A Steady-State Approach to Trend/Cycle Decomposition *

Transcription

1 A eady-ae Approach o Trend/Cycle Decomposon * James Morley Washngon Unversy n. Lous Jeremy Pger Federal Reserve Bank of. Lous February 9 4 ABTRACT: In hs paper we presen a new approach o rend/cycle decomposon. The rend of an negraed me seres s esmaed usng he condonal expecaon of he seady-sae level of he seres. Gven a nonlnear forecasng model hs seady-sae approach can dffer n mporan ways from he relaed long-horzon forecas decomposon proposed by Beverdge and Nelson (98). We use generaed daa from nonlnear regme-swchng processes o demonsrae he advanages of he seady-sae approach. We hen apply he seady-sae approach o esmae he rend and cycle of U.. real GDP mpled by a regme-swchng forecasng model. Our fndngs porray a very dfferen pcure of he busness cycle han mpled by sandard lnear forecasng models. Keywords: eady ae Trend/Cycle Decomposon Nonlnear Regme wchng JEL Classfcaon: C5 C E3 Morley: Deparmen of Economcs Box 8 Elo Hall 5 Washngon Unversy One Brookngs Drve. Lous MO (morley@economcs.wusl.edu); Pger: Research Deparmen Federal Reserve Bank of. Lous 4 Locus.. Lous MO 6366 (pger@sls.frb.org). We would lke o hank Gaeano Annolf and Charles Nelson for helpful commens and suggesons. Responsbly for any errors s our own. Morley acknowledges suppor from he Wedenbaum Cener on he Economy Governmen and Publc Polcy. The vews expressed n hs paper should no be nerpreed as hose of he Wedenbaum Cener he Federal Reserve Bank of. Lous or he Federal Reserve ysem.

2 . Inroducon Trend/cycle decomposon of negraed economc me seres s mporan for boh heorecal and sascal reasons. In hs paper we presen a new approach and compare wh some exsng mehods n he leraure. Unlke radonal mehods such as he Hodrck-Presco (HP) fler (Hodrck and Presco 997) and he Beverdge- Nelson (BN) decomposon (Beverdge and Nelson 98) he approach presened n hs paper accuraely esmaes he permanen and ransory componens of negraed me seres even when he dynamc srucure of he daa nvolves nonlneares such as regme swchng. Our approach o rend/cycle decomposon s based on he premse ha he rend of a me seres s equvalen o s mplc seady-sae level. In a dynamc sysem seady sae s he hypohecal equlbrum ha would occur followng he realzaon of all currenly mpled ransory dynamcs. In pracce he concep of seady sae denfes he rend snce he mpled ransory dynamcs for a me seres can be calculaed from a forecasng model. By employng a forecasng model o measure ransory dynamcs our seadysae approach s closely relaed o he long-horzon forecas decomposon proposed by Beverdge and Nelson (98) and exended o nonlnear processes by Clarda and Taylor (3). Indeed for lnear forecasng models he wo mehods produce equvalen resuls. However ha equvalence does no generally hold for nonlnear models wh regme-swchng parameers. We presen wo examples where he wo mehods produce dfferen resuls and he seady-sae approach s preferable. In he frs example we generae daa from Hamlon s (989) Markov-swchng model. The model assumes ha

3 he mean growh rae of a me seres undergoes dscree regme shfs accordng o a Markov process. For daa generaed from he model he seady-sae approach s preferable because unlke he long-horzon forecas does no nclude expeced fuure regmes n he esmae of he curren level of rend. In he second example we generae daa from Km and Nelson s (999a) pluckng model. Accordng o hs model he level of oupu undergoes negave plucks ha produce ransory dynamcs. Agan he seady-sae approach s preferable o he long-horzon forecas because does no nclude expecaons of fuure ransory plucks n he esmae of rend. Imporanly he seady-sae approach allows for a non-zero mean cycle and produces equvalen resuls o he Kalman fler esmaes of rend and cycle repored n Km and Nelson (999a). There are wo feaures of he seady-sae approach o rend/cycle decomposon ha are worh emphaszng. Frs can be appled gven a wde varey of forecasng models. Thus whle produces equvalen resuls o he Kalman fler for Km and Nelson s (999a) pluckng model s more general snce does no requre a known unobserved componens (UC) represenaon for he me seres of neres. Ths generaly has he desrable mplcaon ha evaluaon of dfferen esmaes of rend and cycle can ulmaely be hough of as a maer of model comparson gven a se of possble forecasng models. econd he concep of seady sae s generally suffcen o denfy he level of he rend. Thus here s no need for he arbrary normalzaon or denfcaon assumpons ha are ofen employed n unobserved componens esmaon of rend and cycle. Also here s no need o mpose heory-based denfcaon assumpons ha can bas or preudge subsequen analyss. For example srucural VARs

4 can denfy permanen and ransory componens hrough heorecal assumpons abou long-run relaonshps (e.g. Blanchard and Quah 989). However s ofen hese heorecal relaonshps ha researchers wan o nvesgae emprcally nsead of assume a pror. The seady-sae approach o rend/cycle decomposon allows such nvesgaon whou preudgng he resuls. The remander of he paper s organzed as follows. econ presens he deals of our seady-sae approach o rend/cycle decomposon. econ 3 demonsraes he advanages of he seady-sae approach when appled o negraed me seres generaed from known regme-swchng processes namely Hamlon s (989) auoregressve Markov-swchng model of oupu growh dynamcs and Km and Nelson s (999a) pluckng model. econ 4 presens an applcaon of he approach o esmae he rend and cycle of U.. real GDP mpled by a regme-swchng forecasng model developed by Km Morley and Pger (3). econ 5 concludes.. Mehod The rend/cycle decomposon mehod proposed n hs paper uses he concep of seady sae o denfy and esmae permanen and ransory componens of negraed me seres ha follow nonlnear regme-swchng processes. In hs secon we presen he deals of he mehod. Frs we nroduce he concepual framework ha relaes he rend of a me seres o s mplc seady-sae level. econd we dscuss he prncples underlyng he calculaon of he condonal expecaon of seady sae gven a forecasng model wh regme swchng parameers. Thrd we compare our mehod o he BN decomposon. Clarda and Taylor (3) make a smlar pon n argung for he use of he BN decomposon. 3

5 . Concepual Framework An negraed me seres { } y can always be hough of as he sum of wo unobserved componens relaed o permanen and ransory nnovaons: y c. () The permanen componen s he accumulaon of permanen η nnovaons: η. () The ransory componen c s a weghed-average of ransory ω nnovaons: c ψ ω (3) where he random MA coeffcens are normalzed by ψ. We gve meanng o he labels permanen and ransory by mposng he followng resrcon ha lnks he permanen componen o he concep of seady sae: lm ψ. (4) 4

6 In words ransory ω nnovaons have no long-run mpac on he me seres { } y. Thus n he hypohecal absence of any fuure nnovaons afer me { y } would converge o. I s n hs sense ha s he seady-sae level of In hs paper we focus our aenon on negraed processes wh regme swchng parameers. In parcular we consder he case where he random MA coeffcens n (3) and/or he means of he nnovaons n () and (3) can ake on dfferen values accordng o an unobservable dscree sae varable wh known dsrbuon whch we denoe as : y. ψ (5) ψ η µ η (6) ω λ ω (7) where η ~ N( σ η ) ω ~ N( σ ω ) and Cov ( η ω ) σηω. The dependence of ψ µ and λ on he sae varable need no be lnear. However condonal on he process for { y } s lnear and assumed o be Gaussan. The seup n ()-(7) s que general. pecfcally he nnovaons o he permanen and ransory componens can be correlaed. Also from (5)-(7) he process can be regme swchng for a varey of reasons. In parcular he regme swchng can be n erms of he dynamcs he nnovaon o he permanen componen he nnovaon I s also possble o consder he case where he varance-covarance parameers depend on he sae vecor. We gnore hs more general case for smplcy of presenaon. 5

7 o he ransory componen or any combnaon of hese. Despe hs generaly he permanen and ransory componens of a regme swchng process can ofen be denfed and esmaed whou pror knowledge of eher he correlaon beween nnovaons or he source of regme swchng. Tha s a UC represenaon need no be known. Insead as dscussed nex he fac ha s he seady-sae level of y provdes a possble means for denfcaon and esmaon even when only a forecasng model of he frs dfference y s avalable.. Esmang eady ae All of he forecasng models for followng sae-space form: y consdered n hs paper can be cas n he [ h h L hn ] y (8) c F~ ~ e (9) where e ~ N( Q). Tha s he observaon y can be represened as a lnear combnaon of he elemens of he n vecor accordng o he weghs h...n. Meanwhle he vecor follows a frs-order vecor auoregressve process where he elemens of he nercep vecor c ~ and he coeffcen marx F~ may depend on a vecor ~ ha conans he curren and possbly lagged values of he unobservable dscree sae varable. The vecor whe nose error e s assumed o be Gaussan and 6

8 uncorrelaed wh. The frs-order form of (9) s more general han may a frs appear snce any hgher order process can always be recas n s frs-order companon form. Also can nclude boh observables such as y and/or s lags and unobservables such as movng-average error erms. Thus he represenaon n (8) and (9) encompasses a wde varey of forecasng models ncludng all unvarae and mulvarae Markov-swchng ARMA models. Esmaon of he permanen and ransory componens of y nvolves usng he model gven n (8) and (9) o calculae he condonal expecaon of. Noe ha n he absence of regme swchng n he underlyng me seres process and he forecasng model he BN decomposon calculaes he condonal expecaon of (see Morley Nelson and Zvo 3). However he presence of regme swchng complcaes denfcaon and esmaon of and he BN decomposon ncludng s exenson o nonlnear models by Clarda and Taylor (3) does no necessarly provde a condonal expecaon of. In calculang he condonal expecaon of we sar wh he premse ha condonal on observng and he sequence { } would be possble o denfy and esmae from a forecasng model usng he fac ha s a seady-sae value. Tha s we can generally calculae E { } Ω ] [ where Ω s nformaon observed a me. Frs gven (9) solve recursvely for he condonal expecaon of fuure values of { }: 3 3 Noe ha f we had allowed for more complcaed nonlneares n (9) such as dependence of regmes on he resduals we would need o use smulaon and numercal negraon as n Clarda and Taylor (3) o 7

9 8 { } { } ] [ ] [ ~ ~ E F c E Ω Ω. () Then from (8) he condonal expecaon of fuure values of { } y s { } [ ] { } Ω Ω n E h h h y y E ] [ ] [ L. () uppose ha for some he followng seady-sae condon holds: { } { } ] [ ] [ l y E y E Ω Ω L. () Tha s he expeced change n he seres remans consan for l perods. Gven () { } ] [ y E Ω provdes a condonal expecaon of seady sae and hus of he rend. To see why noe ha based on () he condonal expecaon of he seres can always be decomposed as follows: { } { } { } ] [ ] [ ] [ c E E y E Ω Ω Ω. (3) calculae hs condon expecaon. The exenson s concepually sraghforward alhough compuaonally burdensome.

10 When E [ c { } Ω ] we would expec ransory dynamcs o work her way ou n fuure perods meanng ha he condon n () would no hold. Conversely he seady-sae condon n () mples [ c { } Ω ] E. 4 The nuon s ha n usng () o esmae fuure values of { } we are seng fuure values of { e } o her expeced value of zero. Ths s equvalen o seng fuure values of he { η } and { } ω shocks n (6)-(7) o her expeced value of zero. As he horzon ges large enough he expecaon s ha fuure realzaons of { y } wll no longer depend on ransory dynamcs due o pas realzaons of { ω }. Thus a some pon any expeced ransory dynamcs ha preven he condon n () from holdng mus be caused by he mean of a fuure ransory nnovaon akng on non-zero values:.e. λ. Meanwhle f he condon n () holds fuure ransory nnovaons due o regme changes are equal o zero for a long enough perod for he expeced mpac of any prevous non-zero ransory nnovaons o de ou. Whle E y { } Ω ] [ he curren seady sae s a condonal expecaon of a seady sae s no. In parcular E y { } Ω ] [ permanen nnovaons mpled by fuure realzaons of { } wll reflec fuure. These nnovaons mus be 4 There s a heorecal possbly ha for some se of random MA coeffcens n (3) and a parcular sequence of ransory shocks he seady-sae condon n () may hold by mere concdence and no because he mpac of pas ransory nnovaons has ded ou. To mnmze hs possbly he number of consecuve perods l ha he seady-sae condon s requred o hold can be se o an arbrarly large number. By he long-run resrcon n (4) a consan bu non-zero mpac of a pas ransory nnovaon mus evenually change. Thus a large enough l wll preven an erroneous esmae of seady sae alhough a he cos of ncreasng he number of perods before he seady-sae condon holds. Dependng on he naure of he process s possble ha he condon n () never holds for l consecuve perods and we canno denfy rend usng he seady-sae approach. However hese problems do no arse n he applcaons consdered n hs paper where we se l and fnd ha resuls are robus o hgher values of l. 9

11 removed from E[ y { } Ω ] n order o measure E[ { } ] Ω. We measure he long-run mpac mpled by each specfc fuure realzaon as { } ] µ E[ y. (4) In parcular he condonal expecaon n (4) calculaes he long-run growh rae whn each regme. Then he condonal esmae of he curren seady sae level s { } Ω ] E[ y { } Ω ] E[ µ. (5) In pracce and he sequence { } are no observed. We only assumed ha hey were avalable condonng nformaon because allowed us o use he concep of seady sae o denfy he rend. 5 Gven he concepual framework above we would be unable o denfy rend whou ha specfc condonng nformaon. However once we have denfed he rend n heory we can use Mone Carlo smulaon mehods o negrae ou he unknown condonng varables n order o calculae he expecaon of he rend condonal only on observed daa. In smulang and he sequence { } we follow he mul-move Gbbssamplng procedure employed by Carer and Kohn (994) and Km and Nelson (998 5 The choce of he condonng se s fundamenal o denfyng seady sae. I also urns ou o be mporan o condon only on nformaon up o me raher han he whole sample. If we condon on fuure daa our nferences abou he fuure seady sae wll reflec fuure permanen shocks bu hese shocks are no observed so we canno easly denfy and remove hem o esmae he curren seady sae.

12 999b). 6 The deals of he procedure are somewha nvolved and are presened n an appendx. However we skech ou he man feaures here. Gven an arbrary nal sequence { ()} J where J s suffcenly large o always nclude and m erae hrough he followng seps: ( m). Draw { } ( m ) from he condonal dsrbuon { } { } Ω. ( m). Draw { } J ( m) from he condonal dsrbuon { } { } Ω 3. Updae m m and reurn o sep unl m > M. 7 For each draw of and he sequence { (m) } (m). we can esmae seady sae usng he condon n () o solve for he expecaon n (5). Then denong he smulaed ) ˆ ( m) ( m) ( m seady-sae esmae as E[ { } Ω ] our esmae of seady sae s M ( m) ˆ ˆ (6) M m whch should converge o he condonal expecaon E Ω ] as he number of smulaons M goes o nfny. pecfcally [ lm M M M m ( m) ( m) [ E[ { } Ω ] Ω ] E[ Ω ] ( m) ˆ E (7) 6 For smplcy of presenaon we assume known parameers. In pracce esmaon of parameers can be done usng eher classcal or Bayesan mehods. Bayesan esmaon smply nvolves he addonal sep n he Gbbs-samplng procedure of drawng model parameers from her condonal poseror dsrbuons. 7 In pracce we only keep rack of draws afer an nal burn n perod n order o ensure draws are no longer mpaced by our nal arbrary sequence for he Markov-swchng sae varable.

13 where he second equaly follows from he law of eraed expecaons. The esmaed ransory componen denoed ĉ s smply he dfference beween he level of he seres and he esmaed rend: cˆ ˆ. y.3 Comparson wh BN Decomposon Gven a lnear forecasng model he seady-sae esmae of rend s always equvalen o he long-horzon forecas (mnus any deermnsc growh) esmae developed by Beverdge and Nelson (98): { E [ y ]} BN lm µ. (8) A lnear model s a model ha can be cas as a specal case of (8) and (9) n whch he nercep vecor c and he coeffcen marx F do no depend on. For such a model we can calculae ˆ E[ Ω ] usng a smplfed verson of he seady-sae approach oulned n ()-(6). A smple example may help llusrae he equvalence of he seady-sae approach and he BN decomposon for lnear models. Consder an negraed me seres ha can be forecas by a saonary AR() model n frs dfferences: y µ φ( µ ) ε (9) y

14 where φ < and ε ~ N( ) σ ε. For hs model he sae-space represenaon s smply he nercep form of he AR() model where he nercep parameer c µ ( φ). Then he expecaon n () smplfes o E y ] E[ y { } Ω ] snce s [ Ω observed ( y ) and he process does no depend on { }. For hs model s sraghforward o solve for hs expecaon usng () and he expeced change n he seres mpled by (9): E [ y Ω ] µ φ ( y µ ) () nce lm φ here wll be a ha sasfes he seady-sae condon n () as : E y Ω ] L E[ y Ω µ. () [ ] l The deermnsc growh µ reflecs he expecaon of fuure permanen nnovaons. Thus n order o calculae he condonal expecaon of he curren seady sae we mus remove he mpac of expeced fuure permanen nnovaons on E[ y Ω ] as s done n (5): E[ Ω ] E[ y Ω µ. () ] 3

15 Gven he forecasng equaon n () and s sraghforward o show ha () s equvalen o φ ˆ y ( y µ ) (3) φ whch s he same as he BN rend for an AR() forecasng model (see Morley ). By conras he seady-sae approach and he BN decomposon are generally no he same for nonlnear forecasng models wh regme-swchng parameers as n (8) and (9). The reason s ha he BN esmae of rend n (8) mplcly ncludes he expecaon of fuure nnovaons ha are no par of he acual curren rend. In he nex secon we demonsrae hs dfference for wo well-known regme-swchng processes. 3. Examples In hs secon we presen wo examples of he seady-sae approach o rend/cycle decomposon appled o negraed me seres generaed from regmeswchng processes. In boh cases we generae daa from unobserved componens models allowng us o compare esmaes of he rend and cycle wh her rue values. We also use he generaed daa o llusrae how he seady-sae approach denfes he level of rend and he mean of he cycle. 3. Hamlon s (989) Markov-wchng Model The frs daa generang process we consder s based on Hamlon s (989) regme-swchng model of U.. real GNP. The orgnal specfcaon s a forecasng 4

16 model of he frs dfferences y. However n order o observe he acual permanen and ransory componens for he generaed daa we consder he followng unobservedcomponens represenaon for y : y µ φ( L) c µ η (4) c ω where η ~ N( σ η ) ω ~ N( σ ω ) cov( η ω ) φ ( L) φl s a k h k order lag polynomal wll all roos ousde he un crcle and {} s a Markovswchng sae varable wh ranson probables Pr[ ] q and Pr[ ] p. The permanen componen of real oupu follows a random walk wh a regme-swchng drf componen. The ransory componen c follows a lnear saonary auoregressve process. In he example we se lag lengh k so ha c s an AR() process. We generae a sample of observaons from he daa generang process n (4) usng he followng parameer calbraon: µ. 8 µ. 3 σ η. 5 σ ω.5 φ. φ. q. 96 and p. 8. In pracce gven hs UC model and parameers he rend and cycle can be esmaed usng he flerng echnques n Lam (99) and Km (994). However we consder he seady-sae approach here n order o evaluae s performance. Of course should be emphaszed ha unlke he flerng echnques he seady-sae approach does no requre knowledge of he UC represenaon or any arbrary normalzaon or denfcaon assumpons ha ofen 5

17 accompany UC models. I can be appled gven a forecasng model for he frs dfferences only. For he seady-sae approach o rend/cycle decomposon we cas a forecasng model no he sae-space form gven by (8) and (9). One way o do so for he UC model n (4) s o solve for correspondng reduced- form represenaon for case s a regme-swchng ARMA() process: y 8 whch n hs ( φ θ L ) ε (5) L φ L )( y µ µ ) ( θl where ε ~ N( σ ε ) and θ and θ are complcaed nonlnear funcons of he parameers n (4). 9 In erms of he sae-space form gven by (8) and (9) he varable vecors are y [ ] [ ε ε y y ] e [ ε ] ε. The nercep vecor s c ~ [ ~ ] α and where ~ α ~ φ L φ L )( µ µ ) whch means ha [ ]. Fnally he ( companon marx whch does no depend on ~ for he model n (5) s 8 I should be noed ha we could have drecly cas he UC model n (4) no he sae-space form gven by (8) and (9). However he pon of he example s o show ha rend and cycle can be esmaed even f only a reduced-form represenaon were known. 9 We calculae values for he movng-average parameers by solvng he sysem of nonlnear equaons ha relaes he reduced-form parameers o he auocovarances mpled by (4). Of he mulple soluons avalable we use he one ha corresponds o real numbers for he parameers and an nverble represenaon for he movng-average componen. 6

18 F θ θ φ φ. Gven he model n hs sae-space form ˆ E[ Ω] can be calculaed usng he seady-sae approach oulned n ()-(6). We perform M 5 smulaons o negrae and he sequence { } ou of he condonal dsrbuon E { } Ω ] [ afer an nal smulaons o ensure convergence of he Gbbs sampler. Fgure dsplays he generaed me seres s rue rend and some esmaes of he rend. In addon o he seady-sae esmae of rend we consder he HP fler esmae wh he smoohng parameer se o 6 and an esmae based on he longhorzon forecas (.e. he BN decomposon) calculaed usng he mehod presened n Clarda and Taylor (3). The frs panel of Fgure shows ha mos of he varaon n he generaed seres reflecs he regme swchng n he rend. The second panel shows ha he seady-sae approach s able o capure hs form of nonlneary. Indeed he seady-sae esmae of rend s vrually ndsngushable from he rue rend. Meanwhle he hrd panel shows ha he HP fler esmae of rend msses he nonlneary n he daa. Insead essenally races ou a smooh lne hrough he seres. Fnally he fourh panel shows ha he long-run forecas esmae of rend s much more varable han he rue rend. In parcular whenever here s a change n regme he Brefly Clarda and Taylor s (3) mehod nvolves generang a large number of smulaed fuure realzaons of he me seres from he condonal dsrbuon mpled by a forecasng model and he observed daa a each pon of me. Then he condonal expecaon of he seres a any fuure horzon s calculaed by averagng he smulaed fuure realzaons a ha horzon. As wh Beverdge and Nelson (98) he esmae of rend s he long-horzon condonal expecaon of he seres mnus any deermnsc growh whch s defned as he uncondonal expecaon of he change n he me seres. 7

19 esmaed rend aduss by more han he seres self due o he perssence of he regmes. For example he perssence of recessonary regmes means ha a ranson from an expansonary regme o a recessonary regme wll grealy ncrease he probably ha fuure regmes are also recessonary. Thus he long-run condonal expecaon of he seres wll be dramacally lowered by he regme shf. 3. Km and Nelson s (999a) Pluckng Model The second daa generang process we consder s based on he pluckng model of U.. real GDP proposed by Km and Nelson (999a). Insead of allowng for regme swchng n he permanen componen Km and Nelson assume ha he regme swchng s n he ransory componen. In parcular he model s y c µ η (6) φ( L) c λ ω where η ω φ (L) and are he same as n (4). The permanen componen of real oupu follows a random walk wh consan drf. The ransory componen c follows a saonary auoregressve process ha depends on he value of. For example f λ < c s plucked downward by he amoun λ when. However he effecs of hs pluck are ransory as hey are worked off hrough he dynamcs of c. Imporanly noe ha hs model mples ha he ransory componen c has a non- The pluckng ermnology s due o Mlon Fredman ( ). 8

20 zero mean gven by λ Pr[ ] E ( c ) where φ() Pr[ q ]. As before we p q se lag lengh k. We generae a sample of sze from he daa generang process n (6) usng he followng calbraon: µ. 8 σ η.5 σ ω. λ φ. φ. 4 p.96 and q. 8. Conssen wh he parameer esmaes of Km and Nelson (999a) σ ω s very small relave o λ meanng mos varaon n c s due o plucks. The reduced-form represenaon for regme-swchng ARMA () process: y mpled by he UC model n (6) s agan a ( φ θ L ) ε (7) L φ L )( y µ ) λ ( θl where ε θ and θ are he same as n (5). The model n (7) can be cas n he saespace form gven by (8) and (9) n much he same way as he Hamlon model above. The only dfference s n he α ~ elemen of he nercep vecor c ~. pecfcally for he model n (7) α φ φ ) µ λ ( ~ whch means ha [ ] ~. Agan ˆ E[ Ω ] can be calculaed usng he seady-sae approach oulned n ()-(6). As before we perform M 5 smulaons o negrae and he sequence { } ou of he condonal dsrbuon E[ { } Ω ] smulaons o ensure convergence of he Gbbs ampler. afer an nal Fgure dsplays he seres he rue rend and some esmaes of he rend. Agan n addon o he seady-sae esmae we dsplay he HP fler esmae and he 9

21 long-horzon forecas esmae calculaed usng he mehod presened n Clarda and Taylor (3). The frs panel of Fgure shows he pluckng naure of he generaed seres. I remans very close o rend excep when s plucked n he downward drecon. As wh he prevous example he second panel shows ha he seady-sae esmae of rend s able o capure he rue rend. The hrd panel shows ha he HP fler esmae of rend msses he nonlnear aspecs of he daa and essenally races ou a smooh lne hrough he seres labelng some of he ransory varaon n he seres as varaon n he rend. Fnally he fourh panel shows ha he long-run forecas esmae of rend moves wh he rue rend bu s shfed downward. Tha s he long-run forecas mehod fals o denfy he level of he rend or equvalenly he mean of he cycle. Ths can be seen more clearly n Fgure 3 whch dsplays he rue generaed cycle for he pluckng model along wh he seady-sae H-P fler and long-run forecas esmaes of he cycle. The ably of he seady-sae esmae o capure he negave mean of he rue cycle s apparen. Meanwhle he H-P fler and long-run forecas echnque produce esmaes of he cycle ha are shfed upward from he rue cycle. Indeed he ably of he seady-sae approach o denfy he level of he rend or equvalenly he mean of he cycle s a key advanage over oher mehods. We urn o hs ssue n more deal nex. 3.3 Idenfyng he Level of Trend I may appear a frs ha he dea of cycle wh a non-zero mean s merely a maer of normalzaon. Tha s he nal level of he rend could always be se o make he resulng cycle have a mean of zero. However he resulng re-normalzed rend may

22 no correspond o seady sae. Pu anoher way he concep of seady sae unquely denfes he level of he rend or equvalenly he mean of he cycle. Km and Nelson s (999a) pluckng model allows for a sraghforward demonsraon of hs pon. Fgure 4 dsplays he level and he frs-dfference of he cycle generaed from he pluckng model. Whle would be possble o shf he level of he cycle upwards o have a mean of zero should be noed ha dong so would have no mpac on he frsdfference of he cycle. Meanwhle s he frs-dfference of he cycle ha denfes when he seres s n seady sae. In parcular he seres can only be hough o be n seady sae when he frs-dfference of he cycle s consan and equal o zero. Oherwse he cycle mus be changng whch from (3) means ha pas ransory nnovaons are sll affecng he cycle and he seres s no n seady sae. Thus we can use he frs-dfference of he cycle and he concep of seady sae o denfy he mean of he cycle or equvalenly he level of he rend. pecfcally seady sae mples ha he cycle s equal o zero and he rend s equal o he acual level of he seres only when he frs-dfference of he cycle s equal o zero. Ths relaonshp holds for he cycle dsplayed n Fgure 4 bu would no hold for a re-normalzed cycle wh a mean of zero. 4. Applcaon eparang ransory or cyclcal varaon n me seres of macroeconomc acvy from permanen or rend varaon has a rch hsory n macroeconomcs. A prmary reason for he aenon hs enerprse has drawn s ha he busness cycle s ypcally measured usng he cyclcal componen of an oupu seres such as real Gross Domesc Produc (GDP) whle he rend componen s used o measure long-run

23 growh. In order o es macroeconomc heores of he busness cycle or long-run growh we need accurae esmaes of hese componens. There are many alernave echnques ha have been used o exrac rend and cycle from real GDP. A popular early echnque was o assume ha he rend s a deermnsc polynomal. More recenly he focus has shfed o sochasc rends n whch he permanen componen s no merely a deermnsc funcon of me bu conans random elemens. Popular echnques o exrac sochasc rends from real GDP nclude he Hodrck-Presco (997) fler he lnear unobserved componens models of Clark (987) and Wason (986) and he Beverdge-Nelson (98) decomposon. Much of hs work s based on he assumpon ha he daa generang process for real GDP s lnear. Indeed he UC models referenced above and he BN decomposon are based on ARIMA forecasng models for real GDP and are herefore explcly lnear. A he same me here s growng evdence ha he me-seres properes of real GDP are well descrbed by models conanng deparures from lnear ARIMA models ncludng hreshold models and regme-swchng models. However he mplcaons of hese nonlnear dynamcs for measurng rend and cycle have been largely gnored. Excepons are generally whn he UC framework where nonlneares are explcly enered no he processes for he rend and cycle (see for example Km and Nelson 999a). In hs secon we apply he seady-sae approach o exrac he rend and cycle from U.. real GDP under he assumpon of a nonlnear daa generang process. We Ths sascal defnon of rend and cycle s no whou problems. For example Blanchard and Quah (989) pon ou ha he srucural rend of oupu may also experence ransory flucuaons and hus rend need no correspond o he permanen componen of oupu.

24 noe ha he seady-sae approach requres a forecasng model. One advanage of he seady sae approach s ha can be appled gven any forecasng model whch reduces he problem of evaluang a parcular decomposon o a maer of model selecon. However performng model selecon across an appropraely large se of forecasng models s beyond he scope of hs paper. Thus we focus here on a parcular nonlnear forecasng model of U.. real GDP developed by Km Morley and Pger (KMP) (3) and leave model selecon o fuure research. The KMP model s a regme-swchng ARIMA model for U.. real GDP: m φ ( L) y µ µ λ ε (8) where ε ~ N( σ ε ) he lag operaor φ (L) s k-h order wh roos ousde he un crcle y s he frs dfference of he logarhm of real GDP and s an unobserved Markov-swchng sae varable ha akes on dscree values of or accordng o ranson probables Pr[ ] q and Pr[ ] p. The saes are normalzed by resrcng µ <. Tha s corresponds o a lower growh regme or f µ µ a conraconary regme. < The model n (8) adms a hree-phase represenaon of GDP growh dynamcs. When oupu grows a he rae µ wh devaons from hs growh rae caused by ε whch are propagaed by he auoregressve dynamcs φ (L). When oupu eners a low growh or conraconary regme and grows a he average rae µ µ. 3

25 However as he recesson progresses and ulmaely ends he erm m λ augmens he growh rae of he forecasng equaon. Ths addonal erm s bes descrbed as a pressure varable obanng larger values as he recesson s longer and more severe. I hen mples a bounce-back effec from he recesson f λ >. Tha s oupu growh wll be above-average for he frs m perods of an expansonary regme. Usng hs model KMP are able o reec boh lneary and he absence of a bounce-back effec n U.. real GDP. Also n a formal model comparson exercse KMP fnd ha hs hreephase model ouperforms lnear ARIMA models and alernave regme-swchng models n mmckng a number of key feaures n he daa. We esmae he KMP model usng he frs dfference of log U.. real GDP from 949:Q o 3:Q. The seres s mulpled by 4 n order o correspond o annualzed growh raes. We se he lag order k and followng he fndngs n KMP m 6 quarers. The model s esmaed va maxmum lkelhood usng he fler gven n Hamlon (989). Table repors he esmaed parameers whle Fgure 5 plos he smoohed probably Pr[ Ω ] along wh he log real GDP seres. As n KMP T µ µ < whch suggess ha corresponds o a conraconary regme. Indeed he smoohed probably Pr[ Ω ] suggess ha he conraconary regme corresponds farly closely o mos NBER recesson daes. Meanwhle λ > suggess ha he quarers followng he end of a conraconary regme correspond o aboveaverage economc growh. T To oban he seady-sae esmae of rend we cas he model n (8) no saespace form. In erms of he sae-space form gven by (8) and (9) he varable vecors are 4

26 y [ ] [ y y ] and [ ] [ ] ~ e ε. The nercep vecor s c ~ α where α ( φ L φ L )( µ µ λ ) whch means ha m ~ ~ [ ] Fnally he companon marx s: φ φ F We se each parameer o s maxmum lkelhood pon esmae lsed n Table. Fnally we calculae ˆ E[ Ω] usng he seady-sae approach oulned n ()- (6). We perform M 5 smulaons o negrae condonal dsrbuon E { } Ω ] [ and he sequence { } ou of he afer an nal smulaons o ensure convergence of he Gbbs sampler. Fgure 6 dsplays he seady-sae esmaes of he rend and cycle for log U.. real GDP. The esmaed cycle has several noceable feaures. Frs clearly has a negave mean suggesng ha s more ypcal for he economy o operae below rend han above. econd urns srongly negave durng mos NBER daed recessons whle s near zero durng expansons. Ths fndng s conssen wh Mlon Fredman s ( ) pluckng model n whch oupu s generally close o rend and s occasonally plucked below rend. We noe however ha for he hree recessons of and mos of he movemen n real GDP s n he rend componen. Tha s accordng o he model hese recessons had largely permanen effecs on real GDP. 5

27 5. Concluson The seady-sae approach o rend/cycle decomposon has many advanages over compeng mehods. I s able o deal wh nonlneares n he daa and denfes he level of he rend or equvalenly he mean of he cycle. The applcaon o U.. real GDP reveals hese advanages and porrays a pcure of he busness cycle ha s very dfferen han wha s mpled by sandard lnear models. We conclude by nong ha our approach by allowng for nonlnear dynamcs provdes a possble reconclaon of he NBER noon of he busness cycle wh he concep of an oupu gap. In parcular accordng o he forecasng model employed n our applcaon he ransory componen of real GDP s very small n expansons and large and negave only durng NBER recessons. Meanwhle he fac ha some NBER recessons correspond only o movemens n he permanen componen suggess ha hese epsodes may reflec fundamenally dfferen economc condons han he oher recessons. Of course even whn he rend/cycle decomposon framework developed n hs paper a more complee nvesgaon of he busness cycle mus nvolve some consderaon of alernave forecasng models. We leave he model selecon ssue and a full resoluon of wha he busness cycle acually looks lke o fuure research. 6

28 Appendx: amplng he Unobserved { } and { } The seady-sae approach o rend-cycle decomposon oulned n econ requres a sep o draw smulaed values of { } dsrbuons { } { } Ω and { } { } Ω and { } J from her condonal. In he followng we descrbe algorhms for hs daa generaon. The presenaon wll closely follow ha n Km and Nelson (999b). A. amplng { } To draw from { } { } Ω we use he mulmove algorhm dealed n Carer and Kohn (994). To begn he sae space form of he forecasng model s wren so ha he frs R x R block of Q denoed * Q s posve defne whereas he remanng elemens of Q are zero. Denoe he frs R elemens of as * and he frs R rows of F~ as F. As an example consder he ARMA() model n (5): * ~ ( φ θ L ) ε L φ L )( y µ µ ) ( θl Ths forecasng model has a sae-space form ha mees he requremens for he Q marx gven above: [ θ ] ( φ θ L φl )( y µ µ ) 7

29 c~ F~ e where [ ε ε ε ] e [ ε ] ~ [ ] ' c F ~ and σ ε Q. Here R so * Q σ ε * s * ε and ~ [ ] F. The dsrbuon of neres { } { } Ω s hen facored as: { } { } Ω { } * ( Ω ) ( { } ) C Ω Ths facorzaon s esablshed by he Markov srucure of. In parcular condonal on and Ω here s no addonal nformaon regardng n and Ω. Also noe ha s condoned on * raher han on. Ths s because he remanng erms n beyond hose n * deermne elemens of wh cerany whch creaes a sngulary ha makes generang mpossble. Gven hs facorzaon a draw from { } { } Ω drawng a realzaon of *( g) recursvely from { } from { } ( Ω ) ( Ω ) can proceed by frs denoed (g) and hen drawng. Operaonally hs s performed by frs runnng he Kalman Fler on he sae-space model o compue and save { } and { P } 8

30 9 where and P are he flered esmae of and s varance-covarance marx respecvely. The Gaussan srucure of he sae-space model mples { } ( ) ( ) P N ~ Ω whch can be used o generae (g). We hen smply need { } ( ) m Ω ) *( o complee he algorhm. Carer and Kohn (994) show ha hs s gven by: { } ( ) ( ) * * ) *( ~ Ω g P N where: { } ) ( ) ( ) ( * ~ * ~ ) *( * *' ~ * ~ *' ~ )* ( * g g F c Q F P F F P E Ω { } P F Q F P F F P P Cov P * ~ * *' ~ * ~ *' ~ * ) ( ) ( * Ω Noe ha for many forecasng models such as he regme-swchng auoregressve model n (8) { } s observed. In hs case we smply condon on he observed { } raher han sample values of { } from s condonal dsrbuon.

31 A. amplng { } J To draw from { } { } Ω we use he mulmove algorhm dealed n Km and Nelson (998). To begn noe ha { } { } Ω can be facored as: { } { } Ω { } ( Ω ) ( { } ) C Ω. Agan hs facorzaon s esablshed by he Markov naure of. In parcular condonal on and Ω here s no addonal nformaon regardng n and Ω. Gven hs a draw from { } { } Ω realzaon of ( g) ( { } Ω ). from { } ( Ω ) can proceed by frs drawng a denoed (g) and hen drawng recursvely from Operaonally hs s performed by frs runnng he Hamlon (989) fler on he forecasng model condonal on { }. For example condonal on { } { ε } and hus on he forecasng model n (5) s an ARMA () for whch he movng average componen θ ε θε s observed. The Hamlon (989) fler produces he flered probables: [ w { } ] { } Pr Ω 3

32 [ ] The fnal flered probably w { } Ω gves us { } whch we can generae whch s gven by: (g) Pr ( Ω ) ( g). We hen draw recursvely from { } from ( Ω ) Pr [ Ω ] [ Ω ] ( g) [ { } ] [ ] { } ( g) Pr w Pr w w Ω ( g) Pr[ w] Pr w { } w ( g) where Pr[ w] are smple funcons of he ranson probables p and q. Fnally gven ( g) Pr w { } ( g) ( g) [ Ω ] Pr[ w ] s agan a smple funcon of he ranson probables p and q and can be used o generae ( g). In urn Pr[ w ] ( g) o generaed { ( g) } J. s hen used o generae ( g). Ths s repeaed 3

33 References Beverdge. and C.R. Nelson 98 A New Approach o Decomposon of Economc Tme eres no Permanen and Transory Componens wh Parcular Aenon o Measuremen of he Busness Cycle Journal of Moneary Economcs Blanchard O. and D. Quah 989 The Dynamc Effecs of Aggregae Demand and upply Dsurbances Amercan Economc Revew Carer C. K. and P. Kohn 994 On Gbbs amplng for ae pace Models Bomerca Clarda R. H. and M.P. Taylor 3 Nonlnear Permanen-Temporary Decomposons n Macroeconomcs and Fnance The Economc Journal 3 C5-C39. Clark P.K. 987 The Cyclcal Componen of U.. Economc Acvy Quarerly Journal of Economcs Fredman M. 964 Moneary sudes of he Naonal Bureau he Naonal Bureau eners s 45 h Year 44 h Annual Repor 7-5 (NBER New York); Reprned n Fredman M. 969 The opmum quany of money and oher essays (Aldne Chcago). Fredman M. 993 The pluckng model of busness flucuaons revsed Economc Inqury Hamlon J.D. 989 A new approach o he economc analyss of nonsaonary me seres and he busness cycle Economerca Hodrck R. J. and E. C. Presco 997 Poswar U.. Busness Cycles: An Emprcal Invesgaon Journal of Money Cred and Bankng 9-6. Km C.-J. 994 Dynamc lnear models wh Markov-swchng Journal of Economercs 6 -. Km C.-J. Morley J. and J. Pger 3 Nonlneary and he Permanen Effecs of Recessons workng paper Federal Reserve Bank of. Lous. Km C.-J. and C.R. Nelson 998 Busness Cycle Turnng Pons a New Concden Index and Tess of Duraon Dependence Based on a Dynamc Facor Model wh Regme wchng The Revew of Economcs and ascs Km C.-J. and C.R. Nelson 999a Fredman s Pluckng Model of Busness Flucuaons: Tess and Esmaes of Permanen and Transory Componens Journal of Money Cred and Bankng

34 Km C.-J. and C.R. Nelson 999b ae-pace Models wh Regme wchng: Classcal and Gbbs-amplng Approaches wh Applcaons MIT Press Cambrdge and London. Lam P The Hamlon Model wh a General Auoregressve Componen: Esmaon and Comparson wh Oher Models of Economc Tme eres Journal of Moneary Economcs Morley J. A ae-pace Approach o Calculang he Beverdge-Nelson Decomposon Economcs Leers Morley J.C. Nelson C.R. and E. Zvo 3 Why are he Beverdge-Nelson and Unobserved-Componens Decomposons of GDP so Dfferen? The Revew of Economcs and ascs Wason M.W. 986 Unvarae Derendng Mehods wh ochasc Trends Journal of Moneary Economcs

35 Table Maxmum Lkelhood Esmaes for KMP Model of U.. Real GDP Parameer Esmae andard Error µ.83.8 µ λ q p.679. φ.38.8 φ.76.8 σ ε 34

36 True Trend Generaed eres True Trend eady ae Measure Fg. Generaed seres rue rend and esmaes of rend for he Hamlon model 35

37 True Trend Hodrck-Presco Fler Measure True Trend Nonlnear Beverdge-Nelson (Clarda-Taylor) Measure Fg. (Connued) Generaed seres rue rend and esmaes of rend for he Hamlon model 36

38 True Trend Generaed eres True Trend eady ae Measure Fg. Generaed seres rue rend and esmaes of rend for he pluckng model 37

39 True Trend Hodrck-Presco Fler Measure True Trend Nonlnear Beverdge-Nelson (Clarda-Taylor) Measure Fg. (Connued) Generaed seres rue rend and esmaes of rend for he pluckng model 38

40 True Cycle eady ae Measure True Cycle Hodrck-Presco Measure Fg. 3 True cycle and esmaes of cycle for he pluckng model 39

41 True Cycle Nonlnear Beverdge-Nelson (Clarda-Taylor) Measure Fg. 3 (Connued) True cycle and esmaes of cycle for he pluckng model 4

42 True Cycle Frs Dfference of True Cycle Fg. 4 The level and he frs dfference of he rue cycle for he pluckng model 4

43 Log Real U.. GDP moohed Probably. Fg. 5 U.. real GDP and smoohed probably of a conraconary regme 4

44 Fg. 6 eady-sae esmaes of rend and cycle for U.. real GDP mpled by he KMP model (NBER recessons shaded) 43