Common stochastic trends, cycles and sectoral fluctuations: a study of output in the UK

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1 Commo sochasic reds, cycles ad secoral flucuaios: a sudy of oupu i he UK Ahoy Garra (Bak of Eglad) ad Richard G. Pierse (Uiversiy of Surrey) February 1996 (revised May 1996) Absrac Two aleraive mehodologies are compared for ideifyig commo reds ad cycles i a se of variables. Oe, followig Harvey, uses a uobserved compoes srucural ime series model. The oher, followig Vahid ad Egle, is based o a mulivariae Beveridge- Nelso decomposiio. Boh approaches are applied o a four secor model of oupu i he UK over he period 1970Q1-1993Q2, producig differe reds ad cycles. The cycles from he uobserved compoes model correspod more closely o a referece cycle for aggregae oupu. JEL Classificaio: C32, E32. Keywords: Coiegraio, commo reds, commo cycles, cofeaures, dissaggregaio. The auhors would like o hak Rob Egle, Clive Grager, Adrew Harvey, João Issler, Siem Koopma, Hashem Pesara, Neil Shephard, Farshid Vahid, ad paricipas a he Seveh World Cogress of he Ecoomeric Sociey i Tokyo for helpful commes. All errors ad omissios are he sole resposibiliy of he auhors.

2 1 Iroducio The idea of decomposig a variable io red ad cycle compoes has a log hisory i ecoomics ad he variable ha has received he mos aeio is oupu. Sice he iflueial work of Nelso ad Plosser (1982), i has geerally bee acceped ha oupu has a ui roo so ha he red is sochasic raher ha deermiisic ad he cycle is prese i he saioary firs differece. Several uivariae empirical sudies have followed: Campbell ad Makiw (1987) esimaed some simple ARIMA processes for US GNP while Harvey (1985), Waso (1986) ad Clark (1987) all used Uobserved Compoes (UC) models o ideify red ad cycle. 1 Oher sudies have adoped a mulivariae approach, usig addiioal iformaio from oher aggregae variables o help ideify he red ad cycle i aggregae oupu; for example Kydlad ad Presco (1988) dereded US GNP ad he price level ad he showed ha here is a egaive relaioship bewee he resulig cycles. For oher examples see Clark (1989), Evas (1989), Blachard ad Quah (1989), Kig, Plosser, Sock ad Waso (1991) ad Evas ad Reichli (1994). Aoher source of addiioal iformaio is ha provided by secoral oupu daa which migh for heoreical reasos be expeced o move ogeher. Log ad Plosser (1983) develop a mulisecoral versio of a real busiess cycle model ad show ha, eve if radom produciviy shocks are idepede across secors, he choices of ages will cause comoveme of aciviy measures for differe secors. Log ad Plosser (1987) decompose US oupu iovaios io uobserved commo facors or aggregae shocks ad a se of idepede disurbaces uique o each secor. Their resuls sugges ha he explaaory power of a commo aggregae disurbace for idusrial oupus is sigifica, bu are o very large for mos idusries. More geerally he imporace of secoral iformaio has bee show, for example, by Pesara, Pierse ad Lee (1993) ad Lee, Pesara ad Pierse (1992) who esimae mulisecoral VAR models of oupu growh for he US ad he UK respecively ad ivesigae he effecs of specific ideified macroecoomic shocks ad uideified secoral shocks o oupu persisece. The joi aalysis of more ha oe variable allows he possibiliy ha reds ad cycles may be commo bewee variables. I fac, as demosraed by Sock ad Waso (1988b), if a se of variables are coiegraed wih r coiegraig vecors, he his implies ha here are -r commo reds bewee hem. I a rece paper, Egle ad Kozicki (1993) have iroduced he geeral cocep of commo feaures, which are defied o be daa feaures ha are prese i idividual series bu abse from some liear combiaio of hose series. Coiegraio is oe example bu aoher commo feaure of paricular ieres is ha of commo serial correlaio ad his implies commo cycles. Egle ad Kozicki develop a es for he cofeaure rak which is aalogous o he Johase (1988) es for he umber of coiegraig vecors. If he cofeaure rak is s, he his implies -s commo feaures. Vahid ad Egle (1993) use he framework of he Beveridge-Nelso-Sock-Waso (BNSW) decomposiio o ideify commo reds ad cycles. Whe he umber of coiegraig vecors plus he umber of commo serial correlaio feaures is equal o he umber of variables, he his framework allows a very easy recovery of red ad cycle compoes. Egle ad Issler (1995) apply his approach o secoral oupu for he US ad Calcagii (1995) o labour produciviy for 6 differe couries. A aleraive approach is provided by he mulivariae UC models of Harvey (1989) ad Koopma e al. (1995) which allow for he imposiio of commo reds ad cycles. Oe disicio bewee he wo approaches lies i he ideifyig resricios used o cosruc he red ad cycle. The UC approach esures ideificaio by imposig zero correlaio bewee red ad cycle whereas i Egle ad Vahid s approach (EV) his corelaio ca be o-zero ad ca acually be esimaed. Oher sigifica differeces lie i he flexibiliy each approach allows for he form ha he red ad cycle ca ake. The UC model allows he red o have a ime varyig slope, whereas he red i he EV 1 A useful survey of his lieraure is provided by Sock ad Waso (1988a). 1

3 model is a radom walk wih drif. However he cycle i he EV model comprises all serial correlaio oce he red compoe has bee removed, whereas he UC cycles used by Harvey have a explici rigoomeric represeaio ad are cosraied so ha he cycle for each variable has he same frequecy ad dampig facor. I his paper hese wo approaches are compared ad boh are applied o a four secor model of quarerly oupu for he UK. The applicaio provides a compariso wih he resuls of Egle ad Issler (1995) who apply he EV model o US secoral daa usig 8 secors ad aual daa. A pla of he paper is as follows: Secio 2 develops boh he EV ad UC models. Secio 3 applies boh models o a four secor model of UK oupu over he period 1970Q1 o 1993Q2. The reds ad cycles ha arise from he wo aleraive mehodologies are compared. Secio 4 preses he coclusios. 2 Mehodology The sarig poi of his aalysis is he red plus cycle model y = µ + ψ, = 1,, T (2.1) where he 1 vecor of variables y is decomposed io wo compoes: a red, µ, ad a cycle, ψ. This represeaio of a series as he sum of a red ad a cycle has some iuiive appeal ad has a log hisory i saisical modellig. However, here is o uique way o accomplish he decomposiio ad so several differe mehods have bee suggesed, based o differe assumpios, ad producig models wih quie differe properies. Two approaches are cosidered here: oe by Egle ad Vahid, heceforh deoed as EV, based o he Sock-Waso-Beveridge-Nelso decomposiio, he oher from a mulivariae Uobserved Compoes model of Harvey (1989) ad Koopma e al. (1995), heceforh deoed as UC. 2.1 The EV model Sock ad Waso (1988a) provide a mulivariae geeralisaio of he Beveridge ad Nelso (1981) decomposiio of a ARIMA model io red ad cycle compoes where he iovaios i he wo compoes are perfecly correlaed. The sarig poi is he ifiie movig average (Wold) represeaio of he saioary firs differece of y : y = C( L) e = C( 1) e + ( 1 L) C ( L) e where, i he secod equaliy, he log ru effec of shocks has bee separaed from he res. Iegraig up his equaio defies he Sock-Waso-Beveridge-Nelso (SWBN) decomposiio: y = C( 1) e + C ( L) e i = 0 i. (2.2) I (2.2) he firs erm is he red ad he secod erm, which is saioary, is he cycle. Iovaios i he wo compoes are perfecly correlaed. 2.2 The Uobserved Compoes Model I coras, i he uobserved compoes srucural models of Harvey (1985, 1989) ad Koopma e al. (1995) he iovaios i red ad cycle are cosruced o be idepede. The sarig poi is slighly more geeral ha (2.1) i ha i allows a addiioal compoe i he equaio: 2

4 y = µ + ψ + ε, = 1,, T (2.3) where ε is a irregular compoe wih covariace marix Σ ε which is o explaied by he model. Separae uobserved compoes models are buil of each of he firs wo compoes of (2.3). The red compoe is modelled by he local liear red µ = µ 1 + β 1 + η (2.4) β = β 1 + ξ (2.5) where µ ad β are 1 vecors represeig he level ad slope of he red respecively, ad η ad ξ are idepede error processes wih covariace marices Σ η ad Σ ξ. I he geeral case he red process is secod order iegraed, (I( 2 )). The slope parameer β allows his red o chage smoohly bu i he special case where Σ ξ is zero, he red reduces o a radom walk wih drif erm β = β = β. If, i addiio, Σ η is zero, he he red becomes deermiisic. 1 The cycle compoe of he model is rigoomeric i form ad cosiss of oe or more cycles defied by he pair of equaios: ψ ψ = ρ cosλ I si λ I si λ I cosλ I ψ ψ 1 1 ω + ω (2.6) where ψ ad ψ are 1 vecors ad ω ad ω are vecor error processes idepede of ε, η ad ξ ad wih he same covariace marix Σ ω = Σ, I is he ideiy marix of dimesio ad he vecor process ψ appears by cosrucio. The scalar parameers ρ ad λ (which saisfy he resricios 0< ρ < 1 ad 0< λ < π ) represe he cycle dampig facor ad frequecy respecively. I he uivariae case ( = 1), equaio (2.6) correspods o a resriced ARMA(2,1) process where he wo auoregressive roos form a complex cojugae pair. 2 Whe here is more ha oe variable, i ca be see from (2.6) ha he cycle dampig facor ad frequecy ρ ad λ are imposed o be he same for each variable. I he ermiology of Koopma e al. (1995), his is he assumpio of similar cycles, ad i implies ha he cycles for differe variables have he same properies - he same auocovariace fucio ad specrum. This is a very srog assumpio ad imposes quie a serious resricio o he model, alhough i does have he advaage of limiig he umber of parameers ha have o be esimaed. ω 2.3 Coiegraio ad Cofeaures Suppose ha he y process i (2.1) is coiegraed, havig r coiegraig vecors give by he ( r) marixα. The, by he defiiio of coiegraio, he vecorα y will be saioary. (See Egle ad Grager (1987)). Furhermore, Sock ad Waso (1988b) showed ha his implies ha he variables share -r commo reds. 2 I is possible o model more complex dyamics by allowig several cycles of differe frequecies, hus allowig a more geeral specificaio for he cycle. This is o pursued furher here. 3

5 Egle ad Kozicki (1993) iroduced he cocep of commo feaures, which are daa feaures ha are prese i idividual series bu abse from a liear combiaio of hose series. I paricular Vahid ad Egle (1993) looked a he feaure of commo serial correlaio amog variables which, if i exiss, implies commo cycles. They used a es developed by Egle ad Kozicki o es he cofeaure rak. This es is based o caoical correlaio aalysis alog he lies of he Johase (1988) es for he umber of coiegraig vecors. If he serial correlaio cofeaure rak is s, he his implies -s commo cycles. 2.4 Commo Treds ad Cycles i he EV Model I equaio (2.2) coiegraio implies haα C( 1 ) = 0 so ha C( 1 ) ca be wrie as he produc C( 1 ) = δ γ where δ ad γ are boh ( r) marices ad α δ = 0. The he firs erm i (2.2) becomes C( 1) e = δ γ e = δ τ (2.7) i i = 0 i = 0 i whereτ is a vecor of commo reds ha follows he radom walk process τ = τ 1 + γ e. Similarly, if he secors exhibi commo serial correlaio feaures he, for some ( s) marix of commo feaures, α ~, i follows ha ~ α C ( L) = 0 ad C ( L) = Φ λ ( L) where Φ is a ( s) marix of coefficies wih α ~ Φ = 0 ad λ ( L ) is a ( s ) marix i he lag operaor. The he secod erm i (2.2) becomes C ( L) e = Φ λ ( L) e = Φ c (2.8) where c is a vecor of commo cycles. The se of cofeaure vecors α ~ mus be liearly idepede of he coiegraio vecors α ad i has o be he case ha r + s. If r + s < he he red ad cycle decomposiio i he Vahid ad Egle framework is o uique. However, i he special case ha r + s =, he marix α ~ A = α is square ad of full rak, wih a coformably pariioed iverse defied by A 1 = [ α ~ α ] ad i follows ha, y = A 1 A y = ~ ~ α α y + α α y = Pδ τ + ( I P) Φc (2.9) where P ~ α α ~ is a idempoe projecio marix ad α α I P. Thus he codiio r + s = allows a simple ad uique decomposiio ha is idepede of he ormalisaio of he coiegraio ad cofeaure vecors. 4

6 2.5 Commo Treds ad Cycles i he UC Model Commo reds i he geeral UC model of (2.4)-(2.6) ca arise eiher hrough commo levels, or commo slopes or boh. For compariso wih he EV model, here oly he special case is cosidered where he slope parameer is fixed so ha Σ ξ =0 ad β = β = β. I his case, he red ad hece y will be iegraed of order oe ad commo reds implies ha µ = Θ µ ~ µ + µ 0 (2.10) where µ ~ is he ( r 1 1) vecor of commo reds, µ 0 is a vecor of fixed values ad Θ µ is a ( r) facor loadig marix saisfyig he resricioα = 0. The variace covariace marix Σ η will also be sigular. Similarly, he exisece of commo cycles i (2.6) implies ha Θ µ ψ = Θ ψ ψ~ (2.11) where ψ ~ is he ( s 1) vecor of commo cycles ad he ( s) facor loadig marix Θ ψ saisfies he resricio α ~ Θ ψ = 0. Noe ha o vecor of fixed values is eeded here because he cycle has zero mea. The sysem (2.3)-(2.6) ca he be rewrie i erms of he commo reds ad cycles as ~ ~ 0, = 1,, T y = Θ µ µ + µ + Θψ ψ + ε ~ ~ ~ µ µ β η ~ = (2.12) ψ ~ ~ ψ = ρ cosλ I si λ I si λ I cosλ I ~ ψ ~ ψ 1 1 ~ ω ~ + ω where he error processes η ~, ω ~ ad ω ~ are of dimesios r 1, s 1, ad s 1 respecively wih osigular covariace marices Σ ~ η ad Σ = Σ. β ~ is a r 1 vecor of ω~ ~ ω drif erms i he commo reds. The sysem (2.12) is he versio of he UC model wih commo reds ad cycles ha is used i he esimaio i Secio 3. The facor loadig marices Θ µ ad Θ ψ i (2.12) are o uiquely defied uless some codiios are imposed o esure ideificaio. The sadard ideificaio resricios impose lower riagulariy so ha Θ ij = 0, for j > i ad Θ ii = 1, for i, ad he associaed covariace marices Σ η ~ ad Σ ~ ε are diagoal. The las codiio esures ha he commo reds (cycles) are all ucorrelaed wih each oher ad he firs wo codiios imply ha oly he firs commo red (cycle) affecs he firs secor, he firs wo commo reds (cycles) he secod secor ad so o for he firs r ( s ) of he secors. These resricios merely esure ideificaio; oce he model parameers have bee esimaed, he commo reds ad cycles ca he be rasformed by premuliplicaio by ay orhogoal marix. This is called facor roaio ad may allow he rasformed commo facors o be give a more useful ierpreaio. 5

7 3. The Resuls The mehodologies oulied i he previous secio are applied o model Gross Domesic Produc (GDP) for he Uied Kigdom disaggregaed io four secors (lised i order of size wih heir 1994 weigh i pareheses): Services (629), Producio (280), Cosrucio (72) ad Agriculure (19). These secors represe he highes level breakdow of he UK Sadard Idusrial Classificaio. The daa come from Table 2.5 of he Ceral Saisical Office (CSO) Blue Book; he observaios are quarerly seasoally adjused idices a cosa facor cos (1990=100) for he period 1970Q1 o 1993Q2 (94 observaios). 3.1 Time series properies of he daa The daa are ploed i aural logarihms as he ubroke lies i Figures 1 ad 2, ad all aalysis uses he logarihmic rasformaio. The larges secor, Services, is he mos smoohly reded series ad he oe ha mos closely resembles aggregae GDP (of which i forms 63%). Producio is oiceably more volaile ha Services ad shows a flaer rajecory for he 1970s, wih clear falls i oupu i he recessio of he early 1980s. 3 Cosrucio is eve more volaile, ad despie havig he geeral appearace of a upward red, shows periods of sharp declie i he lae 1970s ad early 1980s. The smalles secor, Agriculure, is clearly reded bu displays some volailiy. Table 1: Orders of Iegraio: Augmeed Dickey-Fuller Tess Secor Levels Firs differeces Services Producio Cosrucio Agriculure Noes: The firs colum is he 4h order Dickey-Fuller saisic, ADF(4), wihou ime red, he secod wih ime red. The 95% criical value wihou ime red is -2.87, ad wih ime red is The sample period is 1970Q1-1993Q2. The firs sep i a more formal aalysis is o es he order of iegraio of he series hrough augmeed Dickey-Fuller ess. A likelihood raio es o a uresriced VAR suggesed a opimal lag legh of 4 ad all remaiig ess are coruced usig his assumpio. Table 1 repors he resuls of ADF(4) ess for each secor boh i levels ad firs differeces, ad compued boh wih ad wihou a icluded ime red. I all cases he secors fail o rejec a ui roo i levels. I firs differeces, he ui roo ull hypohesis is clearly rejeced i all secors excep Services, which appears o be borderlie I(2). However, ADF saisics for lag leghs oe hrough hree for his secor clearly rejec a ui roo i firs differeces as do ess compued usig loger sample periods. All four secors are hus ake o be I(1). Turig o examie he coiegraig properies of he daa usig he Johase mehodology, he firs sep is o ideify he appropriae model. A LR es of wheher o iclude a cosa i he VAR (equivalely a ime red i he levels formulaio) gave a value of 27.8 (4 degrees of freedom) which rejecs he ull of o cosa. A furher es of he resricio ha he cosa is prese oly iside he error correcio erm gave a LR saisic of 21.2 which rejecs he resricio. The model chose hus icluded a uresriced cosa i he VAR. 3 The wo sharp falls i he 1970s correspod o a miig srike i 1972Q1 ad he hree day week of 1974Q1. To ake accou of hese wo eves, wo dummy variables were creaed ad used i all subseque aalysis. 6

8 Table 2a. Coiegraio Aalysis: Johase Tess Null Aleraive Tes saisic 95% Criical Value 90% Criical Value r = 0 r = r <= 1 r = r <= 2 r = r <= 3 r = Noes: 90 observaios from 1971Q1o 1993Q2. Maximum lag i he VAR=4. Uresriced iercep model. Coiegraio LR es based o maximal eigevalues of he sochasic marix. The resuls of he Johase aalysis are repored i Table 2a. They idicae oe coiegraig vecor ad hece hree commo sochasic reds. The possibiliy of commo (sychroous) cycles was he ivesigaed usig he cofeaure es described i Egle ad Kozicki (1993). This is based o he caoical correlaios of he firs differeces of he daa wih heir lags (four) ad he error correcio erm lagged oce. The squared caoical correlaios ad he value of he es saisic for he umber of cofeaure vecor are repored i Table 2b. Table 2b. Cofeaure Aalysis: Vahid ad Egle ess Null Squared Tes saisic Degrees of p-value correlaio freedom s > s > s > s > Noes: 90 observaios from 1971Q1 o 1993Q2. Maximum lag i he VAR=4. A he 5 per ce level of sigificace, he es suggess wo cofeaure vecors implyig wo commo cycles. The coclusio of he daa aalysis is herefore ha he four secors exhibi hree commo reds ad wo commo cycles. This is a problem sice i does o saisfy he resricio ( r + s = ) ha allows he simple Vahid ad Egle decomposiio ad i order o make a compariso bewee he wo approaches we are herefore obliged o violae oe of he resuls of he daa aalysis. We chose o accep he resuls of he coiegraio aalysis bu o impose a sigle commo cycle. This is cosise wih Egle ad Issler (1995) who fid may secor specific reds bu relaively few cycles for he US. All subsque aalysis is herefore based o he assumpio ha he four secors exhibi hree commo reds ad oe commo cycle Cosrucig Secoral Treds ad Cycles This secio compares he ses of secoral reds ad cycles which resul from he wo models oulied i Secio 2, cosruced imposig he resricio of hree commo reds ad oe commo cycle. I boh models, he red is a radom walk wih drif, eablig a direc compariso of he wo approaches. 4 We also esimaed he UC model wih he daa cosise commo facor resricios of hree commo reds ad wo commo cycles. The resuls were very similar o hose repored i he ex. 7

9 The mai differeces bewee he wo approaches are he ideifyig assumpios ad he defiiio ad specificaio of he cycle EV Model Resuls Figure 1 plos he EV reds alog wih he acual series. The reds i all secors fi reasoably closely wih he possible excepio of Services where he series iself is relaively smooh bu he esimaed red is much more variable. More geerally, as was foud by Egle ad Issler (1995) for US secoral oupu, he esimaed reds are more volaile ha he acual series, which is arguably a udesireable feaure. 6 Figure 3 graphs he esimaed EV cycles. As here is oly oe commo cycle, he cyclical paer is he same i all secors alhough he ampliudes are differe ad i Cosrucio ad Agriculure he paer is ivered. Cosrucio has he highes variace (0.0020), compared o ad for Producio ad Services respecively. This rakig is cosise wih he coveioal view of Cosrucio as he mos volaile secor. Agriculure has a very small variace ( ) which is a lile surprisig. A ieresig quesio is how far do he secoral cycles mach cyclical movemes i aggregae ecoomic aciviy. For he UK, here is o esablished se of daed cycles or urig pois i aggregae oupu (ulike he NBER referece cycle for he US) so ha a umber of referece pois have o be used. Quah (1994) ideified hree peaks a 1974Q3, 1979Q4, ad 1990Q1. Aris, Blade-Hovell ad Zhag (1994) ideified peaks (P) ad roughs (T) i mohs 1972M2(T), 1973M6(P), 1975M8(T), 1979M6(P), 1981M5(T), 1984M1(P), 1984M8(T), 1989M4(P) ad 1992M5(T). The sychroisaio of hese wo ses of peaks is o exac bu does give a reasoable idicaio of urig pois, which ough o some degree be refleced i he disaggregae secor cycles. 7 Geerally he imig of he peaks ad roughs of he EV cycles does o correspod paricularly well wih he referece pois. For example alhough Services ad Producio do show a marked peak i 1974, his correspods o a rough for Cosrucio ad Agriculure. The laer referece peaks of Quah (1994) are o appare i ay of he secors. Focusig o he wo larges secors, Services ad Producio, he EV cycles sugges big roughs i 1976 ad 1983/4 wih a peak i Noe of hese urig pois maches he referece urig pois. I summary, he EV reds perform reasoably well bu he cycles do o pick up referece urig pois. This may be because of he srog assumpio of a sigle commo cycle ha was required o perform he decomposiio UC Model Resuls. For he UC approach he model (2.12) was esimaed imposig hree commo reds ad oe commo cycle. Table 3 repors he resuls of he maximum likelihood esimaio of he model: he coefficies of he facor loadig marices, Θ µ,θ ψ ad µ 0, ad he sadard deviaios of he error erms o red, cycle ad irregular compoes. 5 The calculaios for he EV model were programmed i he GAUSS laguage while he UC model resuls come from Versio 5.0 of he compuer package Samp (Koopma e al. (1995)). We are graeful o Adrew Harvey for allowig us access o a pre-release versio of his program. 6 This is aribuable o a egaive covariace bewee red ad cycle. 7 The approximae daig of hese urig pois is also cosise wih he imig of peaks i he CSO coicide idex of ecoomic aciviy. 8

10 Table 3: Maximum likelihood esimaio of UC Model (2.12) Table 3a: Commo Tred Compoes Secor Θ µ µ 0 σ(η) σ(ε) Services Producio Cosrucio Agriculure Table 3b: Commo Cyclical Compoes Secor Θ ψ σ(ω) Services Producio Cosrucio Agriculure Noes: Esimaio period: 1970Q1-1993Q2; Model log-likelihood is ; Cycle Frequecy= Cycle dampig facor ρ = ; Period of cycle: 4.60 years. Θ µ,θ ψ are esimaed coefficies of facor loadig marices of red ad cycle respecively µ 0 are esimaed coefficies of he red fixed values σ(η), σ(ε) ad σ(ω) are sadard deviaios of he error o he red level, irregular ad cyclical compoes respecively. Figure 2 plos he UC secoral reds agais he acual series. 8 The Services secor is characerised by a smooh ear-liear red, similar bu less variable ha he red from he EV model. The reds for Services ad Cosrucio fi he series almos perfecly whereas here is a close bu less perfec fi for Producio ad Agriculure. Two differeces are appare whe comparig hese reds wih hose of Figure 1. Firsly, hey fi he he acual series more closely ha he EV reds. Secodly, i all secors hey show less variabiliy ha he acual series. The rakig of he variace appears cosise wih previous sylised facs wih Table 3a reporig ha he mos volaile red is Cosrucio, wih a compoe sadard error of ad he smoohes red, Services, has he smalles esimaed sadard error of The UC cycles are ploed i Figure 3, alogside he EV cycles. The commo dampig facor, ρ, is ad he cycles have a period of 4.6 years, which correspods quie well wih a sadard busiess cycle period. The variaces of he cycles vary, bu i wo cases are cosiderably smaller ha hose of he EV cycles ad oly i Agriculure are hey larger. Cosrucio has he larges variace ( ), followed by Agriculure ( ), Producio ( ) ad Services ( ). The relaive orderig of hese variaces is more iuiively plausible ha ha from he EV cycles. Figure 4 preses he UC cycles ogeher wih he EV cycles, rescaled so ha hey boh have he same variace. This eables a direc compariso of he imig of he urig pois i ecoomic aciviy implied by he boh ses of cycles. Sice here is oly oe commo cycle i he model ad he loadig marix coefficies are all posiive, all secors i he UC model exhibi he same cyclical paer, ad his is quie similar o he EV cycles for Producio ad Services (alhough he variace is very differe i he case of he Services secor). Coversely, he paer of he UC cycles for Cosrucio ad Agriculure is he mirror image of ha for he EV cycles. As a resul, wih he UC cycles, all 8 All graphed compoes for he UC model are of smoohed Kalma filer esimaes. 9

11 secors show he marked peak i 1974 correspodig o he referece peak classified by boh Aris, Blade-Hovell ad Zhag ad Quah. This is followed by big roughs i 1976 ad 1983/4 ad he peaks i 1985 ad Noe of hese laer urig pois maches he referece urig pois very closely ad hey miss he 1990Q1 peak i aggregae ecoomic aciviy ideified i Quah (1994) or he 1989M4 peak classified by Aris, Blade-Hovell ad Zhag (1994). I he UC model each secor also has a irregular compoe. However, i all cases, Table 3a shows ha his has a exremely small variace compared wih ha of he oher compoes, suggesig ha he red ad cycle explai almos all of Usig he crieria of closeess of fi o he acual series ad he volailiy of he red compared o he series, he reds derived from he UC approach seem more saisfacory ha hose from he EV approach. This is also refleced i he UC cycles whose variaces are more plausible. The peaks ad roughs are he same for all secors ad correspod a lile more closely o he referece urig pois. The UC model also has a irregular compoe 4 Coclusios This paper has compared he Vahid-Egle ad Uobserved Compoes mehodologies for decomposig a se of variables io red ad cycle compoes, allowig for commo reds ad commo cycles. The approaches were applied o a four secor disaggregaio of oupu for he UK, wih hree commo reds ad oe commo cycle. The resuls, derived usig models wih he same red specificaio, bu wih differe ideifyig resricios, geerae reds ad cycles wih quie differe properies, wih he UC reds fiig he origial series more closely ad he cycles havig he same paer bu differe variaces, wih peaks ad roughs machig some of a se of referece urig pois. The resuls provide evidece o he relaive imporace of permae versus rasiory shocks. For he larges secor Services ad also for Cosrucio ad Agriculure, rasiory shocks domiae permae shocks, wih he raio of he sadard errors of he red iovaios o he cycle iovaios beig 0.04, 0.81 ad 0.46 respecively. However, for he Producio secor, which forms 28% of aggregae GDP, he raio is 1.47 so he mos impora iovaios i Producio oupu are permae i aure. These resuls show ha, o he whole, rasiory shocks are more impora ha permae shocks alhough permae shocks have a sigifica role i oe impora secor. These raios are larger ha he value of 0.9 repored by Clark (1987) usig he UC framework applied o aggregae US GDP bu cosiderably lower ha raios i he rage of 5 o 6 repored by Nelso ad Plosser (1982) usig a Beveridge-Nelso decomposiio. Egle ad Issler (1995), lookig a secoral US GDP, are less clear o he relaive imporace of permae versus rasiory shocks. The focus of his paper has bee deliberaely arrow. A ieresig quesio would be o ivesigae he exe o which he commo reds ad cycles which we fid ca be associaed wih specific macroecoomic eves such as oil price, produciviy ad omial moey shocks. This will be he subjec of furher research. 10

12 Figure 1: Secoral Treds from he Vahid ad Egle Model Figure 2: Secoral Treds from he Uobserved Compoes Model 11

13 Figure 3: Secoral Cycles from he EV ad UC Models Figure 4: Scaled Secoral Cycles from he EV ad UC Models 12

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