COINTEGRATION. by Juan J. Dolado a, Jesús Gonzalo b and Francesc Marmol b

Transcription

1 COINTEGRATION by Juan J. Dolado a Jesús Gonzalo b and Francesc Marmol b a: Deparmen of Economics b: Deparmen of Saisics and Economerics Universidad Carlos III de Madrid C/. Madrid Geafe (Madrid) SPAIN Version : 0 February 999 INTRODUCTION A subsanial par of economic heory generally deals wih long-run equilibrium relaionships generaed by marke forces and behavioral rules. Correspondingly mos empirical economeric sudies enailing ime series can be inerpreed as aemps o evaluae such relaionships in a dynamic framework. A one ime convenional wisdom was ha in order o apply sandard inference procedures in such sudies he variables in he sysem needed o be saionary since he vas majoriy of economeric heory is buil upon he assumpion of saionariy. Consequenly for many years economericians proceeded as if saionariy could be achieved by simply removing deerminisic componens (e.g. drifs and rends) from he daa. However saionary series should a leas have consan uncondiional mean and variance over ime a condiion which hardly appears o be saisfied in economics even afer removing hose deerminisic erms. Those problems were somehow ignored in applied work unil imporan papers by Granger and Newbold (974) and Nelson and Plosser (98) alered many o he economeric implicaions of non-saionariy and he dangers of running nonsense or spurious regressions (see e.g. Granger chaper? in his book). In paricular mos of he aenion focussed on he implicaions of dealing wih inegraed variables which are a specific class of non-saionary variables wih imporan economic and saisical

2 properies. These are derived from he presence of uni roos which give rise o sochasic rends as opposed o pure deerminisic rends wih innovaions o an inegraed process being permanen raher han ransiory. The presence of a leas a uni roo in economic ime series is implied in many economic models. Among hem here are hose based on he raional use of available informaion or he exisence of very high adjusmen coss in some markes. Ineresing examples include fuure conracs sock prices yield curves exchange raes money velociy hyseresis heories of unemploymen and perhaps he mos popular he implicaions of he permanen income hypohesis for real consumpion under raional expecaions. Saisicians in urn following he influenial approach by Box and Jenkins (970) had advocaed ransforming inegraed ime series ino saionary ones by successive differencing of he series before modelizaion. Therefore from heir viewpoin removing uni roos hrough differencing ough o be a pre-requisie for regression analysis. However some auhors noably Sargan (964) Hendry and Mizon (978) and Davidson e al. (978) iner alia sared o criicized on a number of grounds he specificaion of dynamic models in erms of differenced variables only especially because of he difficulies in inferring he long-run equilibrium from he esimaed model. Afer all if deviaions from ha equilibrium relaionship affec fuure changes in a se of variables omiing he former i.e esimaing a differenced model should enail a misspecificaion error. However for some ime i remained o be well undersood how boh variables in differences and levels could coexis in regression models. Granger (98) resing upon he previous ideas solved he puzzle by poining ou ha a vecor of variables all which achieve saionariy afer differencing could have linear combinaions which are saionary in levels. Laer Engle and Granger (987) were he firs o formalize he idea of inegraed variables sharing an equilibrium relaion which urned ou o be eiher saionary or have a lower degree of inegraion han he original series. They denoed his propery by coinegraion signifying co-movemens among rending variables which could be exploied o es for he exisence of equilibrium relaionships wihin a fully dynamic specificaion framework. In his sense he basic concep of coinegraion applies in a variey of economic models including he relaionships beween capial and oupu real wages and labor produciviy nominal exchange raes and relaive prices consumpion and disposable income long and shor-

3 erm ineres raes money velociy and ineres raes price of shares and dividends producion and sales ec. In paricular Campbell and Shiller (987) have poined ou ha a pair of inegraed variables ha are relaed hrough a Presen Value Model as i is ofen he case in macroeconomics and finance mus be coinegraed. In view of he srengh of hese ideas a burgeoning lieraure on coinegraion has developed over he las decade. In his chaper we will explore he basic concepual issues and discuss relaed economeric echniques wih he aim of offering an inroducory coverage of he main developmens in his new field of research. Secion provides some preliminaries on he implicaions of coinegraion and he basic esimaion and esing procedures in a single equaion framework when variables have a single uni roo. In Secion 3 we exend he previous echniques o more general mulivariae seups inroducing hose sysem-based approaches o coinegraion which are now in common use. Secion 4 in urn presens some ineresing developmens on which he recen research on coinegraion has been focusing. Finally Secion 5 draws some concluding remarks. Nowadays he ineresed reader who wans o deepen beyond he inroducory level offered here could find a number of exbooks (e.g. Banerjee e al. 993 Johansen 995 and Haanaka 996) and surveys (e.g. Engle and Granger 99 and Wason 994 on coinegraion) where more general reamens of he relevan issues covered in his chaper are presened. Likewise here are now many sofware packages ha suppor he echniques discussed here (e.g. Gauss-COINT E-VIEWS and PC-FIML). PRELIMINARIES : UNIT ROOTS AND COINTEGRATION A well-known resul in ime series analysis is Wold s (938) decomposiion heorem which saes ha a saionary ime series process afer removal of any deerminisic componens has an infinie moving average (MA) represenaion which in urn can be represened by a finie auoregressive moving average (ARMA) process. However as menioned in he Inroducion many ime series need o be appropriaely differenced in order o achieve saionariy. From his comes he definiion of inegraion : a ime series is said o be inegraed of order d in shor I(d) if i has a saionary inverible non-deerminisic ARMA represenaion afer differencing d imes. A whie noise series and a sable firs-order auoregressive AR() process are well- 3

4 known examples of I(0) series a random walk process is an example of an I() series while accumulaing a random walk gives rise o an I() series ec. Consider now wo ime series y and y which are boh I(d) (i.e. hey have compaible long-run properies). In general any linear combinaion of y and y will be also I(d). However if here exiss a vecor ( β ) () z = y α βy is I( d b) d b ' such ha he linear combinaion > 0 hen following Engle and Granger (987) y and y are defined as coinegraed of order ( d b) denoed y ( y y )'~ CI( d b) called he coinegraing vecor. = wih ( -β) Several feaures in () are noeworhy. Firs as defined above coinegraion refers o a linear combinaion of nonsaionary variables. Alhough heoreically i is possible ha nonlinear relaionships may exis among a se of inegraed variables he economeric pracice abou his more general ype of coinegraion is less developed (see more on his in Secion 4). Second noe ha he coinegraing vecor is no uniquely defined since for any nonzero value of λ ( λ λβ) ' is also a coinegraing vecor. Thus a normalizaion rule needs o be used; for example λ = has been chosen in (). Third all variables mus be inegraed of he same order o be candidaes o form a coinegraing relaionship. Nowihsanding here are exensions of he concep of coinegraion called mulicoinegraion when he number of variables considered is larger han wo and where he possibiliy of having variables wih differen order of inegraion can be addressed (see e.g. Granger and Lee 989). For example in a rivariae sysem we may have ha y and y are I() and y 3 is I(); if y and y are CI( ) i is possible ha he corresponding combinaion of y and y which achieves ha propery be iself coinegraed wih y 3 giving rise o an I(0) linear combinaion among he hree variables. Fourh and mos imporan mos of he coinegraion lieraure focuses on he case where variables conain a single uni roo since few economic variables prove in pracice o be inegraed of higher order. If variables have a srong seasonal componen however here may be uni roos a he seasonal frequencies a case ha we will briefly consider in Secion 4; see Ghysels chaper? in his book for furher deails. Hence he remainder of his chaper will mainly focus on he case of CI( ) variables so ha z in 4

5 () is I(0) and he concep of coinegraion mimics he exisence of a long-run equilibrium o which he sysem converges over ime. If e.g. economic heory suggess he following long-run relaionship beween y and y () y = α + β y hen z can be inerpreed as he equilibrium error (i.e. he disance ha he sysem is away from he equilibrium a any poin in ime). Noe ha a consan erm has been included in () in order o allow for he possibiliy ha z may have nonzero mean. For example a sandard heory of spaial compeiion argues ha arbirage will preven prices of similar producs in differen locaions from moving oo far apar even if he prices are nonsaionary. However if here are fixed ransporaion coss from one locaion o anoher a consan erm needs o be included in (). A his sage i is imporan o poin ou ha a useful way o undersand coinegraing relaionships is hrough he observaion ha CI( ) variables mus share a se of sochasic rends. Using he example in () since y and y are I() variables hey can be decomposed ino an I() componen (say a random walk) plus an irregular I(0) componen (no necessarily whie noise). Denoing he firs componens by µ i and he second componens by u i i = we can wrie (3) y = µ + u (3 ) y = µ + u. Since he sum of an I() process and an I(0) process is always I() he previous represenaion mus characerize he individual sochasic properies of y and y. However if y β y is I(0) i mus be ha µ = βµ annihilaing he I() componen in he coinegraing relaionship. In oher words if y and y are CI( ) variables hey mus share (up o a scalar) he same sochasic rend say µ denoed as common rend so ha µ = µ and µ = βµ. As before noice ha if µ is a common rend for y and y λµ will also be a common rend implying ha a normalizaion rule is needed for idenificaion. Generalizing he previous argumen o a vecor of coinegraion and common rends hen i can be proved ha if here are n r common rends among he n variables here mus be r coinegraing relaionships. Noe ha 0 < r < n since r = 0 implies ha each series in he sysem is governed by a differen sochasic rend and ha r = n implies ha he series are I(0) insead of I(). These 5

6 properies consiue he core of wo imporan dual approaches oward esing for coinegraion namely one ha ess direcly for he number of coinegraing vecors ( r) and anoher which ess for he number of common rends ( n r). However before explaining hose approaches in more deail (see Secion 3) we now urn o anoher useful represenaion of CI( ) sysems which has proved very popular in pracice. Engle and Granger (987) have shown ha if y and y are coinegraed CI( ) hen here mus exis a so-called vecor error correcion model (VECM) represenaion of he dynamic sysem governing he join behavior of y and y over ime of he following form p p 0 i i 3 i i i = i = (4) y = θ + θ z + θ y + θ y + ε p3 p4 0 i i 3 i i i= i= (4 ) y = θ + θ z + θ y + θ y + ε where denoes he firs-order ime difference (i.e. y = y y ) and where he lag lenghs p i i... = 4 are such ha he innovaions ε ( ε ε )' are i. i. d. ( ) = 0 Σ. Furhermore hey proved he converse resul ha a VECM generaes coinegraed CI( ) series as long as he coefficiens on z (he so-called loading or speed of adjusmen parameers) are no simulaneously equal o zero. Noe ha he erm z in equaions (4) and (4 ) represens he exen of he disequilibrium levels of y and y in he previous period. Thus he VECM represenaion saes ha changes in one variable no only depends on changes of he oher variables and is own pas changes bu also on he exen of he disequilibrium beween he levels of y and y. For example if β = in () as many heories predic when y and y are aken in logarihmic form hen if y is larger han y in he pas ( z > 0 ) hen θ < 0 and θ > 0 will imply ha everyhing else equal y would fall and y would rise in he curren period implying ha boh series adjus oward is longrun equilibrium. Noice ha boh θ and θ canno be equal o zero. However if θ < 0 and θ = 0 hen all of he adjusmen falls on y or vice versa if θ = 0 and θ > 0. Noe also ha he larger are he speed of adjusmen parameers (wih he righ signs) he greaer is he convergence rae oward equilibrium. Of course a leas one of 6

7 hose erms mus be nonzero implying he exisence of Granger causaliy in coinegraed sysems in a leas one direcion. Hence he appeal of he VECM formulaion is ha i combines flexibiliy in dynamic specificaion wih desirable long-run properies : i could be seen as capuring he ransiional dynamics of he sysem o he long-run equilibrium suggesed by economic heory (see e.g. Hendry and Richard 983). Furher if coinegraion exiss he VECM represenaion will generae beer forecass han he corresponding represenaion in firs-differenced form (i.e. wih θ = θ = ) 0 paricularly over medium and long-run horizons since under coinegraion z will have a finie forecas error variance whereas any oher linear combinaion of he forecass of he individual series in y will have infinie variance; see Engle and Yoo (987) for furher deails. Based upon he VECM represenaion Engle and Granger (987) sugges a wo-sep esimaion procedure for dynamic modeling which has become very popular in applied research. Assuming ha y ~I() hen he procedure goes as follows : regression (i) Firs in order o es wheher he series are coinegraed he coinegraion (5) y = α + βy + z is esimaed by ordinary leas squares (OLS) and i is esed wheher he coinegraing residuals z$ = y α$ β $ y are I(). To do his for example we can perform a Dickey-Fuller es on he residual sequence { $z } o deermine wheher i has a uni roo. For his consider he auoregression of he residuals (6) $ z = ρ z$ + ε where no inercep erm has been included since he { $z } being residuals from a regression equaion wih a consan erm have zero mean. If we can rejec he null hypohesis ha ρ = 0 agains he alernaive ρ < 0 a a given significance level we can conclude ha he residual sequence is I(0) and herefore ha y and y are CI( ) I is noeworhy ha for carrying ou his es i is no possible o use he Dickey-Fuller ables hemselves since { $z } are a generaed series of residuals from fiing regression (5). The problem is ha he OLS esimaes of α and β are such ha hey minimize he residual variance in (5) and hus prejudice he esing procedure oward finding. 7

8 saionariy. Hence larger (in absolue value) criical levels han he sandard Dickey- Fuller ones are needed. In his respec MacKinnon (99) provides appropriae ables o es he null hypohesis ρ = 0 for any sample size and also when he number of regressors in (5) is expanded from one o several variables. In general if he { $ε } sequence exhibis serial correlaion hen an augmened Dickey-Fuller (ADF) es should be used based his ime on he exended auoregression (6 ) z$ = ρ z$ + ζ z$ + ε i i= p i where again if ρ < 0 we can conclude ha y and y are CI( ). Alernaive versions of he es on { $z } being I() versus I(0) can be found in Phillips and Ouliaris (990). Banerjee e al. (997) in urn sugges anoher class of ess based his ime on he direc significance of he loading parameers in (4) and (4 ) where he β coefficien is esimaed alongside he remaining parameers in a single sep using nonlinear leas squares (NLS). If we rejec ha $z are I() Sock (987) has shown ha he OLS esimae of β in equaion (5) is super-consisen in he sense ha he OLS esimaor $ β converges in probabiliy o is rue value β a a rae proporional o he inverse of he sample size T raher han a T / as is he sandard resul in he ordinary case where y and y are I(0). Thus when T grows convergence is much quicker in he CI( ) case. The inuiion behind his remarkable resul can be seen by analyzing he behavior of $ β in (5) (where he consan is omied for simpliciy) in he paricular case where z ~ i. i. d. ( 0 z ) σ and ha θ 0 = θ = 0 and p3 = p4 = 0 so ha y is assumed o follow a simple random walk (7) y = ε or inegraing (7) backwards wih y 0 = 0 (7 ) y = ε i i = wih ε possibly correlaed wih z. In his case we ge var( ) var( ) exploding as T. Neverheless i is no difficul o show ha T y = ε = σ T y = converges 8

9 o a random variable. Similarly he cross-produc T / T = y z will explode in conras o he saionary case where a simple applicaion of he Cenral Limi Theorem implies ha i is asympoically normally disribued. In he I() case T T = converges also o a random variable. Boh random variables are funcionals of Brownian moions which will be denoed henceforh in general as f ( B ). A Brownian moion is a zero-mean normally disribued coninuous (a.s.) process wih independen incremens i.e. loosely speaking he coninuous version of he discree random walk. See Phillips (987) and Bierens (chaper? in his book). Now from he expression for he OLS esimaor of β we obain y z (8) β $ β = = T T = y z y and from he previous discussion i follows ha (9) T( $β β) T = T T y z = T y = is asympoically (as T ) he raio of wo non-degenerae random variables ha in general is no normally disribued. Thus in spie of he super-consisency sandard inference canno be applied o β $ excep in some resricive cases which are discussed below. (ii) Afer rejecing he null hypohesis ha he coinegraing residuals in equaion (5) are I() he $z erm is included in he VECM sysem and he remaining parameers are esimaed by OLS. Indeed given he superconsisency of β $ Engle and Granger (987) show ha heir asympoic disribuions will be idenical o using he rue value of β. Now all he variables in (3) and (3 ) are I(0) and convenional modeling sraegies (e.g. esing he maximum lag lengh residual auocorrelaion or wheher eiher θ or θ is zero ec.) can be applied o assess model adequacy; see Lükepohl (chaper? in his book) for furher deails. 9

10 In spie of he beauy and simpliciy of he previous procedure however several problems remain. In paricular alhough $ β is super-consisen his is an asympoic resul and hus biases could be imporan in finie samples. For insance assume ha he raes of convergence of wo esimaors are T / and 0 0 T. Then we will need huge sample sizes o have he second esimaor dominaing he firs one. In his sense Mone Carlo experimens by Banerjee e al. (993) showed ha he biases could be imporan paricularly when z and y are highly serially correlaed and hey are no independen. Phillips (99) in urn has shown analyically ha in he case where y and z are independen a all leads and lags he disribuion in (9) as T grows behaves like a gaussian disribuion (echnically is a mixure of normals) and hence he disribuion of he -saisic of β is also asympoically normal. For his reason Phillips and Hansen (990) have developed an esimaion procedure which correcs for he previous bias while achieves asympoic normaliy. The procedure denoed as a fully modified ordinary leas squares esimaor (FM-OLS) is based upon a correcion o he OLS esimaor given in (8) by which he error erm z is condiioned on he whole process { y 0... } = ± and hence orhogonaliy beween regressors and disurbance is achieved by consrucion. For example if z and ε in (5) and (7) are correlaed whie noises wih γ ( ε ) ( ε ) denoed $ β FM is given by = E z var he FM-OLS esimaor of β (0) β$ FM = T = ( γ$ ) y y y T = y where $γ is he empirical counerpar of γ obained from regressing he OLS residuals $z on y. When z and y follow more general processes he FM-OLS esimaor of β is similar o (0) excep ha furher correcions are needed in is numeraor. Alernaively Saikkonen (99) and Sock and Wason (993) have shown ha since ( { }) ( ) E z y = h L y where h(l) is a wo-sided filer in he lag operaor L regression of y on y and leads and lags of y (suiably runcaed) using eiher 0

11 OLS or GLS will yield an esimaor of β which is asympoically equivalen o he FM- OLS esimaor. The resuling esimaion approach is known as dynamic OLS (respecively GLS) or DOLS (respecively DGLS). 3 SYSTEM-BASED APPROACHES TO COINTEGRATION Whereas in he previous secion we confined he analysis o he case where here is a mos a single coinegraing vecor in a bivariae sysem his se-up is usually quie resricive when analyzing he coinegraing properies of an n-dimensional vecor of I() variables where several coinegraion relaionships may arise. For example when dealing wih a rivariae sysem formed by he logarihms of nominal wages prices and labor produciviy here may exis wo relaionships one deermining an employmen equaion and anoher deermining a wage equaion. In his secion we survey some of he popular esimaion and esing procedures for coinegraion in his more general mulivariae conex which will be denoed as sysem-based approaches. In general if y now represens a vecor of n I() variables is Wold represenaion (assuming again no deerminisic erms) is given by () y C( L) = ε where now ε ~ nid( 0 Σ ) Σ being he covariance marix of ε and C(L) an ( n n) inverible marix of polynomial lags where he erm inverible means ha C( L ) = 0 has all is roos sricly larger han uniy in absolue value. If here is a coinegraing ( ) n vecor β' ( β... β ) () ' y = ' C( ) + C( L) = nn hen premuliplying () by β yields ~ [ ] β β ε where C(L) has been expanded around L = using a firs-order Taylor expansion and ~ C( L ) can be shown o be an inverible lag marix. Since he coinegraion propery implies ha β'y is I(0) hen i mus be ha β'c( ) = 0 and hence ( = ) L will cancel ou on boh sides of (). Moreover given ha C(L) is inverible hen y has a vecor auoregressive represenaion such ha (3) A( L) y = ε where A( L) C( L) A = I n being he ( n n) I n ( ) C( ) = 0 implying ha ( ) ideniy marix. Hence we mus have ha A can be wrien as a linear combinaion of he elemens

12 β namely A( ) = αβ' wih α being anoher ( n ) vecor. In he same manner if here were r coinegraing vecors ( 0 < r < n ) hen A( ) ime ( n r) = BΓ' where B and Γ are his marices which collec he r differen α and β vecors. Marix B is known as he loading marix since is rows deermine how many coinegraing relaionships ener each of he individual dynamic equaions in (3). Tesing he rank of A() or C() which happen o be r and n r respecively consiues he basis of he following wo procedures : (i) Johansen (995) develops a maximum likelihood esimaion procedure based on he so-called reduced rank regression mehod ha as he oher mehods o be laer discussed presens some advanages over he wo-sep regression procedure described in he previous secion. Firs i relaxes he assumpion ha he coinegraing vecor is unique and secondly i akes ino accoun he shor-run dynamics of he sysem when esimaing he coinegraing vecors. The underlying inuiion behing Johansen s esing procedure can be easily explained by means of he following example. Assume ha y has a VAR( ) represenaion ha is A(L) in (3) is such ha ( ) he VAR( ) process can be reparameerized in he VECM represenaion as (4) ( ) y = A I n y + ε. If A I A( ) n A L = I A L. Hence = = 0 hen y is I() and here are no coinegraing relaionships ( r = 0 ) whereas if ( ) rank A I = n here are n coinegraing relaionships among he n n series and hence y ~I(0). Thus esing he null hypohesis ha he number of coinegraing vecors (r) is equivalen o esing wheher ( ) n rank A I = r. Likewise alernaive hypoheses could be designed in differen ways e.g. ha he rank is ( r + ) or ha i is n. Under he previous consideraions Johansen (995) deals wih he more general case where y follows a VAR( p ) process of he form (5) y = A y + A y + L+ Ap y p + ε which as in (3) and (3 ) can be rewrien in he ECM represenaion (6) y = D y + D y + L+ D y + Dy + ε. p p+ Where Di = ( Ai + + Ap ) + L i = p... and ( p n ) D = A + L + A I = A( ) n

13 = BΓ'. To esimae B and Γ we need o esimae D subjec o some idenificaion resricion since oherwise B and Γ could no be separaely idenified. Maximum likelihood esimaion of D goes along he same principles of he basic pariioned regression model namely he regressand and he regressor of ineres ( y y ) regressed by OLS on he remaining se of regressors ( y y p+ ) and are... giving rise o wo marices of residuals denoed as $e 0 and $e and he regression model e$ De $ $ o = + residuals. Following he preceding discussion Johansen (995) shows ha esing for he rank of $ D is equivalen o es for he number of canonical correlaions beween $e 0 and $e ha are differen from zero. This can be conduced using eiher of he following wo es saisics n (7) λr ( r) = T ln ( λ$ i ) i= r+ (8) λmax ( r r + ) = T ln ( λ$ r + ) where he λ $ i s are he eigenvalues of he marix S S 0 00 S ordered in decreasing order ( > λ $ > > λ $ > 0) S 0 wih respec o he marix L n where Sij = T e $ ie $ ' = j i j = 0. These eigenvalues can be obained as he soluion of he deerminanal equaion (9) λs S S S = The saisic in (7) known as he race saisic ess he null hypohesis ha he number of coinegraing vecors is less han or equal o r agains a general alernaive. Noe ha since ln( ) = 0 and ln( 0) i is clear ha he race saisic equals zero when all he $ λ i s are zero whereas he furher he eigenvalues are from zero he more negaive is ln ( λ$ i ) and he larger is he saisic. Likewise he saisic in (8) known as he maximum eigenvalue saisic ess a null of r coinegraing vecors agains he specific alernaive of r +. As above if $ λ r+ is close o zero he saisic will be small. Furher if he null hypohesis is no rejeced he r coinegraing vecors conained in marix Γ can be esimaed as he firs r columns of marix $ ( $ $ ) conains he eigenvecors associaed o he eigenvalues in (9) compued as V = v K v n which T 3

14 ( ) λ i S S S S v $ = i = K n i 0 subjec o he lengh normalizaion rule V $ ' S V$ = I n. Once Γ has been esimaed esimaes of he B D i and Σ marices in (6) can be obained by insering $ Γ in heir corresponding OLS formulae which will be funcions of Γ. Oserwald-Lenum (99) has abulaed he criical values for boh ess using Mone Carlo simulaions since heir asympoic disribuions are mulivariae f ( B ) which depend upon: (i) he number of nonsaionary componens under he null hypohesis ( n r) and (ii) he form of he vecor of deerminisic componens µ (e.g. a vecor of drif erms) which needs o be included in he esimaion of he ECM represenaion where he variables have nonzero means. Since in order o simplify maers he inclusion of deerminisic componens in (6) has no been considered so far i is worh using a simple example o illusrae he ype of ineresing saisical problems ha may arise when aking hem ino accoun. Suppose ha r = and ha he unique coinegraing vecor in Γ is normalized o be β' ( β β ) = K nn while he vecor of speed of adjusmen parameers wih which he coinegraing vecor appears in each of he equaions for he n variables is α' ( α α ) ( ) = K nn. If here is a vecor of drif erms µ ' = µ K µ n such ha hey saisfy he resricions µ = α µ (wih α = ) i hen follows ha all y i in (6) are expeced o be zero when y + β y + L + β y + µ = and hence he general soluion for each of he nn n 0 { y i } processes when inegraed will no conain a ime rend. Many oher possibiliies like e.g. allowing for a linear rend in each variable bu no in he coinegraing relaions may be considered. In each case he asympoic disribuion of he coinegraion ess given in (7) and (8) will differ and he corresponding ses of simulaed criical values can be found in he reference quoed above. Someimes heory will guide he choice of resricions; for example if one is considering he relaion beween shor-erm and longerm ineres raes i may be wise o impose he resricion ha he processes for boh ineres raes do no have linear rends and ha he drif erms are resriced o appear in he coinegraing relaionship inerpreed as he erm srucure. However in oher insances one may be ineresed in esing alernaive ses of resricions on he way µ eners he sysem; see e.g. Lükepohl (chaper? in his book) for furher deails i ii 4

15 In ha respec he Johansen s approach allows o es resricions on µ B and Γ subjec o a given number of coinegraing relaionships. The insigh o all hese ess which urn ou o have asympoic chi-square disribuions is o compare he number of coinegraing vecors (i.e. he number of eigenvalues which are significanly differen from zero) boh when he resricions are imposed and when hey are no. Since if he rue coinegraion rank is r only r linear combinaions of he variables are saionary one should find ha he number of coinegraing vecors does no diminish if he resricions are no binding and vice versa. Thus denoing by $ λ i and λ i * he se of r eigenvalues for he unresriced and resriced cases boh ses of eigenvalues should be equivalen if he resricions are valid. For example a modificaion of he race es in he form [ ] r * (0) T ln( i ) ln ( $ λ λi ) i= will be small if he λ i * s are similar o he $ λ i s whereas i will be large if he λ i * s are smaller han he $ λ i s. If we impose s resricions hen he above es will rejec he null hypohesis if he calculaed value of (0) exceeds ha in a chi-square able wih r( n s) degrees of freedom. Mos of he exising Mone Carlo sudies on he Johansen mehodology poin ou ha dimension of he daa series for a given sample size may pose paricular problems since he number of parameers of he underlying VAR models grows very large as he dimension increases. Likewise difficulies ofen arise when for a given n he lag lengh of he sysem p is eiher over or under-parameerized. In paricular Ho and Sorensen (996) and Gonzalo and Piarakis (998) show by numerical mehods ha he coinegraing order will end o be overesimaed as he dimension of he sysem increases relaive o he ime dimension while serious size and power disorions arise when choosing oo shor and oo long a lag lengh respecively. Alhough several degrees of freedom adjusmens o improve he performance of he es saisics have been advocaed (see e.g. Reinsel and Ahn 99) researchers ough o have considerable care when using he Johansen esimaor o deermine coinegraion order in high dimensional sysems wih small sample sizes. Noneheless i is worh noicing ha a useful approach o reduce he dimension of he VAR sysem is o rely upon exogeneiy argumens o consruc smaller condiional sysems as suggesed by Ericsson (99) and Johansen (99a). Equally if he VAR specificaion is no appropriae Phillips (99) 5

16 and Saikkonen (99) provide efficien esimaion of coinegraing vecors in more general ime series seings including vecor ARMA processes. (ii) As menioned above here is a dual relaionship beween he number of coinegraing vecors (r) and he number of common rends ( n r ) in an n-dimensional sysem. Hence esing for he dimension of he se of common rends provides an alernaive approach o esing for he coinegraion order in a VAR//VECM represenaion. Sock and Wason (988) provide a deailed sudy of his ype of mehodology based on he use of he so-called Beveridge-Nelson (98) decomposiion. This works from he Wold represenaion of an I() sysem which we can wrie as in expression () wih ( ) C L j 0 C j L C 0 = I n. As shown in expression () C( L) = j= ~ can be expanded as C( L) = C( ) + C( L)( L) so ha by inegraing () we ge () y = C( ) Y + w~ ~ ~ = ε can be shown o be covariance saionary and Y = i= ε i is a where w C( L) laen or unobservable se of random walks which capure he I() naure of he daa. However as above menioned if he coinegraion order is r here mus be an ( r n) Γ marix such ha Γ'C( ) = 0 since oherwise Γ' y would be I() insead of I(0). This means ha he ( n n) C() marix canno have full rank. Indeed from sandard linear algebra argumens i is easy o prove ha he rank of C() is ( n r) implying ha here are only ( n r) independen common rends in he sysem. Hence here exiss he so-called common rends represenaion of a coinegraed sysem such ha c () y = Φ y + w~ where Φ is an n ( n r) marix of loading coefficiens such ha Γ' Φ = 0 and y c is an ( n r) vecor random walk. In oher words y can be wrien as he sum of ( n r) common rends and an I(0) componen. Thus esing for ( n r) common rends in he sysem is equivalen o esing for r coinegraing vecors. In his sense Sock and Wason s (988) esing approach relies upon he observaion ha under he null hypohesis he firs-order auoregressive marix of y c should have ( n r) eigenvalues equal o uniy whereas under he alernaive hypohesis of higher coinegraion order some of hose eigenvalues will be less han uniy. I is worh noicing ha here are oher 6

17 alernaive sraegies o idenify he se of common rends y c which do no impose a vecor random walk srucure. In paricular Gonzalo and Granger (995) using argumens embedded in he Johansen s approach sugges idenifying y c as linear combinaions of y which are no caused in he long-run by he coinegraion relaionships Γ' y. These linear combinaions are he orhogonal complemen of marix B in (6) y B' B c = B y where B is an n ( n r) full ranked marix such ha ( ) = 0 ha can be esimaed as he las ( n r ) eigenvecors of he second momens marix S S 0 S 0 wih respec o S 00. For insance when some of he rows of marix B are zero he common rends will be linear combinaions of hose I() variables in he sysem where he coinegraing vecors do no ener ino heir respecive adjusmen equaions. Since common rends are expressed in erms of observable variables insead of a laen se of random walks economic heory can again be quie useful in helping o provide useful inerpreaion of heir role. For example he raional expecaions version of he permanen income hypohesis of consumpion saes ha consumpion follows a random walk whils saving (disposable income minus consumpion) is I(0). Thus if he heory is a valid one he coinegraing vecor in he sysem formed by consumpion and disposable income should be β' = ( ) equaion (i.e. α ' ( α ) and i would only appears in he second = 0 ) implying ha consumpion should be he common rend behind he nonsaionary behavior of boh variables. To give a simple illusraion of he concepual issues discussed in his secion le us consider he following Wold (MA) represenaion of he bivariae I() process ( ) y y y ' = y y ( L) ( 0. L) = Evaluaing C(L) a L = yields ( ) C 05 = so ha rank C( ) represenaion 0. 6L 0. 8L 0. L 0. 6L ε ε =. Hence y ~ CI( ).. Nex invering C(L) yields he VAR 7

18 0. 6L 08. L 0. L 0. 6L y y = ε ε where ( ) A = so ha rank A( ) = and ( ) = ( ) 0 A 0. 4 = αβ'.. Hence having normalized on he firs elemen he coinegraing vecor is β' = ( ) leading o he following VECM represenaion of he sysem y y 0 4 =. 0 y + ε.. y ε ( L) ( ) Nex given C() and normalizing again on he firs elemen i is clear ha he common c facor is y = ε + ε whereas he loading vecor Φ and he common rend i= i i= i represenaion would be as follows y y 05 c = y w. + ~ Noice ha β' y eliminaes y c from he linear combinaion which achieves coinegraion. In oher words Φ is he orhogonal complemen of β once he normalizaion crieria has been chosen. Finally o examine he effecs of drif erms le us add a vecor µ ( µ µ ) = ' of drif coefficiens o he VAR represenaion. Then i is easy o prove ha y and y will have a linear rends wih slopes equal o µ + µ and µ 4 + µ respecively. When µ + µ 0 he daa will have linear rends whereas he coinegraing relaionship will no have hem since he linear combinaion in β annihilaes he individual rends for any µ and µ. The ineresing case arises when he resricion µ + µ = 0 holds since now he linear rend is purged from he sysem leading o he resriced ECM represenaion 8

19 y y 0 4 =. 0. * ( L) ( µ ) * where µ = µ y y + ε ε 4 FURTHER RESEARCH ON COINTEGRATION Alhough he discussion in he previous secions has been confined o he possibiliy of coinegraion arising from linear combinaions of I() variables he lieraure is currenly proceeding in several ineresing exensions of his sandard se-up. In he sequel we will briefly ouline some of hose exensions which have drawn a subsanial amoun of research in he recen pas. (i) Higher Order Coinegraed Sysems The saisical heory of I(d) sysems wih d = 3K is much less developed han he heory for he I() model parly because i is uncommon o find ime series a leas in economics whose degree of inegraion higher han wo parly because he heory in quie involved as i mus deal wih possibly mulicoinegraed cases where for insance linear combinaions of levels and firs differences can achieve saionariy. We refer he reader o Haldrup (997) for a survey of he saisical reamen of I() models resricing he discussion in his chaper o he basics of he CI( ) case. Assuming hus ha y ~ CI( ) wih Wold represenaion given by (3) ( L) y = C( L) ε hen by means of a Taylor expansion we can wrie C(L) as C( L) = C( ) C ( )( L) + C( L)( L) * ~ wih C * ( ) being he firs derivaive of C(L) wih respec o L evaluaed a L =. Following he argumens in he previous secion y ~ CI( ) implies ha here exiss a * se of coinegraing vecors such ha Γ' C( ) = Γ' C ( ) = 0 from which he following VECM represenaion can be derived * (4) ( )( ) A L L y = B Γ' y B Γ' y + ε 9

20 wih A * ( 0 ) =. Johansen (99b) has developed he maximum likelihood esimaion of I n his class of models which albei more complicaed han in he CI( ) case proceeds along similar lines o hose discussed in Secion 3. Likewise here are sysems where he variables have uni roos a he seasonal frequencies. For example if a seasonally inegraed variable is measured every half-ayear hen i will have he following Wold represenaion (5) ( L ) y = C( L) ε. Since ( L ) = ( L)( + L ) he { } y process could be coinegraed by obaining linear combinaions which eliminae he uni roo a he zero frequency (-L) and/or a he seasonal frequency (+L). Assuming ha Γ and Γ are ses of coinegraing relaionships a each of he wo above menioned frequencies Hylleberg e al. (990) have shown ha he VECM represenaion of he sysem his ime will be * ' ' (6) A ( L)( L ) y B y B ( y y ) = Γ Γ + + ε wih A * ( 0 ) =. Noice ha if here is no coinegraion in (+L) Γ = 0 and he second I n erm in he righ hand side of (6) will vanish whereas lack of coinegraion in (- L) implies Γ = 0 and he firs erm will disappear. Similar argumens can be used o obain VECM represenaions for quarerly or monhly daa wih seasonal difference operaors 4 of he form ( L ) and ( L ) respecively. (ii) Fracionally Coinegraed Sysems As discussed earlier in his chaper one of he main characerisics of he exisence of uni roos in he Wold represenaion of a ime series is ha hey have long memory in he sense ha shocks have permanen effecs on he levels of he series so ha he variance of he levels of he series explodes. In general i is known ha if he differencing filer ( L) d d being now a real number is needed o achieve saionariy hen he coefficien of ε j in he Wold represenaion of he I(d) process has a leading erm j d (e.g. he coefficien in an I() process is uniy since d = ) and he process is said o be fracionally inegraed of order d. In his case he variance of he series in levels will explode a he rae T d (e.g. a he rae T when d = ) and hen all ha is needed o have his kind of long memory is a degree of differencing d >. 0

21 Consequenly i is clear ha a wide range of dynamic behavior is ruled ou a priori if d is resriced o ineger values and ha a much broader range of coinegraion possibiliies are enailed when fracional cases are considered. For example we could have a pair of series which are I( d ) d > which coinegrae o obain an I( d 0 ) linear combinaion such ha 0 < d <. A furher complicaion arises in his case if he 0 various inegraion orders are no assumed o be known and need o be esimaed for which frequency domain regression mehods are normally used. Exensions of leas squares and maximum likelihood mehods of esimaion and esing for coinegraion wihin his more general framework can be found in Jeganahan (996) Marmol (998) and Robinson and Marinucci (998). (iii) Nearly Coinegraed Sysems Even when a vecor of ime series is I() he size of he uni roo in each of he series could be very differen. For example in erms of he common rend represenaion of a c bivariae sysem discussed above i could well be he case ha y = φ y + w~ and c y = φ y + w~ are such ha φ is close o zero and ha φ is large. Then y will no be differen from w ~ which is an I(0) series while y will be clearly I(). The wo series are coinegraed since hey share a common rend. However if we regress y on y i.e. we normalize he coinegraing vecor on he coefficien of y he regression will be nearly unbalanced namely he regressand is almos I(0) whils he regressor is I(). In his case he esimaed coefficien on y will converge quickly o zero and he residuals will resemble he properies of y i.e. hey will look saionary. Thus according o he Engle and Granger esing approach we will ofen rejec he null of no coinegraion. By conras if we regress y on y now he residuals will resemble he I() properies of he regressand and we will ofen rejec coinegraion. Therefore normalizaion plays a crucial role in leas squares esimaion of coinegraing vecors in nearly coinegraed sysems. Consequenly if one uses he saic regression approach o esimae he coinegraing vecor i follows from he previous discussion ha is beer o use he less inegraed variable as he regressand. Ng and Perron (997) have shown ha hese problems remain when he equaion are esimaed using more efficien mehods like FM-

22 OLS and DOLS while he Johansen s mehodology provides a beer esimaion approach since normalizaion is only imposed on he lengh of he eigenvecors. (iv) Nonlinear Error Correcion Models When discussing he role of he coinegraing relaionship z in (3) and (3 ) we moivaed he EC model as he disequilibrium mechanism ha leads o he paricular equilibrium. However as a funcion of an I(0) process is generally also I(0) an alernaive more general VECM model has z in (3) and (3 ) replaced by g( z ) where [ ] g( z ) is a funcion such ha g( 0) = 0 and E g( z ) exiss. The funcion g( z ) is such ha i can be esimaed nonparamerically or by assuming a paricular parameric form. For example one can include z + = max{ 0 z } and z min{ z } = 0 separaely ino he model or large and small values of z according o some prespecified hreshold in order o deal wih possible sign or size asymmeries in he dynamic adjusmen. Furher examples can be found in Granger and Teräsvira (993). The heory of non-linear coinegraion models is sill fairly incomplee bu nice applicaions can be found in Gonzalez and Gonzalo (998) and Balke and Fomby (997). (v) Srucural Breaks in Coinegraed Sysems The parameers in he coinegraing regression model (5) may no be consan hrough ime. Gregory and Hansen (995) developed a es for coinegraion allowing for a srucural break in he inercep as well as in he slope of model (5). The new regression model now looks like y = α + α D + β y + β y D + z (7) ( ) ( ) 0 0 where D( 0 ) is a dummy variable such ha D( 0 ) 0 = if 0 < 0 and ( ) D 0 = if 0 < T. The es for coinegraion is conduced by esing for uni roos (for insance wih an ADF es) on he residuals $z for each 0. Gregory and Hansen propose and abulae he criical values of he es saisic ADF { ADF( )} * = inf 0 < 0 < T. The null hypohesis of no coinegraion and no srucural break is rejeced if he saisic ADF * is smaller han he corresponding criical value. In his case he srucural break

23 will be locaed a ime * where he inf of he ADF es is obained. The work of Gregory and Hansen is opening an exensive research on analyzing he sabiliy of he parameers of mulivariae possibly coinegraed sysems models like he VECM in (6). Furher work in his direcion can be found in Hansen and Johansen (993) Quinos (994) and Juhl (997). 5 CONCLUDING REMARKS The considerable gap in he pas beween he economic heoris who had much o say abou equilibrium bu relaively less o say abou dynamics and he economerician whose models concenraed on he shor-run dynamics disregarding he long-run equilibrium has been bridged by he concep of coinegraion. In addiion o allowing he daa o deermine he shor-run dynamics coinegraion sugges ha models can be significanly improved by including long-run equilibrium condiions as suggesed by economic heory. The generic exisence of such long-run relaionships in urn should be esed using he echniques discussed in his chaper o reduce he risk of finding spurious conclusions. The lieraure on coinegraion has grealy enhanced he exising mehods of dynamic economeric modeling of economic ime series and should be consider nowadays as a very valuable par of he praciioner s oolki. REFERENCES Balke N and T. Fomby (997) Threshold coinegraion Inernaional Economic Review Banerjee A. J.J. Dolado J.W. Galbraih and D.F. Hendry (993) Co-inegraion error correcion and he economeric analysis of non-saionary daa Oxford Universiy Press Oxford. Banerjee A. J.J. Dolado and R. Mesre (997) Error-correcion mechanism ess for coinegraion in a single-equaion framework Journal of Time Series Analysis Beveridge S. and C. R. Nelson (98) A new approach o decomposiion of economic ime series ino permanen and ransiory componens wih paricular aenion o measuremen of he Business Cycle Journal of Moneary Economics Bierens H. (999) Uni roos chaper? in his book. 3

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