A HIERARCHICAL BAYESIAN APPROACH FOR SPATIAL TIME SERIES MODELING. Yiannis Kamarianakis. REAL 03-T-15 April, 2003

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1 he Regonal Eonom Applaon Laoaoy REAL a oopeave venue eween he Unvey of llno and he Fedeal Reeve Bank of Chago foung on he developmen and ue of analyal model fo uan and egonal eonom developmen he pupoe of he Duon Pape o ulae nemedae and fnal eul of h eeah among eade whn and oude REAL he opnon and onluon eed n he pape ae hoe of he auho and do no neealy epeen hoe of he Fedeal Reeve Bank of Chago Fedeal Reeve Boad of Goveno o he Unvey of llno All eue and ommen hould e deed o Geoffey J D ewng Deo Regonal Eonom Applaon Laoaoy 67 Souh ahew Uana L Phone Fax e page: wwwuuedu/un/eal A ERARCCAL BAESA APPROAC FOR SPAAL E SERES ODELG y ann amaanak REAL 3--5 Apl 3

2 A ERARCCAL BAESA APPROAC FOR SPAAL E SERES ODELG ann amaanak Regonal Analy Dvon nue of Appled and Compuaonal ahema Foundaon fo Reeah and ehnology-ella Valka Vouon PO Box 57 GR 7 eaklon Cee Geee and Regonal Eonom Applaon Laoaoy Unvey of llno 67 S ahew #38 Uana llno Aa Depe he fa ha he amoun of daae onanng long eonom me ee wh a paal efeene ha gnfanly neaed dung he yea he peene of negaed ehnue ha am o dee he empoal evoluon of he ee whle aounng fo he loaon of he meauemen and he neghong elaon vey pae n he eonome leaue h pape how how he eahal Bayean Spae me model peened y kle Belne and Cee Envonmenal and Eologal Sa l998 fo empeaue modelng an e aloed o model elaonhp eween vaale ha have oh a paal and a empoal efeene he f age of he heahal model nlude a e of egeon euaon eah one oepondng o a dffeen loaon oupled wh a dynam pae-me poe ha aoun fo he unlaned vaaon A he eond age he egeon paamee ae endowed wh po ha efle he neghong elaon of he loaon unde udy; moeove he pao-empoal dependene n he dynam poe fo he unlaned vaaon ae eng ealhed Pung hypepo on pevou age paamee omplee he Bayean fomulaon whh an e mplemened n a akov Chan one Calo famewok he popoed modelng aegy ueful n uanfyng he empoal evoluon n elaon eween eonom vaale and h uanfaon may eve fo exe foeang auay

3 noduon Saal and eonome model ha am o dee he empoal evoluon and he neelaonhp eween vaale ha have a paal efeene ae eng efeed a pae-me model Reeah n uh modelng ehnue ha gnfanly neaed dung he la weny yea ne loely elaed o he poge n ompue ehnology and he exene of lage daaae Depe eeahe effo pae-me modelng ehnue do no le n an negaed heoeal famewok lke fo example he ARA mehodology fo me ee; uually he employed ehnue vay aodng o he knd of applaon ha need o e pefomed Clff and Od 975 wee he f o pefom n a egeon famewok a model ha wa akng no aoun oh paal and empoal elaonhp; n he ealy eghe Pfefe and Deuh 98a 98 98a peened he Spae-me Auoegeve negaed ovng Aveage SARA model amng o offe a ool fo pao-empoal modelng analogou o he ARA mehodology fo unvaae me ee he SARA mehodology ha een appled n a wde vaey of applaon angng fom envonmenal Pfefe and Deuh 98a Soffe 986 o epdemologal Pfefe and Deuh 98a eonome Pfefe and Bodly 99 and aff flow amaanak and Paao 3 o name ju a few Daa lmaon uually n he empoal dmenon and modelng need n egonal eonom applaon foed eeahe o develop pae-me model dffeen fom he SARA one A gnfan onuon owad h deon we efe he dynam pae-me model ha nlude an nananeou paal neaon em fo he epone peened y Elho and he Bayean Veo Auoegeve model wh paal po on he paamee LeSage and velyova 999 h pape popoe a heahal Bayean mehod fo modelng a dependen me ee vaale meaued a dffeen loaon elave o a e of ndependen me ee vaale ha may o may no have a paal efeene he afoemenoned elaonhp le n a egeon famewok ha n he ae of ndependen vaale wh paal efeene eemle he

4 Seemngly Unelaed Regeon SUR model nodued y Zellne 96 ha ofen employed n he Bayean famewok ee fo example Gffh he paamee n he egeon model ae endowed wh po ha efle he neghong elaon eween he loaon of he udy; moeove a pao-empoal poe nluded o aoun fo he unlaned vaaon he followed appoah nfluened y he one adoped y kle e al 998 a fo modelng envonmenal poee a fa a he degn of he heahal Bayean mehodology and he peene of he dynam pao-empoal em ae onened; he man dffeene le n he fomulaon of he paal dependene and he egeon pa n he f age of ou model he heahal ep of he Bayean mehodology ae peened n he eon ha follow; he hd eon onan he duonal aumpon ha haaeze eah of he afoemenoned ep and he fomulaon of he paal and pao-empoal elaon n he euel he full ondonal poeo duon of he model paamee ae deved he ffh and la eon onan a duon on he akov Chan one Calo CC mplemenaon of he model va a G ample An ovevew of he eahal Bayean ehodology he heahal Bayean mehodology and he CC emaon appoah deompoe omplaed emaon polem no mple one ha ely on he ondonal duon fo eah paamee n he model h nnovaon make applaon of he Bayean mehodology fa eae han pa appoahe ha eled on analyal oluon of he poeo duon A LeSage ndae a eul of h ha exenle oolk fo olvng lage lae of emaon polem an e developed a oh a heoeal and appled level he popoed mehodology model he elaonhp eween a epone vaale meaued a loaon ndexed y { S} whh may e ae egon pefeue e e and me a h pape peen a model fo he pao-empoal evoluon of a ngle envonmenal poe meaued a e loaed a a gd odelng on he vaou age po aed n he noon of paal akov feld

5 whee { } and p lanaoy vaale whh an e meaued ehe a he ame pao-empoal doman o hey may have no paal efeene a all Fo he ake of mply we peen he eond ae ha an e genealzed n a aghfowad way Poeedng n a mla way a n kle e al 998 a a f age he model fo he epone vaale ondonal on poee { : D} loaon and me pon he geneal model of he fom and a olleon of paamee θ A eah whee epeen a egeon model wh e dependen oeffen; he e of p lanaoy vaale ha may o may no have a paal efeene and epeen he e of paally efeened egeon oeffen fo eah loaon and fo a poe ha aoun fo pao-empoal dependene he epeen he unlaned vaaon a he f age of he modelng poe whh n pnple hould e modeled a a S S ovaane max oweve akng no aoun ha he poe lan muh of he pae-me uue of one mgh aume ha he ae ondonally ndependen andom vaale ha model an e fomally wen a whee θ p n he eond age of he heahal Bayean mehod he and poee ae aumed o e ndependen ondonal on he eond age paamee θ ha an e paoned a θ θ leadng o θ [ θ ] [ θ ][ θ ] and fo a pae me poe whh n geneal an e deed y he model

6 ha nlude [ θ ] [ ] a a peal ae and ha fom ha ha we ue fom now on n h ae a of egeon oeffen and an ndependen euene of zeo mean eo S S max he hd modelng age he pefaon of [ θ θ θ ] hypepaamee e aume a paon θ eah age and a ondonal ndependene elaon [ θ θ θ ] [ θ θ ][ θ ] 3 3 θ3 3 whee θ 3 a olleon of θ 3 3 θ3 no hypepaamee aoaed wh and 3 ndependene aumpon θ an e paoned a θ θ θ and oupled wh fuhe ondonal [ θ θ ] [ θ θ ][ θ θ ] Condonal ndependene alo aumed fo he hypepo [ θ ] [ θ ][ θ ][ θ ] and he fomulaon an e mplfed y akng θ 3 o e ehe empy o known o ha he oepondng em n he aove euaon dop ou

7 3 Duonal aumpon and pao-empoal dynam Euaon fo he poe of nee an e wen a whee eah em a S veo a egeon model wh paamee endowed wh po ha efle paal dependene a dynamal poe ha aoun fo he unlaned pae-me vaaly and an eo em ae ondonally Gauan uh ha 3 [ ] and ae aumed o e muually ndependen ondonal on eond age paamee; he model fo a yem of egeon euaon eah one oepondng o a dffeen loaon p p and fo a p max ha onan nfomaon on p lanaoy vaale n me ee fom ha may o may no have a paal efeene A h pon we have o nodue a max ha efle he neghong elaon eween he loaon whee he oevaon whee aken; denoed y and a nonzeo w l elemen ndae a neghong elaon fo he loaon l h max an e of he neae-negho paal onguy o nvee dane fom Eah modeled a S 3 { } l : l wl l l n he ae of dffeen e of ndependen vaale oepondng o eah loaon we have a lok dagonal S p max

8 and he followng eon hold fo eah S-veo 33 { } he pae-me dynam em modeled a a veo auoegeve VAR poe 34 whee an S S max and he VAR noe em Euaon 34 fomally wen a 35 l w S l l ha he auoegeve paamee fo eah loaon unde ondeaon vae paally a he paamee fo he paal dependene he duonal aumpon fo he poe fomalzed a 36 and a max wh nonzeo elemen a he dagonal and a poon whee he oepondng elemen of ae nonzeo Dependng on he modelng aege on h eond age he mpled model mgh no e denfale eaue and appea only hough he um n A kle e al pon ou: n a Bayean analy wh pope poale on all uane denfaly ue do no poh u fom poeedng hough we hould e aeful n nepeng eul fom undenfed paamee Fo a geneal duon on h ue he neeed eade efeed o Beag e al 995 A ndaed n he eond eon we paon he hd age po and aume ondonal ndependene he S-veo n elaon 3 pefed o e a Gauan andom vaale

9 37 a and he an e gven value ha efle he lak of nfomaon aou he Fo he we follow LeSage and pu a unfom po ove he neval [ λ λ ] mn max whee λ mn λ max epeen he mnmum and he maxmum egenvalue of he paal wegh max ha we e he paamee o feale ange fo ow andadzed 38 U[ λ λ ] mn max Smla pefaon hold fo he paamee ha oepond o he pao-empoal dynam 39 3 Fo he vaane pefed n he f wo age we aume ndependene and ue he onjugae po 3 G 3 G 33 G whee G efe o he nvee Gamma duon 4 Devaon of he full ondonal duon h eon oulne he devaon of he full ondonal duon ha an e ued n he G amplng famewok n geneal full ondonal duon ae deemned y wng he jon duon of all andom uane dvded y he appopae nomalzng onan

10 n heahal model h poe mplfed due o he vaou ondonal ndependene aumpon n paula all omponen of he full jon duon ha do no funonally depend on he uany anel fom he numeao and denomnao of he full ondonal duon he followng devaon egn afe hee mplfaon have een ondeed he gene noaon A and A ued o epeen he ondonal duon fo A gven all ohe andom uane hould e noed ha he majoy of he poeo peened hee ae modfed veon of he one peened a kle e al 998 Fom 3 36 fo he followng elaonhp hold whee we need he nal ondon hu Smlaly fo whh a aove lead o

11 f Σ Σ µ µ he full ondonal duon fo gven y Σ µ Σ µ µ whh lead o Σ Σ Σ µ Ung he duon n 3 and 33 we an deve p p p hu p

12 n h ae we ue 33 and 37 and deve hu he poeo ake he fom Fom 33 and 38 we have max mn λ λ mn max λ λ hu Fom he VAR uue we an we he followng deompoon

13 dag dag whee he max wh he man dagonal eplaed y zeo hen ung dag dag dag dag and dag dag dag Fom 35 we an we he deompoon whee he he max wh he elemen oepondng o nonzeo elemen of he h ow of he max eplaed y zeo and J whee J an S S max wh one fo elemen of he h ow ha oepond o nonzeo elemen of he h ow of he max hen ung 36 3

14 Fom 3 3 S and S G Fom 36 and 3 S and S G Fom we oan

15 Γ S S and S G 5 Bayean Emaon: G Sample he ole of CC emaon aed on ondonal poeo o podue onluon ha ae unondonal h aomplhed y amplng ove value of he ondonng paamee ahe han negang whh he fomal poedue fo nveng ondonal duon o unondonal n ou ae ne he fom of he ondonal duon known we an ue he G o alenang ondonal amplng appoah Gven nal value fo he paamee of ou polem we an daw one oevaon fom eah fom ue hee when amplng fom o podue a f daw fom he p S- veo ake daw fo he S-veo ung n he poeo he daw aken n he pevou ep and o on A a eond ep we updae y amplng fom poeo ha now ue nfomaon fom he f daw we ook fo and eah hen we updae mlaly he and o on h poe of alenang amplng fom he ondonal duon onnued unl a lage ample of daw ha een olleed h no an ad-ho poedue a fomal mahemaal demonaon povded y Geman and Geman 984 a well a Gelfand and Smh 99 how ha he oha poe epeenng ou paamee a akov han wh he oe eulum duon

16 hle heoy mple ha he akov han guaaneed o onvege o he appopae aonay duon mplemenaon ue ae n pae One mu make hoe elaed o he nfluene of ang value how long o un he han efoe onvegene and how e o mono he han and pefom he deed emaon A ommon poedue o delee he oevaon aken fo he model paamee ha oepond o he nal eaon of he akov han when onvegene no ye eahed Fo onvegene dagno we an ue a eon lke he one povded y Gelman and Run 99 Fnally due o oelaon of CC ample he one Calo andad eo hould e emaed y he ah mean appoah deed n Roe 996 wh he ah ze deemned fom examnaon of he lag auooelaon plo of eveal paamee a oaned fom plo ample 6 Refeene BESAG J GREE P GDO D and EGERSE 995 Bayean ompuaon and oha yem Saal Sene 3-66 CLFF AD and ORD J 975 Spae-me modelng wh an applaon o egonal foeang anaon of he nue of Bh Geogaphe ELORS JP Dynam model n pae and me Geogaphal Analy GELA A and RUB DB 99 nfeene fom eave mulaon ung mulple euene Saal Sene GEA S and GEA D 984 Soha elaxaon G duon and he Bayean eoaon of mage EEE anaon on Paen analy and ahne nellgene GRFFS E Bayean nfeene n he eemngly unelaed egeon model Compue-Aded Eonome edo D Gle ael Dekke AARAAS and PRASACOS P Spae-me modelng of aff flow ehod of paal analy paal me ee analy ERSA Poeedng AARAAS and PRASACOS P 3 Foeang aff flow ondon n an uan newok: A ompaon of unvaae and mulvaae poedue Jounal of he anpoaon Reeah Boad LESAGE JP Applaon of ayean mehod o paal eonome wokng pape Depamen of Eonom Unvey of oledo LESAGE JP and RVELOVA A 999 A paal po fo Bayean veo auoegeve model Jounal of Regonal Sene PFEFER PE and BODL SE 99 A e of pae-me ARA modelng and foeang wh an applaon o eal eae pe nenaonal Jounal of Foeang

17 PFEFER PE and DEUSC SJ 98a A hee-age eave poedue fo pae-me modelng ehnome 98 denfaon and nepeaon of F-Ode Spae-me ARA odel ehnome 3 98a Vaane of he Sample-me Auooelaon Funon of Conempoaneouly Coelaed Vaale SA Jounal of Appled ahema See A 4 98 Seaonal Spae-me ARA modelng Geogaphal Analy 3 98 Spae-me ARA odelng wh onempoaneouly oelaed nnovaon ehnome 3 4 ROBERS GO 996 akov han onep elaed o amplng algohm akov Chan one Calo n Pae Chapman and all London SOFFER DS 986 Emaon and nepeaon of Spae-me ARA model n he peene of mng daa Jounal of he Amean Saal Aoaon LE C BERLER L and CRESSE 998 eahal Bayean pae-me model Envonmenal and Eologal Sa ZELLER A 96 An effen mehod of emang eemngly unelaed egeon and e of aggegaon a Jounal of he Amean Saal Aoaon