Viereljahrshefe zur Wirschafsforschung 7. Jahrgang, Hef 3/2 S. 352 363 Trend and Cycle in he Euro-Area: A Permanen-Transiory Decomposiion Using a Coinegraed VAR Model By Chrisian Schumacher* Summary This paper invesigaes he Euro-area business cycle using a mulivariae auoregressive ime series model wih coinegraion. The coinegraion resricions help o idenify permanen and ransiory shocks which form he sochasic par of rend and cyclical GDP, respecively. The idenificaion allows for a hisorical decomposiion of Euro-area GDP ino rend and cycle. Furher, he relaive imporance of boh srucural shocks is examined wih forecas error variance decomposiions. The resuls show ha permanen shocks accoun for a significan fracion of oupu flucuaions so ha he sochasic rend of Euro-area GDP has considerable variabiliy.. Inroducion The measuremen of business cycles is a widely discussed opic in he lieraure. A wide range of alernaive models sems from he fac ha he business cycle is a purely heoreical concep and i is no clear how i should be measured unless appropriae definiions are made. In his sense, he business cycle is no a direcly observable concep. This implies a variey of possible heories and empirical mehods. To allow for comparisons, empirical conribuions should make clear on which assumpions hey rely on. In his paper, a raher radiional definiion of he business cycle is used. I is assumed ha he economy flucuaes along a rend. The difference beween observed oupu and such a rend is defined o be he cycle or he business flucuaions. The cyclical flucuaions are allowed o be persisen bu have o be ransiory so he cycle is a saionary ime series. In he concep applied here, he rend and cycle of oupu are he resul of shocks hiing he economy. The shocks are divided ino wo groups: permanen and ransiory shocks. Following he widely known definiion of Blanchard and Fisher (989), he permanen shocks deermine he rend whereas he ransiory shocks form he cycle. The permanen-ransiory decomposiion (PT) employed here allows o idenify hese wo ypes of shocks and derive he appropriae cycle plus sochasic rend decomposiion. The mehod has also some background in he recen heoreical lieraure. For example, Yun (995) and Kimball (996) propose raional expecaion models wih imperfec compeiion and price saggering ha show business flucuaions around a sochasic rend due o imperfec price adjusmen. The rend represens a siuaion where prices rigidiies are absen. In his class of models he disincion of rend and cycle has sric microfoundaions. In heory, i is no clear how high he variabiliy of he rend is. For example, he Real Business Cycle (RBC) baseline model aribues almos all flucuaions o flucuaions of he rend because no rigidiies or marke imperfecions are allowed. Hence, under his model he variance of he permanen par is nearly equal o he variance of oupu and here is no room for a cyclical componen defined as above. Hence, he role of permanen shocks or he rend on he one hand and he ransiory shocks on he oher is an empirical quesion. The permanen-ransiory decomposiion employed here is able o address his quesions. The purpose of he paper is wofold: Euro-area GDP is decomposed ino rend and cycle and he relaive imporance of permanen and ransiory shocks is invesigaed. From a mehodological poin of view, he mehod should be disinguished from oher approaches. For example, he rend cycle decomposiion does no ake ino accoun * HWWA Insiue of Inernaional Economics, Neuer Jungfernsieg 2, 2347 Hamburg, e-mail: schumacher@hwwa.de The seminal papers reaing hese quesions empirically are Blanchard and Quah (989) and King e al. (99). 352
business cycle asymmeries explicily. Noneheless, he cycle will no follow a sric cyclical paern in he sense of a rigonomeric funcion. As has been poined ou before, he oupu flucuaions are he resul of shocks hiing he economy. Since hese shocks are sochasic, parly asymmeric paerns of cyclical flucuaions may arise alhough asymmeries are no modeled explicily. Moreover, he approach employed here doesn consider muliple rends such as empirical models wih regime shifs. The work presened here is mos closely relaed o he VAR lieraure ha uses srucural idenificaion schemes o idenify poenial oupu, for example Dupasquier e al. (999), Asley and Yaes (999) and Funke (998). In one way or anoher all hese papers use a-priori long-run idenifying resricions o decompose oupu ino a permanen and ransiory par. In comparison wih hese papers he approach chosen here relies on only wo assumpions: firs, coinegraion mus hold, second, he groups of permanen and ransiory shocks are uncorrelaed. Hence, if coinegraion is found in a mulivariae seing, only one addiional resricion is needed for a unique decomposiion. The imposiion of furher a-priori resricions can be avoided. This is an advanage over he exising mehods especially when higher dimensional sysems are invesigaed. The paper proceeds as follows: In secion 2 he mehod is explained sep-by-sep from esimaion of he mulivariae ime series process o he derivaion of he permanen and ransiory shocks and he resuling hisorical decomposiion of oupu ino rend and cycle. The mehod is relaed o comparable approaches in secion 3. Secion 4 reas he empirical model and presens esimaes of he Euro-area business cycle as well as robusness checks over he ime axis. The role of permanen and ransiory shocks is examined in secion 5. The las secion concludes. 2. The Permanen-ransiory (PT) Decomposiion Behind he PT decomposiion used here sands he general belief ha behind shor-run movemens, he economy evolves along a growh pah, which is inerpreed as he rend. The economy is being affeced by wo ypes of shocks: permanen and ransiory shocks. The permanen shocks are mainly aleraions of echnology and improvemens of produciviy ha have a long-run effec on oupu. The PT decomposiion defines ha par of oupu as rend oupu ha is due o he permanen shocks (see Blanchard and Fisher, 989, 8). The shor-run flucuaions of oupu are deermined by he ransiory shocks. These shocks have no long-run effec on oupu so ha he ransiory componen is a saionary variable. The permanen and ransiory shocks canno be measured direcly. Insead, he PT decomposiion recovers hem by idenificaion. The role of echnological shocks is widely discussed in he heoreical lieraure. The Real-Business-Cycle (RBC) baseline model aribues mos of he variaions in oupu o permanen shocks, ransiory shocks play no role. In recen models wih opimizing behavior, monopolisic compeiion and price saggering, echnology only deermines he growh pah. Shor-run flucuaions are affeced by imperfec price adjusmens. The applicaion of he PT decomposiion can shed some ligh on he quesion of he relaive imporance of permanen and ransiory shocks. Saring poin of he derivaion is he esimaion of a vecor error correcion model (VECM) where X is he m-dimensional vecor of endogenous variables ha include oupu as he variable of main ineres. The variables are assumed o be inegraed of order one and hence ener he model in firs differences, = X X. The auoregressive lag order of he model is k. µ is an unresriced consan. The coinegraion rank and he number of coinegraion relaions in he model is r < m. The coinegraion propery is modeled as a linear combinaion of he levels of X, β X, where β is he (m r) marix of consans ha forms he coinegraion relaionships. Since he variables are assumed o be inegraed of order one, he linear combinaions β X should be saionary. In he (m r) marix α are he loadings ha show how he sysem reacs o coinegraion errors. The ε are he error erms of he sysem and are assumed o have mean zero and variance covariance marix Ω. The VEC model can be esimaed wih he reduced rank regression mehods developed by Johansen (988, 99). Since we are ineresed in idenifying shocks, i is necessary o find an expression for he VECM ha is dependen of he residuals. Laer hose will be ransformed ino shocks. The VEC model can be invered o obain he moving average (MA) represenaion k = Γ i i= = τ + A(L) = µ + A, ε ε + A ε + αβ X + µ + ε + A ε 2 2 +K Here, is linked o he error erms hrough he lag polynomial A(L) = Σ j= A j Lj, where L is he lag operaor so ha LX = X. In he MA represenaion he vecor of endogenous variables only depends on he residuals which will be ransformed ino permanen and ransiory shocks laer. Due o his similariy, each parameer marix A can be inerpreed as a muliplier. An error or shock in period, ε, has an impac on of A. Afer one period he shock in causes + o change wih A, afer wo periods i has an impac on +2 of A 2 and so on. 353
Since he vecor X can be undersood as he sum of cumulaed differences saring from an iniial value, ha is ignoring a saring value for simpliciy and redefining he lag polynomial. To summarize, he PT decomposiion de- X + n = X + + + + K+ + n he long-run impac of a shock in is he sum of he parameer marices A j X n + n = A j. ε j= Leing he forecas inerval n become very large, we ge he long-run muliplier j= A j = A() = A + A + A This long-run effec has a naural inerpreaion as i provides a ime series measure of long-run equilibrium. A() is he value of X due o shocks ha is reached afer all ransiional dynamics have died ou. Since he permanen shocks are defined o have a non-zero longrun effec on oupu, heir derivaion sars a A(). To obain he permanen par of he model, one divides he marix polynomial ino a long-run and a shor-run par, ha is A(L) = A() + A ~ (L)( L), + K where A ~ (L) is simply A ~ (L) = (A(L) A()) ( L). The firs difference of he endogenous variables can hen be expressed as = τ + A(L) ε = τ + A() ε k i i= where β (α (l Σ i= Γ i ) β ) is he long-run effec of he k (m r) permanen shocks α ε. 2 β and α are full rank (m (m r)) orhogonal complemens o he coinegraion vecors β and he marix of he loadings α, respecively. The orhogonal complemen is defined as α α =. For laer use, we call he permanen shocks ε p = α ε wih dimension (m r). I mus be noed ha he permanen shocks are idenified as a group of shocks. Individual shocks are no idenified because we only seek for heir overall impac on oupu which was defined o be he rend. One mus find an expression where he MA represenaion is relaed o he permanen shocks. The moving average represenaion is o be decomposed ino 2 + A ~ (L)( L) ε = τ + β ( α (I Γ ) β ) = τ + A p = τ + A (L ) ε (L ) ε p + A r., α ε + A ~ (L)( L) ε, (L ) ε 2 For a deailed derivaion, see Johansen (995, 4). r The aim is now o find he lag marices A p (L) and A r (L) as well as he ransiory shocks ε r, while he permanen shocks are already idenified in he VEC model. Yang (998) shows how o obain he unknown marices. He defines = τ + α α ε A (L ) A (L ) using he unknown marices α γ and γ wih appropriae dimensions, (m (m r)), (m r) and (m r), respecively. The marix g ransforms he residuals ino he ransiory shocks such ha ε r = γ ε is an r-dimensional vecor. Again, he only hings we know are he permanen shocks α ε and he MA lag polynomial A(L). All unknown marices mus now be consruced so ha he permanen and ransiory par add up o he MA polynomial, ha is A (L ) ε = α α ε A (L ) A (L ) so ha rend and cycle sum up o he whole ime series process. Afer summarizing erms, his adding-up resricion implies α α ( α γ) = I or ( α γ) =. γ γ Hence, if we know he marix γ, we know he lef hand side and can deermine he res of he unknown marices. Because here is no furher informaion lef o idenify γ, one has o impose a furher resricion. Following he majoriy of he shock huning lieraure, Yang (998) assumes ha permanen and ransiory shocks are uncorrelaed, ha is p r E ε ε! E ( )( ). α ε γ ε This resricion is fulfilled by he marix which is only in erms of known marices. Given his marix, he differen srucural groups of shocks and heir mulipliers are idenified. One can now derive he he permanen par of oupu which is inerpreed as he rend. In he PT decomposiion used here, oupu minus rend oupu is he ransiory par. In erms of he srucural mulipliers and shocks i is, γγ ε, γγ ε + + γγ ε = = γ = α α X = τ + ( α Ωα ) α Ωα p p p A (L) A(L) ε = τ + r r r = A (L ) ε = A(L ) i= i= α, α ε, 354
scribed here is based on wo main assumpions: Firs, coinegraion is a valid resricion of he empirical model and second, he groups of srucural shocks are assumed o be uncorrelaed. Wih hese assumpions i is possible o uniquely idenify permanen and ransiory shocks which can be used o decompose a vecor of ime series ino rend and cycle. 3. Comparison wih oher Mehods The proposed mehod is compared wih oher rend and cycle decomposiions in he lieraure. Especially is relaionship o oher measures based on VAR models is worh menioning. Evans and Reichlin (994), for example, propose he so-called mulivariae Beveridge-Nelson (MBN) decomposiion. In he MBN, he rend is resriced o be a random walk. This implies ha shocks ha have a permanen effec on oupu immediaely aler rend oupu wih heir full long-run impac measured by he long-run muliplier A(). This definiion of rend ignores possible parial adjusmens afer a permanen shock occurred. This assumpion is in sark conras o he widely held view ha echnological innovaions have ransiional dynamics. Lippi and Reichlin (994) declare he random walk assumpion of rend oupu as inconsisen wih sandard views abou he dynamics of produciviy shocks ha are jusified wih adjusmen coss on capial and labor, learning-by-doing processes and ime o build. The PT decomposiion applied here allows for more general adjusmen processes afer he occurrence of a srucural shock. Anoher widely applied ool o decompose oupu ino rend and cycle is he srucural VAR (SVAR) approach. 3 Here, a VAR model wihou coinegraion is esimaed. The model is also invered ino MA form. Then, ofen resricions on he long-run marix of shocks A() mus be implemened. For example, a srucural shock has no long-run effec on an endogenous variable and hence is a ransiory shock. 4 The srucural shocks are usually assumed o be muually uncorrelaed. Bu in higher dimensional sysems i is problemaic o idenify shocks and find an economic meaning for hem. Moreover, he idenificaion schemes are no unique. When coinegraion is found, no such idenifying resricions are needed and he PT decomposiion above should be applied wihou he need for furher idenificaion. Since he daa se we will use laer in he empirical applicaion shows common rends, he PT decomposiion seems o be he appropriae mehod. In oher VAR based rend-cycle decomposiions, he coinegraion resricions are someimes no fully aken ino consideraion. The approach of Dupasquier e al. (999) uses VAR models o deermine he permanen par of oupu under consideraion of he ransiory dynamics of permanen shocks, oo. They call heir approach LRRO, because long-run resricions are imposed on shocks o oupu. This is in general also in accordance wih he PT decomposiion, bu wha differs from his paper is he way in which he long-run resricions are imposed. Dupasquier e al. (999) sugges o esimae he VAR in a resriced form when coinegraion is presen. In heir paper, a wosep sraegy is used. A he firs sep, he coinegraion vecors are deermined using for example preliminary esimaions. In he second sep, oupu in firs differences, he coinegraion errors and oher variables ener a new vecor of endogenous saionary variables ha is used o form a VAR model. Then, afer he inversion direc resricions on he long-run marix of shocks serve o idenify permanen and ransiory shocks. Here, Dupasquier e al. (999) use a riangularizaion of he muliplier marix A() so i has full rank. One objecion can be saed agains his idenificaion scheme. If here is coinegraion in he se of variables, here are less permanen shocks han he number of variables and he long-run muliplier marix has reduced rank. Hence, he coinegraion resricions of he firs sep of he LRRO approach is no correcly aken over ino he second sep. In he PT decomposiion applied here, he coinegraion resricions are fully aken ino accoun. Once he coinegraion vecors are esimaed, he assumed non-correlaion of permanen and ransiory shocks leads o a uniquely defined permanen par of oupu. The wo-sep procedure of he LRRO approach is less efficien han he PT decomposiion applied here, because he explici resricions in he firs sep model are no aken ino accoun in he resriced VAR esimaion. Alhough i is possible o resric he long-run marix of a resriced VAR correcly in principle, he PT mehod applied here is more direc. Of course, his advanage holds only if coinegraion can be found. If no, a-priori resricions have o be used o idenify permanen and ransiory shocks as in he SVAR approach. Anoher group of models ha provide useful decomposiions ino rend and cycle is he group of sae-space models wih unobserved componens (UC). These models can be analyzed using he Kalman filer and esimaed wih maximum likelihood where rend oupu and he cycle are unobserved componens. In UC models, an addiional equaion o define rend oupu mus be supplied. Trend oupu is ofen resriced o follow a random walk, someimes wih noise. The UC approach in general has he poenial o implemen richer rend dynamics. Bu mulivariae rends as in VAR or VEC models are no possible due o idenificaion problems. Anoher difference in comparison wih he PT approach is he more resriced modeling sraegy, since a general-o-specific procedure is no applicable due o he compuaional burden of he 3 A recen applicaion is Asley and Yaes (999). 4 See he famous example from Blanchard and Quah (989) for a bivariae VAR model. 355
ieraive Maximum Likelihood esimaion. The major advanage of he UC approach is ha i is he only mehod a he momen ha can implemen economic heory and he esimaion of rend and cycle componens in one sep. Alhough i is possible o esimae VEC models wih economic conen, he PT decomposiion is essenially a wosep procedure ha needs an esimaion sep and afer ha an idenificaion sep where he permanen and ransiory pars are derived from he model s parameers as shown above. To conclude, in relaionship o he VAR mehods discussed above, he PT decomposiion employed here relies on weaker assumpions concerning he ime series properies of he rend par and a more direc idenificaion sraegy when coinegraion is given. These advanages over he exising VAR based rend measures moivae he measuremen of Euro-area rend oupu wih he PT decomposiion in his paper. In comparison wih he UC models i is no clear wheher he higher flexibiliy of he VEC models overcompensaes he deficiencies of he wo-sep idenificaion procedure. These approaches can hardly be compared because of heir differen modeling philosophies. Moreover, he unobservabiliy of he rend and cyclical componens in general should lead one o use hese models as complemens. 4. Esimaion of a Euro-Area VECM We now follow closely Evans and Reichlin (994) and Dupasquier e al. (999) who decompose U.S. oupu ino rend and oupu gap wih similar mehods, o esimae he PT decomposiion for he Euro-area empirically. In boh of hese papers, VAR models wih coinegraion resricions are esimaed. Evans and Reichlin (994) include ime series ino he daa se which are expeced o be good forecasers of GDP. This sems from he fac ha he longrun value of permanen shocks, as represened by he long-run muliplier marix A(), can be inerpreed as he long-run forecas of he underlying series (see Evans and Reichlin, 994, 234). The use of variables ha help o predic oupu movemens may herefore improve he PT decomposiion. In heir paper, Evans and Reichlin (994) use preliminary Granger causaliy ess o idenify possible variables ha help o explain GDP flucuaions. As a resul, hey find consumpion, he unemploymen rae, a composie leading indicaor and a coinciden indicaor as suiable indicaors. 5 Because of daa limiaions, some of hese variables are no available for an invesigaion of Euro-area rend oupu. Noneheless, we apply he same modeling sraegy using a se of Euro-area variables and perform parial Granger causaliy ess o check for he explanaory power of various ime series for oupu. Hence, he selecion of variables in he menioned paper is no based on heoreical grounds explicily. The reason for ha sems from he fac ha a lo of heoreical models have a good fi for US daa bu no for European daa (see Neusser, 99). For example, preliminary esing shows ha he long-run resricions of he neoclassical growh model don hold for he Euro-area daa se employed here. This is in line wih earlier ess for single counries, bu cerainly no saisfying from a heoris s poin of view. To invesigae he predicive power of various variables, one-sided granger causaliy ess are performed. I is esed wheher a given number of lags of a possible indicaor variable helps o explain a porion of oupu variabiliy significanly. For his purpose, we use a quarerly daa se which has been compiled from various sources because no official daa wih a sufficien sample size is available a he momen. Deailed informaion abou he daa is given in he daa appendix. Table presens he resuls of he causaliy ess. Table Variable Granger causaliy ess Lag order 2 3 4 5 Dlc.5.68.92.97 Dlgfcf.34.25.55.43 Dsr.83.28..3 Dlli.... Noes: Figures in he able are P-values for he null of no explanaory power of he various indicaors. Dependen variable in each case is oupu. The explanaory variables in firs differences are Dlc = privae consumpion in logs, Dlgfcf = gross fixed capial formaion in logs, Dsr = shor-erm ineres rae, Dlli = OECD leading indicaor in logs. The ess sugges ha only he shor-erm ineres rae as well as he OECD leading indicaor have a significan impac on oupu. Oher variables such as invesmen have no explanaory power for oupu. Hence, in he following a rivariae VAR model wih oupu, he shor-erm ineres rae and he leading indicaor is esimaed. Now, informaion crieria as well as goodness-of-fi saisics are used o deermine he lag lengh of he VAR model. The informaion crieria indicae differen lag lenghs. The Akaike crierion gives 5 lags, whereas he Schwarz crierion indicaes wo and he Hannan-Quinn crierion 3 lags. Since he crieria show no unique resul, addiionally he residual saisics of he VAR models are invesigaed. A model wih four lags has he bes overall properies. Is residual properies are presened in he following able 2. For he goodness-of-fi saisics, he VAR is esimaed wih full rank. 5 The daa se used by Dupasquier e al. (999) combines GDP, consumpion and a shor-erm ineres rae o decompose oupu. 356
Table 2 Goodness-of-fi saisics Mulivariae ess Auocorrelaion Normaliy LM( or 4), CHISQ 9 DH, CHISQ 6 Lag,.2 (.33) 4.63 (.59) Lag 4, 3.78 (.3) Univariae ess Heeroscedasiciy Normaliy ARCH, CHISQ 4 DH, CHISQ 2 Lgdp.4 (.85) 2.35 (.3) Sr 4.85 (.3).47 (.48) Lli 2.4 (.7).8 (.4) Noes: P-values are in parenheses. Each es is chisquared disribued where he index denoes he degrees of freedom. The VAR in levels is esimaed using 4 lags and an unresriced consan as well as he following impulse dummies: dum79, dum82, dum9, dum922, dum923 which are one in he quarer menioned where he firs o digis assign he year and he las digi he corresponding quarer. The dummies are zero elsewhere in he sample. The variables are Lgdp = oupu in logs, Sr = shor-erm ineres rae and Lli = OECD leading indicaor in logs. For all he presened ess, he null hypohesis is absence of specificaion error. 6 Hence, high P-values in brackes end o suppor he absence of misspecificaion. The ess indicae sufficien saisical properies of he VAR model. There is no sign of auocorrelaion, non-normaliy or heeroscedasiciy. However, as denoed in he able, several impulse dummies mus correc for ouliers o fulfill especially he requiremen of normal disribued residuals alhough he resuls concerning he cycle and rend decomposiion are only slighly alered by he inclusion of hese addiional variables. We can now perform a coinegraion rank es o deermine he number of coinegraion vecors in he sysem. Table 3 shows he race saisic and he corresponding criical values. 7 Table 3 Coinegraion rank 95 percen criical values H: r Trace Saisic Asympoic Boosrap 55.55 29.68 32.42 9.23 5.4 7.2 2 2.85 3.76 4.95 Noes: The asympoic criical values are from Oserwald-Lenum (992), he boosrap criical values are generaed wih,5 replicaions of he model. The es procedure begins o assume ha here are zero coinegraion resricions under he null. If he null is rejeced, he coinegraion rank is increased and he new null of a coinegraion rank of one has o be esed. The esing procedure proceeds his way and sops unil he null canno be rejeced he firs ime. The criical values don follow sandard disribuions and have been simulaed in he lieraure using asympoic disribuions (see, for example, Oserwald-Lenum, 992). The corresponding criical values are deermined for alernaive specificaions of he deerminisic par of he VAR model, because he asympoic disribuions are alered by he inclusion of ime rends, consans or inervenion dummies. Because our model includes various impulse dummies, he criical values may be differen from he abulaed ones in he lieraure. Though, heir impac may be small since each impulse dummy eliminaes only one poin of informaion from a relaively large daa se. Noneheless, in addiion o he asympoic criical values, a boosrap simulaion exercise is performed where he criical values are derived on he basis of heir observed empirical disribuion. This procedure has he advanage of considering he differen deerminisic erms of he models as well as he finie sample size of he daa se. The asympoic values insead are derived under he assumpion of an infinie sample size. However, here is no a big difference beween he criical values as he able of he race saisic shows. The criical values obained from he boosrap exercise are only slighly larger han heir asympoic counerpars wihou he inclusion of dummies and only an unresriced consan. The race saisics indicae in boh cases he exisence of wo coinegraion relaionships. So we can conclude ha a coinegraion rank of wo is a saisically suppored resricion for he model. Moreover, he imporan requiremen of coinegraion for he applicabiliy of he PT decomposiion is given. 5. Trend and Cycle in he Euro-Area Wih he esimaed VEC model we can now derive he permanen and ransiory par of oupu. Therefore, in addiion o he coinegraion resricions he second assumpion of he uncorrelaedness of he srucural shocks is imposed, so he groups of permanen and ransiory shocks can be idenified as well as heir mulipliers. According o he definiion of Blanchard and Fisher (989), he rend of oupu for he Euro-area is derived as ha par of oupu ha is due o he permanen shocks and he cycle as ha par deermined by he ransiory shocks. In figure, he cycle in levels and he rend wih oupu in firs differences are displayed. 6 The ess are described in deail in Hansen and Juselius (995, 72 76). 7 The race saisic is derived for example in Johansen (988). 357
Figure 2. Percen Cycle Level.5..5. -.5 -. -.5-2. 977 979 98 983 985 987 989 99 993 995 997 999 2. Percen Trend and oupu s difference.5..5. -.5 -. -.5 978 98 982 984 986 988 99 992 994 996 998 Oupu Trend The cyclical componen as shown in he upper panel lies beween.5 percen and.8 percen. I no a smooh funcion of ime bu wih a clearly persisen shape. Bu as he use of he PT decomposiion suggess, he cycle is saionary over he sample. The second panel of he figure shows he firs differences of oupu and he rend. Here, rend variabiliy is considerable. In some periods he rend follows oupu flucuaions in he same direcion, alhough i is clearly less volaile. The relaive variance of he rend in relaion o he variance of oupu is 28 percen. Hence, one fourh of oupu flucuaions are due o movemens of he rend. For applied business cycle usage, i is imporan wheher he obained esimaes of rend and cycle are sufficienly sable when new ime series informaion becomes available. When longer ime series are published, boh componens shouldn change dramaically afer a reesimaion of he model because his would disor heir usefulness as indicaors for moneary policy (see Camba- Méndez and Palenzuela, 2). To check he sabiliy of he rend oupu and cyclical esimaors, we now perform a recursive esimaion where he sample is divided ino various subsamples saring from 993:. The sample size is increased sep-by-sep by one quarer, he whole 358
Figure 2 3 Percen Cycle Level 2.5 Percen Trend and Oupu s difference 2 2..5..5 -. -2 Final Recursive -.5-3 -. 993 994 995 996 997 998 999 993 994 995 996 997 998 999 3 Percen Cycle Level 2.5 Percen Trend and Oupu s difference 2 2..5..5 -. -2 -.5-3 977 978 98 983 985 987 989 99 -. 977 978 98 983 985 987 989 99 VEC model is esimaed and he srucural componens are derived for each subsample. Each ime he las esimae of rend oupu growh and he cyclical componens are saved. This gives ime series of mos recen PT componens which can be compared wih he final esimae. Moreover, for each of he subsamples rends and cycles are sored for he pas ime span. These esimaes indicae he variabiliy of pas business cycle esimaes and wheher he judgemen abou pas business cycles varies wih he occurrence of new informaion. Theoreically, he above measures of reliabiliy overesimae he robusness of he esimaors, because in realiy, no only he model s parameer change, bu also he recen daa poins are revised by he saisical offices. Since his is a problem each rend-cycle measure has o face and usually such revisions are no regularly documened for European ime series, his is no furher invesigaed here. In he upper panel, he wo graphs show end-of-he-subsample esimaors of he cycle and rend. The rend and cycle componens are quie sable a he end of each subsample. In 993, he cyclical downurn was somewha overesimaed in absolue erms. The final esimae for he full sample is no ha pessimisic in ha ime span. The rend esimaion is relaively more reliable han he cycle. For he whole ime span under consideraion he recursive esimaes don deviae far from he final one. Concerning he judgemen of pas business cycles he recursive esimaes are quie sable and give an impression of a quie sable business cycle measure. Bu again, he rend measures show less variabiliy han heir cyclical counerpars. 359
6. The Relaive Imporance of Permanen and Transiory Shocks To rea he quesion of how imporan he wo groups of srucural shocks are, a forecas experimen is underaken. The model predics fuure oucomes of he variables based on pas informaion. Fuure shocks ha hi he economy lead o forecas errors. The saisic derived below shows how much variance can be aribued o permanen and ransiory shocks. I provides a naural measure of he relaive imporance of shocks. Saring poin o derive he forecas error decomposiion is he MA represenaion of he VEC model X = τ + A(L) ε. A projecion h periods ino he fuure gives he forecas error FE = = A ε + A ε + K + A ε, + h h + h h + h + h + h h + so forecas errors are funcions dependen on fuure shocks ha hi he economy. The variance of his forecas error and is decomposiion ino a permanen and ransiory par is given in he following. FEV = E ( +h +h ) ( +h +h ) +h h = A i E(ε +i ε +i )A i ' i= h α α ' = A i (α γ )( ) E (ε +i ε' +i )( ) (α γ )' A i ' i= γ γ h Ω p = (A i p A i r ) E( ) (A i p A i r )' Ω r From he second o he hird line, he add-up resricion of he permanen and ransiory componens was imposed. In he hird line i is again assumed ha permanen and ransiory shocks are uncorrelaed. Wriing his ou gives Figure 3 Relaive forecas error variance of oupu aribuable o permanen shocks Percen, oupu in levels Percen, oupu in s difference 75 75 5 5 25 25 5 5 2 25 3 Quarers 5 5 2 25 3 Quarers 36
FEV + h h = h p p p r r r A i Ω A i + A i Ω A i i= p r FEV + FEV + h h + h h = Hence, he uncorrelaedness of he srucural shocks implies ha he forecas error variance can be decomposed ino he forecas error variance of he permanen and ransiory par addiively. In he lieraure, ofen he relaive imporance of he permanen and ransiory shocks is invesigaed. This is simply he proporion of he forecas error variance due o permanen shocks from he overall oupu variance. So far, he forecas error variance has been decomposed for he firs difference of he endogenous variables. These values converge o he variances of he rend and cycle in he long-run so he forecas error variance decomposiion should end o replicae he variance raios of he hisorical decomposiion of oupu in he second panel of figure asympoically. In comparable sudies, ofen variance decomposiions are ofen applied for a furher invesigaion of he relaive imporance of permanen and ransiory shocks. In deail, i is esed how high he relaive forecas error variance of he permanen shocks is a business cycle frequencies, for example up o 2 quarers. The forecas error variances in levels are obained by simply cumulaing he appropriae muliplier marices. The empirical forecas error decomposiions for he Euro-area oupu in levels and firs differences are presened in figure 3. The figures show he proporion of forecas error variance of oupu ha is due o a permanen shock in he iniial period. Confidence bands are calculaed via boosrap wih,5 replicaions. The relaive forecas error variance of oupu in levels aribuable o he permanen shocks is displayed in he lef graph. Since he permanen shocks are by definiion he only ones ha have a long-run effec on oupu, he forecas error variance should be explained fully by he permanen shocks in he long-run. Hence, a he end of he simulaion horizon, he relaive forecas error variance should be percen. This is he case in he figure, where he relaive variance has a endency owards he upper bound. Bu for business cycle analysis he higher shor- or medium run frequencies are more of ineres. Here, wihin a ime span up o five years or 2 quarers, he permanen shocks accoun for a lo of he forecas error variance. For example, afer five years he relaive forecas error variance of he permanen shock is approximaely 8 percen increasing from nearly 3 percen in he iniial period. Hence, in accordance wih he above resuls from he hisorical decomposiion, he permanen shock plays a significan role a business cycle frequencies. The relaive forecas error variance in firs differences converges o he relaive variances of he permanen par of oupu very fas. The resuls show ha afer a shor ime period of adjusmen, he permanen shocks accoun for 28 percen of oupu variabiliy so one fourh of oupu variaions can be aribued o flucuaions of he rend and hree fourhs are aribuable o cyclical flucuaions. 7. Conclusions The empirical resuls give an impression of Euro-area business cycle flucuaions. The measured rend of oupu in he Euro-area has some variabiliy. According o he empirical resuls, more han one fourh of oupu variabiliy is due o variabiliy of he rend. When oupu decreases in a cyclical downurn he rend may move in he same direcion. Hence, he variabiliy of he rend is higher han ha of a simple linear deerminisic rend. On he oher hand, permanen shocks don accoun for all of he oupu flucuaions. So he resuls obained here doesn suppor exreme views of he business cycle. If one equals he permanen or rend par of oupu wih he supplied producion of he economy, one has o conclude he supply side of he economy is quie flexible bu far away from fully explaining oupu flucuaions. Daa Appendix In he daa se used in his paper, ime series for GDP, privae consumpion, gross fixed capial formaion, a shor-erm ineres rae and a leading indicaor are employed. Since daa limiaions are presen especially for he Euro-area, his appendix explains how he various ime series are obained. The used ime series are quarerly and seasonally adjused excep he ineres rae. The sample range is from 977: o 999: 4. The main daa source for aggregaed Euro-area naional accouns daa is Eurosa. This insiuion provides ime series in accordance wih he European Sysem of Naional Accouns. Unforunaely, he ime series of he quarerly naional accouns are only available from 99 up o now. This daa series would imply only a very small sample size for economeric esing. To ge longer ime series and herefore increase he degrees of freedom, one soluion of his problem is o aggregae naional ime series. For he naional accouns 36
ime series, we use daa from OECD, Main Economic Indicaors. Since hese ime series are measured in naional currencies, he ime series of he member counries mus be convered. Aggregaion requires a conversion of each member counries GDP ino a single currency. Here, he mehod of he ECB ha uses fixed exchange raes o aggregae pas money daa is employed (ECB, 999, 42). Oupu a PPP exchange raes for 997 of he OECD are used o consruc he weighs of he naional series. A deailed discussion of he alernaive weighing schemes can be found in Fagan and Henry (999). The resuling series is hen linked wih he GDP series of Eurosa. For esimaion, he GDP and consumpion ime series are ransformed ino naural logarihms. The OECD leading indicaor for he Euro-area is already available for a sufficien ime span and is direcly used for esimaion afer aking logarihms. The money marke rae is he 3-monh deposis ineres rae provided by he ECB. The series sars in he firs quarer of 994. A longer ime series can be generaed by aggregaion again. Naional shor-erm ineres raes are provided by he IMF, Inernaional Financial Saisics. We use he money marke rae (line 6B).The weighs for aggregaion are he same as for he naional accouns daa. The resuling series is linked wih he 3-monh deposis ineres rae. References Asley, M., T. Yaes (999): Inflaion and real disequilibria. Bank of England Working Paper 3. Blanchard, O., and S. Fisher (989): Lecures on Macroeconomics. Cambridge. Blanchard, O., and D. Quah (989): The Dynamic Effecs of Aggregae Demand and Supply Disurbances. In: American Economic Review, 79 (4), 655 673. Camba-Méndez, G., and D. Palenzuela (2): Assessmen crieria for oupu gap esimaes. ECB Working Paper 54. Duspaquier, C., A. Guay, and P. S.-Aman (999): A Survey of Alernaive Mehodologies for Esimaing Poenial Oupu and he Oupu Gap. In: Journal of Macroeconomics, 2, 577 595. European Cenral Bank (999): Euro area moneary aggregaes and heir role in he Eurosysems moneary policy sraegy. In: ECB Monhly Bullein, February, 29 46. Evans, G., and L. Reichlin (994): Informaion, Forecass, and Measuremen of he Business Cycle. In: Journal of Moneary Economics, 33, 233 254. Fagan, G., and J. Henry (998): Long-run Money Demand in he EU: Evidence for Area-wide Aggregaes. In: Empirical Economics, 23 (3), 483 56. Funke, M. (997): Supply Poenial and Oupu Gaps in Wes German Manufacuring. In: Inernaional Journal of Forecasing, 3, 2 222. Hansen, H., and K. Juselius (995): CATS in RATS Coinegraion Analysis of Time Series. Evanson. Johansen, S. (995): Likelihood-based Inference in Coinegraed Vecor Auoregressions. Oxford. Johansen, S. (99): Esimaion and Hypohesis Tesing of Coinegraing Vecors in Gaussian Vecor Auoregressive Models. In: Economerica, 59, 55 58. Johansen, S. (988): Saisical Analysis of Coinegraing Vecors. In: Journal of Economic Dynamics and Conrol, 2, 23 254. Kimball, M. (995): The quaniaive analyics of he basic neomonearis model. In: Journal of Money, Credi, and Banking, 27, 24 278. King, R., C. Plosser, J. Sock and M. Wason (99): Sochasic Trends and Economic Flucuaions. In: American Economic Review, 8 (4), 89 84. Lippi, M., and L. Reichlin (994): Diffusion of echnical change and he decomposiion of oupu ino rend and cycle. In: Review of Economic Sudies, 6, 9 3. Neusser, K. (99): Tesing he Long-run Implicaions of he Neoclassical Growh Model. In: Journal of Moneary Economics, 27, 3 37. Oserwald-Lenum, M. (992): A Noe wih Quaniles of he Asympoic Disribuion of he Maximum Likelihood Coinegraion Rank Tes Saisics. In: Oxford Bullein of Economics and Saisics, 54 (3), 46 47. Yang, M. (998): On idenifying permanen and ransiory shocks in VAR models. In: Economics Leers, 58, 7 75. Yun, T. (996): Nominal price rigidiy, money supply endogeneiy, and business cycles. In: Journal of Moneary Economics, 37, 345 37. 362
Zusammenfassung Trend und Zyklus im Euroraum: Eine permanen-ransiorische Zerlegung uner Verwendung eines koinegrieren VAR-Modells Der Beirag unersuch den Konjunkurzyklus im Euroraum uner Verwendung eines mulivariaen Zeireihenmodells mi Koinegraion. Die Koinegraionsresrikionen helfen, permanene und ransiorische Schocks zu idenifizieren. Die permanenen Schocks bilden den sochasischen Trend, während die ransiorischen Schocks den zyklischen Teil der Produkion besimmen. Die Idenifikaion erlaub eine hisorische Zerlegung der Produkion im Euroraum in Trend und Zyklus. Zudem wird mi Hilfe von Prognosefehler- Varianzzerlegungen die Bedeuung der srukurellen Schocks unersuch. Die Ergebnisse zeigen, dass die permanenen Schocks einen signifikanen Aneil an den Schwankungen in der Produkion haben, so dass der Trend der Produkion im Euroraum eine deuliche Variabiliä aufweis. 363