DOCUMENTOS DE ECONOMÍA Y FINANZAS INTERNACIONALES. Working Papers on International Economics and Finance

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1 DOCUMENTOS DE ECONOMÍA Y FINANZAS INTERNACIONALES Workig Papers o Ieraioal Ecoomics ad Fiace DEFI 8- Julio 8 Time-varyig coiegraig regressio aalysis wih a applicaio o he log-ru ieres rae pass-hrough i he Euro Area Afoso-Rodríguez, Julio A. Saaa-Gallego, María Asociació Española de Ecoomía y Fiazas Ieracioales ISSN:

2 Time-varyig coiegraig regressio aalysis wih a applicaio o he log-ru ieres rae pass-hrough i he Euro Area Afoso-Rodríguez, Julio A. Deparme of Applied Ecoomics ad Quaiaive Mehods Uiversiy of La Lagua Camio La Horera, s/. Campus de Guajara, C.P. 387 La Lagua, Teerife, Caary Islads Teléfoo: 9374, jafosor@ull.es Saaa-Gallego, María Deparme of Applied Ecoomics Uiversiy of he Balearic Islads Edificio Gaspar Melchor de Jovellaos, Campus UIB Cra. Valldemossa, km 7.5, C.P. 7 Palma (Illes Balears) Teléfoo: , maria.saaa@uib.es Absrac This paper sudy he mechaism of rasmissio bewee he moey ad he reail credi markes 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 (Europea Moeary Uio). This mechaism, kow 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 works, he aalysis is performed o he basis o a ime-ivaria log-ru 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 markes 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 pass-hrough. Keywords ad phrases: reail ieres raes, moeary policy, coiegraio aalysis, srucural isabiliy, ime-varyig coiegraio JEL classificaio: E5, F36, C

3 . Iroducio Moeary rasmissio is a key 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 marke ieres raes, are rasferred o he bak 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 ake as a example of a bak-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 framework of aalysis of a o-liear model for log-ru relaioship allowig for a ime-varyig relaioship bewee he moey marke ad he reail ieres raes for a se of EMU couries seleced by he crierio of havig he loges available series. The srucure of he paper is as follows: Secio briefly reviews he moivaio ad relevace of IRPT ad some empirical fidigs i his lieraure, while ha i secio 3 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 rak ime-varyig error-correcio model (ECM), ad discuss some of heir mai feaures. Secio 4 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 (, 4), 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. Fially, some heoreical developmes are preseed i Appedixes A o C, while he mai empirical resuls are preseed i Appedixes D o F.. Moeary rasmissio pass-hrough Sice he origi of he Ecoomic ad Moeary Uio i Europe (EMU), a large umber of empirical sudies have ried o fid evidece of possible differeces i he impac of moeary policy chages o oupu ad iflaio raes amog See, e.g., Kwapil ad Scharler () ad he refereces cied o earlier empirical sudies for EMU couries ad differe periods of ime.

4 Europea couries. Also, give ha he mechaism of seig policy raes ca be viewed as he sadard ool of moeary policy, he implemeaio of he moeary policy hrough ope marke operaios ries o esure ha policy raes are rasmied o he ieres raes a which fiacial isiuios refiace. The characerisics of he process of rasmissio of moeary policy rules, from chages i he policy raes o reail bak raes, has cosequeces o he sabiliy properies of some heoreical models icorporaig shor ad log-ru relaioship bewee moeary policy arges o ieres rae o bods ad (expeced) iflaio raes ad oupu (see, e.g., Kwapil ad Scharler (), ad Kobayashi (8)). A he reail level, may of such sudies have bee coduced i a aemp o esimae he degree of ieres rae pass-hrough (IRPT) i he EMU sysem. I his lieraure, he erm IRPT geerally has wo meaigs: loa rae pass-hrough ad deposi rae pass-hrough. Bak decisios regardig he paids o heir asses ad liabiliies have a impac o he expediure ad ivesme behaviour of deposi holders ad borrowers ad hus o real ecoomy aciviy. However, he chael of rasmissio of policy raes o ledig ad deposi raes (IRPT) ca suffer from some ypes of failures give ha i is affeced by various facors, as could be he case i periods of low ecoomy aciviy where fiacial iermediaries may require higher compesaios for risk ad hece chages i he policy rae would oly parially be passed o o firms or households. Also, he ime ad degree of pass-hrough of official ad marke ieres raes o reail bak ieres raes codiio he effeciveess of moeary policy rasmissio ad hus may affec price sabiliy. Furhermore, price se by baks ifluece heir margis ad profiabiliy ad hece he solvecy of he bakig sysem ad hus fiacial sabiliy. However, o he basis of formal heoreical models of moeary rasmissio mechaisms wih sicky prices ad he evidece of loa rae sickiess i he shor ru, i.e., chages i shor-erm marke ieres raes are o immediaely fully refleced i reail bak ieres raes, have araced grea aeio i he EMU sysem because of heir sharp coras wih he US case. Also, de Bod, e.al. (5a) empirically suppor, hrough a Grager causaliy aalysis, he relevace of focusig o loas ieres raes ad marke ieres raes give he appare lack of relevace of bak deposi raes for reail ledig raes for a wide se of couries. The empirical lieraure o he rasmissio of moeary policy is profuse, paricularly i rece years, ad alhough may rece sudies o loa rae passhrough differ i erms of esimaio mehods, daa used ad periods aalyzed, here is a cerai amou of commo evidece abou a icomplee degree of shor-ru pass-hrough eve afer corollig for differeces i bak solvecy, credi risk ad he slope of he yield curve, while here is o geeral cosesus abou he degree of IRPT i he log-ru. Some iiial refereces are Berake ad Blider (99), ha ivesigae he respose of credi aggregaes o moeary policy shocks, while ha Coarelli ad Kourelis (994) ad Borio ad Friz (995) focus o he pass-hrough of policy raes o ledig raes. A review of some more rece sudies ca be fouded i, e.g., Sorese ad Werer (6), de Bod (5b), Kobayashi (8) ad Belke, e.al. (). Oe addiioal releva pheomeo ha could parially explai some of he empirical resuls relaed o differeces i he degree of pass-hrough for differe couries ad ime periods is he fragmeaio of he fiacial ad bak sysem i

5 he EMU couries. Give he evidece provided by he Syheic Idicaor of Fiacial Fragmeaio (BBVA Research) ad he observed icrease i ieres raes for ew credi o firms sice he ieraioal fiacial crisis, Ferádez de Lis, e.al. (5) have sudied he process of price formaio for firm credis i he eurozoe ad foud ha a subsaial compoe of hese prices is aribuable o he qualificaio of he coury s credi risk. Despie he ecoomic ad moeary iegraio process i he EMU sysem, each member s bakig srucure remais very specific. This heerogeeiy seems o be a key facor i explaiig he degree of moeary policy rasmissio i erms of IRPT across EMU couries, bu his could be also iflueced by grea differeces over couries i he relaive exposiio o he moeary ad fiacial isiuios i erms of deb accumulaio, as ca be see i Figure. for households. Figure.. Gross deb-o-icome raio of households EMU9 Spai Frace Irelad Germay Ialy The Neherlads A rough idicaor o he effeciveess of he moeary policy rasmissio is he ledig spread, i.e., he differece bewee ledig ad policy raes. However, as Illes ad Lombardi (3) idicae, he icrease i he ledig spreads does o cosiue sufficie evidece of a deerioraio of he chael of pass-hrough, give ha he ledig spread is expeced o vary over ime as a fucio of he busiess cycle ad some oher facors affecig he rasmissio mechaism. To obai some isigh of such facors, hese auhors cosider he decomposiio of he ledig spread io hree compoes, each represeig a differe aspec of ris amely, a measure of credi risk (based o he spread bewee ledig ad goverme bod ieres raes), a measure of risk o goverme bods (give by he spread bewee he yield of a oe-year goverme bod ad he overigh ierbak rae), ad he spread bewee he overigh ierbak rae ad he policy rae. For several euro area couries, he bulk of he ledig spread is explaied by he credi risk measure, which ca be aribued o he role of he overigh ierbak raes as arge raes for moeary policy where misaligmes ca sigal sresses i he ierbak moey marke, icludig ay credi or liquidiy risk ivolved i ledig o baks. 3

6 Figure.. -moh Euribor mohly raes (jauary -jauary 6) Figure. shows he evoluio of he Euribor ieres rae, a measure for marke ieres rae o morgages ha ca also be cosidered as a proxy for policy rae i he euro area. However, for our empirical aalysis we will use he Euro OverNigh Idex Average (EONIA), as a global ad harmoized idicaor of he moey marke rae i he euro area. The empirical evidece o differeces i he speed ad degree of pass-hrough i he eurozoe has bee more proouced durig ad before he rece global fiacial crisis (see, e.g., Blo ad Labodace (3), ad Hrisov, e.al. (4)). I Hrisov, e.al. (4), he esimaed decrease i he IRPT is relaed o a chage i he srucural parameers characerizig he ecoomies ad a subsaial icrease i he average size of srucural shocks, ad makig use of simulaios based o he DSGE model by Gerali, e.al. (), hese auhors fid ha a icrease i he fricios faced by bak sysem ca explai he decrease i he reail bak IRPT. Give he possible srucural isabiliy i he relaio bewee reail bak ieres raes ad policy raes, i he ex secio we propose o use a quie geeral ad flexible mehodology i he coex of sochasically redig variables, possibly coiegraed. 3. Ecoomeric aalysis We cosider he case where he osaioary observed (k+)-dimesioal ime series z = ( y, x ) =,...,, where x = ( x,,..., x ) k, is geeraed as z = z + ε, where ε = ( ε,, ε ) is a zero-mea (k+)-dimesioal weakly depede ad saioary error sequece, bu i is assumed ha i ca also be embedded io he followig geeral Error-Correcio Model (ECM) represeaio I his model, he spread bewee wholesale raes o loas ad o deposis is liked o he degree of leverage (wholesale loas-o-bak capial raio) hrough a coefficie measurig he cos payig by baks for divergece bewee capial-o-asses raio from is arge value. Uder he assumpio ha reail baks are moopolisic compeiors o boh he loa ad he deposi markes, he spread bewee reail raes o loas ad he policy rae is icreasig i he policy rae, ad proporioal o he wholesale spread deermied by he bak s capial posiio. 4

7 Φɶ ( L)[( L) I ( α ) λκ L] z = e (3.) k+ where ɶ m j Φ( L) = Φ ( L ) Mk+,, wih Φ( L) = Ik+ 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 k ), κ = (, β ) ad he k+ square marix M k+, ca be eiher he ideiy marix or, more geerally, be defied as a ime-varyig roaio marix of he form M k+, β = k I k (3.) hus preservig he equivalece of he roos of each lag polyomial, wih β he k- dimesioal sigle ime-varyig coiegraig vecor. For he las erm i (3.), e = ( e, e ), i is assumed o be a zero-mea iid sequece wih fiie covariace, + marix Σ = E [ e e ] >, ad E[ e δ ] < for some δ >. This is a modified versio e 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 kow ad ime-ivaria coiegraig vecor, β = β =,...,. Takig (3.), equaio (3.) ca also be rewrie as κ z = ( α ) κ λ κ z + ξ x λk (3.3) wih ξ = ( υ, ε ) = C( L) e, j 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 j= j j he 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) k( r) = = B Ω B W ξ ε wih Ωξ = C() Σ C() he log-ru covariace marix of ξ, ad Wξ( r) = ( Wυ ( r), W k( r)) a k+-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 (3.3) 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 ime-ivaria coiegraig vecor, i.e. κ = κ = (, β k ), 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 e 5

8 y = x + u β (3.4) wih a coiegraig error erm of he form u = α u + υ + α x β (3.5) ad x = x + ( α ) λ κ z + ε (3.6) k Uder ime-ivariace of he coiegraig relaio, equaio (3.5), 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 he righ-had side of equaio (3.5) 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 break poi as = + ( ) kh τ β β λ (3.7) where H ( τ ) ( [ ]) = I > τ wih τ (,) he sadard sep fucio, (b) a marigale process as i Hase (99), give by β = β + υ (3.8) wih υ a zero-mea error iid sequece wih fiie covariace marix E [ υ υ ] = Σ υ ad β k a k-vecor of fixed values, ad (c), assumig ha he coiegraig vecor varies i a smooh way, i ca be represeed as a liear combiaio of Chebishev ime polyomials (CTP) (see, e.g., Bieres ad Maris () exedig he earlier proposal by Bieres (997)), so ha i ca be wrie as m β = b G ( ) = b G ( ) + b G ( ) = β ( m) + ( m) (3.9) kj, j, kj, j, kj, j, j= j= j= m+ where G,( ) =, G j,( ) = cos( jπ(.5)/ ), for j =,,,, m <, ad he ime-ivaria vecor of coefficies of he liear combiaio, b kj,, are defied as bkj, = = G j, ( ) β, for j =,,,, wih b = β, ad = 6

9 b = β = i he case of cosacy of he coiegraig vecor, i.e. β kj, k = G j,( ) k = β for all =,,, by he properies of he CTPs (see Appedix A for more k deails). The pracical use of (3.9) requires he choice of he approximaio order, m, i such a way ha β,( m) is flexible eough o approximae he paer of k smooh variaio of β, implyig ha he remaiig erm ( m) could vaish asympoically. 3 Firs, uder he saioariy codiio o he regressio error erm give by α < he scaled parial sum of u ca be wrie as u u u α [ r] [ r] [ r] / / / / = ( [ r] ) υ + α + α x = = = β (3.) / where / j= β jx j is O p() i he cases (a) ad (c), while ha i is O ( ) p i he case (b), ad hece divergig wih he sample size. Specifically we obai [ r] [ r] [ τ] / / / β x = λ k x[ r] ε ε I( r > τ) = = = [ τ] / = λ k x + ε I( r > τ) = x I( r ), / = λk [ τ] > τ (3.) [ r] [ r] / / / x = x = p = = β υ O ( ) (3.) wih [ r] r x υ B k Bυ + k ε hυ = h= ( s) d ( s) r E[ ] ad j H H r [ r] m [ r] / 3/ / β x = π b kj, j, ( ) x j, ([ ]) x[ r] = j= = m [ r] 3/ + O( ) b kj, x j= = (3.3) 3 See Lemma i Bieres ad Maris (), where i is precisely iroduced he smoohess codiio o β esurig he qualiy of he approximaio by he fiie-order liear combiaio β ( m) = b G ( ) for some fixed aural umber m <. See also Theorem i Maris i m j= kj, j, (3) for a echical codiio of his ype whe his approach is used o esig for coefficie cosacy i a saioary uivariae AR() process. 7

10 respecively, where H,( r) = si( jπ( 5)/ ) i (3.3) for j =,, m. O he j oher had, uder o saioariy of he regressio error erm i (3.4) wih α =, we ge he represeaio / / / / = + υ j + jx j j= j= u u β (3.4) / / [ 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.)- (3.3). Thus, assumig he validiy of he ime-varyig ECM represeaio i (3.), 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 (3.6), he assumpio ha x are o muually coiegraed ad have roos ha are kow a priori o be equal o oe (i.e. x are k iegraed bu o-coiegraed regressors), correspods o he resricio λ = (which implies λ = ), ad hece (3.6) mus be replaced by he usual k k represeaio as a k-dimesioal iegraed process 4 x = x + ε (3.6 ) which also resuls i he case of o coiegraio, i.e. whe α = i (3.5), irrespecive of he value of λ k. A secod form of he model is he ECM represeaio (3.) 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 rak ECM of he form z = ( α ) Φ() λκ z + Λ ( L) z + e (3.5) by makig use of he BN decomposiio * j, j+ j+ * Φ( L) = Φ() + ( L) Φ ( L) where * m * j ( L) = j j L * m m j Φ = Φ ad Φj = i = j Φ, i wih he lag polyomial Λ( L) = j = Λ j, L ad ime-varyig coefficies Λ = Φ ( α ) λκ + Φ, uless κ = κ or, aleraively, α = irrespecive of he behaviour of κ. Uder he resricios cosidered we 4 By recursive subsiuios, equaio (4.6) ca also be wrie as xk, = x + j= ε k, j + ( α ) z κ z = u + κ z, wih λ k j= κ j j, where he las erm is decomposed as j = j j j= j j= j j 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 k, jεk, j ( β k, xk, β xk, ). I / he fixed parameer case ad uder coiegraio we ge xk, Bk ( r) + ( α ) λ k Bu ( r), wih = [r], r (,], ad B ( r) = ( α ) B ( r) as, while ha uder o coiegraio he weak limi u υ / is xk, B k ( r), implyig a differe behavior i each siuaio ha is ulikely i ay real aalysis. 8

11 have φ ( ) () ( ) () α Φ λ = α φk() where we have pariioed Φ ( L) afer he firs row ad colum, so ha x is weakly exogeeous for β if ad oly if Φ () is block upper riagular (i.e. φ () k = k), 5 i which case oly he firs row of equaio (3.5) icludes he κ z = u + κ z, which equals he usual error correcio erm u = κ z oly uder cosacy of he coiegraig vecor. Noe ha (3.5) 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 (3.5), 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 (3.4) as ( ) () ( L) α, z = α Φ λκ z + Λ z + e (3.6) where he error erm i (3.6) 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 likelihood raio es for ime-ivaria coiegraio from (3.5), 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 (3.5), he res of he paper ress o he aalysis of he ime-varyig coiegraig regressio model i (3.4) for a paricular choice of a smooh mechaism drivig he coefficies i β = ( β,,..., β ), =,...,, which allows o capure smooh chages i he model s parameers i a very simple way, ad o develop appropriae esig procedures for esig he ull hypohesis of coiegraio. 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 (3.4) give by y = α + x + u β (3.7) wih a possibly ime-varyig deermiisic red fucio. Wihou icorporaig ay paricular mechaism of ime variaio, ad uder o saioariy of he regressors, his model is closely relaed o he coiegraig regressio wih imevaryig coefficies proposed by Park ad Hah (999), where β = β k( / ) is a smooh fucio defied o [,] ad is depede upo he sample size,. If β k( ) 5 From his, he resricio λ k = k implies weak exogeeiy of he iegraed regressors i he paricular cases where Φ ( L ) = Ik + or, more geerally, Φ( L) = diag( φ( L), Φ kk ( L)). 9

12 is sufficiely smooh, he esimad Πm( βk) = ( β k( r),..., β k( rm)), wih rj [,], j =,, m, m, may be approximaed by a series of polyomial ad/or rigoomeric fucios o [,]. The auhors propose he rigoomeric pairs (cos( πj/ T),si( π j/ T)), j =,,, m, o obai his approximae represeaio ad formulae a echical codiio esurig a rae of covergece for sufficiely large m. 6 We follow he same idea bu propose he use of a se of differe approximaig fucios based o he Chebyshev ime polyomials. 4. 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 Belke e.al. (3), whe aalyzig aggregaed micro daa from may baks, 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 marke rae i he Euro Area, ad credi caegories wih various mauriies. Maroa (9) cosiders he possibiliy of allowig for muliple ukow srucural breaks i he coiegraig relaioship based o uharmoized reail raes for several EMU couries, ad foud 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 marke 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. 4.. The daa ad some iiial basic resuls Followig Belke 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 Bak (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 6 See Lemma, p. 668, i Park ad Hah (999).

13 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 Spai As he moey marke rae for all he couries we cosider he EONIA, 7 because i seems o beer reflec he sace of he moeary policy. Figure 4. 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 markebased sysem as opposed o he bak-based sysem for he Euro Area, for he period jauary 999 o december 4. 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.4,.4 ad.33 for lags -3 of he EFFR series. Figure 4.. Mohly EONIA ad Effecive Federal Fuds Raes ( ) i levels (lef) ad i firs differeces (righ) EONIA FedFuds rae EONIA FedFuds rae Nex, figures show he ime paer of he reail ieres raes for each of he seve couries for boh ypes of credi caegories ad he hree mauriies 7 EONIA is he effecive overigh referece rae for he euro compued as a weighed average of all overigh usecured ledig rasacios i he ierbak marke uderake i he EMU ad Europea Free Trade Associaio (EFTA) couries.

14 cosidered i he aalysis. Figure 4.. 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 Figure 4.3. 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 4.4. 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 4.5. 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

15 Figure 4.6. 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 Figure 4.7. 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 4.8. 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 markes 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 shorerm 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 logru 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 3

16 aalysis of he mechaism of rasmissio of he moeary policy from he moey o he reail markes is a simple regressio model of he form y = α +β x + u (4.) 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, 8 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 marke rae. Table E. i Appedix E preses he resuls of a variey of esig procedures for coiegraio, roughly idicaig he o exisece of a sable log-ru relaioship i all he cases whe based o he ime-ivaria regressio model (4.). Oe possible explaaio for hese resuls could be aribued o he exisece of a ime-varyig sable relaioship omied i (4.), as is appare from he resuls of Hase s (99) ess for parameer isabiliy i coiegraio regressios wih iegraed regressors. 9 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 ake i he res of his secio. 4.. 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 (, 4) o he coiegraio aalysis, we propose he followig geeralizaio of equaio (4.) as y = α ( m) + β ( m) x + u (4.) where he ime-varyig iercep ad slope are defied as m α ( m) = a G ( ) (4.3) j, j, j= ad 8 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 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). 4

17 m β ( m) = b G ( ) (4.4) 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 imevaryig srucure of he model parameers, as show i he aalysis of Appedix B. The aalysis performed i his secio based o (4.)-(4.3) requires he OLS esimaio of he coefficies aj,, b j, j =,,..., m for a paricular choice of m <, ad he compuaio of he es saisics ˆ ( ) K ˆ m = u( m) ˆ ( q ) = = j ωu, j (4.5) ad where of u, wih CSˆ ( m) = max uˆ ( m) ω ˆ ( q ) (4.6) =,..., j u, j= u, u, u, ω ˆ ( q ) = σ ˆ + λ ˆ ( q ) is a kerel-ype esimaor of he log-ru variace ˆ σ, ˆ ( ) u = = u m ad some weighig fucio w( ) ad badwidh auocovariaces of he residuals m ˆ ˆ uˆ( m) = u (( aj, aj, ),( bj, bj, )) G j, ( ) x j= λ ˆ ( q ) = w( h/ q ) uˆ ( m) uˆ ( m) for u, h= = h+ h q / = o( ), based o he The saisic (4.5) 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 (4.6) is he Xiao ad Phillips () es saisic adaped o he residuals from (4.) as has bee cosidered by Neo (4). 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 5

18 (4) 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 (4). 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 (4.), 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 marke 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 -4. Fially, based o hese resuls, Appedix F shows he esimaed values of he logru 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 markes for he series ad models cosidered, alhough here is some idicaio ha he pass-hrough is icomplee. 5. Summary ad coclusios I his paper we sudy a paricular represeaio for a coiegraig regressio model wih ime varyig coefficies, ad evaluae is empirical performace i he aalysis of he possible log-ru relaioship bewee he moey ad reail credi markes, i he mechaism kow as ieres rae pass-hrough, for a subse of couries of he EMU. For modellig he ime variaio i he model parameers, which are assumed o vary i a smooh way, we propose he use of a fiie umber of Chebyshev ime polyomials, as i Bieres ad Maris (), Neo (, 4), ad Maris (3), exedig he idea origially developed by Bieres (997). The echical appedixes of he paper prese a umber of ew heoreical resuls for his model which prove he validiy of some exisig procedures o esig agais he aleraive hypohesis of parameer isabiliy i coiegraio regressios (Appedix A), provides some useful simplificaios o impleme he FM-OLS esimaio (Appedix B), ad propose wo very simple o compue, modelfree ad asympoically ormal saisics for esig he ull hypohesis of saioary coiegraio i his ime-varyig coiegraig regressio model. There are some remaiig releva quesios ha we lef for fuure work i his framework. Amog ohers, he choice i pracice of he approximaio order m, ad he effecs of a wrog choice of his quaiy o he ew proposed ifereial procedures. For he empirical applicaio, we ca highligh he overall evidece of a sable log-ru ime-varyig relaioship bewee he marke ad reail credi 6

19 ieres raes cosidered i his sudy. However, he poi esimaes of he logru IRPT show evidece of icomplee pass-hrough bewee hese series. Refereces Belke, A., J. Beckma, 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(), -4. Berake, B.S., A.S. Blider (99). The federal fuds rae ad he chaels of moeary rasmissio. America Ecoomic Review, 8(4), 9-9. 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(), 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), Blo, C., F. Labodace (3). Bak ieres rae pass-hrough i he eurozoe: moeary policy rasmissio durig he boom ad sice he fiacial crash. Ecoomics Bullei, 33(), Borio, C.E.V., W. Friz (995). The respose of shor-erm bak ledig raes o policy raes: a cross-coury perspecive. BIS Workig Paper No.7. Coarelli, C., A. Kourelis (994). Fiacial srucure, bak ledig raes, ad he rasmissio mechaism of moeary policy. IMF Saff Papers, 4(4), de Bod, G.J. (5b). Ieres rae pass-hrough: empirical resuls for he euro area. Germa Ecoomic Review, 6(), de Bod, G.J., B. Mojo, N. Valla (5a). Term srucure ad he sluggishess of reail bak ieres raes i euro area couries. ECB Workig Paper No.58. ECB (9). Rece developmes i he reail bak ieres rae pass-hrough i he euro area. ECB Mohly Bullei, augus, Ellio, G., M. Jasso, E. Pesaveo (5). Opimal power for esig poeial coiegraig vecors wih kow parameers for osaioariy. Joural of Busiess ad Ecoomic Saisics, 3(), Ferádez de Lis, S., J.F. Izquierdo, A. Rubio (5). Deermiaes del ipo de ierés del crédio a empresas e la Eurozoa. BBVA Research, Workig Paper No.5/9. Gerali, A., S. Neri, L. Sessa, F.M. Sigorei (). Credi ad bakig i a DSGE model of he euro area. Joural of Moey, Credi ad Bakig, 4(6), 8-4. Hase, B.E. (99). Tess for parameer isabiliy i regressios wih I() processes. Joural of Busiess ad Ecoomic Saisics, (3), Hrisov, N., O. Hülsewig, T. Wollmershäuser (4). The ieres rae pass-hrough i he Euro area durig he global fiacial crisis. Joural of Bakig ad Fiace, 48, 4-9. Hsu, C.C. (8). A oe o ess for parial parameer sabiliy i he coiegraed sysem. Ecoomics Leers, 99(3), Illes, A., M. Lombardi (3). Ieres rae pass-hrough sice he fiacial crisis. BIS Quarerly Review, Sepember, Kobayashi, T. (8). Icomplee ieres rae pass-hrough ad opimal moeary policy. Ieraioal Joural of Ceral Bakig, 4(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 breaks i he ledig ieres rae pass-hrough ad he 7

20 euro. Ecoomic Modellig, 6(), 9-5. Maris, L.F. (3). Tesig for parameer cosacy usig Chebyshev ime polyomials. The Macheser School, 8(4), 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, 34(6), Neo, D. (4). The FMLS-based CUSUM saisic for esig he ull of smooh imevaryig coiegraio i he presece of a srucural break. Ecoomics Leers, 5(), 8-. Par J.Y., S.B. Hah (999). Coiegraig regressios wih ime varyig coefficies. Ecoomeric Theory, 5(5), 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(4), Shi, Y. (994). A residual-based es of he ull of coiegraio agais he aleraive of o coiegraio. Ecoomeric Theory, (), 9-5. Sorese, K., T. Werer (6). Bak ieres rae pass-hrough i he Euro Area: a cross coury compariso. ECB Workig Paper No.58. Xiao, Z., P.C.B. Phillips (). A CUSUM es for coiegraio usig regressio residuals. Joural of Ecoomerics, 8(),

21 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 = kj, j, j= k k β α G ( ) (A.) α kj, = = β G j,( ) by he orhogoaliy propery of Chebyshev polyomials G,( ), i.e., G ( ) G ( ) = I( i = j), wih j = j, i, G j, ( ) = j = jπ(.5) = 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 = kj, j, j= (A.) β ( m) α G ( ) (A.3) for some fixed aural umber m <, ad he fac ha for he remaiig erm ( m) b α G ( ) we have ha lim b ( m) b ( m) = ad = j = m + kj, 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 m x y = β ( m) x + u = α kj, x G j, ( ) + u = ( α, A km, ) + u (A.4) j= Xkm, wih Akm, = ( α,..., α km, ), Xkm, = ( x,..., x km, ), ad xkj, = 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 weakly saioary error sequece wih fiie variace, ad defiig he parial sum process of u as U( r ) = whe r [, [ r] ], ad U( r) = = u for r [,], he we ge ad [ r] r r / u u u (A.5) = F( / ) u F( s) db( s) = F() B() f( s) B( s) ds [ r] [ r] r 3/ / = u = = (A.6) F( / ) U ( / ) F( / )( U ( / )) F( s) B( s) ds where Bu( r ) is a Browia moio process characerizig he weak limi of / U ( ) r, for ay differeiable real fucio o [,], F( ), wih derivaive f( ). / / [ r] Also, give ha x [ r] = x + B( r), wih ( r) B = j= ε j for r [,], ad B, ( r ) = for r [, ], he we ge ha = x x ad k k kj, 9

22 = kj, kj, x x are boh O p(). Takig ow β = α, he TV coiegraig regressio ca be rewrie as y = β x + u + α x G ( ) = β x + v (A.7) kj, j, j= where v = u + X km, A km,, where he OLS esimaio error of β is give by βˆ (/ +κ) ( κ) βk = x x x v = = (/ +κ) ( κ) x x x X km, Akm, = = = ( u + ) / wih x = / x, X = X = ( x,..., x ), ad x, = x, G,( ), j =, km, km, km, kj k j..., m. Give ha xkj, = G j, ( ) ε + ( G j, ( ) G j, ( )) x ( ), i comes ha he variace of x kj, is o cosa sice i depeds o, bu as j, ( ) j, ( ) ( / ) j, ( ) ( ) (A.8) G + h G = hjπ H + O wih H,( ) = si( jπ(.5)/ ) for each j =,,... he secod erm becomes asympoically egligible ad hece Var[ x ] G ( ) E [ ε ε ] as, ha depeds o oly hrough he kj, j, Chebyshev polyomial. The scaled OLS esimaio error is he give by /+κ ˆ ( κ ) /+κ ( β β k ) = Q k u + km, km, x x X A (A.9) = = where Q = = x x, wih he idex κ akig he values κ = / uder k coiegraio, ha is whe he error erm u is saioary, ad κ = / uder o coiegraio, so ha he secod erm bewee brackes will domiaes he behavior of ( βˆ β k) uder coiegraio whe Akm, km. From his resul, he -h OLS residual is give by / ˆ / vˆ = v κ x [ +κ ( β β )] = uɶ + d A (A.) k km, km, where he wo erms composig hese residuals are ad κ ( κ) = k j j j= uɶ u x Q x u (A.) km, = km, km, j j k 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) ˆk, k, k, km, km, where sɶ = x uɶ = Op() uder coiegraio, so ha ˆ S ˆ = j = s j ca be wrie as j

23 ˆ S s x d A = ( Vɶ + V A ) / = ɶ j + j km, j km, j= j= k( km), km, (A.4) which implies ha ˆ ( ) = ɶ S V = Op whe Akm, = km, where S ˆ k, has a well defied weak 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. km k, where Mk = = x x ad ω v. k = ω v ωkvωkk ω kv 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 wih ˆ ˆ ˆ ˆ = k k ˆ =, ˆ ωv = ωv, = Lˆ S M S ( S ) Q ( S ) (A.5) ˆ v, ω a cosise esimaor of ω v uder parameer sabiliy ad sric exogeeiy of he iegraed regressors, i.e. δ = ad ω kv = k respecively, usually give by a kerel-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 kerel (weighig) v, = h= = h+ h fucio ad badwidh ca be wrie as q / = o( ). Give ha he residual covariace of order h / h = ɶɶ h + A km d ɶ km, h + d ɶ km, ( h) = h+ = h+ = h+ vˆ vˆ u u ( u u) ˆ v, he ω ca be decomposed as wih + A d d A km km, km, ( h) km = h+ km km, km, ( / ) km, km, ( h) km = h= = h+ (A.6) / / ω ˆ ˆ v, = ω u, + A km dkm, uɶ + w( h/ q) ( dkm, uɶ h + dkm, ( h) uɶ ) = h= = h+ + A d d + w h q d d A u, = h= = h+ h (A.7) ω ˆ = uɶ + w( h/ q ) uɶɶ u a cosise esimaor of he p log-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 brackes we have q dkm, d km, 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 Akm, = km), he es

24 saisic is give by where = ω k ˆ ɶ ɶ u, = Lˆ V Q V (A.9) Vɶ = sɶ V( r) = B ( s) dv ( s) + ( ri Q ( r) Q ()) as / [ r] r [ r] = k k u, k k kk kk ku, wih V, ( r ) he weak limi of u k uɶ uder saioariy, give by / [ r] = r Vu, k( r ) = Bu( r) v k( r) Qkk() B k( s) dbu( s), wih vk( r) = B k( s) ds, r Qkk( r) = Bk( s) B k( s) ds, for < r, ad ku = h = E[ ε hu] he oe-sided logru covariace bewee pas values of ε ad u. Also, akig io accou ha ˆ p ω ω = E [ u u ], he ˆ ( ) ( ) L q ω V r Q () V ( r ). This limi u, u h= h u k kk k disribuio cao be used i pracice i he geeral case due o he presece of he measure of weak edogeeiy of he regressors hrough ku ad he fac ha E[ B ( s) V ( s)] uder edogeeiy of he iegraed regressors, which implies k u, k k r ha he compoe B k( s) dvu, k( 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 ) Qk ( S ) = V ɶ Qk V ɶ = = + V ɶ Qk Vk( km), Akm, = + A km, k( km), k k( km), km, V Q V A = so ha i is domiaed for he las erm, implyig ha i is O ( ) p (A.), so ha Lˆ ( q ) = O ( / q ), ad hece i will diverge a he give rae i he case of a smooh p 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 he compuaio of he eleme γˆ ˆ ˆ kv, = Ω k ω kv,, where Ωˆ ˆ ˆ k = k + Λ k ad ωˆ ˆ ˆ kv, = kv, + Λ vk,, we use cosise kerel-ype esimaors of he log-ru covariace marix of x = ε ad of he log-ru covariace vecor of ε ad v, respecively, wih compoes ˆ ˆ = = x x + Λ, ˆ Λ = w( h/ q ) x x, ˆ k k = w( h/ q ) x kv, h= = h+ h vˆ, ad he FM-OLS esimaor of β k is defied as k h= = h+ h ˆ Λ ˆ v =h = w( h/ q) = h+ x v h. Thus, ˆ β + = ( x x ) ( x y + ˆ + ), = = kv, + where y ˆ = y γ kv, x ad ˆ + ˆ ˆ ˆ kv, = kv, k γ kv,. I he case of parameer isabiliy of he ype cosidered, he we have ha y + = x β k + v + wih + + / ˆ v = v γˆ x = u + ( X x Ω Ω ˆ ) A kv, km, k k( km), km

25 + where u ˆ = u x γ ku,, ˆ ωˆ ˆ kv, = ωku, + Ω k( km), Akm ad ˆ γˆ ˆ ˆ kv, = γ ku, + Ωk Ω k( km), Akm, / wih ˆ ˆ Ωk( km), = k( km), +h = w( h/ q) = h+ x d km, ( h), ad ˆ k ( km ), = + x d. Also, akig io accou ha, is decomposed as ˆ + ˆ ˆ kv, = + ku, + + k( km), Akm, wih ˆ + ˆ ˆ ˆ ku, = ku, k γ ku,, ad ˆ + = ˆ ˆ ˆ Ω Ω ˆ, he he scaled FM-OLS esimaio error is / h w( h/ q) = = h+ h km, k( km), k( km), give by k k k( km), /+κ ˆ + ( κ ) + (/ κ ) ( ˆ + k ) = x x x kv, = = β β v (A.) where he las erm bewee parehesis ca be expressed as ( κ ) + (/ κ ) ˆ + x kv, = v ( κ ) + (/ κ ) ˆ + x ku, = = u + /+κ = x X km, ˆ kv / ˆ ˆ ˆ + x x Ωk Ωk( km), k( km), Akm = (A.) which implies cosise esimaio uder coiegraio uder parameer sabiliy, i.e. Akm = km. If we defie he FM-OLS esimaed scores as ˆ + ˆ ˆ v s = x kv,, so ha ˆ = s = he FM-OLS versio of he es saisic ˆ+ ˆ + ˆ + L ( ˆ = ωv. ) = S Mk S, wih ω ˆ ˆ ˆ ˆ ˆ v. = ωv, ω ku, Ωk ω ku, 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 = k =,..., ωˆ S V S v, wih V = M M M M, where he sup-f es is give by k k k k k SFˆ (, ) max τ τ = F, = [ τ],...,[ τ] ˆ < τ < τ < ad allows o es for a sigle srucural chage a a ukow break poi, while ha he mea-f es, which is defied as [ 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. 3

26 Appedix B. OLS esimaio of a ime-varyig coiegraig regressio model via Chebyshev polyomials uder a srucural break 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 km) + v (B.) Xkm, where β = β ( m) is as i (A.3), so ha he OLS esimaio error of ( α A km) is αˆ αk (/ +κ) ( κ) ˆ ( m) = X X( m) X( m) v (B.) Akm, Akm = = wih X( m ) = ( x, X km, ). However, he rue mechaism drivig he ime-varyig coiegraig vecor is give by a permae ad abrup chage a he break poi of he sample γ = [ τ ] such as β = αk + λ ( ) kh γ, wih H ( γ ) ( ) = I > γ ad break fracio τ (,), where λ k = ( λ,..., λk ) is he k-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) k k which implies ha he regressio error erm v i (B.) ad (B.) is give by v = u + λ x H( γ ) A X, (B.4) k km km, so ha he las erm i he righ-had side of (B.) ca be decomposed as where ( κ) ( κ) X( ) = X( ) = = m v m u /+κ X m x H λk X km, Akm = + ( )( ( γ ) ) X( ) x ( γ ) = X( ) x = = [ τ ] + m H m I k I k = X( m) X ( m) = ( Q( m) Q[ τ] ( m)) = [ τ ] + km, k km, k ad km km X( m) X km, = X( m) X ( m) = Q( m) (B.7) = = Ikm, km Ikm, km [ ] where [ ] ( ) τ Q τ m = = X( m) X ( m) ad Q ( m ) is Q ( ) [ τ] m wih τ =. Thus, equaio (B.) ca be rewrie as αˆ / αk +κ /+κ λk ( κ) ˆ = + Q ( m) ( m) u X Akm, Akm Akm = (B.5) (B.6) /+κ λk Q ( m) Q[ τ] ( m) km, k which gives ˆ /, ( / ) +κ αk αk + λk +κ ( κ) λ k ˆ = Q ( m) X( m) u Q[ τ] ( m) Akm, = km, k (B.8) 4