Integrated Modified OLS estimation and residualbased tests of cointegration. New evidence for the Taylor rule and present value models

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1 Iegaed Modified OLS esimaio ad esidualbased ess of coiegaio. New evidece fo he Taylo ule ad ese value models Julio A. Afoso Rodíguez Deame of Isiuioal Ecoomics Ecoomic Saisics ad Ecoomeics Faculy of Ecoomics ad Busiess Admiisaio. Uivesiy of La Lagua Camio La Hoea s/. Camus de Guajaa 387 Sa Cisóbal de La Lagua. Teeife. Caay Islads Tfo.: Fax: Absac: I his ae we discuss he asymoically efficie esimaio of a uivaiae saic coiegaig egessio elaioshi whe we ae io accou he deemiisic sucue of hei sochasically iegaed comoes i a slighly moe geeal famewo ha cosideed by Hase (992. Afe eviewig he oeies of OLS ad Fully Modified OLS (FM-OLS esimaio i his famewo we coside he ecely oosed Iegaed Modified OLS (IM-OLS esimao by Vogelsag ad Wage (2 of he coiegaig veco ad oose a ew oe secificaio of he iegaed modified coiegaig egessio equaio. This aleaive mehod of bias emoval has he advaage ove he exisig mehods ha does o equie ay uig aamees such as eels badwidhs o lags. Also based o he sequece of IM- OLS esiduals we oose some ew es saisics based o diffee measues of excessive flucuaio fo esig he ull hyohesis of coiegaio agais he aleaive of o coiegaio. Fo hese es saisics we deive hei asymoic ull ad aleaive disibuios ad sudy hei fiie samle efomace hough a localo-uiy aoach o he ull of coiegaio. Fially we coside hei alicaio o wo diffee ecoomeic models ieeed as oeially coiegaed egessio models he Taylo ule of moeay olicy ad he aioal ese value model (PVM fo soc ices usig US daa. Keywods ad hases: coiegaio asymoically efficie esimaio OLS FM- OLS IM-OLS edig iegaed egessos JEL Classificaio: C3 C32 E52 G2

2 . Ioducio Coiegaio aalysis is widely used i emiical macoecoomics ad fiace ad icludes boh he esimaio of coiegaig elaioshis ad hyohesis esig ad also esig he hyohesis of coiegaio amog osaioay vaiables. I he ecoomeic lieaue hee ae may coibuios i hese wo oics some of which deals wih hese wo quesios simulaeously. Give he usual liea secificaio of a oeially coiegaig egessio a fis cadidae fo esimaio is he mehod of odiay leas squaes (OLS ha deemies suecosise esimaes of he egessio aamees ude coiegaio. Howeve wih edogeous egessos he limiig disibuio of he OLS esimao is coamiaed by a umbe of uisace aamees also ow as secod ode bias ems which edes ifeece oblemaic. Cosequely hee has bee oosed seveal modificaios o OLS o maes sadad asymoic ifeece feasible bu a he cos of ioducig he choice of seveal uig aamees ad fucios. These mehods iclude he fully modified OLS (FM-OLS aoach of Phillis ad Hase (99 he caoical coiegaig egessio (CCR by Pa (992 ad he dyamic OLS (DOLS aoach of Phillis ad Loea (99 Saioe (99 ad Soc ad Waso (993. This ae deals wih he aalysis of a ew asymoically efficie esimaio mehod of a liea coiegaig egessio ecely oosed by Vogelsag ad Wage (2 ha does o equie ay addiioal choice moe ha he iiial sadad assumios o he model secificaio maig i a vey aealig aleaive. This mehods which is called he iegaed modified OLS (IM-OLS esimao wos ude a simle asfomaio of he model vaiables ha asymoically oduces he same coecio effec as he commoly used esimaio mehods cied above. A imoa issue which is ofe is o ae io accou ad ha ca subsaially

3 affec he efomace ad oeies of hese esimaio ocedues is he aue ad sucue of he deemiisic comoe if ay of he geeaig mechaism of he model vaiables ad is elaio wih he deemiisic comoe if is cosideed i he secificaio of he coiegaig egessio. Followig he wo by Hase (992 we geealize is fomulaio by allowig fo deemiisically edig iegaed egessos wih a ossibly diffee sucue fo hei deemiisic comoes ad oose a simle ule fo a oe secificaio of he deemiisic ed fucio i he coiegaig egessio ha simulaeously coec fo hei effecs. Give he aicula asfomaio of he model vaiables equied fo efomig he asymoically efficie IM-OLS esimaio we show ha a oe accommodaio of hese comoes mus be based o a evious asfomaio of he model vaiables i aicula he OLS deedig. Wih hese coeced obsevaios we efom he IM- OLS esimaio of he coiegaig egessio ad deive he limiig disibuios of he esulig esimaes ad esiduals boh ude he assumio of coiegaio ad o coiegaio. Based o hese ew asymoically efficie esimaos of he veco aamees i he coiegaig egessio we sudy he buildig of some simle saisics fo esig he ull hyohesis of coiegaio by usig diffee measues of excessive flucuaio i he IM-OLS esidual sequece ha cao be comaible wih he saioaiy assumio of he eo sequece. These es saisics ae based o he saisics oosed by Shi (994 Xiao ad Phillis (22 ad Wu ad Xiao (28 wih he same objecive as ous ad ha use wo basic measues of excessive flucuaios he Camé-vo Mises (CvM ad Kolmogoov-Smiov (KS meics. We deive hei limiig ull ad aleaive disibuios ad evaluae hei owe behavio i fiiesamles hough a simulaio exeime. 2

4 To illusae hei efomace i acice we coside he alicaio of hese ew esig ocedues o he aalysis of he coiegaio hyohesis udelyig wo diffee ecoomeic models he Taylo ule of moeay olicy ad he aioal ese value model (PVM fo soc ice deemiaio. 2. The model OLS ad FM-OLS esimaio of he liea coiegaig egessio wih edig egessos We assume ha he vaiables of iees he scala Y ad he -dimesioal veco X = ( X... X come fom he followig daa geeaig ocess (DGP Y α τ η X = + = A τ η... (2. whee η = ( η η is he sochasic ed comoe ha saisfy he fis ode ecuece elaio η = η + ε wih ε = ( ε ε a + veco zeo mea sequece of eo ocesses. Also we coside he geeal case whee boh Y ad each eleme of he veco X = ( X... X coais a deemiisic ed comoe give by a olyomial ed fucio of a abiay ode i i = ha is d i = α i τ wih i i α ad τ = (... i. To mae his assumio comaible i = ( α... i i α i αi i i wih he sadad fomulaio i (2. whee all he deemiisic ed comoes aeas as if i wee of he same ye ad ode we have o wie τ i α τ = ( α : = α τ i i i i i i τ i i i = (2.2 wih = max(... ad i a ( i veco of zeoes so ha 3

5 α τ α A τ = = τ α τ α Wih his fomulaio we ioduce he saic oeially coiegaig egessio equaio bewee he uobseved sochasic ed comoes of he elemes i Z as η = η + u which gives Y = α τ + X + u =... (2.3 wih α = α A. Associaed o he deemiisic comoe we ioduce he olyomial ode ed ad samle size deede scalig maix Γ give by Γ = diag(... which deemies ha τ = Γ τ τ ( = (... uifomly ove [] as. Also we have ha τ τ ( s ds ad [ ] = = τ τ = Q = Q Q = τ ( s τ ( s ds <. I ode o comlee he secificaio of ou daa geeaig ocess we ex ioduce a quie geeal ad commo assumio o he eo ems ivolved i (2.3. Assumio 2.. We assume ha he eo em i he coiegaig egessio u saisfy he fis-ode ecuece elaio u = α u + υ wih α whee he zeo mea (+-dimesioal eo sequece ξ = ( υ ε veify ay of he exisig codiios ha guaaee he validiy of he fucioal ceal limi heoem (FCLT aoximaio of he fom υ B ( = = BM ( = W ( [ ] / 2 Bυ ( / 2 ( = ε B Ω Ω wih W( = ( Wυ ( W ( a +-dimesioal sadad Bowia moio ad Ω he covaiace maix of B( which is assumed o be osiive defiie ad ha ca also be 4

6 ieeed as he log-u covaiace maix of he veco eo sequece ξ ha is Ω = E[ ξ ξ ] + ( E[ ξ ξ ] + E[ ξ ξ ] which ca be decomosed as Ω = + Λ j= j j wih = Σ + Λ he oe-sided log-u covaiace maix whee Σ = E [ ξ ξ ] ad Λ = j= E[ ξ ξ j ]. The assumio of osiive defiieess of Ω excludes coiegaio amog he iegaed egessos X (subcoiegaio wih B ( = BM( Ω Ω >. Give he ue iagula Cholesy decomosiio of he maix Ω we he have ha Bυ( = Bυ. ( + γ υb ( wih Bυ. ( = ω υ. Wυ ( ad B ( = Ω W ( whee / 2 γ Ω ω ad ω 2 = 2 2 E[ B ( ] = E[ B ( B ( ] = ω ω Ω ω is he = υ υ υ. υ. υ. υ υ υ υ codiioal vaiace of Bυ ( give B ( which gives E[ B ( Bυ. ( ] =. Fo he iiial values η ad u we ioduce he vey geeal codiios η / 2 = o ( ad u = o which iclude he aicula case of cosa fiie / 2 ( values. I he case of a saioay eo em u wih α < we he have ha / 2 [ ] = u Bu ( = ( α Bυ ( wih Bu ( = Bu. ( + γ B ( γ = Ω ω u E[ B ( ] = ω = ( α ω u u υ E[ B ( ] = ω = ω γ Ω γ ad E[ B ( B ( ] = u. u. u u ω u = ( α ω υ while ha i he case of osaioaiy ha is whe α = he / 2 u Bu Bυ [ ] ( = ( wih ω = ω. Wih hese esuls he we have 2 2 u υ [ ] ( v ( v Bu ( v /2 U = = [ ] u Ju ( = = Bu ( s ds v = /2 wih v =/2 ad v = /2 idicaig esecively he saioay ad osaioay cases. Give he secificaio of he liea saic coiegaig egessio equaio (2.3 he sadad aoach o esimaig he veco aamees α cosiss i he use of he 5

7 OLS esimaio which gives α τ τ ( = τ X u X = X = α ˆ α τ τ ˆ Taig io accou he sucue fo he deemiisic ad sochasic ed comoes of he obseved ocesses Y ad X i (2. we ca wie τ Γ τ Γ + τ = = / 2 X A Γ τ + η A Γ I η τ τ = W / 2 = W / 2 η η (2.4 so ha α ˆ α v τ / 2 ˆ ( (/ / 2 ( = W τ η = η ( v τ / 2 u = η wih he owe v aig values ±/2 deedig o he sochasic oeies of he eo sequece u ad deemiig he ode of cosisecy of he OLS esimaes ha is v Γ ˆ ˆ [( α α + A ( ] / 2+ v ( ˆ τ / 2 ( v τ = (/ / 2 ( / 2 u τ η = η = η (2.5 The usual esul i his coex is as i (2.5 bu wih A = + which coesods o he case whee he iegaed egessos have o deemiisic comoe ad i ou fomulaio he deemiisic em ha aeas i he coiegaig equaio coesods o he oe icluded i Y. Hase (992 has sudied a simila siuaio bu assumig ha Y = η wih d α τ ad i = m i =... wih = α τ = 2 τ <...< m ad scalig maix Γ (... m m = diag 2 (... m m = 6

8 (see Theoem (a b.93. The mai diffeeces wih ou aoach ae he o iclusio of a cosa em ad he iclusio of a a codiio o he coefficie maix A m aiculaly a( A m = m. The fom (2. we have ha Γ [( A A A ] X = τ + Γ [( A A A ]( η / 2 m m m m [ ] m[ ] m m m m [ ] ( / 2 = τ m[ ] + O ( τ m( which allows he ossibiliy o develo a sequece of weighs which yield a odegeeae desig limiig maix whe esimaig (2.3 by OLS ude he esicio α m = m. Howeve as ca see fom he evious esul his oly yields cosise esuls whe ad hee is o cosa em i he egessio eihe i he olyomial ed fucio. Ude he assumio of coiegaio (v = /2 he he limiig disibuio of he las em i (2.5 is give by τ τ ( s [ ] ( v + / 2 u dbu ( s ( s + = η B u τ ( s τ ( s + = dbu. ( s + d ( s + ( ( s s B γ B B u wih u = j= E[ ε ju ] give by he obabiliy limi of u = = E[ η u ]. 2 This limiig disibuio coais he secod-ode bias due o he coelaio bewee B u ( ad B ( (edogeeiy of he sochasic ed comoes of he egessos ad he o-cealiy bias ha comes fom he fac ha he egessio eos ae seially coelaed hough he aamee u. Fo he fis em above we have ha / 2 B u. = ωu. Ω W u. ( s db ( s ( s dw ( s whee give ha B ( ad B. ( ae ideede codiioig o B ( (o u See also Hassle (2 fo a elaed sudy i he case whee he secificaio of he coiegaig egessio equaio does o iclude ay deemiisic em bu he iegaed egessos X coai a cosa em. 2 The esul [ ] u is obaied by wiig [ ] ( [ ] ([ ] [ = E η u = ] E[ η u ] so ha u = = ( = [([ ] E[ η u ] + (([ ] E[ ε u ]] [ ] [ ] [ ] [ ] u = j= = j+ j ad he use of he iiial codiio η ad Assumio 2. o he oeies of he eo ems. 7

9 W ( ca be used o show ha his em is a zeo mea Gaussia mixue of he fom ( s dw ( s = N( dp( = ( s ( s W u. G G G W W G > The secod em i he exessio bewee baces is a maix ui oo disibuio aisig fom he sochasic eds i X which is cacelled ude sic exogeeiy of he egessos ha is whe ω u =. Usig ow (2.5 ad he decomosiio fo X i (2. we have ha he sequece of OLS esiduals is give by uˆ ˆ ˆ ( = u τ ( α α X ( = u τ { Γ [( α ˆ α + A ( ˆ ]} ( η [ ( ˆ ] / 2 whee he fis eleme comoe i (2.5 ca be wie as which gives Γ [( α ˆ α + A ( ˆ ] / 2 = Q ˆ τ u Q τ η = = ( [ ( ] = τ Q τ j j j= uˆ ( u u / 2 η ˆ τ Q τ j η j [ ( ] j= = u η [ ( ˆ ] (/ 2 + v / 2+ v (2.7 so ha he OLS esiduals ae fee of he ed aamees ad ae decomosed i ems of he deeded vesios of u ad η as defied i (2.7. These OLS esiduals ca be used as he basis fo buildig some simle saisics fo esig he ull hyohesis of coiegaio agais he aleaive of o coiegaio give ha uˆ ( = O ( whe v = /2 ad / 2 ˆ ( ( u = O whe v = /2. This diffeece i behavio ude he ull ad he aleaive ca be exloied by seachig fo excessive flucuaios i he sequece of scaled aial sum of esiduals Bˆ ( = uˆ ( hough seveal / 2 [ ] [ ] = global measues such as a Camé-vo Mises (CvM measue of flucuaio as i Shi 8

10 (994 o a Kolmogoov-Smiov (KS measue as i Xiao ad Phillis (22 ad Wu ad Xiao (28. 3 The CvM-ye es by Shi (994 is based o a global measue of flucuaio give by S Bˆ 2 ( = (/ = ( ( while ha he KS-ye es saisic oosed by Wu ad Xiao (28 is based o he ecusive ceeed measue of maximum flucuaio R ˆ ˆ ( = max =... B ( ( / B (. Xiao ad Phillis (22 cosides a o ceeed vesio of his es saisic give by CS ( = max Bˆ ( which is he same as R ( whe based o OLS =... esiduals ad he deemiisic comoe coais a cosa em. The mai oblem wih his aoach is ha uless coeced he ull disibuio of all hese es saisics ae lagued of uisace aamees due o edogeeiy of egessos ad he seial coelaio i he eo ems ha cao be emoved by simle scalig mehods. Thee exis some diffee mehods which ae ow as asymoically efficie esimaio mehods o emove hese aamees ad ha diffe i he eame of each souce of bias. Amog he exisig esimaio mehods he hee mos commoly used ae he Dyamic OLS (DOLS esimao oosed by Phillis ad Loea (99 Saioe (99 ad Soc ad Waso (993 he Caoical Coiegaig Regessio (CCR esimao by Pa (992 ad he Fully-Modified OLS (FM-OLS esimao by Phillis ad Hase (99. These hee esimaos ae asymoically equivale ad as was oved by Saioe (99 efficie. The coeced es saisic oosed by Shi (994 maes use of he DOLS esiduals while ha he es saisics cosideed i Xiao ad Phillis (22 ad Wu ad Xiao (28 ae based o FM-OLS esiduals. Fo a ece eview ad comaiso of hese hee aleaive esimaio mehods see e.g. Kuozumi ad Hayaawa (29 ad efeeces heei. I ode o esablish he basis 3 The es saisic oosed by Shi (994 is he geealizaio of he KPSS saisic fo he ull of saioaiy by Kwiaowsi e.al. (992 while he es saisics cosideed i Xiao ad Phillis (22 ad Wu ad Xiao (28 ae he geealizaios of he KS es saisic fomulaed by Xiao (2. 9

11 fo ou oosal i he ex secio we coside a aleaive o he coiegaig egessio equaio (2.3. By alyig he aiioed OLS esimaio o he egessio equaio (2.3 wih esec o he ed aamees we have ha his model ca also be wie as whee Yˆ = Xˆ + u =... (2.8 Yˆ = Y τ Q τ Y j= j j ˆ X = X j= X j τ jq τ ad u = u τ Q τ u deoe he deeded obsevaios of he model vaiables j= j j obaied by OLS fiig of hei oigial obsevaios o a h-ode olyomial ed fucio whee is chose accodig o he ule max(... i he case whee he olyomial ed fucios i Y ad each comoe of X diffe i hei odes. The ex Poosiio 2. deemies he effeciveess of his ocedue o mae he esimaio esuls ivaia o he ed aamees i (2.. Poosiio 2.. Give (2.-(2.2 whe cosideig he OLS deedig of Y ad X by fiig a olyomial ed fucio of ode = max(... o each of hese vaiables he we have ha ˆ = η = η τ Q τ η Y j = j j ad ˆ X (... = η = η j= η j τ jq τ = η η whee η ad η ae geealized deeded asfomaios of η ad η wih / 2 η[ ] B B B τ Q τ η ( = ( ( s τ ( s ds τ ( (2.9 a (+-ode deeded asfomaio of B (. Accodig o Lemma A.2 i Phillis ad Hase (99 B ( = BM( Ω v( is a full a Gaussia ocesses wih v ( a scala fucio of ad τ (. Poof. See Aedix A. By OLS esimaio of he coiegaig veco comoe i (2.8 we have (/ 2 + v ˆ ( ˆ ˆ = ˆ X X X = = / 2 / 2 ( v / 2 = (/ ( η ( η ( η u = = u

12 which gives he same limiig esuls as befoe ude coiegaio ha is wih v = /2 o ha / 2 j= τ ju j τ ( s dbu ( s ad hus u = u + O. I ode o / 2 ( comlee he above esuls we ex coside he elaioshi bewee he FM-OLS ad OLS esimaos of α ad i (2.3 which is give by + τ τ + + = ( Y + τ X + X = X = u αˆ τ τ ˆ (2. whee Y = Y X γ + = u u γ ad γ = Ω ω u. 4 Agai as i (2.-(2.2 he ey is o coside he followig geeal decomosiio fo Z = X + X = A τ + η = Φ τ + ε = ( Φ τ : + ε = Φ Γ τ + ε τ (2. wih he maix of ed coefficies Φ give by a liea combiaio of he coesodig elemes of A. Poosiio 2.2. Give (2.-(2.4 ad he FM-OLS esimao of (2.3 i (2. he we have ha (a + ˆ + Γ ˆ ˆ ˆ [ α + A ] Γ [ α + A Φ γ ] = ˆ + ˆ + M τ ε Q Q η ε γ M Q Q u = = + M η ε Q Q τ ε γ + M u = = wih FM-OLS esiduals such ha + (buˆ ( = uˆ ( ε γ / (/ ( η M (/ η ε γ + u = M = Q Q Q Q Q = = τ η Q = = η η = (/ M = = η η ad ε = ε j= ε j τ jq τ whee M M wih Poof. See Aedix B. (2.2 (2.3 ad ε ε ε τ τ. 4 I ca be show ha he coecio em fo Y is associaed wih he coecio fo he edogeeiy bias while + u elimiaes he o-cealiy bias.

13 Rema 2.. The FM-OLS esimao i (2. as well as he esuls i (2.2 ad (2.3 is o feasible sice i is defied i ems of he uow quaiies γ ad + = γ. The feasible vesio is obaied by elacig hese elemes by u u oaameic eel esimaes of he comoes of Ω based o he OLS esiduals i (2.7 ha ae cosise ude he assumio of coiegaio ad equies he choice of he badwidh o esue he oe asymoic coecio fo seial coelaio ad edogeeiy. 5 Rema 2.2. Usig (2. we have ha Z = X = Φ τ + ε. By OLS deedig we have ˆ = j= j τ j τ = ε j= ε j τ j τ = ε Z Z Z τ Q τ ε ε τ Q τ ε. If we defie ow Y + + as Y ˆ = Y Z γ = Y ε γ as idicaed by Hase (992 (age 93 he he FM-OLS esimao of α ad is give by + αˆ ˆ α Q Q τ + ˆ ( ˆ = + W + ε γ W + Q Q = η u + which gives exacly he same exessios as befoe fo ˆ + ( α ˆ ad he FM-OLS esiduals. Fo lae use we defie he aial sum of he deeded eos i he coiegaig egessio (2.3 o (2.8 as U = j= u j wih [ ] [ ] [ ] ( v ( v ( v ( v [ ] = = τ Q τ = = = = U u u u whee asymoically we have 5 Similaly sice he CCR esimao oosed by Pa (992 is defied as he OLS esimao bewee * he modified deede vaiable Y Y ( ˆ ( ˆ * = Σ + γ ξ ( ad ( τ X wih * X ˆ = X Σ ξ ( ˆ ( ( ( ξ = u Z ad = j = E[ ε ξ j ] = j = E[ ε ( u j ε j ]. This mehod uses he same icile as he FM-OLS mehod o elimiae he edogeeiy bias while i deals wih he o-cealiy aamee i a diffee mae bu also elies o cosise esimaes of he quaiies Σ ad γ which deed o some uig aamees. 2

14 V ( v /2 ( = = α < ( s ds v = /2 ( v u U[ ] Ju ( Bu (2.4 wih V ( a geealized (+h-level Bowia bidge ocess give by u u ( u ( τ ( ( ( Q τ u V B τ s ds τ s db s (2.5 = wih vaiace E V = ω b whee 2 2 [ u ( ] u ( b ( = τ ( s ds Q τ ( s ds ad Bu ( a (+h-ode deeded Bowia moio ocess defied as u ( u( τ ( Q τ ( ( u B = B τ τ s B s ds (2.6 as he sochasic limis i ( Thus ude he assumio of o coiegaio whe α = ad v = /2 we ge he usual esul ( ˆ ( ( ( ( B s B s ds s dju s B whee dju ( = Bu (. Fially maig use of (2.5 ad he elaio B ( = B ( + γ B ( we he have ha V ( ca be decomosed as u u. u u = u. τ Q τ u. V ( B ( ( s ds ( s db ( s { } B ( db ( s ( s Q ( s ds + γ τ τ = V ( + γ V ( u. (2.7 whee by cosucio i is veified E[ V ( Vu. ( ] = E[ B ( Vu. ( ] = wih B ( defied i (2.9 ad 2 2 E[ Vu. ( ] = ω u. b (. 6 Exlici exessios fo hese wo limiig ocesses Vu ( ad Bu ( ca be obaied i he leadig cases of = (cosa ad = (cosa ad liea ed. Secifically we have ha ( ( ( Vu = Bu Bu ad V ( = B ( + (2 3 B ( 6 ( B ( s ds fo he fis ad secodlevel Bowia bidge while ha u u u u B ( = B ( B ( s ds ad u u u B ( = B ( + 2(3 2 B ( s ds u u u 2(6 3 sbu ( s ds ae he aicula exessios fo he demeaed ad demeaed ad deeded Bowia ocesses esecively. 3

15 3. IM-OLS esimaio wih edig egessos I his secio we coside he ew esimao of a saic coiegaig egessio model lie (2.3 ecely oosed by Vogelsag ad Wage (2. Fo his esimao hese auhos show ha a simle asfomaio of he model comoes is used o obai a asymoically ubiased esimao of wih a zeo mea Gaussia mixue limiig disibuio bu whe he assumed DGP is as i (2. wih A = +. Lie FM-OLS he asfomaio has wo ses bu eihe se equies he esimaio of ay of he comoes of Ω ad so he choice of badwidh ad eel is comleely avoided. Thus comuig he aial sum of boh sides of (2.3 gives he so-called iegaed coiegaig egessio model as S = α S + S + U =... (3. wih U = j= u j S = j= Yj = α j= τ j + j= η j = α S + h ad S = τ = Γ τ = Γ S (3.2 j= j j= j S = X = A S + η = A Γ S + H (3.3 j= j j= j Taig ogehe (3.2 ad (3.3 we have S Γ + S S = 3/ 2 = W 3/ 2 S A Γ I H H wih S [ ] τ ( s ds ( 3/ 2 g = H [ ] B ( s ds (3.4 as. The he IM-OLS esimao of (3. which ca be wie as α ad is give by he OLS esimao i 4

16 whee ( v v α α Γ [( α α + A ( ] W = / 2+ v ( S 3/ 2 = (/ ( 3/ 2 S H = H S (v (/ 3/ 2 U = H (3.5 S ( v (/ U ( ( 3/ 2 Ju d = g H (3.6 wih g ( give i (3.4 J ( = B ( ude he assumio of coiegaio v = /2 u u ad Ju( = Bu ( s ds ude o coiegaio ha is whe v = /2. Vogelsag ad Wage (2 Theoem 2 cosides his case whe A = + which coesods o he case of iegaed egessos wihou deemiisically edig comoes so ha he edig aamees i he secificaio of he coiegaig egessio mus be associaed o he deemiisic comoe of he deede vaiable. Give ha he limiig esul i (3.6 does o coai he addiive em u aial summig befoe esimaig he model hus efoms he same ole fo IM-OLS ha ( + + u lays fo FM-OLS bu his sill leaves he oblem ha he coelaio bewee u ad ε ules ou he ossibiliy of codiioig o B ( o obai a codiioal asymoic omaliy esul. The soluio oosed by hese auhos equies ha X be added as a egesso o he aial sum egessio (3. as S = α S + S + γ X + ζ =... (3.7 which ca be called ow he iegaed modified (IM coiegaig egessio whee ζ = U γ X. The by OLS esimaio of (3.7 we have ha 5

17 v Γ ( α α / 2+ v 3/ 2 3/ 2 / 2 ( = (/ ( / 2+ v S S S S X = / 2 γ X S 3/ 2 ( v (/ S U / 2 = X (3.8 which gives a well defied limi esul ad fee of uisace edig aamees ude he assumio of coiegaio ad o deemiisic comoe i he DGP (2. fo X as i Theoem 2 i Vogelsag ad Wage (2. Howeve whe usig / 2 X = A Γ τ + ( η fom (2. ad (2.4 wih τ = S S ( he we ca wie S ( Γ S + 3/ 2 S = A Γ ( S + ( H / 2 ( ( ( ( X A Γ S + η A Γ S Γ + + S = 3/ 2 A Γ I H / 2 A Γ I η S + ( 3/ 2 + H / 2 A Γ η ha is S S S ( 3/ 2 3/ 2 S = W2 H W22 H / 2 / 2 X η η S S ( 3/ 2 3/ / 2 / 2 η η = W H W W H whee he las em bewee baces is give by S ( + 3/ = / 2 η A Γ ( S ( W W H 6

18 which divege wih he samle size eve i he case of a cosa em ( =. Aleaively if we edefie he IM egessio model (3.7 i ems of he IM-OLS deeded vaiables we have whee Sˆ = Sˆ + γ Xˆ + ζ =... * * * * ˆ * S S S j ˆ * S = S S j S j S S j j S * j= j= ˆ X X X j α ad S as i (3.3 we he have ha Give S = S + h * * Sˆ h h j h = * j j j = ˆ * S S S S j= j j= S H H H which ae fee of ed aamees while ha ˆ * X is give by wih * = η η j j j j j= j= Xˆ S S S S + A S S S S = + A * * τ τ j j j j η τ j= j= * / 2 / 2 η = ( η (/ ( η j ( S j j= (/ ( S j( S j ( S j= ad * A τ = A Γ τ (/ τ j( S j j= = * (/ ( S j ( S j ( S A Γ τ j= which deemies ha Xˆ = η + A ( Γ τ. Fo = / 2 * / 2 * / 2 * / 2 ˆ * X = 7

19 / 2 * / 2 * / 2 * / 2 η + A τ = η + O( so ha he deemiisic comoe is asymoically ieleva while fo we have ha X ˆ = η + O ( / 2 * / 2 * / 2 which imlies ha deemiisic comoe domiaes he sochasic oe yieldig icosise esuls. Thus o deal wih his geeal case fom (2.8 ad maig use of he esul i Rema 2.2 fo he OLS deeded obsevaios of X Z = ε we ge he followig augmeed vesio of (2.8 ˆ Yˆ = Xˆ + γ Zˆ + u γ Zˆ = Xˆ + γ Zˆ + z =... which gives he followig coeced vesio of he IM coiegaig egessio equaio Sˆ = Sˆ + γ Zˆ + ζ = Sˆ + γ Tˆ + ζ =... (3.9 j j= Sˆ Xˆ η ad ζ ˆ = j= z = U γ T whee wih = j= j = j= j ad ˆ 3/ 2ˆ 3/ 2 S ˆ I S S = ˆ = W / 2 ˆ / 2 ˆ T I T T (3. [ ] [ ] [ ] ˆ [ ] = = ε ε j τ j τ = = j= = Tˆ Z Q = η + η ε τ τ [ ] / 2 / 2 [ ] j j Q (/ j= = which gives asymoically a -dimesioal Bowia bidge of ode (+ such ha / 2 T [ ] V = B B τ Q τ ˆ ( ( d ( s ( s ( s ds o moe comacly ˆ ( 3/ 2 S [ ] B s ds g ( / 2 = ˆ T V [ ] ( (3. Also fom (2.4 (2.7 ad (3. i ca be easily veified ha ude coiegaio ( α < i Assumio 2. he scaled eo em i he IM coiegaig egessio (3.9 behaves asymoically as 8

20 ζ = U Tˆ V (. / 2 / 2 / 2 u. γ (3.2 The we defie he IM-OLS esimao of he coefficie veco ( γ based o OLS deeded obsevaios as Sˆ ˆ ˆ ˆ S = ( ˆ S S T ˆ ˆ = T = T γ (3.3 wih IM-OLS esidual sequece give by ˆ ˆ ˆ ζ ( = S ( S T γ =... (3.4 Nex oosiio esablish he mai esul i his secio elaed o he wea covegece of IM-OLS esimaos ad esiduals ude he assumio of coiegaio ha is whe he eo em sequece u i he oigial coiegaig egessio equaio (2.3 is osaioay wih α = i Assumio 2.. Poosiio 3.. Give (2. ad (2.2 ad ude Assumio 2. he IM-OLS esimaio of he coiegaig egessio model i (2.3 based o he IM egessio (3.9 wih OLS deeded obsevaios he equaio (3.3 deemie ha: ad / 2+ v 3/ 2 ( ˆ S 3/ 2 ˆ / 2 ( a = (/ ( ˆ / 2 / 2 + v S T ˆ = γ T 3/ 2 Sˆ ( v (/ U / 2 ˆ = T ( ( b ( ( ( d ( Vu. ( d g g g γ γ = ω ( ( u. Π g g g u. ( ( d ( W ( d u. Π g g G G u. = ω ( ( d [ ( ( ] dw ( ( c ( ( / 2+ v ( v ( v 3/ 2 ˆ / 2 ˆ ζ = ζ S T / 2+ v ( γ γ ( / 2 ( ( u. u. ( g ( g ( g ( ( d ζ ω W s s ds [ G ( ( ] G s dwu. ( s = ωu. R ( 9

21 whee he esuls i (b ad (d ae esablish ude he assumio of coiegaio ha is wih v = /2 wih Vu. ( = ω u. Wu. ( g ( = Π g ( G ( = g ( s ds = Π G ( ad Poof. See Aedix C. Π = diag( Ω Ω. / 2 / 2 Rema 3.. As ca be see i (b ad (d above fo ifeeial uoses elaed o hyohesis esig hese limiig esuls deeds oly o 2 ω u. ad Ω as uisace aamees. Secially eleva whe usig he IM-OLS esiduals i (d is he quesio of ossible cosise esimaio of he codiioal log-u vaiace ω based o he 2 u. fis diffeeces of ζ ( ζ (. As is discussed i he ex secio ad i Vogelsag ad Wage (2 he sadad aoach based o he use of a oaameic eel-ye esimao deemie icosise esimaio of ω 2 u.. Fo his easo fo ou uoses i his ae we follow a aleaive aoach. 4. IM-OLS esidual-based es fo he ull of coiegaio I his secio we oose some ew saisics based o he sequece of IM-OLS esiduals as has bee defied i secio 3 fo esig he ull hyohesis of coiegaio agais he aleaive of o coiegaio by looig fo excessive flucuaios i he samle ahs of his esidual sequece. These ew es saisics ae aially isied by he oaameic vaiace-aio saisic oosed by Beiug (22 fo esig he ui oo ull hyohesis agais saioaiy i a uivaiae ime seies i he sese ha ou saisics ae oally fee of uig aamees. I ou case we loo fo a ui oo-lie behavio i he esidual sequece ζ ( which is comaible wih he saioaiy of he eo em z i he augmeed coiegaig egessio amog he OLS deeded vaiables. Fis of all we coside he case of he IM coiegaig egessió (3.7 wih α = + 2

22 ad = + A. Thus fom (3.8 we have ha he IM-OLS esimaos of ad γ ca be wie as whee / 2+ v 3/ 2 ( S 3/ 2 / 2 / 2 + v (/ / 2 ( = S X γ = X 3/ 2 S ( v (/ / 2 U = X ( S X = ( H η g( fo = [] wih g ( = 3/ 2 / 2 3/ 2 / 2 ( B ( s ds B ( as. Ude he assumio of coiegaio he limiig disibuio of hese esimaes is as i Poosiio 3.(b wih g ( ad W. ( elaced by g ( ad Wu. ( esecively. Wih he associaed sequece of IM-OLS esiduals ζ ( = S ( S + X γ =... we defie he followig mai comoes of ou flucuaio es saisics u 2 (4. = F ( = (/ ((/ ζ ( ad F ( = max(/ ζ ( (4.2 2 =... 3 =... F ( = max(/ ζ ( ( / ζ ( (4.3 Taig io accou he esul (d i Poosiio 3. we have ha asymoically ( ( ude he coiegaio assumio wih / 2 ζ ωu. R ( R ( = W ( g( g( s g( s ds [ G( G ( s] dw ( s (4.4 u. u. I ode o elimiae he uisace aamee ω fom he limiig ull disibuios of 2 u. hese saisics we defie he adom eleme (4.5 v ( = (/ ζ ( 2 2 which gives he omalized vesio of he above flucuaio saisics ad F ( v ( F ( = (

23 F ( = v ( F ( j = 2 3 (4.7 j j Taig io accou he flucuaio saisics F ( i (4.-(4.3 as well as he j omalized squaed eo P 2 ( ca also be wie as 2 v ( v 2 F ( = (/ ( ζ ( = F = ζ ( 2 v/2 ( v 2 ( max ( =... v = (/ [ ( ζ ( ] = [ ζ ( ] 2 v ( v 2 2 v ( v 2 ad similaly fo F3 ( he boh he umeao ad he deomiao of he omalized es saisics i (4.6 ad (4.7 ae of he same ode of magiude ude he ull hyohesis of coiegaio as well as ude he aleaive of o coiegaio (whe v = /2 bu wih vey diffee limiig disibuios i each case. Similaly i he case of he IM-OLS esimaio of he coiegaig egessio model based o OLS deeded obsevaios of he vaiables as was ioduced i secio 3 he we defie he coesodig omalized flucuaio es saisics as ad F ( v ( F ( = (4.8 2 whee F ( = v ( F ( j = 2 3 (4.9 j j (4. v ( = (/ ζ ( 2 2 wih he flucuaio measues F ( j = 2 3 defied as i (4.-(4.3 wih he IM-OLS esiduals ζ ( give i (3.4. j Nex oosiio esablish he asymoic ull ad aleaive disibuio of all hese es saisics. 22

24 Poosiio 4.. Ude he ull hyohesis of coiegaio ha is whe α = i Assumio 2. wih v = /2 he: 2 2 ( a F ( ω R ( s ds u. F ( ω su R ( F ( ω su R ( R ( 2 u. 3 u. [] [] v ( ωu. R ( 2 Fj j = 2 3 ad v ( wih ( ad similaly fo ( R elaced by R ( as has bee defied i esul (d of Poosiio 3.. Also ude he aleaive hyohesis of o coiegaio ha is whe α < i Assumio 2. wih v = /2 he: whee [] [] v ( J ( ( b F ( J ( s ds F ( su J ( F ( su J ( J ( ( J ( J ( g( g( s g( s ds g ( s J ( s ds = u u wih Ju ( = Bu ( s ds ad similaly fo 2 F ( 2 2 v ( wih J ( elaced by J ( defied as ( Fj J ( J ( g ( g ( s g ( s ds g ( s J ( s ds = u u whee Ju ( = Bu ( s ds. Poof. See Aedix D. ( j = 2 3 ad Rema 4.. As cied above ude he aleaive of o coiegaio hese es saisics ae o cosise i he usual way because hei limiig disibuios ae obaied wihou fuhe omalizaio of he comoes i he umeao ad deomiao. Howeve hese disibuios diffe fom he ull disibuios i he sese ha hey ae shifed o he lef ad moe coceaed. This imlies ha a ejecio of he ull of coiegaio agais o coiegaio is egiseed fo small values of ay of hese es saisics which meas ha his is a lef ailed es ha ejecs he ull hyohesis of coiegaio fo values of F ( smalle ha he asymoic ciical value c j α ( give by he αh-lowe quaile of he asymoic ull disibuio. Fom he esuls i a (a of Poosiio 3. i is evide ha he asymoic ull disibuio of all hese es saisics ae fee of uisace aamees ad oly deeds o he j 23

25 combiaio of ad i he case of usig OLS deeded obsevaios. Tables 4. ad 4.2 below ese he ciical values fo he es saisics F ( ad Fj ( fo = comued via diec simulaio based o 2 ideede elicaios wih 2 obsevaios ad ξ = ( u ε iidn( + I + = Rema 4.2. I he defiiio of all hese es saisics isead of usig he simle j omalizaio faco defied i (4.5 ad (4. o elimiae he uisace aamee ω 2 u. i he flucuaio measues F ( we could coside he commoly used j oaameic eel esimao ω 2 ( m based o he fis diffeeces of he IM-OLS esiduals ζ ( which is defied as 2 ω ( m = w( j/ m ζ ( ζ j ( j=( = j + s = w ζ ( ζ s ( = s= m (4. wih badwidh m ad eel fucio w(. Iesecive of he choice of he eel he cosisecy of his esimao elies o he magiude of he badwidh aamee m. Ude saioaiy ad wih / 2 ( m = o which icludes boh he case of a samlesize deede deemiisic badwidh choice ad a daa-deede sochasic oe usually we mus obai he cosisecy esul ω ( m ω 2 2 u. bu his oio equies he deemiaio of a aicula value fo his aamee. I his seu ad fom esul (c i Poosiio 3. we have ha he fis diffeece of he IM-OLS esiduals ca be wie as / 2+ v v 3/ 2 ˆ / 2 ˆ ( ζ ( = ζ ( S T / 2+ v ( γ γ / 2+ v v 3/ 2 ˆ / 2 ˆ ( = ζ ( X Z / 2+ v ( γ γ 24

26 whee ζ ˆ = z = u γ Z wih Xˆ = η ad Zˆ = ε so ha ζ ( = u γ ε ( η [ ( ] / 2v / 2 / 2 / 2+ v / 2v / 2+ v ε [ ( γ γ ] Ude he assumio of coiegaio we have ha ζ = + / 2 ( u γ ε ε ( γ γ O( = z ε + O / 2 ( ( γ ( γ give ha u = u + O ad / 2 ( / 2 ε = ε + O ( wih z ad ε zeo-mea saioay ocesses ha ae asymoically ucoelaed by cosucio so ha he log-u covaiace maix of ( z ε is diag( ω Ω. Wih hese ad usig he 2 u. esul (b i Poosiio 3. above i moe comac fom as / 2 ( d Ω d ω u. Π = ωu. / 2 γ γ dγ Ω dγ he we ge ω ( ω ( + d d 7 which is a adom limi ad is give by he 2 2 m u. γ γ adom veco d γ deemiig he limiig ull disibuio of γ γ γ. I his case usig ω 2 ( m wih some simle ule fo deemiig he badwidh ude saioaiy we cojecue ha his will oduce cosise es saisics. Fomally give ha ude he assumio of o coiegaio we have ζ ( = u ( η ( / 2 / 2 / 2 ε γ γ + = [ ( ] O ( O ( ad hus ζ ( = O ( usig (C.6 i Aedix C he 2 2 ω ( m = ( γ / w( j/ m ε ε j ( γ / + o( j=( = j + wih a well defied sochasic limi so ha ω ( m = O (. Aleaively ad Fo a moe deailed demosaio of his esul see age 32 i Vogelsag ad Wage (2. 25

27 followig he idea develoed by Kiefe ad Vogelsag (25 ad fuhe aalized by Su Phillis ad Ji (28 we could coside he so called fixed-b esimaio heoy of a log-u vaiace based o a badwidh ha is simly ooioal o he samle size as m = b wih b (]. The esuls i his case wee exeded by Vogelsag ad Wage (2 o models wih osaioay egessos bu he asymoics ae elaively moe comlex ad o eaed hee. A aicula case ha ca be eaed wihou ay addiioal develome is whe b = so ha he badwidh is se equal o he samle size m =. By usig Lemma i Cai ad Shiai (26 fo he Bale eel w(x = x fo x we ca wie (4. as follows 2 2v ( v 2 ω ( = 2 ( ζ ( = ( v ( v ( v 2 ( ζ ( ( ζ ( + ( ζ ( = whose asymoic disibuio is ooioal o 2 u. (4.2 ω ude he coiegaio assumio ad as fo he simle eleme v ( is of he same ode of magiude as 2 he flucuaio measues (4.8 ad (4.9 boh ude coiegaio ad o coiegaio esulig i icosise es saisics. Alhough all hese ohe oios seems o oduce icosise es saisics we exloe hei use i fuue eseach. Table 4. Asymoic lowe ciical values fo he coiegaio es based o he flucuaio es saisic Fj ( j = 2 3 Sigificace level α = Tes saisic F ( Tes saisic F2 ( Tes saisic F3 (

28 Table 4.2 Asymoic lowe ciical values fo he coiegaio es based o he flucuaio es saisic Fj ( j = 2 3 Demeaed case = Demeaed ad deeded case = Sigificace level α = = F ( F2 ( F3 ( Nex i ode o evaluae he owe of hese es saisics we use a local-o-uiy aoach o coiegaio i fiie samles whee he eo em i he coiegaig egessio equaio follows he AR( ocess u = α u + υ wih α = c/ c as i Phillis (987 which gives he followig esul. Coolay 4.2. Ude he local-o-uiy aoach o he ull of coiegaio ad / 2 ( s c Assumio 2. he we have ha u B ( = e db ( s ad ( ( 3/ 2 ζ[ ] Jc [ ] c whee Jc ( is as J ( i a (b of Poosiio 4. wih Ju ( elaced by J ( = B ( s ds ad B ( s he deeded Osei-Uhlebec ocess B ( s. c c c Nex Table 4.3 shows he owe esuls fo samle sizes = ad 5 comued by simulaio wih 5 elicaios fo values of c = ad fo he flucuaio-ye es saisic F3 ( wih = ad = 5. The fis emaable evidece is ha of icosisecy of he oosed es saisic ad he vey low owe dislayed iesecive of he value of c which idicaes he eed of a moe deely ivesigaio of hese esig ocedues. υ c 27

29 Table 4.3 Fiie-samle owe of he es saisic F3 ( = ude he local-o-uiy aoach o saioaiy (coiegaio a he 5% omial level Case = F3 ( Case = F3 ( Samle size = 5 = 5 c = = c = = c = 2.5 = c = 5 = c = = Alicaios This secio is devoed o he alicaio of he above cosideed saisics fo esig he ull hyohesis of coiegaio agais o coiegaio i a liea saic coiegaig egessio model i wo leadig cases a aicula fom of he Taylo ule of moeay olicy ad he ese value model of soc ice deemiaio i ems of is fudameal value. 5. Taylo ule Taylo s (993 oigial fomulaio of he so-called Taylo s ule of moeay olicy is e e give by i = + π + fπ ( π π + f y y wih i he ceal ba olicy ae e he equilibium eal iees ae π he welve moh iflaio ae e π he iflaio age of he ceal ba ad y he ouu ga. Whils hee ae seveal ossible 28

30 secificaios of he Taylo ule Taylo (999 ad Öseholm (25 coside a modified emiical vesio of his equaio wih coemoay vaiables give by i = + π + y + u =... (5. e e wih = ( π. No oly is his vesio closes o Taylo s oigial fomulaio bu i has bee used fequely i emiical wo ad is a secial case of moe elaboae models hus sevig as a useful bechma. Isead of focusig o he esimaed values of he esose coefficies we ae maily coceed i he emiical evidece of a sable log-u elaioshi. A imoa quesio deemiig he validiy of hese ifeeial ocedues is ha of edogeeiy of he egessos. As idicaed by Öseholm (25 modellig iflaio ad ouu ga as fowad looig is oe aioale fo aguig ha hese vaiables may o be edeemied. To illusae hese esuls we use quaely daa fo he US whee he full samle is 96: o 999:4 wih = 6 obsevaios. All he daa wee obaied fom he Fedeal Reseve Ba of S. Louis ad gahed i Figue Fedeal fuds ae CPI iflaio Ouu ga Figue. Fedeal fuds ae CPI iflaio ad (omial ouu ga fo US 96: o 999:4 Befoe efomig coiegaio aalysis fom he secified egessio equaio (5. Table 5. eses he esuls fo wo oaameic esig ocedues o emiically -4 29

31 deemiae he degee of esisece of he idividual seies. VR efes o he vaiaceaio es saisic of Beiug (22 o esig he ull of a ui oo agais he aleaive of saioaiy while ha KPSS efes o he es saisic by Kwiaowsi e.al. (992 fo esig he evesed hyohesis ha is he ull of saioaiy agais he aleaive of a ui oo. Fo he comuaio of he OLS esidual-based flucuaio es saisics as well as fo he FM-OLS esimaes ad esiduals we use a oaameic eel esimao of he log-u vaiace ad covaiace maix ad vecos equied wih he Bale eel ad samle-size deede badwidh give by he commoly used ule m = [2 (/ /4 ] o comue he esimao of he codiioal log-u vaiace ω 2 u. cosise ude coiegaio. Table 5. Alicaio o he vaiables of he Taylo ule model. Resuls of oaameic ess of saioaiy. VR KPSS (a (b (c (a (b (c Samle 96:-999:4 i ** ** π *.592 *.96 * y (eal y Samle 96:-979:2 i *.663 *.66 *.83 π *.32 y (eal y Samle 979:3-999:4 i **.628 *.32 π ** y (eal y * Samle 987:4-999:4 i *.25.3 π *.86 * y * (eal y * Noe: Labels (a idicaes o deemiisic comoe (b OLS demeaed obsevaios ad (c OLS demeaed ad deeded wih a liea ime ed. * Sigifica a he 5% omial level. ** Sigifica a he % omial level. Iesecive of he assumed deemiisic comoe he emiical evidece sogly suo he (ea ui oo hyohesis fo he sochasic ed comoe udelyig each of hese idividual seies ad fo all he subsamles cosideed which emiically jusified he subseque coiegaio aalysis. Table 5.2 eses he esuls of seveal 3

32 esidua-based ess fo he ull hyohesis of coiegaio agais he aleaive of o coiegaio whe usig he full samle daa. We coside fo uoses of comaiso he flucuaio-ye es saisics ha we oose i secio 4 which ae based o he IM-OLS esimaio wih OLS demeaed ad deeded obsevaios of he model vaiables ad he flucuaio-ye es saisics oosed by Shi (994 Xiao ad Phillis (22 ad Xiao (999 ad Wu ad Xiao (28 based o OLS ad FM- OLS esiduals. Wih he exceio of he esuls based o he CvM measue of excessive flucuaios of he Shi s (994 vesio of he KPSS es saisic whee he evidece is o fully coclusive fo diffee cofiguaios of he deemiisic comoe we foud sogly suo fo o ejecio of he ull hyohesis of coiegaio. The esuls of ou ew esig ocedues ae comleely agee wih his coclusio. Table 5.2 Alicaio o he vaiables of he Taylo ule model. Resuls of oaameic flucuaio ess fo he ull of coiegaio OLS FM-OLS IM-OLS (a (b (c (a (b (c (a (b (c Sˆ ( CSˆ ( Rˆ ( F ( F2 ( F3 ( Noe. Labels (a idicaes o deemiisic comoe (b OLS demeaed obsevaios ad (c OLS demeaed ad deeded wih a liea ime ed. Ciical values fo he S ˆ ( es saisic ae ae fom Shi (994 while ha fo he flucuaio es saisic R ˆ ( ae give i Xiao (999. Fo he flucuaio CUSUM-ye es saisic CS ˆ ( hese ae give i Xiao ad Phillis (22 fo he case of o deemiisic comoe ad i Hao ad Ide (996 fo he cases of a cosa em ad a liea ed. 5.2 Pese value model fo US soc eus Sice he semial aes by Flood ad Gabe (98 Shille (98 ad Blachad ad Waso (982 amog ohes hee exis a well fouded heoy deemiig he 3

33 elaioshi bewee soc ices ad fudameals ha hese ices ae suosed o eese. Howeve hee exiss a lage emiical evidece of icosise elaio bewee soc ices ad divideds o exlai some of he sylized facs abou soc eus which jusify he usual aoach of ioducig a seculaive bubble ocess io asse icig models. Accodig o Cambell ad Shille (988 he log liea aoximaio o he sadad aioal execaios model fo soc ices P = + γ E P + D deemies he soc mae ese value model (PVM ( [ + + ] = α + d + u (5.2 whee ad d ae logaihms of P he eal soc ice ad D he divided aid o he owe of he soc bewee ad + whee (+γ is he discou faco. A commo aoach o esig bubbles is o examie he saioaiy of he eos u i (5.2 hough he use of ui oo ess based o he esimaed eos. By eaig he osaioay bubble ocess if ese as a ui oo ocess he coveioal ui oo based ocedues es he ull hyohesis ha he soc ice coais a bubble. Howeve i he case of a eiodically collasig bubble ocess sice i is o a coveioal ui oo ocess he coveioal ui oo ess will ejec he ull of a bubble leadig o he icoec coclusio ha hee is o bubble i mae ices. Fo his easo Wu ad Xiao (28 cosides moe aoiae o use some id of flucuaio-ye es saisic o deecig bubbles isead of he adiioal ui oo ess. As meioed i Wu ad Xiao (28 his ye of ess mus have owe agais boh collasible bubbles ad o-collasible bubbles o because of he collase effec bu because of he exlosive behavio deemied by he submaigale oey of aioal bubbles. Howeve sucual isabiliy of he elaioshi i fom of sucual chage ime-vayig discou aes ad is-emium effecs may be ohe ossible souces of excessive flucuaios alhough he eveual collase of he bubble ocess 32

34 if i exiss is geeally diffee fom a coveioal sucual bea. We efom coiegaio aalysis of his model (5.2 usig US mohly daa o Sadad & Poos (S&P soc ices ad comosie divideds fo he eiod 87: o 2:2 which ae aed fom Robe Shille s websie. 8 The daa ae gahed i Figue 2 ad ude he ossibiliy of deemiisic edig behavio we ca exed model (5.2 icooaig a liea ed Mohly SP5 log Pice Mohly log divided Figue 2. S&P log soc ices ad comosie divideds 87: o 2:2 Nex Table 5.3 shows he esuls of he oaameic vaiace-aio (VR ad KPSS ess o he idividual seies boh fo he full samle ad fo wo subsamles befoe ad afe Wold Wa II. The emiical evidece fo ay ossible secificaio of he deemiisic comoe clealy idicaes he osaioaiy of he idividual seies. Give his evidece we efom a coiegaio aalysis. Table 5.4 eses he esuls fo he OLS FM-OLS ad IM-OLS esidual-based flucuaio-ye ess fo he ull hyohesis of coiegaio cosideed i his ae ad fo he model based o he full samle. 8 Robe Shille s websie is h://aida.eco.yale.edu/ shille. 33

35 Table 5.3 Alicaio o US soc mae daa. Resuls of oaameic ess of saioaiy. VR KPSS (a (b (c (a (b (c Samle 87:-2: ** 5.96 **.25 ** d ** 6.4 **.22 ** Samle 87:-946: *.89 ** 3.45 **.9 d ** 3.49 **.329 ** Samle 947:-2: ** 3.93 **.45 ** d ** **.239 ** * Sigifica a he 5% omial level. ** Sigifica a he % omial level. As ca be see fom he esuls i Table 5.4 ad exce fo he case of he KPSS-ye es S ˆ ( by Shi (994 all he ohe esuls coicide i he geeal evidece i favou of he coveioal sochasic coiegaio hyohesis. Table 5.4 Alicaio o US soc mae daa. Resuls of oaameic flucuaio ess fo he ull of coiegaio OLS FM-OLS IM-OLS (a (b (c (a (b (c (a (b (c Sˆ ( CSˆ ( Rˆ ( F ( F2 ( F3 ( Noe. Labels (a idicaes o deemiisic comoe (b OLS demeaed obsevaios ad (c OLS demeaed ad deeded wih a liea ime ed. Ciical values fo he S ˆ ( es saisic ae ae fom Shi (994 while ha fo he flucuaio es saisic R ˆ ( ae give i Xiao (999. Fo he flucuaio CUSUM-ye es saisic CS ˆ ( hese ae give i Xiao ad Phillis (22 fo he case of o deemiisic comoe ad i Hao ad Ide (996 fo he cases of a cosa em ad a liea ed. 6. Coclusios ad some exesios The ese ae is devoed o he aalysis of he asymoically efficie esimaio of a liea saic coiegaig egessio model by maig use of a ew ecely oosed esimaio mehod by Vogelsag ad Wage (25 he so-called iegaed modified OLS esimao (IM-OLS ha has he mai advaage ha does o equie he choice of ay uig aamee whe we deal wih deemiisically edig iegaed 34

36 egessos. We show ha his mehod mus be modified o coecly accommodae he sucue of he deemiisic comoe of he egessos ad o avoid ossible icosisecies i he esimaio esuls. As a byoduc of hese esuls we oose he use of he IM-OLS esiduals o build some ew simle saisics o esig he ull hyohesis of coiegaio agais he aleaive of o coiegaio. While he mai comoe of hese ew es saisics seems o wo well i deecig excessive flucuaios i he esidual sequece ude o coiegaio i is o ye clea how o obai ivoal es saisics fee of uisace aamees ad cosise ess give he difficulies i obaiig a oe esimao of a log-u vaiace. This ceal quesio will be sudied i fuue wo as well as he cosideaio of moe comlex deemiisic comoes ad hei eame i he coex of he IM-OLS esimaio. Refeeces Blachad O.J. M.W. Waso (982. Bubbles aioal execaios ad fiacial maes. I: Cisis i Ecoomic ad Fiacial Sucue: Bubbles Buss ad Shocs edied by Paul Wachel. Lexigo: Lexigo Boos. Beiug J. (22. Noaameic ess fo ui oos ad coiegaio. Joual of Ecoomeics 8( Cai Y. M. Shiai (26. O he aleaive log-u vaiace aio es fo a ui oo. Ecoomeic Theoy 22( Cambell J.Y. R.J. Shille (988. The divided-ice aio ad execaios of fuue divideds ad discou facos. The Review of Fiacial Sudies ( Flood R.P. P.M. Gabe (98. Mae fudameals vesus ice-level bubbles: he fis ess. Joual of Poliical Ecoomy 88( Hase B.E. (992. Efficie esimaio ad esig of coiegaig vecos i he esece of deemiisic eds. Joual of Ecoomeics 53( Hao K. B. Ide (996. Diagosic es fo sucual chage i coiegaed egessio models. Ecoomics Lees 5( Hassle U. (2. The effec of liea ime eds o he KPSS es fo coiegaio. Joual of Time Seies Aalysis 22( Kiefe N.M. T.J. Vogelsag (25. A ew asymoic heoy fo heeosedasiciyauocoelaio obus ess. Ecoomeic Theoy 2( Kuozumi E. K. Hayaawa (29. Asymoic oeies of he efficie esimaos fo coiegaig egessio models wih seially deede eos. Joual of Ecoomeics 49( Kwiaowsi D. P.C.B. Phillis P. Schmid Y. Shi (992. Tesig he ull hyohesis of saioaiy agais he aleaive of a ui oo. How sue ae we ha 35

37 ecoomic ime seies have a ui oo?. Joual of Ecoomeics 54( Öseholm P. (25. The Taylo ule: a suious egessio?. Bullei of Ecoomic Reseach 57( Pa J.Y. (992. Caoical coiegaig egessios. Ecoomeica 6( Phillis P.C.B. (987. Towads a uified asymoic heoy fo auoegessio. Biomeia 74( Phillis P.C.B. B.E. Hase (99. Saisical ifeece i isumeal vaiables egessio wih I( ocesses. The Review of Ecoomic Sudies 57( Phillis P.C.B. M. Loea (99. Esimaig log-u ecoomic equilibia. The Review of Ecoomic Sudies 58( Saioe P. (99. Asymoically efficie esimaio of coiegaio egessios. Ecoomeic Theoy 7( -2. Shille R.J. (98. Do soc ices move oo much o be jusified by subseque chages i divideds? The Ameica Ecoomic Review 7( Shi Y. (994. A esidual-based es of he ull of coiegaio agais he aleaive of o coiegaio. Ecoomeic Theoy ( 9-5. Soc J.H. M.W. Waso (993. A simle esimao of coiegaig vecos i highe-ode iegaed sysems. Ecoomeica 6( Su Y. P.C.B. Phillis S. Ji (28. Oimal badwidh selecio i heeosedasiciy-auocoelaio obus esig. Ecoomeic 76( Taylo J.B. (993. Disceio vesus olicy ules i acice. Caegie-Rochese Cofeece Seies o Public Policy 39( Taylo J.B. (999. A hisoical aalysis of moeay olicy ules. I Taylo J.B. (ed. Moeay Policy Rules Chicago: Chicago Uivesiy Pess. Vogelsag T.J. M. Wage (2. Iegaed Modified OLS esimaio ad fixed-b ifeece fo coiegaig egessios. Woig Pae No.263 Isiue fo Advaced Sudies (IHS Viea. Wu G. Z. Xiao (28. Ae hee seculaive bubbles i soc maes? Evidece fom a aleaive aoach. Saisics ad is Ieface Xiao Z. (999. A esidual based es fo he ull hyohesis of coiegaio. Ecoomics Lees 64( Xiao Z. (2. Tesig he ull hyohesis of saioaiy agais a auoegessive ui oo aleaive. Joual of Time Seies Aalysis 22( Xiao Z. P.C.B. Phillis (22. A CUSUM es fo coiegaio usig egessio esiduals. Joual of Ecoomeics 8( Aedix A. Poof of Poosiio 2.. By OLS deedig of he obseved ocesses Y ad X as defied by (2. ad (2.2 we have ha Yˆ Y Yj = j =... ˆ τ Q τ X j= X j X Each of he comoes above ca be decomosed as η + α ( τ τ τ Q τ i =... whee i i i i j= i j j i i j= i j τ j τ ad η = η η Q 36

38 τ τ τ τ τ = τ τ τ τ i ( : i i j j Q i i j i j i j Q j= j= τ i Q i Q i i ( i τ i = τ ( : ( i Q i Q i i i Q i ( i Q ( i ( i τ i = τ ( I i : τ i = Q whe i < fo all i =. i + i + i + i τ i + i give he bloc sucue fo he ivese of Obviously he same esul diecly holds whe i = while ha if ay i > he we have α ( τ τ τ Q τ = α ( τ Q Q τ which does i i i j= i j j i i i ( i o vaish ad i is of ode O( i. B. Poof of Poosiio 2.2. Fis give ha we ca wie τ Q Q ( τ η = η = Q Q M M Q Q = M Q Q M he usig (2. we have ha M M Q Q τ X M Q Q M = η M M Q Q Q Γ Φ τ = + ε M Q Q M Q Γ Φ = η M τ ε Q Q η ε Γ Φ = = = + M η ε Q Q τ ε = = ad + M M Q Q + M Q Q u W + = + M Q Q M u M u wih W give i (2.4. Taig hese esuls ogehe we ge (2.2. Secod give he sequece of FM-OLS esiduals defied by uˆ + ˆ ˆ ( = Y + ( τ X ( α + + wih + Y = Y ( τ Γ Φ + ε γ ca be wie as uˆ ( = uˆ ( ε γ τ M τ ε Q Q η ε γ Q Q u = = + + η M η ε Q Q τ ε γ + u = = o i moe comac fom as i (2.3 whe usig M τ ε Q Q η ε = Q τ ε Q M η ε = = = = g 37

39 ad η ε = η ( ε τ Q τ ε. = = = j j C. Poof of Poosiio 3.(a b. Paial summig fom (2.8 gives Sˆ = Sˆ + U =... so ha (C. 3/ 2 ˆ S ( v 3/ 2 ˆ / 2 = ( (/ ( ˆ + W / 2 ˆ S T γ = T 3/ 2 Sˆ ( v (/ U / 2 ˆ = T ad hus / 2+ v ( v ( W = / 2+ v γ γ 3/ 2 ˆ 3/ 2 S ˆ 3/ 2 ˆ / 2 ˆ S ( v = (/ ( (/ U / 2 ˆ / 2 S T ˆ = = T T (C.2 Maig use of he covegece esuls i (2.4 (2.5 ad (3. he ude he coiegaio assumio ha is whe v = /2 we have ha ( ( g ( g ( d ( Vu ( d g γ ( { } g ( g ( d ( Vu. ( d ( ( d g g V γ = + (C.3 whee he las wo ems ae based o he decomosiio i (2.7. Fo he las em above as i Vogelsag ad Wage (2 (equaio (43 age 27 we ca wie ( ( d = ( ( d ( ( d = g V γ g g γ g g I γ so ha ( ( Π ( ( d ( Vu. ( d g g g γ γ o equivalely ( ( Π ( ( d [ ( ( ] dvu. ( g g G G γ γ whee he las equaliy comes fom defiig G ( = g ( s ds = Π g ( s ds wih Π = diag( Ω Ω ad g ( = Π g (. Also by defiig / 2 / 2 wih W. ( = BM ( b ( he we have u ( (C.4 2 u. u. u. g V ( = ω W ( ( ωu. Π g ( g ( d [ ( ( ] dwu. ( G G (C.5 γ γ As i equaio (24 i Vogelsag ad Wage (2 codiioal o B ( he above limiig disibuio (C.5 is N( 2 Θ 2 wih Θ 2 a well defied codiioal asymoic 38

40 sochasic covaiace maix. Ude o coiegaio ha is wih v = /2 ad osaioaiy of he eo sequece u he we have ( ( ( ( d Ju ( d g g g (C.6 γ whee Ju ( = Bu ( s ds. As ca be see fom (C.5 ad (C.6 he covegece aes fo he IM-OLS esimao of ae he same as whe usig OLS o ay of he asymoically equivale ad efficie esimaio mehods. Poof of Poosiio 3.(c d. Give he IM-OLS esidual sequece i (3.4 he IM coiegaig egessio equaio i (3.9 ad (C.2 we ca wie ζ ( as / 2+ v v 3/ 2 ˆ / 2 ˆ ( ζ ( = ζ ( S T =... / 2+ v ( γ γ Ude he coiegaio assumio maig use of (3. (3.2 ad he wea covegece of he IM-OLS esimaos of ad γ he esul (d he follows by he coiuous maig heoem. g D. Poof of Poosiio 4.(a. I follows diecly fom he esuls i (b ad (d fom Poosiio 3. ad he coiuous maig heoem. Poof of Poosiio 4.(b. Fom esul (c i Poosiio 3. wih v = /2 we have 3/ 2 3/ 2 3/ 2 ˆ / 2 ˆ ζ ( = ζ ( S T γ whee 3/ 2 3/ 2 / 2 ˆ 3/ 2 ζ = U γ ( T = U + O ( so ha usig (C.6 above ad he coiuous maig heoem we have ha ( 3/ 2 ζ ( J ( = Ju ( g ( g ( s g ( s ds ( s Ju ( s ds g wih Ju ( = Bu ( s ds as i (2.4 which gives he idicaed esuls. g 39