear. This comoe of ime series is easi redicabe ad shoud be removed from he origia ime series before aig GC grah ad redicio mode. Seasoai es is reseed i he foowig subsecio o ideif exisece of seasoa comoe of auomobie saes. The secod issue is osaioari of variabes. Due o saioar requireme of VR mode, a variabes i VR mode mus be saioar variabes. This secio aso rovides saioar es o es saioari of variabes i he ssem. Deais ad some discussios regardig hese issues are rovided i he foowig 3

4 uomobie Saes Modeig usig Grager-Causai Grah wih PROC VRMX i SS 9.3, coiued subsecios.. Seasoai Tes Time series usua cosiss of four comoes: red, seasoa, ccica ad irreguar comoes. s he seasoa comoe awas haes wih he same magiude durig he same ime eriod each ear, so i is ofe cosidered uieresig ad ca cause a series o be ambiguous. I case of a ambiguous series, i is ver difficu o disiguish oe comoe from ohers. hough, origia series icudig a four comoes refecs he acua curre daa, ad ca be rediced usig forecasig echiques direc, difficu usua arises whe we cosider wha comoe coribues o he redicio resu of a observed ime series. B removig he seasoa comoe, i is easier o focus o oher comoes. The F-es for sabe seasoai [6, 7] is seeced o ideif seasoa comoe i ime series. The mehod assumes ha a ime series is comosed of hree comoes which are reaed i a muiicaive form as show i Eq. Z C S I where Z is ime series, S is he seasoa comoe, I is he irreguar comoe ad C is a combiaio of he og-erm red ad og-erm ccica comoes. B aig movig average echique, idividua comoe ca be smoohed ou ad ideified. The, he F-es for sabe seasoai is erformed o he seasoa-irreguar comoes of he series, SI S I. The series, SI ij i,,..., N; j,,...,, is used o comued he sabe seasoa for each moh as show i Eq. SI N j SI ij N i for j,,..., ssumig ha he seasoa cce reeas ever mohs, he es for sabe seasoai ess he hohesis H0 : SI SI... SI SI, agais he aeraive of iequai, where SI is he grad mea of he seasoa-irreguar comoes. This es ca be erformed i SS usig PROC X See edix.. Saioar es Due o auoregressive aure of VR mode, saioar assumio mus hod whe a VR o he ssem of variabes. To es saioar of variabes, SS offers muie ui roo ess, such as Phiis- Perro es, a radom-wak wih drif es, augmeed Dicke-Fuer es DF ec. I his aer, he ugmeed Dicke ad Fuer DF [8, 9] is seeced o ideif osaioari codiio of variabes. Two versios of he es used i his aer are show i Eq. 3 ad Eq The u hohesis is 0 agais he aeraive of 0. Eq.3 is he es for a ui roo wih drif sige mea ad Eq. 4 is a ui roo wih drif ad deermiisic ime red red mode. I case of exisig osaioari, differecig echique mus be aied o osaioar variabes. The DF es ca be erformed i SS usig PROC RIM See edix. METHODOLOGY The mehodoog framework o deveo GC grah ad VR modes cosiss of hree mai ses as show i Fig. 3. The firs se is he daa re-rocessig se. I his se, he daa described i he daa secio is subjeced o screeig ess o fier ou he effecs of seasoai, ad osaioar. Nex se is GC grah se. Grager-Causai es is used o ideif casua reaioshis of auomobie saes ad hohesized variabes. The resus of causai es is he used o buid a GC grah of saes ad seeced variabes. The as se is mode buidig & adaaio se. I his se, he VR modes are deveoed based o he resus from he secod se. The reduced forms of VR mode is esed for reasoabe GC grah. Fia, he seeced fia mode is esimaed ad comariso of redicio is made o access he erformace of redicio accurac of GC grah ad VR mode. The foowig 4

5 uomobie Saes Modeig usig Grager-Causai Grah wih PROC VRMX i SS 9.3, coiued 5 subsecios rovide he deais of GC grah, causai es ad VR mode.. VR ad VRX Modes Vecor uoregressive Mode VR is a aura exesio of he uivariae auoregressive R modes. I reas each variabe i he ssem smmerica. The evouio of each edogeous variabe i he ssem ca be exaied b is ow ags ad he ags of a oher edogeous, ad exogeous variabes i case of VRX. VR aso ca hade feedback from each edogeous variabe i which ca be used o exai a srucura reaioshi of variabes i he ssem. Figure 3: Mehodoog Framework Le,...,,,..., deoe a ime series vecor of -dimesioa ssem. h order vecor auoregressive, deoed as VR, ca be rereseed as show i Eq. c c c where i c arameer rereseig ierce erms s ij auoregressive coefficies i whie oise disurbaces VR rocess ca aso be affeced b exogeous variabes. I he case of Vecor uoregressive wih Exogeous Variabes VRX, he VRX,m ca be rereseed i a sime form as show i Eq. m x c 0 *

6 uomobie Saes Modeig usig Grager-Causai Grah wih PROC VRMX i SS 9.3, coiued where c deoes a vecor of cosas ad deoes a marix of auoregressive coefficies for,,...,. The vecor is a vecor wih smmeric defiie marix. The x x,..., x w is a w-dimesioa exogeous ime series vecor ad is a w marix of coefficies. For saioar assumio of VR modes, he saioar codiio is saisfied if a roos of Z 0 ie ouside he ui circe. If he saioar codiio is o saisfied, a osaioar mode a differeced mode is more aroriae.. GC Grah For a defiiio of GC grah, firs we cosider grah G give b a ordered air G = V, E where V is a se of eemes caed verices or odes ad E is a se of direced or udireced edges which beog o he casses { a b a, bv, a b} ad { a b a, bv, a b}, resecive. For he verex se, each verex a i he grah rerese oe variabe or comoe of ime series vecor. B his defiiio, he causa srucure of a ssem of ime series vecor, Y, ca be described b wo casses as foows: i Cass { a b a, bv, a b} : This cass of GC grah rereses he direced edges corresodig o direc causa reaios bewee he comoes of Y which ca be ideified b Grager Causai es. ii Cass { a b a, bv, a b} : This cass of GC grah rereses he udireced edges corresodig o he coemoraeous codiioa associaio bewee he variabes. Sice our rimar ieres i his aer is o deveo a redicio mode of auomobie saes ad ecoomic idicaors from GC grah, we wi focus our aasis io fidig he causa reaios bewee variabes o buid a GC grah wih direced edges of auomobie saes ad ecoomic idicaors. The causa reaios bewee variabes of he ssem of auomobie saes ad hohesized ecoomic variabes ca be ideified usig Grager Causai es as described i he foowig: The coce of Grager Causai is ha some variabes are usefu i forecasig ohers. B defiiio, is said o Grager-cause x if ca be used o he redic some sage i he fuure of x. Cosiderig he exame of bivariae VR mode wih coefficies s ij for i, j, ad s,..., as foows * c c `... ` 3 The variabe is said o cause Grager, bu do o cause Grager s s if 0 s. This mode srucure imies ha if 0, is ifueced o b is ow as vaues ad o b he as of. For a arger VR mode >, he Grager Causai ca be used o es wheher oe variabe is ifueced o b isef ad o b oher variabes i he mode ssem. Le be -dimesioa ime series vecor which is arraged ad ariioed i subgrous ad wih dimesios ad, resecive : ha is,, wih he corresodig whie oise rocess,. Equivae rereseaio of VR mode i Eq. ca be wrie as c c 4 Cosider esig H : C, where C is a s marix of rak s ad c is a s-dimesioa vecor where s 0 c. The Wad saisic ca be obaied from T C ˆ c[ C ˆ ˆ C ] ˆ d C c s 5 If he u hohesis cao be rejeced, we ca excude ha variabe from VR mode or rea i as exogeous variabe i VRX mode. Sequeia ess of Grager Causai ca be erformed. Coseque, he direced GC grah ca be bui from he resus of his es. Exames of GC grah 6

7 uomobie Saes Modeig usig Grager-Causai Grah wih PROC VRMX i SS 9.3, coiued aers are show i Fig. 4. These exames rese four es of GC grah aers, icudig direc causai, direc feedback, idirec causai ad surious causai. The VR mode ad Grager Causai es ca be erformed i SS usig PROC VRMX See edix. a b c d Figure 4: Exames of GC grah aers a direc causai, b direc feedback, c idirec causai, d surious causai IMPLEMENTTION DETILS ND RESULTS Tabe reses a resu of he F-es for sabe seasoai o auomobie saes. The robabii eve Pvaue of his es is 0.7. Sice he u hohesis i he sabe seasoai es is o rejeced a he 0% sigificace eve 0.0, he auomobie saes series is o seasoa, ad he ideifiabe seasoai is o rese. Thus, he auomobie saes series is used wihou seasoa adjusme i he subseque aasis for deveoig a redicio mode. Tabe : Seasoai Tes Resu Sum of Squares DF Mea Square F-Vaue Bewee Mohs Residua Toa Grager-Causai Wad Tes Tes Grou Variabe Grou Variabes DF Chi-Square Pr > ChiSq Saes HS CPI GP EMP VG <0.000 HS Saes CPI GP EMP VG CPI Saes HS GP EMP VG < GP Saes HS CPI EMP VG EMP Saes HS CPI GP VG VG Saes HS CPI GP EMP Tabe 3: Grager-Causai Wad Tess Grager-Causai Wad Tes Tes Grou Variabe Grou Variabes DF Chi-Square Pr > ChiSq Saes CPI GP EMP VG <0.000 CPI Saes GP EMP VG < GP Saes CPI EMP VG EMP Saes CPI GP VG VG Saes CPI GP EMP Tabe 4: Grager-Causai Wad Tess Rees Grager-Causai Wad Tes Tes Grou Variabe Grou Variabes DF Chi-Square Pr > ChiSq Saes CPI GP VG 45.4 <0.000 CPI Saes GP VG < GP Saes CPI VG VG Saes CPI GP Tabe 5: Grager-Causai Wad Tess Rees From he DF es resus of saioari, saes ad five hohesized variabes i he daase have ui roos a 5% sigificace eve for ui roo wih drif mode. Differecig echique is used for hese variabes. The resus show ha he exhibi saioar characerisic afer firs differeced. The ex se is o a Grager Causai es o hese variabes. Tabe 3 shows he resus of Grager-Causai Wad ess. % sigificace eve, he u hohesis of o causai cao be rejeced for HS. Reesig Grager Causai wih o rejecig variabes Tabe 4 shows ha he u hohesis cao be rejeced for EMP a % sigificace eve. Excudig a o-rejecig variabes from he mode, Tabe 5 shows resus of Grager Causai ess wih 4 variabes. The resus cofirm ha four variabes Saes, 7

8 uomobie Saes Modeig usig Grager-Causai Grah wih PROC VRMX i SS 9.3, coiued CPI, GP ad VG have causa reaioshis, ad ca be used o buid VR mode i he ex se. Esimaig VR ad VRX modes idicaes ha exogeei of EMP is sigifica for auomobie saes. Fia, a GC grah of auomobie saes mode is bui ad reseed i Fig. 5. GC grah is o fu coeced grah due o exogeei of EMP. Figure 5: Grager-Causai Grah of uomobie Saes ad Ecoomic Idicaors From he resu of GC grah, VRX mode wih 4 edogeous Saes, GP, VG ad CPI ad exogeous variabes EMP is esimaed. Ou-of-same daa are radom seeced o vaidae he VRX mode. Four ime series modes are seeced o comare wih VRX mode. The are RM, RIMX, Sewise auoregressive ad Exoeia Smoohig modes as show i Tabe 6. Cosiderig forecasig erformace of auomobie saes redicio, VRX of auomobie saes ssem, rereseed b GC grah i Fig. 5, ca imrove a redicio accurac b 40%, 47%, 36% ad 34%, comared o RIM, Sewise uoregressive, Exoeia Smoohig ad RIMX modes i erms of RMSE as show i Tabe 6. These RMSEs vaues are quaified o -se ahead redicio of auomobie saes. Tabe 6: Mode Comariso CONCLUSION Forecasig Performace Comariso Mode RMSE -se ahead redicio RIM.0936 Sewise uoregressive.377 Exoeia Smoohig.0095 RIMX VRX The advaage of Grager-Causai GC Grah o mode auomobie saes is ha i rovides a grahica aroach for modeig, ideificaio ad visuaizaio of he causa reaioshis bewee he comoes of auomobie saes ad ecoomic idicaors. This aer reses a mehodoog o ideif causa reaioshis bewee variabes o form Grager-Causai GC grah ad VRX modes wih PROC VRMX i SS 9.3. This aer aso addresses issues eraiig o he redicio of auomobie saes wih he aid of GC grah, icudig seasoai ad osaioari of variabes. From he resus, GC grah of auomobie saes ad hohesized ecoomic idicaors is ideified. Based o ideified GC grah, he emirica resus show ha VR mode wih seeced edogeous ad exogeous variabes ca sigifica imrove a redicio accurac of auomobie saes for og erm redicio. PPENDIX /*-- Seasoai es --*/ ods ouu D8#=SaesD8_; ods ouu D8#=SaesD8_; ods ouu D8#3=SaesD8_3; ods ouu D8#4=SaesD8_4; roc x daa=gm.saes dae=dae; var saes; x mode=add; ru; ie Sabe Seasoai Tes; roc ri daa=saesd8_ LBEL; ru; 8

9 uomobie Saes Modeig usig Grager-Causai Grah wih PROC VRMX i SS 9.3, coiued ie Noarameric Sabe Seasoai Tes; roc ri daa=saesd8_ LBEL; ru; ie Movig Seasoai Tes; roc ri daa=saesd8_3 LBEL; ru; oio odae oumber; ie Combied Seasoai Tes; roc ri daa=saesd8_4 LBEL NOOBS; var _NME_ Name Labe cvaue; ru; /*-- Saioar Tes --*/ PROC RIM daa=gm.saes; ideif var=saes saioari=adf=4; ideif var=hs saioari=adf=4; ideif var=cpi saioari=adf=; ideif var=emp saioari=adf=6; ideif var=vg saioari=adf=5; ideif var=gp saioari=adf=6; RUN; /*-- Grager Causai Tes --*/ PROC VRMX daa=gm.saes; mode saes g ci em avg hs/ =4 ; causa grou=saes grou=g ci em avg hs; causa grou=g grou=saes ci em avg hs; causa grou=ci grou=saes g em avg hs; causa grou=em grou=saes g ci avg hs; causa grou=vg grou=saes g ci em hs bc; causa grou=hs grou=saes g ci em avg; RUN; /*-- Mode Comariso-RIM,,0 --*/ PROC RIM daa=gm.saes; ideif var=saes; esimae =; forecas ead=; ru; /*-- Mode Comariso-RIMX --*/ PROC VRMX daa=gm.saes; mode saes= ci /= oi xag=4 ; ouu ead=; ru; /*-- Mode Comariso-VRX4,4 --*/ PROC VRMX daa=gm.saes; mode saes ci em avg g= ci /=4 oi xag=4 agmax = ; ouu ead=; ru; REFERENCES []. Sa-gasoogsog, "Log-Term Demad Predicio usig Log-Ru Equiibrium Reaioshi of Irisic Time-Scae Decomosiio Comoes," i 0 Idusria ad Ssems Egieerig Research Coferece, Orado, FL, 0. [] J. R. Ladwehr, e a., "Gu Likig for he Ordiar: Icororaig Desig Fuec Imroves uomobie Saes Forecass," MRKETING SCIENCE, vo. 30, , 0. [3] M. G. Dekime, e a., "Log-ru effecs of rice romoios i scaer markes," Joura of Ecoomerics, vo. 89,. 69-9, 998. [4] K. Pauwes, e a., "New Producs, Saes Promoios, ad Firm Vaue: The Case of he uomobie Idusr," The Joura of Markeig, vo. 68,. 4-56, 004. [5] V. R. Nijs, e a., "The Caegor-Demad Effecs of Price Promoios," MRKETING SCIENCE, vo. 0,. -, 00. [6] K. Pauwes, e a., "The Log-Term Effecs of Price Promoios o Caegor Icidece, Brad Choice, ad Purchase Quai," Joura of Markeig Research, vo. 39, , 00. [7] C. W. J. Grager, "Ivesigaig Causa Reaios b Ecoomeric Modes ad Cross-secra Mehods," Ecoomerica, vo. 37, , 969. [8] M. Eicher, "Grager causai ad ah diagrams for muivariae ime series," Joura of Ecoomerics, vo. 9

10 uomobie Saes Modeig usig Grager-Causai Grah wih PROC VRMX i SS 9.3, coiued 37, , 007. [9] M. Eicher, "Grahica Modeig of Damic Reaioshis i Muivariae Time Series," i Hadbook of Time Series asis, ed: Wie-VCH Verag GmbH & Co. KGa, 006, [0] M. Eicher, "Grahica modeig of muivariae ime series," Probabii Theor ad Reaed Fieds, [] G. D. Marik ad M. H. Domiique, "Time-series modes i markeig: Pas, rese ad fuure," Ieraioa Joura of Research i Markeig, vo. 7,. 83, 000. [] M. G. Dekime ad D. M. Hasses, "The Persisece of Markeig Effecs o Saes," Markeig Sciece, vo. 4,. -, 995. [3] M. G. Dekime, e a., "Time-Series Modes i Markeig: Hadbook of Markeig Decisio Modes." vo., B. Wierega, Ed., ed: Sriger US, 008, [4] S. Johase, "Esimaio ad Hohesis Tesig of Coiegraio Vecors i Gaussia Vecor uoregressive Modes," Ecoomerica, vo. 59, , 99. [5] C.. Sims, "Macroecoomics ad Reai," Ecoomerica, vo. 48,. -48, 980. [6] R. Vaugh, e a., "Ideifig seasoai i fuures rices usig X-," Joura of Fuures Markes, vo.,. 93-0, 98. [7] B. C. Suradhar, e a., " sime es for sabe seasoai," Joura of Saisica Paig ad Iferece, vo. 43, , 995. [8] D.. Dicke ad W.. Fuer, "Disribuio of he Esimaors for uoregressive Time Series Wih a Ui Roo," Joura of he merica Saisica ssociaio, vo. 74, , 979. [9] D.. Dicke, e a., "Tesig for Ui Roos i Seasoa Time Series," Joura of he merica Saisica ssociaio, vo. 79, , 984. CONTCT INFORMTION Your commes ad quesios are vaued ad ecouraged. Coac he auhors a: Name: kkarao Sa-gasoogsog Eerrise: Okahoma Sae Uiversi ddress: 45N. Uiversi Pace#30 Ci, Sae ZIP: Siwaer, Okahoma, 74075, US E-mai: akkara@oksae.edu Name: Saish T.S. Bukkaaam Eerrise: Okahoma Sae Uiversi ddress: 3 Egieerig Norh Ci, Sae ZIP: Siwaer, Okahoma, 74078, US E-mai: saish..bukkaaam@oksae.edu kkarao Sa-gasoogsog is a PhD sude i Schoo of Idusria Egieerig ad Maageme a Okahoma Sae Uiversi. He has hree ears of rofessioa exerieces as R&D Egieer. He has eared wo SS cerificaios: SS Cerified Base Programmer for SS 9 ad Cerified Predicive Modeer usig SS Eerrise Mier 6.. He was s award wier of he M00 coferece s Daa Miig Shooou, ad reciie of SS ambassador hoorabe meio award SGF 0 Saish T.S. Bukkaaam is T&T rofessor i Schoo of Idusria Egieerig ad Maageme a Okahoma Sae Uiversi. His research ieds over a hudred eer-reviewed ubicaios. He was a reciie of ha Pi Mu/ Omega Rho Ousadig Teacher of he Year i Idusria Ssems Egieerig, ad Ousadig Youg Maufacurig Egieer ward from he Socie of Maufacurig Egieers. SS ad a oher SS Isiue Ic. roduc or service ames are regisered rademarks or rademarks of SS Isiue Ic. i he US ad oher couries. idicaes US regisraio. Oher brad ad roduc ames are rademarks of heir resecive comaies. 0