Interest rate pass-through in the Euro Area: a timevarying cointegration approach

Transcription

1 Ieres rae pass-hrough i he Euro Area: a imevaryig coiegraio approach Afoso-Rodríguez Julio A. Deparme of Applied Ecoomics ad Quaiaive Mehods Uiversiy of La Lagua Faculy of Busiess ad Ecoomics Camio La Horera s/. Campus de Guajara C.P. 387 La Lagua Teerife Caary Islads Teléfoo: jafosor@ull.es Saaa-Gallego María Deparme of Applied Ecoomics Uiversiy of he Balearic Islads Edificio Gaspar Melchor de Jovellaos Campus UIB Cra. Valldemossa m 7.5 C.P. 7 Palma (Illes Balears) Teléfoo: maria.saaa@uib.es Prelimiary versio Absrac This paper sudy he mechaism of rasmissio bewee he moey ad he reail credi mares saed i erms of he log-ru relaioship bewee he harmoized ieres raes for differe credi caegories ad for a subse of couries of he EMU. This mechaism ow as he ieres rae pass-hrough (IRPT) pheomeo has bee aalyzed i may empirical sudies usig a variey of ecoomeric echiques for differe samples of couries ad periods of ime ad he geeral coclusio is ha he pass-hrough seems o be icomplee i he log-ru. Excep for a few rece wors he aalysis is performed o he basis o a ime-ivaria logru relaioship which may o be appropriae i his case ad could codiio his resul. To evaluae he robusess of hese fidigs we exed he aalysis hrough a o-liear model for he log-ru relaioship bewee he moey ad he reail mares ha icorporaes i a very flexible form ad wih miimum requiremes o uig parameers he olieariy i he form of ime-varyig parameers. To ha ed we follow he approach iiiaed i Bieres (997) ad also propose some ew ools o es for he exisece of a sable ime-varyig coiegraio relaioship. The resuls obaied seems o suppor he former evidece of a icomplee passhrough. Keywords ad phrases: reail ieres raes moeary policy coiegraio aalysis srucural isabiliy ime-varyig coiegraio JEL classificaio: E5 F36 C

2 . Iroducio Moeary rasmissio is a ey issue whe aalyzig moeary policy decisios. I his sese he rasmissio of moeary policy relies o how policy rae chages measured as chages i moey mare ieres raes are rasferred o he ba sysem via chages i he reail raes for each possible credi caegory i he ecoomy which is called he ieres rae pass-hrough (IRPT) effec. This mechaism is impora for achievig he aims of moeary policy such as achievig price sabiliy ad ifluecig he pah of he real ecoomy hrough ifluecig aggregae demad a leas o some exe. This pheomeo is closely relaed o he aalysis of he sabiliy properies of moeary policy rules i erms of givig rise o a uique ad sable equilibrium if he implied respose of he omial ieres raes o iflaio chages is sufficiely srog (Taylor priciple). A icomplee IRPT could violae he Taylor priciple ad moeary policy would fail o be sabilizig i he sese ha reail ieres raes do o respod sufficiely o esure ha real raes are sabilizig. This appears o be paricularly impora for he Euro Area usually ae as a example of a ba-based fiacial sysem for which he empirical evidece seems o idicae a limied IRPT (reail ieres raes respodig less ha oe-o-oe o policy raes). This paper coribues o he empirical aalysis of measurig he magiude of he adjusme i he framewor of aalysis of a o-liear model for log-ru relaioship allowig for a ime-varyig relaioship bewee he moey mare ad he reail ieres raes for a se of EMU couries seleced by he crierio of havig he loges available series. Thus i secio we coribue o he aalysis of he specificaio of a geeral imevaryig coiegraig model boh i he form of a ime-varyig coiegraig regressio model or aleraively as a reduced ra ime-varyig error-correcio model (ECM) ad discuss some of heir mai feaures. Secio 3 iroduce he empirical aalysis based o evaluaig he exisece of a ime-varyig coiegraio relaioship bewee he seleced ieres rae series adopig he mehodology iroduced i Bieres (997) ha propose o model he parameers as smooh fucios of ime hrough a weighed average of Chebyshev ime polyomials. This mehodology has bee used before i Bieres ad Maris () ad i Neo ( ) bu we propose some ew ools o empirically assess he sabiliy of he o-liear relaioship allowig for cosise esimaes of he isaaeous ad ime-varyig magiudes of he IRPT. Some heoreical developmes are preseed i Appedixes A o C while he mai empirical resuls are preseed i Appedixes D o F.. Ecoomeric aalysis We cosider he case where he osaioary observed (+)-dimesioal ime series z = ( y x ) =... where x = ( x... x ) is geeraed as z = z + ε where ε = ( ε ε ) is a zero-mea (+)-dimesioal wealy depede ad saioary error sequece bu i is assumed ha i ca also be embedded io he followig geeral Error-Correcio Model (ECM) represeaio Φ ɶΦ ( L)[( L) I ( α ) λκ L] z = e (.) + See e.g. Kwapil ad Scharler () ad he refereces cied o earlier empirical sudies for EMU couries ad differe periods of ime.

3 m j where Φ ɶΦ ( L) = Φ ( L ) M + wih Φ ( L) = I + j= Φ jl a saioary marix polyomial of fiie order m i he lag operaor L (i.e. Φ ( z) = has roos ouside he ui circle) λ = ( λ λ ) κ = ( β ) ad he + square marix M + ca be eiher he ideiy marix or more geerally be defied as a ime-varyig roaio marix of he form β M + = (.) I hus preservig he equivalece of he roos of each lag polyomial wih β he - dimesioal sigle ime-varyig coiegraig vecor. For he las erm i (.) e = ( e e ) i is assumed o be a zero-mea iid sequece wih fiie covariace + δ marix Σ e = E [ e e ] > ad E[ e ] < for some δ >. This is a modified versio of he ECM represeaio used i Ellio e.al. (5) o derive a family of opimal esig procedures for coiegraio i he case of a ow ad ime-ivaria coiegraig vecor β = β =.... Taig (.) equaio (.) ca also be rewrie as κ z κ λ = ( α ) κ z + ξ (.3) x λ j wih ξ = ( υ ε ) = C( L) e C( L) = Φ ( L) = j= C jl ad u = κ z = y β x he coiegraig error erm. Uder he summabiliy codiio jtr( C C ) < ad he j= j j properies of he error sequece e he process ξ saisfy a mulivariae ivariace priciple such as [ r] [ r ] / / B ( r) / υ υ ξ = ξ( r) = = ξ( r) ( r) = = B Ω B W ξ ε wih Ω = C() e C() ξ Σ he log-ru covariace marix of ξ ad Wξ ( r) = ( Wυ ( r) W ( r)) a +-variae sadard Browia process. Give ha κ z = u κ z ad κ z = u + κ z wih κ = ( β ) he he firs compoe of he vecor i (.3) allows o represe u as a ime-varyig AR() process u = ρ ( u + κ z ) + υ where he ime-varyig auoregressive coefficie is give by ρ = + ( α ) κ λ = α + ( α)( κ λ ) ha becomes fixed i he case of a imeivaria coiegraig vecor i.e. κ = = κ ( β ) or aleraively uder he ormalizaio resricio κ λ = i which case ρ = α. Uder his las codiio we obai he followig saic ime-varyig coiegraig regressio model (a geeralizaio of he so-called Phillip s riagular model) give by y = β x + u (.) wih a coiegraig error erm of he form u = α u + υ + α β x (.5) ad x = x + ( α ) λ κ z + ε (.6) Uder ime-ivariace of he coiegraig relaio equaio (.5) ogeher wih he assumpio o he saioariy of υ allows o differeiae bewee he exisece of a coiegraio relaioship amog y ad x whe α < ad he absece of such a sable log-ru relaioship (o coiegraio) whe α =. The exra erm appearig i

4 he righ-had side of equaio (.5) is osaioary i geeral due o he iclusio of he iegraed regressors x excep i he case of a ime-ivaria coiegraig vecor wih β = β or rivially whe α = so ha u = υ ad all he serial correlaio i he regressio error erm is hrough he dyamics i υ alhough is behavior properies ad ifluece will deped o he paricular mechaism ha deermies he chages i β. As examples we cosider hree very differe mechaisms: (a) he case of a sigle discree chage possibly affecig o all he coefficies i β a a give brea poi as β = β + λ H ( τ ) (.7) where H ( τ ) = I( > [ τ ]) wih τ () he sadard sep fucio (b) a marigale process as i Hase (99) give by β = β + υ (.8) wih υ a zero-mea error iid sequece wih fiie covariace marix E [ υ υ ] = Σ υ ad β a -vecor of fixed values ad (c) a ime-varyig coiegraig vecor via Chebishev ime polyomials proposed by Bieres ad Maris () exedig he resuls i Bieres (997) which is give by m β = b G ( ) (.9) j j j= where G ( ) = ad G j ( ) = cos( jπ(.5)/ ) j = m (see Appedix A for more deails). Firs uder he saioariy codiio o he regressio error erm give by α < he scaled parial sum of u ca be wrie as where [ r ] [ r] [ r ] / / / / u = υ + α ( u u[ r ]) + α x = α = = / β x is O () i he cases (a) ad (c) while ha i is j= j j p β (.) he case (b) ad hece divergig wih he sample size. Specifically we obai [ r ] [ r] [ τ ] / / / β x = λ x [ r ] ε ε I( r > τ) = = = [ τ ] / = λ x + ε I ( r > τ) = / = λ x I( r > τ ) wih ad [ τ ] [ r ] [ r ] / / / β x = x υ = p = = / Op( ) i (.) β υ O ( ) (.) [ r] r x υ B Bυ + ε h υ = h= ( s) d ( s) r E[ ] j H H r m [ r ] 3/ + O( ) b j x j= = [ r ] m [ r] / 3/ / β x = π b j j ( ) x j ([ ]) x [ r] = j= = (.3) 3

5 i each case where H ( r) = si( jπ( 5)/ ) i (.3) for j = m. O he j oher had uder o saioariy of he regressio error erm i (.) wih α = we ge he represeaio / / / / = + υ j + jx j j= j= u u β (.) / / [ r ] / where ε = x wih u[ r ] = = υ + Op( ) Bυ( r) uder ime ivariace of he regressio coefficies ad he usual assumpio of he iiial value / u = O p () where he las erm j= β jx j is give as i (.)-(.3). Thus assumig he validiy of he ime-varyig ECM represeaio i (.) he coiegraio assumpio implies he exra codiio α = while ha uder o coiegraio he disequilibrium error erm u coais a addiioal erm icorporaig he chages i he values of β wheever i has a clear defiiio. Also as ca be see from (.6) he assumpio ha x are o muually coiegraed ad have roos ha are ow a priori o be equal o oe (i.e. x are iegraed bu o-coiegraed regressors) correspods o he resricio λ = (which implies λ = ) ad hece (.6) mus be replaced by he usual represeaio as a -dimesioal iegraed process x = x + ε (.6 ) which also resuls i he case of o coiegraio i.e. whe α = i (.5) irrespecive of he value of λ. A secod form of he model is he ECM represeaio (.) wih ɶΦ ( L) replaced by Φ ( L) i.e. Φ ( L) z = ( α ) Φ ( L) λκ z + e ha ca be wrie as a ime-varyig reduced ra ECM of he form z = ( α ) Φ () λκ z + Λ ( L) z + e (.5) * by maig use of he BN decomposiio Φ ( L) = Φ () + ( L) Φ ( L) where m j Φ ( ) = Φ ad Φ * * L j j L = Φ ime-varyig coefficies = = * m j i j i Φ wih he lag polyomial * j j+ j+ j Λ ( L) = Λ L ad m j= Λ j Λ = Φ ( α ) λκ + Φ uless κ = κ or aleraively α = irrespecive of he behaviour of κ. Uder he resricios cosidered we have φ() ( α ) Φ () λ = ( α ) () φ where we have pariioed Φ ( L) afer he firs row ad colum so ha x is wealy By recursive subsiuios equaio (3.6) ca also be wrie as x = x + j= ε j + ( α ) λ j= κ jz j where he las erm is decomposed as j = κ jz j = j= u j + j= κ jz j wih u = κ z. Wih a fixed coiegraig vecor κ = κ κ z = while ha wih a ime-varyig j = j j coiegraig vecor we have he represeaio j = κ jz j = j= β j ε j ( β x β x ). I he / fixed parameer case ad uder coiegraio we ge x B ( r) + ( α ) λ Bu ( r) wih = [r] r ( ] ad B ( r) = ( α ) B ( r) as while ha uder o coiegraio he wea limi is u υ x B ( r) implyig a differe behavior i each siuaio ha is uliely i ay real aalysis. /

6 exogeeous for β if ad oly if () Φ is bloc upper riagular (i.e. φ () = z = u + κ ) 3 i which case oly he firs row of equaio (.5) icludes he κ κ z which equals he usual error correcio erm u = κ z oly uder cosacy of he coiegraig vecor. Noe ha (.5) is he ad hoc specificaio of a ime-varyig ECM proposed by Bieres ad Maris (9 ) excep for he fac ha he fiie order lag polyomial Λ ( L) is assumed o be ime-ivaria. If isead of (.5) ad give he decomposiio κ z = u + κ z wih u = κ z we cosider he aleraive ECM represeaio icludig he lagged valued of he error correcio erm resulig from (.) as ( ) Φ() λκ Λ ( L) α z = α Φ λκ z + Λ z + e (.6) where he error erm i (.6) is give by eα = e + ( α ) Φ () λ β x which behaves as a osaioary sequece for α < uder ime-varyig coiegraio. Bieres ad Maris (9 ) propose a lielihood raio es for ime-ivaria coiegraio from (.5) wih a fixed lag polyomial Λ ( L) agais he aleraive of a smoohly varyig coiegraig relaioship over ime. Isead of relyig o he use of he ECM i (.5) he res of he paper ress o he aalysis of he ime-varyig coiegraig regressio model i (.) for a paricular choice of a smooh mechaism drivig he coefficies i β = ( β... β ) =.... Also if we cosider he iclusio of some deermiisic ime reds i he geeraig mechaism of he observaios of z = ( y x ) such as z = d + η where d = ( d d ) η = ( η η ) ad η = η + ε he we ca obai a augmeed versio of (.) give by y = α + x + u β (.7) wih a possibly ime-varyig deermiisic red fucio. 3. Empirical aalysis I his secio we focus o he aalysis of he ime-varyig coiegraig relaioship amog he reail ieres raes for differe mauriies ad wo defiiios of credi variables amely credis for house purchase ad loas for cosumpio o evaluae he magiude of he log-ru IRPT for a subse of couries i he Euro Area for which he loges ad complee series is available. As argued i Bele e.al. (3) whe aalyzig aggregaed micro daa from may bas each of hese isiuios migh face differe iformaio ad rasacio coss a smooh rasiio paer seems o be a plausible mechaism. These auhors use a smooh rasiio regressio o icorporae differe paers of olieariy i he adjusme ad shor-ru dyamics for he relaioship bewee he Euro OverNigh Idex Average (EONIA) as a global idicaor of he moey mare rae i he Euro Area ad credi caegories wih various mauriies. Maroa (9) cosiders he possibiliy of allowig for muliple uow srucural breas i he coiegraig relaioship based o uharmoized reail raes for several EMU couries ad foud 3 From his he resricio λ = implies wea exogeeiy of he iegraed regressors i he paricular cases where Φ ( L ) = I + or more geerally Φ ( L) = diag( φ( L) Φ ( L)). 5

7 differe esimaes of he equilibrium pass-hrough idicaig a slow adjusme o he moeary regimes. From hese resuls ad he evidece preseed i ECB (9) idicaig o evidece for a srucural chage i he IRPT mechaism durig he rece period of he fiacial crisis isead of relyig of hese paricular choices for explaiig he possible variabiliy of he magiude of he log-ru relaioship bewee he reail ad he mare ieres raes we cosider he more flexible ad geeral approach based o he assumpio of ime-varyig parameers i he coiegraig regressio model modelled as a weighed average of Chebyshev ime polyomials wih deermiisic weighs followig he proposal by Bieres (997). For a formal descripio of his approach ad some impora resuls arisig from fiig a ime-varyig coiegraig regressio model via Chebyshev ime polyomials see Appedix A. Nex we describe he daa used i he empirical aalysis ad he srucure of he ecoomeric sudy. 3.. The daa ad some iiial basic resuls Followig Bele e.al. (3) for he harmoized reail raes daa we use he harmoized ieres rae series from he Moeary Fiacial Isiuios (MFI) ieres rae saisics of he Europea Ceral Ba (ECB) for he seve couries ad periods appearig i Table represeig he loges series for which complee daa are available. All daa refer o loas for households ad o-profi isiuios ad are mohly averages ad exclusively ew busiess. For he credi caegories we cosider credis for house purchase ad loas for cosumpio wih shor medium ad log mauriies (up o year over ad up o 5 years ad over 5 years respecively). Table. Mohly reail raes by coury Credis for house purchase Loas for cosumpio Coury Period Period Ausria Belgium Filad Frace Germay Ialy.3-. Spai.-. 8 As he moey mare rae for all he couries we cosider he EONIA because i seems o beer reflec he sace of he moeary policy. Figure shows mohly series of he EONIA ad he hree-moh Effecive Federal Fuds Raes (EFFR) as a proxy for he policy rae i he US ecoomy which ca be described as a mare-based sysem as opposed o he ba-based sysem for he Euro Area for he period jauary 999 o december. The ime pah of boh series closely resembles displayig a appare osaioary behavior bu wih a cerai ime delay i he respose of he EONIA raes o chages i he EFFR. Cross coemporaeous correlaio bewee boh series i firs differeces is.358 while cross auocorrelaios are.. ad.33 for lags -3 of he EFFR series. EONIA is he effecive overigh referece rae for he euro compued as a weighed average of all overigh usecured ledig rasacios i he ierba mare uderae i he EMU ad Europea Free Trade Associaio (EFTA) couries. 6

8 Figure. Mohly EONIA ad Effecive Federal Fuds Raes (.999-.) i levels ad i firs differeces EONIA FedFuds rae EONIA FedFuds rae Nex figures -8 shows he ime paer of he reail ieres raes for each of he seve couries for boh ypes of credi caegories ad he hree mauriies cosidered i he aalysis. Figure. Reail raes of credis for house purchase (lef) ad loas for cosumpio (righ): Ausria up o year over ad up o 5 years over 5 years up o year over ad up o 5 years over 5 years 7

9 Figure 3. Reail raes of credis for house purchase (lef) ad loas for cosumpio (righ): Belgium up o year over ad up o 5 years over 5 years up o year over ad up o 5 years over 5 years Figure. Reail raes of credis for house purchase (lef) ad loas for cosumpio (righ): Filad up o year over ad up o 5 years over 5 years up o year over ad up o 5 years over 5 years Figure 5. Reail raes of credis for house purchase (lef) ad loas for cosumpio (righ): Frace up o year over ad up o 5 years over 5 years up o year over ad up o 5 years over 5 years Figure 6. Reail raes of credis for house purchase (lef) ad loas for cosumpio (righ): Germay up o year over ad up o 5 years over 5 years up o year over ad up o 5 years over 5 years 8

10 Figure 7. Reail raes of credis for house purchase (lef) ad loas for cosumpio (righ): Ialy up o year over ad up o 5 years over 5 years up o year over ad up o 5 years over 5 years Figure 8. Reail raes of credis for house purchase (lef) ad loas for cosumpio (righ): Spai up o year over ad up o 5 years over 5 years up o year over ad up o 5 years over 5 years Simple visual ispecio of all hese series reveals a very differe behavior of he reail credi mares i each coury i he sample wih a cerai homogeeiy amog mauriies for each ecoomy ad credi caegory. Paricularly ieresig i he case of Spai where he series of ieres raes for house purchase ad shor-erm mauriy has experieced a wide growh from 3 while he shor-erm rae of loas for cosumpio displays a sharp fall a he ed of ad has remaied sice he bewee 6 ad 8% which is he highes value for he seve couries. These differe behaviours seems o aicipae he differeces ecouered i pracice i he esimaio of he magiude of he shor ad log-ru IRPT measures bu ca also serve as a jusificaio for he use of a flexible modellig such as he oe cosidered i his paper. The mai ool proposed o he aalysis of he mechaism of rasmissio of he moeary policy from he moey o he reail mares is a simple regressio model of he form y = α + β x + u (3.) ha falls io he class of coiegraig regressio models give he o saioary behavior of he series ivolved wih he depede variable y give by he reail rae for differe mauriies of he credi for house purchase ad loas for cosumpio as credi caegories ad explaaory variable x give by he EONIA ieres rae. The resuls of he aalysis of iegraio ad saioariy for all he series are o preseed here 5 bu srogly idicaes ha he variables are o saioary hus supporig he aalysis of he regressio model as a way of represeig he coiegraig relaioship amog he reail ieres raes ad he moey mare rae. Table E. i Appedix E preses he resuls of a variey of esig procedures for coiegraio roughly 5 The aalysis was performed based o he usual ADF ad PP es saisics for he ull of iegraio agais he aleraive of saioariy ad he KPSS es saisic for he hypohesis i reverse order. I all he cases he saioariy hypohesis is rejeced a he 5% level of sigificaio. 9

11 idicaig he o exisece of a sable log-ru relaioship i all he cases whe based o he ime-ivaria regressio model (3.). Oe possible explaaio for hese resuls could be aribued o he exisece of a ime-varyig sable relaioship omied i (3.) as is appare from he resuls of Hase s (99) ess for parameer isabiliy i coiegraio regressios wih iegraed regressors. 6 Appedix A coais he heoreical aalysis of he cosisecy of his esig procedure agais he aleraive of a ime-varyig coiegraig regressio where he paer of chages i he parameers is modelled via Chebyshev ime polyomials. This is he approach ae i he res of his secio. 3.. Time-varyig coiegraig regressio aalysis Wih he aim o explore he capabiliy of he approach proposed by Bieres (997) ad exeded by Bieres ad Maris (9 ) ad Neo ( ) o he coiegraio aalysis we propose he followig geeralizaio of equaio (3.) as y = α ( m) + β ( m) x + u (3.) where he ime-varyig iercep ad slope are defied as ad m α ( m) = a G ( ) (3.3) j j j= m β ( m) = b G ( ) (3.) j j j= respecively wih G ( ) = G j ( ) = cos( jπ(.5)/ ) j =... m m. This geeral specificaio allows o obai hree aleraive models give by ad Model. No iercep ad TV slope a j = j =... m Model. Fixed iercep ad TV slope a j = j =... m Model 3. TV iercep ad slope. This model does o allow o capure i geeral srucural chages i he coiegraig relaioship sice he fucios α ( m) ad β ( m) are assumed o be smooh ad slow ime-varyig deermiisic fucios of ime. However here exiss he possibiliy of easily combie he proposed formulaio wih a srucural brea uless he magiude shifs be small eough o be subsumed by he ime-varyig srucure of he model parameers as show i he aalysis of Appedix B. The aalysis performed i his secio based o (3.)-(3.3) requires he OLS esimaio of he coefficies a j b j j =... m for a paricular choice of m < ad he compuaio of he es saisics ad ˆ K ( ) ˆ m = u ( m) ˆ ( q ) = = j ωu j (3.5) ˆ CS ( m) = max uˆ ( m) ω ˆ ( q ) (3.6) =... j u j= 6 Quios ad Phillips (993) also propose a umber of relaed procedures o es he ull hypohesis of ime-ivaria coiegraio agais specific direcios of deparures from he ull icludig he possibiliy o es he sabiliy of a subse of coefficies. For more resuls relaed o esig for parial parameer isabiliy i coiegraig regressios see also Kuo (998) ad Hsu (8).

12 where u wih ω ˆ ( q ) = σ ˆ + λ ˆ ( q ) is a erel-ype esimaor of he log-ru variace of u u u σ ˆ = uˆ ( m) ad u = λ ˆ ( q ) = w( h/ q ) uˆ ( m) uˆ ( m) for u h= = h+ h / some weighig fucio w( ) ad badwidh q = o( ) based o he auocovariaces of he residuals m ˆ ˆ ˆ u ( m) = u (( a j a j )( bj bj )) G j ( ) x j= The saisic (3.5) is he so-called KPSS es saisic for esig he ull hypohesis of saioariy for he regressio error erm u ad hece coiegraio while ha CSˆ ( m ) i (3.6) is he Xiao ad Phillips () es saisic adaped o he residuals from (3.) as has bee cosidered by Neo (). I he case of edogeeous regressors he OLS versio of hese es saisics cao be used i pracical applicaios give ha heir limiig ull disribuios deped o some uisace parameers ad hece mus be compued o he basis of residuals from some asympoically efficie esimaio such as he FM-OLS mehod (see Neo () ad Appedix B). Table D. i Appedix D coais he fiie-sample upper criical values for Models -3 wih oe iegraed regressor ad m = 5 hus geeralizig he resuls i Neo (). The limiig ull disribuio of hese es saisics is model-depede i he sese ha he criical values differ for each model ad value of m. To avoid his depedece o he model specificaio ad dimesio whe esig for ime-varyig coiegraio we also propose he use of he es saisics proposed by McCabe e.al. (6) (MLH) described i Appedix C which also have he advaages of relyig oly o he OLS esimaio of he ime-varyig coiegraig regressio (3.) eve uder edogeeiy of he iegraed regressors. The resuls of hese esig procedures preseed i Table E. ad E.3 i Appedix E are mixed boh for each coury ad for he differe mauriies of he wo ypes of credi caegories aalyzed i erms of he sabiliy of he log-ru relaioship bewee he reail ad he mare raes wih differe coclusios depedig o he order of approximaio of he ime-evolvig parameers give by m. However whe based o he resuls of he MLH ess he overall coclusio is ha of saioary ime-varyig coiegraio for almos all he cases paricularly whe focus o he resuls of he saisic labelled MLH (see equaio (C.6) i Appedix C) ad for moderae values of m ragig from -. Fially based o hese resuls Appedix F shows he esimaed values of he log-ru IRPT for he series of each coury ad for Models ad 3 wih values of m ragig from o. From hese esimaes we cao coclude a clear evidece o he degree of adjusme of he reail ad moey mares for he series ad models cosidered alhough here is some idicaio ha he pass-hrough is icomplee. Refereces Bele A. J. Becma F. Verheye (3). Ieres rae pass-hrough i he EMU New evidece from oliear coiegraio echiques for fully harmoized daa. Joural of Ieraioal Moey ad Fiace 37() -. Bieres H.J. (997). Tesig he ui roo wih hypohesis agais oliear red saioariy wih a applicaio o he US price level ad ieres rae. Joural of Ecoomerics 8() 9-6. Bieres H.J. L.F. Maris (9). Appedix: Time varyig coiegraio. hp://eco.la.psu.edu/ hbieres/tvcoint_appendix.pdf. Bieres H.J. L.F. Maris (). Time-varyig coiegraio. Ecoomeric Theory 6(5) ECB (9). Rece developmes i he reail ba ieres rae pass-hrough i he euro area. ECB Mohly Bullei augus 93-5.

13 Ellio G. M. Jasso E. Pesaveo (5). Opimal power for esig poeial coiegraig vecors wih ow parameers for osaioariy. Joural of Busiess ad Ecoomic Saisics 3() 3-8. Hase B.E. (99). Tess for parameer isabiliy i regressios wih I() processes. Joural of Busiess ad Ecoomic Saisics (3) Hsu C.C. (8). A oe o ess for parial parameer sabiliy i he coiegraed sysem. Ecoomics Leers 99(3) Kuo B.S. (998). Tes for parial parameer isabiliy i regressios wih I() processes. Joural of Ecoomerics 86() Kwapil C. J. Scharler (). Ieres rae pass-hrough moeary policy rules ad macroecoomic sabiliy. Joural of Ieraioal Moey ad Fiace 9() Maroa G. (9). Srucural breas i he ledig ieres rae pass-hrough ad he euro. Ecoomic Modellig 6() 9-5. McCabe B. S. Leyboure D. Harris (6). A residual-based es for sochasic coiegraio. Ecoomeric Theory (3) Neo D. (). Tesig ad esimaig ime-varyig elasiciies of Swiss gasolie demad. Eergy Ecoomics 3(6) Neo D. (). The FMLS-based CUSUM saisic for esig he ull of smooh ime-varyig coiegraio i he presece of a srucural brea. Ecoomics Leers 5() 8-. Phillips P.C.B. B.E. Hase (99). Saisical iferece i isrumeal variables regressio wih I() processes. The Review of Ecoomic Sudies 57() Phillips P.C.B. S. Ouliaris (99). Asympoic properies of residual based ess for coiegraio. Ecoomerica 58() Quios C.E. P.C.B. Phillips (993). Parameer cosacy i coiegraig regressios. Empirical Ecoomics 8() Shi Y. (99). A residual-based es of he ull of coiegraio agais he aleraive of o coiegraio. Ecoomeric Theory () 9-5. Xiao Z. P.C.B. Phillips (). A CUSUM es for coiegraio usig regressio residuals. Joural of Ecoomerics 8() 3-6. Appedix A. Hase s ess for parameer isabiliy uder ime-varyig coiegraig regressio via Chebyshev ime polyomials Le us assume ha he ime-varyig coiegraig vecor β = ( β... β ) i he specificaio of he ime-varyig (TV) coiegraig regressio model y = β x + u wih x = ε is give by where = j j j= β α G ( ) (A.) α β by he orhogoaliy propery of Chebyshev j = = β G j ( ) polyomials G ( ) i.e. j = G j Gi I i j ( ) ( ) = ( = ) wih G j ( ) = j = jπ(.5) (A.) = cos j =... which implies ha = G j ( ) = for ay j =... Also give ha β ca be wrie as β = β ( m) + b ( m) wih m = j j j= β ( m) α G ( ) (A.3) for some fixed aural umber m < ad he fac ha for he remaiig erm ( ) b m = α G ( ) we have ha lim b ( m) b ( m) = ad j= m+ j j q = m m m + m = lim b ( ) b ( ) ( ) for q ad m (see Lemma i Bieres ad Maris ()) he our TV coiegraig regressio model is give by

14 m x y = β ( m) x + u = α x G ( ) + u = ( α A ) + u (A.) j j m X j= m wih Am = ( α... α m ) Xm = ( x... x m ) ad xj = x G j ( ) j =... m. The ecessary ools required for he asympoic aalysis of he esimaio resuls arisig from his specificaio are provided by Bieres (997) (see Lemmas A.-A.5) ad Bieres ad Maris (9 ). Thus uder he assumpio ha he regressio error erm u is give by u = α u + υ wih α ad υ a zero-mea wealy saioary error sequece wih fiie variace ad defiig he parial sum process of u as U ( ) r = whe ad [ r ] ad U ( r) = u for r r [ ] = [ ] he we ge [ r] / r r u u u (A.5) = F( / ) u F( s) db ( s) = F() B () f ( s) B ( s) ds [ r ] [ r ] 3/ / r = u = = (A.6) F( / ) U ( / ) F( / )( U ( / )) F( s) B ( s) ds / where Bu ( r ) is a Browia moio process characerizig he wea limi of U ( ) r for ay differeiable real fucio o [] F( ) wih derivaive f( ). Also give ha / / [ r] x [ r ] = x + B ( r) wih ( r) B = j= ε j for r [ ] ad B ( r ) = for r [ ] he we ge ha = j x x ad = j j x x are boh O (). Taig ow β = α he TV coiegraig regressio ca be rewrie as y = β x + u + α x G ( ) = β x + v (A.7) j j j= where v = u + X m A m where he OLS esimaio error of β is give by wih β ˆ (/ +κ) ( κ) β = x x x v = = (/ +κ) ( κ) = x x x u + X m Am = = x = x / ( ) / X = X = ( x... x ) ad x = x G ( ) j = m m m j j... m. Give ha xj = G j ( ) ε + ( G j ( ) G j ( )) x ( ) i comes ha he variace of x j is o cosa sice i depeds o bu as G + h G = hjπ H + O wih H ( ) = si( jπ(.5)/ ) for j ( ) j ( ) ( / ) j ( ) ( ) each j =... he secod erm becomes asympoically egligible ad hece Var[ x ] G ( ) E [ ε ε ] as ha depeds o oly hrough he Chebyshev j j polyomial. The scaled OLS esimaio error is he give by / +κ ˆ ( κ ) / +κ ( β β ) = Q u + m m x x X A (A.9) = = where Q = = x x wih he idex κ aig he values κ = / uder coiegraio ha is whe he error erm u is saioary ad κ = / uder o coiegraio so ha he secod erm bewee braces will domiaes he behavior of ( β ˆ β ) uder coiegraio whe Am m. From his resul he -h OLS j p (A.8) 3

15 residual is give by κ vˆ = v x / +κ [ ( β ˆ β / )] = uɶ + d A (A.) m m where he wo erms composig hese residuals are ad κ ( κ) = x Q x j j j= ɶ (A.) u u u m = m m j j j= d X X x Q x (A.) Hase (99) proposed a se of saisics o es for parameer isabiliy i regressio models wih iegraed regressors ha are based o differe measures of excessive variabiliy of he parial sums of he esimaed scores from he model fiig. Firs we cosider he case of he OLS esimaed scores sˆ ˆ = x v ha ca be decomposed as / s = sɶ + x d A (A.3) ˆ m m where sɶ = x uɶ = O () uder coiegraio so ha ˆ S ˆ = j= s j ca be wrie as p S ˆ / = s ɶ j + x d j m j A m j= j= (A.) = ( Vɶ + V ( m) Am ) which implies ha S ˆ ( ) = V ɶ = Op whe Am = m where S ˆ has a well defied wea limi. Uder he geeral assumpio ha he model parameers β follow a marigale process β = β + η wih E [ η η ] = δ G ad G = ωv. M where M = = x x ad ω v. = ω v ω v Ω ω v is he codiioal log-ru variace of v give he sequece of error erms drivig he iegraed regressors ε he he OLS versio of he LM-ype es saisic is give by ˆ ˆ ˆ ˆ L ˆ = ( ) ( ) ˆ S M S = ω ωˆ S Q S (A.5) wih ˆ v v = v = ω a cosise esimaor of ω v uder parameer sabiliy ad sric exogeeiy of he iegraed regressors i.e. δ = ad ω v = respecively usually give by a erel-ype esimaor based o he sequece of sample serial covariaces of he OLS residuals such as ω ˆ = vˆ + w( h/ q ) vˆ vˆ wih w( ) he erel (weighig) fucio ad badwidh order h ca be wrie as ˆ v v = h= = h+ h q o ˆ ˆ h = ɶɶ h = h+ = h+ / + A m dm u h + dm ( h) u = h+ v v u u + A d d A he ω ca be decomposed as m m m ( h) m = h+ / = ( ). Give ha he residual covariace of ( ɶ ɶ ) (A.6)

16 wih / / ω ˆ ˆ v = ω u + A m dm uɶ + w( h/ q) ( dm uɶ h + dm ( h) uɶ ) = h= = h+ + A m dm d m + w( h/ q) d m dm ( h) Am = h= = h+ u = h= = h+ h (A.7) ω ˆ = uɶ + w( h/ q ) uɶɶ u a cosise esimaor of he log- p ru variace of u uder coiegraio give ha h uɶɶ = + u h E[ uu h ]. The secod erm o he righ had side is also O p () uder he assumpio of coiegraio while ha for he las erm o he righ had side bewee braces we have q dm d m + w ( h / q ) + op () = Op ( q ) = q (A.8) h= so ha ω ˆ = O ( q ) uder ime-varyig coiegraio of he ype cosidered. v p Oherwise uder ime-ivaria coiegraio (i.e. whe Am = m ) he es saisic is give by Lˆ = ω ˆ Vɶ Q Vɶ (A.9) u = / [ r ] r where V ɶ = s ɶ V ( r) = B ( s) dv ( s) + ( ri Q ( r) Q ()) as [ r ] = u u wih V ( r ) he wea limi of u uɶ uder saioariy give by V ( r ) = / [ r ] = r r Bu ( r) v ( r) Q () B ( s) dbu ( s) wih v ( r) = B ( s) ds Q ( r) = B ( s) B ( s) ds for < r ad u = h= E[ ε hu ] he oe-sided log-ru covariace bewee pas p values of ε ad u. Also aig io accou ha ωˆ u ω u = h= E[ u hu ] he ˆ ( ) ( ) L q ωu V r Q () V ( r ). This limi disribuio cao be used i pracice i he geeral case due o he presece of he measure of wea edogeeiy of he regressors hrough ad he fac ha E[ B ( s) V ( s)] uder edogeeiy of he u u r iegraed regressors which implies ha he compoe B ( s) dvu ( s) has o a mixed Gaussia disribuio. However despie his resul give ha uder ime-varyig coiegraio we have ha he umeraor of he es saisic ca be wrie as ˆ ˆ ( S ) Q ( S ) = Vɶ Q V ɶ = = + V Q V ( m) Am = + A m V ( m) Q V ( m) Am = Op = ɶ (A.) so ha i is domiaed for he las erm we ge ha L ˆ ( q ) = O ( / q ) ad hece i will p diverge a he give rae i he case of a smooh ime-varyig coiegraio relaio as described by he represeaio based o Chebyshev ime polyomials. I order o circumve he problems associaed wih he use of he OLS versio of he es saisic uder he ull of ime-ivaria coiegraio wih edogeeous regressors Hase (99) propose a modified versio based o a asympoically efficie esimaor such as he Fully-Modified OLS (FM-OLS) esimaor by Phillips ad Hase (99). From ( ) u 5

17 he compuaio of he eleme ˆ γ ˆ ˆ v = Ω ω v wih Ω ˆ ˆ ˆ = + Λ ad ω = ˆ + Λ ˆ cosise erel-ype esimaors of he log-ru covariace marix ˆ v v v of = x ε ad log-ru covariace vecor of ε ad v respecively wih compoes ˆ ˆ = = x x + Λ ˆ Λ = w( h/ q ) x x v h= = h+ h h= = h+ h ˆ = w( h/ q ) x vˆ ad ˆ Λ = w( h/ q ) x vˆ FM-OLS esimaor of v h= = h+ h he β is ˆ β + = ( x x ) ( x y + ˆ + ) where = = v + y ˆ = y γ v x ad ˆ + ˆ ˆ ˆ v = v γ v. I he case of parameer isabiliy of he ype cosidered he we have ha y + = x β + v + wih / ˆ v + = v γ ˆ x = u + + ( X x Ω Ω ˆ ) A v m ( m) m + where u ˆ = u x γ u ˆ ω ˆ ˆ v = ω u + Ω ( m) Am ad ˆ γ ˆ ˆ ˆ v = γ u + Ω Ω ( m) Am / wih ˆ ˆ Ω ( m) = ( m) + h= w( h/ q) = h+ x d m ( h) ad ˆ ( m ) = + x d. Also aig io accou ha is decomposed as ˆ + ˆ + ˆ + v = u + ( m) Am wih ˆ + ˆ ˆ ˆ u = u γ u ad ˆ + ˆ ( m) = ( m) ˆ ˆ Ω Ω ˆ he he scaled FM-OLS esimaio error is give by / h w( h/ q) = = h+ h m ( m) ˆ v / +κ ˆ + ( κ ) + (/ κ ) ( ˆ + β β ) = x x x v = = β β v (A.) where he las erm bewee parehesis ca be expressed as ( κ ) + (/ κ ) ˆ + ( κ ) + (/ κ ) ˆ + x v = x u = = / +κ x Xm = v u + / ˆ ˆ ˆ + x x Ω Ω ( m) ( m) Am = (A.) which implies cosise esimaio uder coiegraio uder parameer sabiliy i.e. Am = m. If we defie he FM-OLS esimaed scores as sˆ + ˆ ˆ = x v + + v so ha + sˆ = he FM-OLS versio of he es saisic ˆ+ L ( ˆ ) ˆ + ˆ + = ω S M S = v. = wih ˆ ω ˆ ˆ ˆ ˆ v. = ωv ω u Ω ω u ad ˆ + + S ˆ = j= s will provide similar resuls o wha obaied whe usig he OLS esimaes ad residuals. Also similar cosisecy resuls are obaied for he sup-f ad mea-f ess based o ˆ ˆ F ˆ = =... ωˆ S V S v wih V = M M M M where he sup-f es is give by SFˆ ( ) max τ τ = F = [ τ ]...[ τ ] ˆ < τ < τ < ad allows o es for a sigle srucural chage a a uow brea poi while ha he mea-f es which is defied as 6

18 [ τ ] ˆ τ τ = F [ τ ] [ τ ] + = [ τ ] MFˆ ( ) is also desiged o es agais a marigale mechaism guidig he variabiliy of he regressio coefficies. Appedix B. OLS esimaio of a ime-varyig coiegraig regressio model via Chebyshev polyomials uder a srucural brea i he coiegraig vecor Le us assume ha he ime-varyig coiegraig regressio model is specified as x y = β ( m) x + u = ( α A m) + v (B.) Xm where β = β ( m) is as i (A.3) so ha he OLS esimaio error of ( α A m) is ˆ α α (/ +κ) ( κ) ˆ = X ( m) X ( m) X ( m) v (B.) Am Am = = wih X ( m ) = ( x X m ). However he rue mechaism drivig he ime-varyig coiegraig vecor is give by a permae ad abrup chage a he brea poi of he sample γ = [ τ ] such as β = α + λ H ( γ) wih H ( γ ) = I ( > γ ) ad brea fracio τ () where λ = ( λ... λ ) is he -vecor coaiig he shif magiudes. Uder his assumpio he correc specificaio of he coiegraig regressio is as follows y = α x + λ x H ( γ ) + u (B.3) which implies ha he regressio error erm v i (B.) ad (B.) is give by v = u + λ x H ( γ ) A X (B.) m m so ha he las erm i he righ-had side of (B.) ca be decomposed as where ( κ) ( κ) X ( ) = X ( ) = = / +κ + X m x H γ λ Xm Am = m v m u ( )( ( ) ) X ( ) x ( γ ) = X ( ) x = = [ τ ] + I I = X ( m) X ( m) = ( Q( m) Q[ τ ]( m)) = [ τ ] + m m m H m ad m m m m (B.7) where Q X X ad ( ) Q wih τ =. Thus m m X ( ) X m = X ( ) X ( ) = Q( ) I = = m m Im m [ ] [ ]( ) τ τ m = ( m) = ( m) Q m is ( ) [ τ ] m equaio (B.) ca be rewrie as α ˆ / α +κ / +κ λ ( κ) ˆ = + Q ( m) ( m) u X Am Am Am = / +κ λ Q ( m) Q[ τ ]( m) m which gives (B.5) (B.6) 7

19 α ˆ / / ( ) α + λ +κ +κ ( κ) λ ˆ = Q ( m) X ( m) u Q[ τ ]( m) Am = m Uder he assumpio of coiegraio wih he idex κ aig he value κ = / he u is give by u = ( αl) υ = c ( L) e = u eɶ wih u () = c e ad eɶ = cɶ ( ) L e where he lag polyomial c ( L) is j c ( L) = ( α L) d ( L) = c L wih j= j j cɶ ( L) = j= cɶ jl ad cɶ j = i = j + c i. The o obai he limi for each compoe of he m-vecor / = x u / / u ( m) u = x X = = / = m u x we have ha / / xj = j ( ) x = = / = G j ( ) x u + G j ( + ) ε eɶ + = = + ( G j ( + ) G j ( )) x eɶ = / + ( G j ()) x eɶ G j ( + )) x ( + ) eɶ ) u G u for each j = m wih / = j j u (B.8) (B.9) G ( ) x u B ( r) db ( r) where B ( r) = G ( r) B ( r) G ( ) r = ad G ( r) = cos( jπ r) j = m while ha for he j j secod erm i he righ-had side of (B.9) we have j j ( + ) ε ɶ + = [ ε ɶ + ] j ( ) = = / / + G j + ε eɶ + E ε + eɶ + O = / = E[ ε + eɶ ] I( j = ) + Op ( ) G e E e G p wih ε + u h= ε h ( )( [ ]) ( ) (B.) E[ ε eɶ ] = E[ ε u ]. Also give ha G ( + ) = ad G + = jπ H + O for j = m we obai j ( ) ( / ) j ( ) ( ) / ( j ( + ) j ( )) x ɶ = ( π/ ) j ( ) x ɶ = = / + O x eɶ = Op = G G e j H e ( ) ( ) so ha i is asympoically egligible. Taig ogeher hese resuls we obai where / ( m) ( m) ( ) ( ) u( ) + u = (B.) X m u m r db r (B.) m ( r) = ( B ( r) B ( r)) wih ( m) ( m) ( m) u u B ( r) = ( G ( r) B ( r)... G ( r) B ( r)) ad ( m) m = (... ). Fially from (B.8) uder he assumpio of a srucural chage 8

20 of decreasig magiude wih he sample size a he rae λ = O( ) whe κ = / (ha is uder coiegraio) he we have α ˆ ( ) α + λ ( κ) λ ˆ = Q ( m) X ( m) u Q[ τ ]( m) Am = m ( m) ( m) λ Q ( m) m ( r) dbu ( r) + u Q( τ m) m (B.3) r ( m) ( m) wih Q( r m) = m ( s) m ( s) ds so ha he limiig disribuio uder coiegraio of he parameers i he ime-varyig regressio model will coai as uisace parameers hose measurig he magiude of he shifs λ. If we iroduce he ( ) sroger codiio λ = O( +δ ) for some δ > he α ˆ ( ) α + λ ( κ) δ ˆ = Q ( m) X ( m) u + Op( ) (B.) Am = ad hece he abrup chage ca be capured by he smooh fucio of ime characerizig he specificaio of he esimaed regressio i (B.). Observe ha he same resul is obaied for he esimaor of β i he ime-ivaria specificaio of he coiegraig regressio y = β x + v alhough (B.) has he advaage of providig supercosise esimaes of he ime chagig parameers uder coiegraio. Appedix C. McCabe e.al. (MLH) (6) ess for sochasic coiegraio based o OLS residuals Le us assume ha he +-dimesioal vecor ζ = ( u ε ) follows a liear process such as ζ = C( L) e = j= C je j wih j= j C j < ad e = ( e e ) iid( Σ e). If we ow defie he augmeed +-dimesioal vecor ξ = ( u υ ε ) wih υ = u u σ uder he above assumpio we ge B ( r) ξ ( r) B ( r) Ω ( r) (C.) [ r] u / / ξ B ξ = υ = Ω ξ W ξ = B ( r) where B ξ ( r) is a +-vecor Browia process wih covariace marix Ω ξ give by ωu ωuυ ωu Ω = ω ξ υ ω υ (C.) Ω ad W ( r) = ( Wu ( r) Wυ ( r) W ( r)) ξ a +-vecor sadard Browia process. Taig he upper Cholesy decomposiio of Ω ξ Ω ω ω = ω ω Ω Ω ωuυ ω u Ω ω υ ωuυ ω u Ω ω υ / u. u. ω u Ω ωu. ωυ. ωu. ωυ. / / ξ υ. υ / (C.3) 9

21 wih ω ad u. u ad υ codiioal o ε give by ω υ. he log-ru variaces of ω υ = ωυ ω υ Ω ωυ = ωυ ρυ ρ = ω ω Ω ω υ υ υ υ ω u. = ωu ω u Ω ω u = ωu ( ρu ) ad. ω Ω ω ( ) wih ρ u = ω u ω u Ω ω u ad ω Ω ω he squared log-ru correlaio coefficies bewee hese error erms he he limiig process i (C.) ca be expressed as B. ( ).. ( ). ( ) u ( r) ωu Wu r ρ uυ + ωu Wυ r ρ uυ + ω u Ω B r B ( r) υ = ω υ. Wυ ( r) + ω υ Ω B ( r) (C.) / B ( r) ( r) Ω W where ρ = ω ω ω ω Ω ω. uυ. (/ u. υ. )( uυ ωu Ω ωυ) C.. The case of saioary coiegraio uder ime ivariace of he coiegraig vecor Taig Am = m i (A.) so ha he correcly specified coiegraig regressio has fixed coefficies he sequece of OLS residuals is give by v ˆ ˆ = uɶ = u κ x Θ / ( ) where ˆ +κ ( ˆ κ Θ = β β ) = Q x u. To es he ull hypohesis of j= j j saioary coiegraio agais o coiegraio McCabe e.al. (6) have proposed he es saisics ˆ / H ˆ ˆ v ( q) = vv s ω ˆ (C.5) ( q ) ad v = s + ˆ H ( q ) ( vˆ ˆ ) (C.6) ω υ 3/ = σv ˆ υ ( q) = wih ω ˆ v ( q) ad ω ˆ υ ( q) esimaes of ω u ad ω υ such as he erel-based heerosedasiciy ad auocorrelaio cosise (HAC) esimaors give by ω ˆ ˆ ˆ v ( q) = h= ( ) w( h/ q) = s + h + aa h where aˆ ˆ ˆ = vv s = s ad ˆ ˆ ω ˆ υ ( q ) = h= ( ) w( h/ q) = h + bb h wih bˆ ˆ ˆ = v σ v. Give ha he umeraor of Hˆ ( q ) i (C.5) ca be decomposed as v / / κ ˆ / s = s + Θ x ( s ) + x s = s + = s + = s + / κ ˆ + Θ ˆ x x ( s ) Θ = s + vˆ vˆ u u ( u u ) uder he coiegraio assumpio wih κ = / we ge / / / s = s + p = s + = s + (C.7) vˆ vˆ u u O ( ) / where E[ uu s ] as s wih s = O( ). Thus by Theorem i McCabe e.al. (6) ad uder he addiioal codiio o he badwidh parameer q = o( s) q < s ˆ ( ) d H () v q N. For he es saisic H ˆ ( ) υ q i (C.6) by defiig σ ˆ = vˆ ad v = σ = u he sequece of squared ad ceered OLS u = residuals vˆ σ ˆ is decomposed as v

22 vˆ σ ˆ = u σ + Θ ˆ ( x x Q ) Θ ˆ κ v u ˆ κ κ ( κ) Θ x u x ju j j= so ha he erm i he umeraor of H ˆ ( ) υ q is give by 3/ 3/ ( ˆ ˆ σ v ) = ( σu ) = = / κ ˆ + + Θ ˆ ( / ) x x Q = Θ v u / ˆ κ ( κ) + ( κ) Θ ( / ) x u x ju j = j= Agai uder he assumpio of saioary coiegraio wih κ = / we ge give ha 3/ 3/ / ( ˆ ˆ σ v ) = ( σ u ) + p ( ) = = / + / = u σu σu σ u + Op = / ( + ) / = ( u σ u ) + Op ( ) = / ( = u σ u ) + Op ( / ) = v u O / = u p ( ) [( ) ( )] ( ) (C.8) (/ ) ( u σ ) = O ( ). For he mai erm above we have / = ( / /)( u σ u ) = ( r /) dbυ ( r) where Bυ ( r) is he parial sum process of υ = u σ ha wealy coverges o he Browia moio process B ( r) u so ha uder coiegraio υ ( vˆ σˆ ) ( r /) db ( r) = N((/) ω ). 3/ d d = v υ υ Thus uder he same codiio as above for he badwidh parameer q we ge ˆ ( ) d H () υ q N. C.. The case of saioary coiegraio uder a ime-varyig coiegraig vecor via Chebyshev ime polyomials From he esimaio of he ime-varyig coiegraig regressio i (B.) wih m = m he rue or proper order of he Chebyshev polyomial approximaio he (B.8) is ow give by α ˆ / α +κ ( κ) ˆ = Q ( m) X ( m) u (C.9) Am Am = so ha he OLS residuals uˆ ˆ ( m) = y β ( m) x ca be wrie as ˆ κ α / α +κ uˆ ( m) = u ( x X m ) ˆ (C.) Am Am / which implies ha uˆ ( m) = u + O ( ) uder coiegraio ad hece all he above p resuls are verified. Also aig he parial sum of hese OLS residuals uder coiegraio i he ime-varyig seup we obai

23 which gives wih [ r ] [ r] [ r ] ˆ / / α α uˆ ( m) = u ( x X m ) ˆ = = = Am Am [ r] / ( m) ωu u = { } r ( m) ( m) ( m) u( ) m ( ) Q ( ) m ( ) u( ) u B r s ds m s db s + uˆ ( m) W ( r) (C.) (C.) r ( m) ( m) ( m) u u u = W ( r) W ( r) m ( s) dsq ( m) m ( s) dw ( s) (C.3) uder sric exogeeiy of he sochasic iegraed regressors.

24 Appedix D. Fiie sample upper criical values for KPPS-ype es by Shi (99) ad CUSUM-ype es by Xiao ad Phillips () uder ime-varyig coiegraio via Chebyshev polyomials of order m = 5. Table D.. Upper criical values for KPSS-ype ad Xiao-Phillips es saisics uder ime-varyig coiegraio for models -3 wih = sochasic iegraed regressor KPSS-TYPE XIAO-PHILLIPS = m = 3 5 m = 3 5 Model m = 3 5 m = 3 5 Model m = 3 5 m = 3 5 Model KPSS-TYPE XIAO-PHILLIPS = 5 m = 3 5 m = 3 5 Model m = 3 5 m = 3 5 Model m = 3 5 m = 3 5 Model KPSS-TYPE XIAO-PHILLIPS = m = 3 5 m = 3 5 Model m = 3 5 m = 3 5 Model m = 3 5 m = 3 5 Model Noe. Model idicaes he specificaio of he ime-varyig coiegraig regressio wihou iercep while ha Models ad 3 icorporae his compoe. Model assumes a fixed iercep ad a imevaryig (TV) slope coefficie while ha Model 3 idicaes ha boh he iercep ad slope parameers are TV ad give by a weighed sum of Chebyshev polyomials up o degree m =... 5 wih deermiisic weighs. 3

25 Appedix E. Table E.. Tess for coiegraio i regressios wih a cosa erm Coury Phillips-Ouliaris Shi s es Hase s Lc es Ausria Mauriy Z Z OLS DOLS FMOLS OLS FMOLS Credis for house purchase shor-erm medium-erm log-erm Loas for cosumpio shor-erm medium-erm log-erm Phillips-Ouliaris Shi s es Hase s Lc es Belgium Mauriy Z Z OLS DOLS FMOLS OLS FMOLS Credis for house purchase shor-erm medium-erm log-erm Loas for cosumpio shor-erm medium-erm log-erm Phillips-Ouliaris Shi s es Hase s Lc es Filad Mauriy Z Z OLS DOLS FMOLS OLS FMOLS Credis for house purchase shor-erm medium-erm log-erm Loas for cosumpio shor-erm medium-erm log-erm Phillips-Ouliaris Shi s es Hase s Lc es Frace Mauriy Z Z OLS DOLS FMOLS OLS FMOLS Credis for house purchase shor-erm medium-erm log-erm Loas for cosumpio shor-erm medium-erm log-erm Phillips-Ouliaris Shi s es Hase s Lc es Germay Mauriy Z Z OLS DOLS FMOLS OLS FMOLS Credis for house purchase shor-erm medium-erm log-erm Loas for cosumpio shor-erm medium-erm log-erm Phillips-Ouliaris Shi s es Hase s Lc es Ialy Mauriy Z Z OLS DOLS FMOLS OLS FMOLS Credis for house purchase shor-erm medium-erm log-erm Loas for cosumpio shor-erm medium-erm log-erm Phillips-Ouliaris Shi s es Hase s Lc es Spai Mauriy Z Z OLS DOLS FMOLS OLS FMOLS Credis for house purchase shor-erm medium-erm log-erm Loas for cosumpio shor-erm medium-erm log-erm Noes. Asympoic criical values for Z (ormalized esimaio error) ad Z (pseudo-t raio) es saisics by Phillips ad Ouliaris (99)are give by 7.39(%).935(5%) 8.38(%) ad 3.657(%) 3.365(5%) 3.968(%) respecively. For he Shi s es he criical values are.3(%).3(5%) ad.533(%) while ha for Hase s es for sabiliy of he coiegraio relaioship are.5(%).575(%) ad.898(%).

26 Table E.. Tess for coiegraio i ime-varyig regressios wih a cosa erm Coury Credis for house purchase Loas for cosumpio Ausria Model. Fixed iercep TV slope Model 3. TV iercep ad slope Model. Fixed iercep TV slope Model 3. TV iercep ad slope Mauriy TV-KPSS TV-XP HQC TV-KPSS TV-XP HQC TV-KPSS TV-XP HQC TV-KPSS TV-XP HQC shor m = medium m = log m = Noe. HQC is he esimaed value of he Haa-Qui iformaio crierio defied by HQC(m) = log(ssr(m)/) + (+)(m+)log(log()) wih =. The criical values for esig he ull of ime-varyig coiegraio for each model ad es saisic (TV-KPSS ad TV-XP) are give i Table D. 5