ÁREA 3 - MACROECONOMIA, ECONOMIA MONETÁRIA E FINANÇAS

Transcription

1 ARE BUSINESS CYCLES ALL ALIKE IN EUROPE? Márco Anôno Salvao João Vcor Iler ℵ Angelo Mon alverne Duare Reumo: Invegamo nee argo e o cclo econômco êm um comporameno emelhane a parr do cálculo do cuo de bem-ear do cclo econômco para a Unão Européa como uma olução do problema propoo por Luca (987. a paíe foram ecolhdo porque pouem uma longa radção de negração e comérco, e comporando com um expermeno naural para nvegar o quão mlar ão o cuo de bem-ear do cclo econômco enre o paíe. Admndo preferênca do po CES e uma forma reduzda razoável para o conumo, emamo o cuo de bem-ear uando rê méodo alernava para decompoção endênca-cclo, conudo focando o exercíco obre a decompoção de Beverdge-Nelon mulvarada. Noo reulado moram que o cuo de bem-ear ão muo dferene enre o paíe da Unão Européa e enre ee e o Eado Undo, endo uma fore evdênca de que o cclo econômco não êm comporameno emelhane na Europa. Clafcação JEL: E3, C3, C53 Palavra-chave: Cclo Econômco, Cuo de Bem-ear, Conegração, Decompoção de Beverdge- Nelon. Abrac: We nvegae wheher bune cycle are all alke compung he welfare co of bune cycle for European-Unon (EU a he oluon of he problem propoed by Luca (987. Becaue hee counre have a long radon of negraon and rade, a naural expermen o nvegae how mlar her welfare co of bune cycle are. Ung andard aumpon on preference and a reaonable reduced form for conumpon, we compued welfare co ung hree alernave rend-cycle decompoon mehod, bu focung on he mulvarae Beverdge-Nelon decompoon. Our reul how ha welfare co are very dfferen acro EU counre and beween US and EU counre, and hu a rong evdence ha bune cycle are no alke n Europe. JEL Code: E3, C3, C53 Keyword: Bune cycle, welfare co, conegraon, Beverdge-Nelon decompoon. ÁREA 3 - MACROECONOMIA, ECONOMIA MONEÁRIA E FINANÇAS PUCMna e EPGE/FGV-RJ, alvao@pucmna.br ℵ EPGE/FGV-RJ BACEN

2 Are Bune Cycle All Alke n Europe?. Inroducon A dcuon wh a long radon n macroeconomc on wha generae bune cycle. Accordng o one vew, whch we label he nuonal vew, bune cycle are generaed by large and nfrequen hock ha h macroeconomc varable, leadng hem o flucuae abou her rend. Becaue nuonal eng vary from counry o counry, hee hock are dfferen acro counre and bune cycle are no all alke. Accordng o a dfferen vew, whch we label he dynamc-ochac-general-equlbrum DSGE vew, bune cycle are generaed by mall and frequen whe-noe hock ha h macroeconomc varable, whch have a dynamc pah qualavely well approxmaed by a dynamc ochac general equlbrum model. Becaue hee hock are mall (low varance, and here no reaon o beleve ha propagaon mechanm vary from counry o counry, bune cycle are all alke. Indeed, Luca (977, n h openng aemen of h clac paper, ak: Why ha, n capal econome, aggregae varable undergo repeaed flucuaon abou rend, all of eenally he ame characer? Of coure, no rval o nvegae f bune cycle are all alke. Fr, one ha o defne n whch ene hey hould be alke and dfferen way o meaure mlare. A fr approach, followed by Blanchard and Waon (986, o look drecly no hock hemelve, nvegang wheher hey are mall or large, a well a her naure. Uually h done ung a rucural economerc model. Snce here no conenu on how hock denfcaon hould be performed, and everal hock-denfcaon echnque have been crczed on dfferen ground, hard o come ou wh a afacory anwer once h drec approach followed. he horcomng of he drec approach can be overcome f nead of focung drecly on hock, one ue an ndrec approach, focung on a fundamenal dfference n he naure of bune cycle enaled by hee wo ype of hock. A concep ha ha receved ome aenon recenly, and ha can be ued o nvegae wheher bune cycle are alke he welfare co of bune cycle. he dea raghforward: Luca (987 calculae he proporon of exra conumpon, n all dae and ae of naure, a raonal conumer would requre n order o be ndfferen beween an nfne equence of conumpon under uncerany and a ceran equence whch cycle free. h proporon labelled he welfare co of bune cycle, and can be drecly compued ung conumpon daa and a paramerc veron of he uly funcon; ee he varan n Imrohoroglu (989, Obfeld (994, Van Wncoop (994, Akeon and Phelan (995, Pemberon (996, Dolma (998, allarn (, Orok (, and Franco, Gullen and Iler (3. If hock are frequen and mlar acro counre, n whch hey have a low varance, and f he propagaon mechanm mlar n naure o ha n dynamc ochac general equlbrum model, one hould fnd ha he welfare co of bune cycle acro econome are all mlar. However, f nuonal facor are mporan, hock wll be dfferen n naure and he welfare co of bune cycle wll be dfferen acro econome. Of coure, one can alway fnd a e of counre ha have mlar nuonal eng. For hem, fndng mlar welfare co of bune cycle may u be a conequence of mlar nuon. However, f he oppoe rue for h e of counre, hen hard o argue for he DSGE vew. In h paper, we nvegae wheher bune cycle are all alke compung he welfare co of bune cycle for an mporan ube of European counre European-Unon (EU counre: Aura, Belgum,

3 Denmark, Fnland, France, Germany, Grea Bran, Greece, Ireland, Ialy, Luxembourg, Porugal, Span, Sweden and he Neherland. A well known, hee counre have a long radon of negraon and rade dang well before he common-currency Euro area wa mplemened. Becaue of h feaure, a naural expermen o nvegae how mlar her welfare co of bune cycle are, n he ene ha we wll be urpred o fnd maor dfference beween hem. In compung he welfare co of bune cycle for EU counre we ue he echnque n Beverdge and Nelon (98 o decompoe (he log of conumpon n a rend and a cyclcal componen. In h cae, he rend wll be ochac and modeled a a random walk. h choce rele on a zable amoun of economerc evdence avalable on conumpon, or, alernavely, on he amoun of auhor ha have ued he un-roo pecfcaon, e.g., Hall (978, Nelon and Ploer (98, Campbell (987, Kng e al. (99, Cochrane (994, ner ala. Moreover, o make our reul comparable o prevou work, we alo modeled he rend a eher a deermnc lnear proce (wh and whou a break or followng a lowly evolvng ecular proce capured by he Hodrck and Preco (997 fler. Our reul how ha he welfare co of bune cycle are very dfferen acro EU counre. Ung he Beverdge and Nelon decompoon, and plauble value for he rk averon coeffcen and he dcoun rae of fuure uly, we fnd ha he welfare co of Span (4.% of conumpon almo en me ha of he UK (.45% of conumpon medan of.85%. Maor dfference n welfare co are alo found when alernave rend-cycle decompoon mehod are employed, alhough hey are no a pronounced a he one obaned ung he Beverdge and Nelon decompoon. he paper dvded a follow. Secon provde a heorecal and acal framework o evaluae he welfare co of bune cycle. Secon 3 provde he emae ha are ued n calculang hem. Secon 4 provde he calculaon reul, and Secon 5 conclude. here alo an Appendx provdng he economerc background neceary o mplemen he calculaon carred ou n he paper. Luca (987, pp. -3, foonoe explcely conder he pobly ha he rend n conumpon ochac a n Nelon and Ploer (98.

4 . he Problem Luca (987 propoed a way o evaluae he welfare gan of cycle moohng. Suppoe an agen ha chooe a conumpon equence { c } ha maxmze neremporal uly, U, ubec o a budge conran: U E β u ( c ( where E ( E ( Ω he condonal expecaon operaor of a random varable, ung Ω a he nformaon e, and β (, a conan dcoun facor. He worked wh a cla of conumpon ream wh rend and cycle componen uch a: c ( α α exp z,,,k ( z a aonary ochac proce wh a aonary drbuon gven by ln( z ~ N (,. where { } Cycle-free conumpon wll be he equence { c } Noce ha { }, where z α ( α nce exp( z c [ ] E. c he reulng equence when we replace he random varable c wh uncondonal mean. Hence, for any parcular me perod, c repreen a mean-preervng pread of c. Rk avere conumer prefer { } c o { c }, o he co of he economc nably can be meaured by calculang λ whch olve he followng equaon: E E β u (( λ c β u ( c (3 hen λ he compenaon requred by conumer ha make hem ndfferen beween he unceran c and he ream { c }. Noce ha uncerany here come n he form of ochac bune cycle, nce he rend n conumpon deermnc. ream { } Luca (987 aumed ha he uly funcon n CES cla: φ c u ( c, (4 φ where φ > he conan coeffcen of relave rk-averon and u ( c converge o ln( c a φ. I calculaed λ ha afe (3 for ome value of β and φ ung US daa for po-war perod. Obvouly here are oher form of c bede (. If we uppoe c dfference aonary hen can be decompoed a he um of a deermnc rend, a random walk rend and a aonary cycle (ARMA proce, a hown n Beverdge and Nelon (98, Noce ha Luca (987 ue he uncondonal mean operaor nead of he condonal mean operaor n λ. he ame problem can be propoed ung he condonal expecaon nead. h exacly how we proceed n h paper. 3

5 ln( c ω ln( α.ln( α ε µ ω ln( α ( α ln( X ln( Y where ln( α ( α exp( ω he deermnc erm, ln( X ε he random walk componen, ln( Y µ he MA ( repreenaon of he aonary par (cycle, and ω he condonal varance of ln( c ranory hock µ are aumed o have a b-varae normal drbuon a follow, ε ~ dn, µ (5. he permanen hock ε and he.e., hock are ndependen, hu erally uncorrelaed, bu conemporaneouly correlaed f. Calculang he welfare co of bune cycle for he dfference-aonary cae requre fr a dcuon on how o deal wh he fac ha now uncerany come boh n he rend and he cyclcal componen of ln( c. Moreover, nce he rend componen ha a un roo, uncondonal mean and varance are no defned. Noce ha, n he exerce propoed by Luca, all he cyclcal varaon n ln( c elmnaed, whch equvalen o elmnang all varably, nce he rend deermnc. Here, h equvalence lo, becaue he rend ochac a well. o deal wh h ue, we follow Obfeld (994 n conderng he condonal expecaon operaor E o ( n (3, n pe of he uncondonal expecaon operaor E (. In h cae, c now redefned a c E o ( c. herefore, we are aumng ha poble o offer he conumer an ceran conumpon ream c (wh no rend and cyclcal varaon baed on nformaon avalable a he oue of he problem. Of coure, he alernave for he conumer o face c, whch ha a condonal varance ha depend on ω. Conumpon ha now a un roo and o ω, a (alhough ω < for all fne. Hence, uncerany can ge relavely large a he horzon ncreae, whch may be balanced by he fac ha here dcounng n he welfare funcon. A n Obfeld (994, he problem we propoe olvng here E β u (( λ c β u ( E ( c. (7 Under (4, (5 and (6, and ung he propere of he momen of log-normal drbuon, we can calculae (7. Apar from an rrelevan conan erm, lef-hand de gven by φ [ α ] ( λ φ ( φ φω E β u (( λ c [ β ( α ] exp. (8 φ (6 4

6 ( φ φ Noce ha, (8 converge f β ( α exp < Calculang he condonal mean of c yeld φ. c ω Hence, apar from an rrelevan conan erm, he rgh-hand de of (7 φ α φ β u ( c [ β ( α ] φ (9 φ whch converge f β ( α. < Gven he parameer defnng he procee { } λ( φ, β φ [ β ( α ] φ [ β ( α ] E ( c α ( α exp E ( X Y α ( α. c and { } ( φ φω exp c, λ ( φ, β (φ. ( In he defnon of ω n (, we replace and by her repecve uncondonal counerpar, ~ and ~ (whch may be a reaonable approxmaon even for relavely mall, and a very good approxmaon for nermedae and large, makng ω ~ ~.. Aumng ha he condon for (8 and (9 hold, ( converge o 3 ( ( ( exp ~ φ φ φ ~ β α φ exp φ β ( α λ( φ, β β exp ~ ~, β (φ, f φ f φ whch how he way we choe o emae λ ( φ, β n h paper. 4 In ubecon. we dcu a mehodology for calculang λ ( φ, β emae andard error. I' raghfoward o ee ha λ ( φ, β ncreang n β, 5 hu welfare co of flucuaon a large a agen are paen. ( 3 Equaon ( for φ derved on appendx A. 4 In our reul we have oberved ha, for all value of (, β φ ( φ φ he cae ha β ( α exp <, nce he erm large enough a o preven he convergence condon o hold. φ φ we condered here, β ( α. However, wa no alway exp ( φ φ < wa alway greaer han uny, and omeme 5

7 We now urn o oher poble way of modellng he rend componen. If he rend modeled a a deermnc funcon of me, a n (, hen he analy done a orgnally propoed by Luca (987. In pe of he fac ha Luca ha propoed he analy a n (3 above, he acually mplemened n a dfferen way (ee Luca, 987, foonoe, p. 3, removng he rend n conumpon ung he flerng procedure propoed n Hodrck and Preco (997. he fler wo ded,.e., ue pa and fuure conumpon value o ge he lowly-movng rend. In prncple, he rend removed ung uch a procedure hould be reaed a a random varable. However, for mplcy, Luca reaed he rend a deermnc, whch we alo do here. Hence, when ung he Hodrck and Preco rend, our reul hould be vewed a a lower-bound for he welfare co of bune cycle. o mplemen he calculaon n h cae, we compued he deermnc growh rae preen n he Hodrck and Preco rend, reang he cyclcal componen a n (7 above. Hence, c α ( exp( z / z α, ln( z ~ N (, z and c α ( α, where α and α are now he deermnc componen aocaed wh he Hodrck-Preco rend, and z he redual cyclcal componen aocaed wh. We may oberve ha for lnear and Hodrck-Preco rend, ~, and o λ n equaon ( doe no depend of β and α and monooncaly ncreang n φ.. Sandard Erro of λ ( φ, β Emae Le Ω he varance-covarance marx of he permanen hock ε and he ranory hock µ of he log of conumpon, a preened n equaon (6, and Ωˆ $ he maxmum lkelhood emaor of Ω, 6 hu, ˆ, d ˆ, N, ( ˆ, Le ˆα a conen and aympocaly normally-drbued emaor of α,.e., d ˆ α α N (3 ( (, α Le θ ( α,,, and ˆ θ ( ˆ, ˆ, ˆ, ˆ,,,, α. From ( and (3 and applyng Dela Mehod, we have, [ ] ˆ d α λ( θ λ( θ,c( C( N θ θ (4 Ω where ( θ he vecor of paral dervave of λ wh repec o θ. C 3. Reduced Form and Long-Run Conran ( 5 φ φ he erm exp alway greaer han uny. 6 See Hamlon (994, page

8 Le y (ln(,ln( c I a vecor conanng he logarhm of conumpon and dpoable ncome. 7 Aume ha boh ere ndvdually conan a un-roo, and are generaed by a p-h order vecor auoregreon (VAR, y π y π y L π p y p ε, or, Π(Ly ε p where Π(L I n π L π L L π p L. Decompong Π (L a Π(L Π( L p ( L Γ( L leadng o he vecor error-correcon model (VECM y Γ y Γ y L Γp y p Π( y p ε, (4 γ, α he conegraon vecor and γ a where Π( α, Γ I π,,, K, p conan vecor. n Conegraon beween he logarhm of conumpon and ncome may be explaned ung he heory of permanen-ncome. In h heory, conumpon can be vewed a proporonal o he expeced preen dcouned value of all ncome ream. Hence, he expeced preen value of conumpon and ncome are equal, and boh ere are proporonal n he long run, 8 moreover, he conegrang vecor wll be α (-,. We urn now o he dcuon of how o exrac rend and cycle from (4. Fr, pu he yem (4 n ae-pace form, a dcued n Proe (997, y Zf (5 f f Z ε y y Γ L Γ p γα γ where f M, I ( p, Z [ I ( p ]. y ( p p α α y p From he work of Beverdge and Nelon (98, and Sock and Waon (988, gnorng nal condon and deermnc componen, he ere n y can be decompoed no a rend ( τ and a cyclcal componen (, a y τ where, k τ y lm E [ y ] and lm E [ y ] k k (6 k I raghforward o how ha τ a mulvarae random-walk. Ung he ae-pace repreenaon (5, we can compue he lm above. he cyclcal and rend componen wll be, repecvely, 9 Z τ y [ I ] m f (7 7 A full dcuon of he economerc model employed here can be found n Beverdge and Nelon (98, Sock and Waon (988, Engle and Granger (987, Campbell (987, Campbell and Deaon (989, and Proe ( See Campbell (987 and Campbell and Deaon ( See appendx B for cycle and rend equaon dervaon. 7

9 where mp, or, ung formula (6 and (7 n Proe (997, K Γ ( L y Py, and (8 τ K ε (9 where K ( I P ( Γ( γ α and P ( Γ( γα γ [ α ( Γ( γα γ ]α N are proecon marce. We can alo ue (5 o foreca rend and cyclcal componen a any horzon no he fuure. he foreca of, gven nformaon up o, ˆ E [ ] KΓ ( L Zf Py PZ f and he foreca of τ, gven nformaon up o, ˆ τ τ nce he be foreca of a random walk perod ahead mply value oday. o fully characerze he elemen n (, we need o compue he varance and he covarance of foreca of rend and cyclcal componen. Recall ha he condonal expecaon of a log-normal random varable u a funcon of he mean and varance of he normal drbuon aocaed wh. Hence, o compue he varance of hee foreca, we have u o apply andard reul of ae-pace repreenaon. I raghforward o how ha: E ( ( KQK τ ˆ τ τ τˆ. where E [ ε ε ] Q, and ha, ( ˆ ( ˆ VQV P W ( QW ( P E E KQV K QW τ ˆ τ ˆ ( V P KΓ (, a compued n he appendx C. and ( ( P where [ ] Baed on hee la hree covarance marce, he correlaon beween rend and cyclcal componen of he daa can be fully characerzed. Hence, o ge he correpondng elemen of mean, varance, and covarance aocaed wh ln( c, one ha mply o chooe he approprae elemen of hee vecor and marce. 4. Daa European Unon (EU-5 counre annual daa for real ncome and populaon were obaned from Penn World able (Summer and Heon from 95 o. Annual daa for houehold conumpon were exraced from EUROSA, Sac Sweden and Penn World able from 95 o. A preen European Unon compoed by 5 counre: Aura, Belgum, Denmark, Fnland, France, Germany, Grea Bran, Greece, Ireland, Ialy, Luxembourg, Porugal, Span, Sweden and he Neherland. Daa for Greece wa avalable from 95 o and for Germany from 97 o. 8

10 We eed conegraon beween ere of log of per capa conumpon and ncome of each counry and EU-5. able preen reul of he Johanen (988, 99 conegraon e. he hypohe of no conegraon equaon wa reec and he hypohe of a mo one conegraon equaon wa no reeced a 5% gnfcance, excep o Denmark, France, Germany, Greece, Ialy and he Neherland. Condonng on he exence of one conegrang vecor, we eed he rercon ha wa equal o (-, ung he lkelhood-rao e n Johanen (99. h hypohe wa no reeced for Aura, Ireland, Luxembourg, Sweden and Uned Kngdom (UK. Reul are repored on able. he preence of un roo wa nvegaed n conumpon and ncome ere for hoe counre whch ere do no conegrae. A 5%, he un roo hypohe wa no reeced n all cae ung he ADF e; ee he ame reul obaned ung he KPSS e. 9

11 5. Emprcal Reul A ph-order vecor error-correcon model (VECM wh an unrerced conan erm for he log of conumpon and ncome wa fed ung daa for each counry where we found conegraon. Oherwe, a vecor auoregreon model for he fr dfference of hoe ere wa emaed. We eleced lag lengh by he ue of nformaon crera, coupled wh dagnoc e reul. Baed on VECM emae we mplemened he mulvarae Beverdge and Nelon decompoon a uggeed n Proe (997. We compue rend and cycle componen of conumpon ung eher equaon (8 and (9 or equaon (6. Welfare co of bune cycle ( λ for EU-5 and EU counre wa compued ung equaon ( conderng Beverdge-Nelon decompoon, lnear me rend and Hodrck-Preco rend. A a benchmark, we alo compued he welfare co of bune cycle for he USA ung aggregaed conumpon daa from 95 o. Reul for reaonable preference parameer and dcoun value ( β.97, φ are preened n able 4. Sandard error were calculaed ung Dela Mehod a dcued above and, a we may oberve, hey are neglgble f compared o λ. hu, welfare co emae are acally dfferen from zero a % of gnfcance. Reul for β {.95;.97;.985} and φ {;5;;} are preened n Appendx D.

12 On he one hand, for Beverdge-Nelon decompoon welfare co for mo EU counre much greaer han ha for EU-5 a a whole and for he USA. Number for UK (.45% and Sweden (.8% are of he ame order magnude a for USA (.75%. However, he reul for he EU-5 a a whole even maller. On he oher hand, here a group of counre whoe welfare co are more han.5%: Span (4.%, Fnland (3.7%, Germany (3.9%, Greece (3.6%, Belgum (.9%, Ialy (.85% and Porugal (.8%. Comparng wh Franco, Gullen and Iler (3 reul for USA for po-wwii perod, our reul hree me greaer. Ung Hodrck-Preco Flerng we were able o reproduce Luca (987 and Franco, Gullen and Iler (3 reul for USA,.e. λ USA.4%. Welfare co for EU-5 a a whole (.% lower han ha for USA. Reul for France (.3%, UK (.4%, Belgum (.5% and Ialy (.6% are mlar o ha of he USA. For he remanng EU counre, parcularly Porugal (.3% and Luxembourg (.7%, λ beween 4 and 8 me ha of he USA. hey ue non-durable and ervce annualy daa from 947-.

13 Summarly, welfare co are very dfferen acro EU counre and beween US and EU counre, and hu a rong evdence ha bune cycle are no alke n Europe. Dfference n nuonal eng from counry o counry, and conequenely he effec of hock n he econome, are good explanaon for varaon n bune cycle. hu, our reul a conrary evdence of he dynamc-ochac-generalequlbrum vew. 6. Concluon In h paper, we nvegae wheher bune cycle are all alke compung he welfare co of bune cycle for an mporan ube of European counre -- European-Unon (EU counre: Aura, Belgum, Denmark, Fnland, France, Germany, Grea Bran, Greece, Ireland, Ialy, Luxembourg, Porugal, Span, Sweden and he Neherland. A well known, hee counre have a long radon of negraon and rade dang well before he common-currency Euro area wa mplemened. Becaue of h feaure, a naural expermen o nvegae how mlar her welfare co of bune cycle are, n he ene ha we wll be urpred o fnd maor dfference beween hem. In compung he welfare co of bune cycle for EU counre we ue he echnque n Beverdge and Nelon (98 o decompoe (he log of conumpon n a rend and a cyclcal componen. In h cae, he rend wll be ochac and modeled a a random walk. Moreover, o make our reul comparable o prevou work, we alo modeled he rend a eher a deermnc lnear proce (wh and whou a break or followng a lowly evolvng ecular proce capured by he Hodrck and Preco (997 fler. Our reul how ha he welfare co of bune cycle are very dfferen acro EU counre. Ung he Beverdge and Nelon decompoon, and plauble value for he rk averon coeffcen and he dcoun rae of fuure uly, we fnd ha he welfare co of Span (4.% of conumpon almo en me ha of he UK (.45% of conumpon medan of.85%. Maor dfference n welfare co are alo found when alernave rend-cycle decompoon mehod are employed, alhough hey are no a pronounced a he one obaned ung he Beverdge and Nelon decompoon. 7. Bblography Akeon, A. and Phelan, C., 995, Reconderng he Co of Bune Cycle wh Incomplee Marke, NBER Macroeconomc Annual, 87-7, wh dcuon. Beverdge, S. and Nelon, C.R., 98, A New Approach o Decompoon of Economc me Sere no a Permanen and ranory Componen wh Parcular Aenon o Meauremen of he Bune Cycle, Journal of Moneary Economc, 7, Blanchard, O., and Waon, M., 986, Are Bune Cycle all Alke?, NBER Workng Paper Sere, 39. Campbell, J. 987, Doe Savng Ancpae Declnng Labor Income? An Alernave e of he Permanen Income Hypohe, Economerca, vol. 55(6, pp Campbell, John Y. and Deaon, Angu 989, Why Conumpon So Smooh?, he Revew of Economc Sude 56:

14 Cochrane, J.H., 994, Permanen and ranory Componen of GNP and Sock Prce, Quarerly Journal of Economc, 3, Dolma, J., 998, Rk Preference and hewelfare Co of Bune Cycle, Revew of Economc Dynamc,, Engle, R.F. and Granger C.W.J., 987, Conegraon and Error Correcon: Repreenaon, Emaon and eng, Economerca, 55, Hall, R.E., 978, Sochac Implcaon of he Lfe Cycle-Permanen Income Hypohe: heory and Evdence, Journal of Polcal Economy, 86, Hamlon, J., 994, me Sere Analy, Prnceon Unvery Pre. Hodrck, R.J. and Preco, E.C., 997, Powar U.S. Bune Cycle: An Emprcal Invegaon, Journal of Money, Cred, and Bankng, 9, 6. Imrohoroglu, Aye, 989, Co of Bune Cycle Wh Indvble and Lqudy Conran, Journal of Polcal Economy, 97 (6, Iler, J.V. and Vahd, F.,, Common Cycle and he Imporance of ranory Shock o Macroeconomc Aggregae, Journal of Moneary Economc, 47, Franco, A. R., Gullen, O., and Iler, J.V., 3, On he Welfare Co of Bune Cycle n he h Cenury, Enao Econômco da EPGE, 48. Johanen, S., 988, Sacal Analy of Conegraon Vecor, Journal of Economc Dynamc and Conrol,, pp Johanen, S., 99, Emaon and Hypohe eng of Conegraed Vecor n Gauan Vecor Auoregreon, Economerca, vol. 59-6, pp Kng, R.G., Ploer, C.I., Sock, J.H. and Waon, M.W., 99, Sochac rend and Economc Flucuaon, Amercan Economc Revew, 8, Kuzne S., 96, Capal n he Amercan Economy: I formaon and fnancng, Prnceon Unvery Pre. Luca, R., 977, Undeandng Bune Cycle, Carnge-Rocheer Sere n Publc Polcy, vol. 5, ed. Karl Brunner and Allan Melzer, pp Luca, R., 987, Model of Bune Cycle, Oxford: Blackwell. Nelon, C.R. and Ploer, C., 98, rend and Random Walk n Macroeconomc me Sere, Journal of Moneary Economc,, Obfeld, M., 994, Evaluang Rky Conumpon Pah: he Role of Ineremporal Subuably, European Economc Revew, 38, Orok, C.,, On Meaurng he Welfare Co of Bune Cycle, Journal of Moneary Economc, 47, 6-9. Pemberon, J., 996, Growh rend,cyclcal Flucuaon,and Welfare wh Non-Expeced Uly Preference, Economc Leer, 5, Proe,., 997, Shor-run Dynamc n Conegraed Syem, Oxford Bullen of Economc and Sac, 59 (3, Sock, J.H. and Waon, M.W., 988, eng for Common rend, Journal of he Amercan Sacal Aocaon, 83,

15 allarn Jr.,.D.,, Rk-enve Real Bune Cycle, Journal of Moneary Economc, 45, Vahd, F. and Engle, R.F., 993, Common rend and common Cycle, Journal of Appled Economerc, 8, Waon, M.W., 986, Unvarae Derendng Mehod wh Sochac rend, Journal of Moneary Economc, 8, Van Wncoop, E., 994, Welfare Gan From Inernaonal Rkharng, Journal of Moneary Economc, 34, 75-. Appendx Appendx A - Convergence of λ( φ, β for φ (φ φ ( φ φ φ( ~ ~ β ( α exp exp A and B ( φ, β φ, hen λ( φ, β A ( φ. B ( φ, β. β ( α Rewrng B ( φ, β. B ( φ, β Le ( φ B ( φ, β [ B ( φ, β ] B ( φ, β where φ ( φ φ β ( α exp φ φ β β ( α B (,, B ( φ, β and φ β ( α φ β ( α B ( φ φ exp φ ( φ hu β lm B ( φ, β φ β. Applyng L Hopal Rule n B ( φ lm B β ( φ, β. φ ( β So, poble o reume we have lm ( φ B. φ Snce ha lm B ( φ, β, applyng he defnon of e number (bae of he naural logarhm we have, φ lm β ( φ, β exp ( φ β B and β lm (, exp ~ ~ λ φ β φ β 4

16 Appendx B - Dervng rend and Cycle Formulae from Space-ae Form Space-ae form: y f Zf f Z ε rend-cycle repreenaon from Beverdge and Nelon (98: k y τ where τ y lm E [ y ] and lm E [ y ] k Solvng he pace-ae form recurvely we ge, y k k Z f Z Z ε Applyng E on boh de and ummng up from o we have, E y where m p. Z f Z f Z he cyclcal and rend componen wll be, repecvely, Z I f τ y [ ] m I raghforward o ee ha [ m ] τ Z I Z ε [ I m ] f,.e., τ a mulvarae random-walk. Appendx C - Compung Condonal Covarance From Propoon n Proe (997, I P ( Γ( γα Γ ( L y Py, ( N τ ( I N P ( Γ( γα Γ( L y τ ( ( ( I N P Γ γα ε where, P ( Γ( γ α γ [ α ( Γ( γα γ ]α and ( I Γ L L Γ ( L Γ ( ( L Γ (, where Γ ( L Γ, n he preen conex. L Γ, or Γ, whch decompoed a So, we have, τ τ ( I N P ( ( γα ε whch mple ha ˆ τ ( ˆ τ. Denong K ( I P ( Γ( γα N, and ( τ ˆ τ K ε we have, where E [ ε ] Q Q ε ( τ ˆ ( K Q K τ τ ˆ τ. KQK E. On he oher hand, we can wre, E τ 5

17 6 Py y L K Γ ( bu, Zf y ε, whch mple ha Zf y ε. However, Zf y y ε, whch mple ha N Z Z I f Z y y ε ε. Hence, ( f PZ Py Zf K E Γ ( ˆ, whch mple ha [ ] Γ ì N Z Z I K P ( ˆ ε ε. Denong [ ] ( Γ K P V and Z Z I W N ( we have, ( ( P QW W P VQV E ( ( ˆ ˆ and ( ( P QW K KQV E ( ˆ ˆ τ τ

18 Appendx D - able 7

19 8

20 9

21