AN ECONOMETRIC CHARACTERIZATION OF BUSINESS CYCLE DYNAMICS WITH FACTOR STRUCTURE AND REGIME SWITCHING * Marcelle Chauvet 1


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1 AN ECONOMETRIC CHARACTERIZATION OF BUSINESS CYCLE DYNAMICS WITH FACTOR STRUCTURE AND REGIME SWITCHING * Marcelle Chauve Deparmen of Economics Universiy of California, Riverside 5 Universiy Avenue Riverside, CA A dynamic facor model wih regime swiching is proposed as an empirical characerizaion of business cycles. The approach inegraes he idea of comovemens among macroeconomic variables and asymmeries of business cycle expansions and conracions. The firs is capured wih an unobservable dynamic facor and he second by allowing he facor o swich regimes. The model is esimaed by maximizing is likelihood funcion and he empirical resuls indicae ha he combinaion of hese wo feaures leads o a successful represenaion of he daa relaive o exan lieraure. This holds for wihin and ouofsample and for boh revised and real ime daa. Running Head: Business Cycle Dynamics KEY WORDS: Asymmeries, Business cycles, Comovemens, Dynamic facor model, Kalman filer, Markov swiching. JEL Classificaion: C32, C5, E32 * Manuscrip submied in January 996. This paper was wrien wih financial suppor from CNPq  Brazilian Council for Scienific and Technological Research. This maerial is based on my docoral disseraion from he Universiy of Pennsylvania. I am graeful o my advisor Frank Diebold for his invaluable advice during all sages of his research. I also hank James Hamilon for helpful suggesions. The auhor bears full responsibiliy for any errors.
2 . INTRODUCTION A grea deal of aenion has focused on measuring business cycles and idenifying heir urning poins. The possibiliy of a se of indicaors providing early signals of change in aggregae economic aciviy is imporan o any business or governmen affeced by expansions and conracions. Based on he early work of Burns and Michell (946), he U.S. Deparmen of Commerce consrucs economic indicaors ha are widely used o predic business cycle urning poins. Burns and Michell provide a careful saisical descripion of he cyclical aspecs of various ime series, and classify macroeconomic variables as lagging, leading or coinciden wih economic aciviy. However, he analysis does no provide a formal mahemaical reamen of business cycle measuremen. In paricular, an explici probabiliy model generaing he aggregae ime series is lacking. As a resul, he Deparmen of Commerce indicaors do no conain all he informaion necessary o characerize business cycle dynamics a he ime he evens occur and are consanly revised expos. The NBER daing of business cycle urning poins is also based on a poseriori inferences of coinciden variables. Since revisions of he indicaors can be subsanial, real ime assessmen of economic condiions may be severely compromised. This paper proposes a heoreical framework in which a formal underlying probabiliy model is used o represen business cycles and o generae coinciden indicaors and inferred probabiliies of expansions and recessions. The resuls are reproducible, and he mehod enables analysis of business cycles in real ime. For example, he imminence of a recession saring in a cerain monh can be deeced by he inferred probabiliies or by he implied coinciden indicaor a he same ime he macroeconomic variables are signaling he recession. In he proposed model, business cycles are empirically characerized by a dynamic facor model wih regime swiching. The dynamic facor is an unobservable variable ha
3 summarizes he common cyclical movemens of some coinciden macroeconomic variables. This facor is subjec o discree shifs in order o capure he asymmeric naure of business cycle phases  expansions are gradual and display a high mean duraion while recessions are shorer and seeper. Hence, he approach used in his paper models he idea of business cycles as he simulaneous movemen of economic aciviy in various secors by using an unobserved dynamic facor. In addiion, he asymmeric naure of expansions and conracions is capured by assuming ha he underlying facor swiches regimes according o a Markov process. Boh of hese ideas were fundamenal elemens of Burns and Michell s (946) research. The wo elemens, however, have been sudied separaely. On he one hand, Sock and Wason (989, 99, 993) develop a model where business cycles are measured by comovemens in various componens of economic aciviy in order o obain an alernaive index o he Deparmen of Commerce indicaors. Recessions (expansions) are generaed by negaive (posiive) symmeric shocks o a linear and dynamically sable ime series sysem. Unforunaely, Sock and Wason s model fails o accoun for he 99 recession using a recession index exraced from heir nonswiching dynamic facor represenaion. The lineariy imposed by heir model implies a builin symmery which forces expansions and conracions o have he same magniude, duraion, and ampliude. In addiion, Sock and Wason s model does no ake ino accoun changes over ime in he sochasic srucure of he economy, such as shifs in policy, while an analysis of he long erm rend of aggregae macroeconomic ime series such as employmen, sales, and oupu indicaes several srucural changes over his cenury. On he oher hand, Hamilon (989) considers nonlineariies in business cycles by assuming ha he growh rae of quarerly GNP follows a nonlinear saionary process and incorporaes occasional discree variaions in he dynamic feaures of his ime series wih a Markov 2
4 swiching model. He finds evidence of asymmeries in cyclical expansions and conracions and ascerains he differences in he dynamics of business cycle phases. However, Hamilon s model, since i is univariae, can no capure he noion of economic flucuaions corresponding o comovemens of many aggregae and secoral variables. In addiion, exensions of Hamilon s analysis o monhly growh raes fail o accoun for several of he hisorical recessions as deermined by he NBER. I is possible ha all underlying business cycle informaion can no be exraced from only one coinciden variable. Also, individual coinciden variables display movemens ha do no correspond o business cycle dynamics bu insead o noise inheren in monhly daa. Diebold and Rudebusch (996) sugges inegraing he wo ideas by fiing a univariae Markov model o he Deparmen of Commerce coinciden index and o is componens. They show evidence of he suiabiliy of a swiching dynamic facor, alhough hey do no fully esimae i. The exising lieraure has no found a generally acceped framework ha provides a unified explanaion of business cycle asymmeries and he comovemens of economic aggregaes. In his paper, we consruc an inegraed represenaion of wha Sock and Wason and Hamilon pursued in separae frameworks, as suggesed in Diebold and Rudebusch (996). The idea is ha he inegraed approach migh capure in large par wha Burns and Michell and he NBER have in mind in heir descripion of business cycle comovemens and asymmeries. The novel aspec of his paper wih respec o he exising lieraure is ha we fully esimae a dynamic facor model wih regime swiches by maximizing is likelihood funcion. Mehods for esimaing his model were no available unil recenly. To esimae he model i is necessary o make inferences abou boh he unobserved nonlinear facor and he laen 3
5 Markov sae. Hamilon s (989) paper popularized he use of Markov regime swiches, bu he nonlineariy inroduced by his precluded he esimaion of mulivariae unobservable dynamic models. In paricular, Hamilon s nonlinear esimaion algorihm can no handle models wih a regime swiching dynamic facor. The dynamic facor model proposed by Sock and Wason s (989, 99, 993) is governed by a linear sochasic process, which implies ha he esimaion can be implemened by applying he Kalman filer. The esimaion procedure underaken in his paper consiss of a combinaion of Hamilon s algorihm and a nonlinear discree version of he Kalman filer, as proposed by Kim (994). The goals in building a dynamic facor model wih regime swiching are o obain opimal inferences of business cycle urning poins, and o consruc alernaive coinciden indicaors o he Deparmen of Commerce coinciden index. The empirical resuls indicae ha he combinaion of a dynamic facor model wih Markov swiches leads o a successful represenaion of he sample daa, relaive o he exising lieraure, along several dimensions. This holds boh wihin and ouofsample, and for boh revised and real ime daa ses. In paricular, he resuls corroborae previous evidence abou asymmeries of business cycle phases. The inferred probabiliies are srongly correlaed wih he NBER business cycle daes and all recessions are well characerized for boh quarerly and monhly daa. In addiion, he exraced coinciden index is srikingly similar o he Deparmen of Commerce coinciden indicaor. However, he advanages of he presen framework in comparison o radiional approaches are ha i allows a more rigorous and imely mehod for real ime assessmen of he economy, and resuls can be consisenly reproduced. In conras, he Deparmen of Commerce and he NBER mehodologies require expos revision in order o obain resuls ha we are able o aain in real ime. 4
6 The paper is organized as follows: he second and hird secions presen he model and discuss he esimaion procedure employed in our sudy of he swiching dynamic facor framework. In he fourh secion, he empirical resuls are presened and inerpreed for boh quarerly and monhly daa. In he fifh secion, he model is esed for ouofsample performance and he sixh secion concludes and suggess direcions for fuure research. 2. THE MODEL A vecor of macroeconomic variables displaying comovemens wih aggregae economic condiions is modeled as composed of wo sochasic auoregressive processes  a single unobserved componen, which corresponds o he common facor among he observable variables, and an idiosyncraic componen. 2 A sochasic rend is no included in he dynamic facor based on evidence ha each of he series sudied migh be inegraed bu no coinegraed. 3 Therefore, he empirical analysis is underaken using he log of he firs difference of he observable variables. The model is: () Y i = λ i (L) F + v i i =,...,n nx nx x nx (2) F = α S + α 2 + φ(l) F  + η s S =, x x x x x x (3) v i = D i (L) v i + ε i i =,...,n nx nxn nx nx The assumpions of he model are: 2 The coinciden variables considered are sales, personal income, employmen and producion. 3 We performed a DickeyFuller es (979) for he presence of uni roos in each of he coinciden variables and i was no able o rejec he null hypohesis of inegraion agains he alernaive of saionariy a he % level. We also esed wheher he four coinciden variables are no coinegraed agains he alernaive of coinegraion using Sock and Wason s (988) es, and i failed o rejec he null. 5
7 η s ~ N(, σ 2 ηs ) ε i ~ i.i.d. N(,Σ) H s ~ NI Σ 2 σ ηs D i (L) = diag(d (L),..., d n (L)) M p ij = Prob[S =j S  =i], p = i for M saes which implies ha F and v i, for i =,..., n are muually uncorrelaed a all leads and lags. The vecor Y i is he log of he endogenous observable variables, he parameers λ i are he facor loadings, which measure he sensiiviy of he i h series o he business cycle, and F is he common facor. In addiion, he variables v i are he idiosyncraic erms, he ε i are he measuremen errors, and η is he ransiion or common shock. The funcions λ(l), φ(l) and D(L) are finie lag polynomials of orders l, f and b, respecively, where L is he lag operaor and =L. A nonlinear srucure is inroduced in he unobserved index in he form of a firs order wosae Markov swiching process. The mean growh rae of he dynamic facor is direcly calculaed from he nonlinear filer and i is subjec o sporadic discree regime shifs. Tha is, he economy can be eiher in a sluggish growh sae, S =2, or in an acceleraed expansion period, S =, wih he alernaion beween regimes ruled by he oucome of a Markov process. The facor mean or he facor variance swich beween saes, governed by he ransiion probabiliies of he Markov process, p ij. For example, Prob[S = S  =] = p is he probabiliy of an expansion given ha he economy is expanding, and Prob[S =2 S  = 2] = p 22 is he j= ij 6
8 probabiliy of recession given ha he economy is in a recession. 4 In he proposed model, cycles are generaed from common shocks o he dynamic facor, η s, and all idiosyncraic movemens arise from ε i. The only source of comovemens among he observable variables comes from he dynamic facor, which can be inerpreed as he business cycle. 5 The idenificaion of he model is discussed in an appendix ha is available from he auhor on reques (see also Chauve 995). A paricular sae space represenaion for he swiching dynamic facor ()(3), wih an AR(2) process for he facor and an AR() for he idiosyncraic erm, is: 4 We also consider anoher specificaion for he facor mean suggesed by Hamilon (994), in which he shifs depend on he dynamics of he auoregressive process, ha is, φ(l) ( F  µs) = ηs, for S =,. This specificaion requires ha he order of he Markov swiching process be a leas as long as he degree of he polynomial φ(l). Boh models were esimaed and compared using he same number of lags for he Markov process. However, anicipaing he empirical resuls, his equaion was slighly dominaed by model (2), boh by specificaion ess and in erms of performance in predicing business cycle urning poins. Oher variaions on he basic models were also inroduced, such as allowing he facor variance and mean o swich regimes, or holding he mean consan wih a swiching variance for he facor. 5 The number of facors underlying he variables was verified by he usual procedure underaken in exploraory facor analysis. Tha is, we check he eigenvalues of he correlaion marix conaining he par of he oal variance ha is accouned for by he common facors. The magniude of he eigenvalues for each facor, which conveys informaion abou how much of he correlaions among he observable variables a paricular facor explains, indicaed srong evidence for he single facor srucure. 7
9 8 Measuremen Equaions: Y Z ξ (4) Y Y Y Y F F v v v v F = λ λ λ λ * Transiion Equaions: ξ α ξs T ξ  u (5) F F v v v v F S d d d d F F v v v v F s = α α φ φ η ε ε ε ε + * or (4 ) Y = Z ξ (5 ) ξ = α ξs + T ξ  + u Noice ha he erm F  is included in he sae vecor o allow esimaion of he dynamic facor in levels from he ideniy F  = F   F ESTIMATION PROCEDURE We simulaneously esimae he dynamic facor model wih regime swiching by maximizing is likelihood funcion. In order o esimae he model i is necessary o make inferences abou boh he unobserved nonlinear facor and he laen Markov sae. The nonlineariy imposed by he regime shifs precluded he esimaion of a mulivariae dynamic
10 model unil he esimaion mehods by Alber and Chib (993), Shephard (994) and Kim (994) were developed. 6 We use mehod developed by Kim (994) o esimae our model. In ha paper, Kim exended Hamilon s Markovswiching model o a general dynamic linear saespace framework where boh he measuremen and he ransiion equaions are allowed o swich regimes. The parameers of he model are dependen upon a sae variable S, which follows a sochasic process. A nonlinear discree version of he Kalman filer is combined wih Hamilon s nonlinear filer in one algorihm. This permis esimaion of he unobserved sae vecor as well as he probabiliies associaed wih he laen Markov sae. An ineresing aspec of Kim s filer for he presen analysis is ha his algorihm combines he esimaion mehods underaken by SockWason and Hamilon, while he proposed model inegraes he frameworks underlying heir models. The objecive of Kim s nonlinear filer is o form forecass of he unobserved sae vecor and he associaed mean squared error marices. The forecass are based on informaion available up o ime , I  [ Y , Y 2,..., Y ], on he Markov sae S aking on he value j, and on S  aking on he value i: 7 6 Alber and Chib (993) and Shephard (994) proposed, independenly, a modified version of he Gaussian filering and smoohing procedures. They employ simulaion and Gibbs sampling o obain he exac likelihood funcion of a nonlinear ime series. Basically, a Gaussian saespace is used o analyze nongaussian frameworks hrough simulaion echniques, and he approach may be used o find maximum likelihood esimaes of he Markov swiching model. However, his mehod is sill very cosly in erms of compuaion ime. 7 In he empirical exercise, we chose o esimae he model including an addiional sae variable, s2, o obain (h,i, more efficien esimaes. In his case, equaions (6) and (7) become, respecively: ξ j)  =E(ξ I, S=j, S=i, S2=h) (h,i, and P j)  = E[(ξ  ξ )(ξ  ξ )' I, S=j, S=i, S2=h)]. 9
11 (i, j) (6) ξ  (i, j) (7) P  = E( ξ I , S = j, S  = i) = E[( ξ  ξ  )( ξ  ξ  )' I , S = j, S  = i)]. For z lagged saes condiioning he forecass and M regimes, he algorihm calculaes M z forecass for each dae, corresponding o every possible value of S z. The filer uses as inpus he join probabiliy of he Markovswiching saes a ime 2 and  condiional on informaion up o , {Prob(S 2 =h, S  =i I  )}; an inference abou he sae vecor using i, j informaion up o , given S 2 =h and S  =i, ha is, {ξ   }; and he mean squared error i, j marices, { P   }. The oupus are heir onesep updaed values. For he probabiliies we use as an iniial condiion he probabiliies associaed wih he ergodic disribuion of he Markov chain. For he sae vecor, is uncondiional mean and uncondiional covariance marix are used as iniial values. 8 The nonlinear filer racks he course of he sae vecor, which is calculaed using only observaions on Y, and compues recursively onesepahead predicions and updaing equaions of he sae vecor and he mean squared error marices. The par of he filer ha corresponds o a nonlinear discree version of he Kalman filer, applied o he paricular saespace (4)(5) is: ( 8) ( 9) ( ) ( ) (i, j) ξ = α + Tξ  (i, j)  j i   i   P = TP T' + H ξ (i, j) (i, j) (i, j) = ξ + K N  P = (I  K Z)P (i, j) (i, j) (i, j)  j (i, j)  (predicion equaions) (updaing equaions) 8 i The uncondiional mean and variancecovariance marix of he sae vecor are, respecively, ξ =E(ξ) and i P i =T P T +σ 2 ηs. The uncondiional mean of he probabiliies are {Prob(S2=h, S=i I}=Prob(S=i)=πi, i=,2, where πi is he ergodic probabiliy.
12 (i, Here, K j) (i, j) (i, j) (i, = P Z'[Q ] is he Kalman Gain, N j) (i, j)  = Y  Z ξ  is he condiional forecas (i,j) (i, j) error of Y, and Q = ZP Z' is is condiional variance. The nonlinear filer allows recursive calculaion of he prediced equaions, given he parameers in T, Z and H and iniial condiions forξ j and P j. In he second par of Kim s filer, he probabiliy erms are compued using Hamilon s nonlinear filer. This provides he condiional probabiliy of he laen Markov sae a ime. The condiional likelihood of he observable variable is evaluaed as a byproduc of he algorihm a each, which allows esimaion of he unknown model parameers. The log likelihood funcion is: T M M n/2 (i, j) / 2 (i, j)' (i, j) (i, j) (2) LogL= log f( Y T, Y T,... I ))= log exp( )} = j= {(2π Q N i= Q N 2 The filer evaluaes his likelihood funcion, which can be maximized wih respec o he model parameers using a nonlinear opimizaion algorihm. The parameers esimaed and he sample daa are hen used in a las applicaion of he filer o draw inferences abou he dynamic facor and probabiliies based on informaion available a ime. 9 For each dae he nonlinear filer compues M z forecass, which implies ha a each ieraion he number of cases is muliplied by M, where M is he number of regimes and z is he number of saes condiioning he forecass. Thus, if he filer does no reduce he number of erms a each ime, i becomes compuaionally unfeasible even in he simples wosae case. Kim proposed an approximaion inroduced hroughξ j and P j for >, based on he work of Harrison and Sevens (976). The approximaion consiss of a weighed average of he 9 We also obain full sample inferences abou he sae vecor and unobserved regimes using Kim s (994) smoohing algorihm.
13 updaing procedures by he probabiliies of he Markov sae, in which he mixure of M z Gaussian densiies is collapsed, afer each observaion, ino a mixure of M z densiies. Tha is: ξ M (i, j) Prob[ S ξ j = i,s = j I ] i= = Prob[ S = j I ] P M (i,j) j (i, j) j (i,j) Prob[ S = i,s = j I ]{ P + ( ξ ξ )( ξ ξ )'}. Prob[ S = j I ] j i= =, 4. EMPIRICAL RESULTS Daa The empirical analysis focuses on boh quarerly and monhly daa. For he monhly sudy, he daa sample is from o 993.3, and for he quarerly, from o The goals are o obain opimal inferences of business cycle urning poins and o consruc alernaive coinciden indicaors o he Deparmen of Commerce coinciden index. Thus, we focus on he same four secoral variables uilized by he NBER and he Deparmen of Commerce, which were seleced based on he work of Burns and Michell (946). The series This filer is an opimal esimaor in he sense ha no oher esimaor based on a linear funcion of he informaion se yields a smaller mean squared error. Smih and Makov (98) examine he naure of his approximaion hrough simulaions o verify is performance in erms of jump esimaion and deecion as well as is fi o he opimal soluion. Using an exensive range of iniial condiions for he inpus and saring parameer values, Smih and Makov conclude ha he approximaion performs well in erms of minimizing he sum of he squared errors compared wih oher nonlinear approximaion mehods. Also i is he mehod ha racks mos closely he rue observaions and i is he bes in esimaing he jumps. The sample range was chosen o be close o he one used by Hamilon, in order o faciliae comparisons. Esimaions were also performed for daa prior o 952 and he resuls are similar o he ones obained here. One difference is ha he probabiliy of saying in a recession is smaller, which may be explained by he magniude of sudden changes in he growh of he variables sudied during he 4s. 2
14 used include manufacuring and rade sales in 982 dollars (MTS), oal personal income less ransfer paymens in 987 dollars (PILTP), employees on nonagriculural payrolls (ENAP), and indusrial producion (IP). As alernaives, we also examine gross domesic produc (GDP), hours of employees on nonagriculural payrolls (HENAP), oal civilian employmen (TCE), and nonagriculural civilian employmen (NACE). 2 In boh quarerly and monhly sudies, he daa are ransformed by compuing one hundred imes he firs difference of he logarihm of each series. 3 The swiching facor coinciden index (SFC) esimaed in his paper is compared o he Composie Coinciden Index of he Deparmen of Commerce (CCI 982=) and he coinciden index proposed by Sock and Wason (SW). The series SW uses he same variables as CCI wih he excepion of employees on nonagriculural payrolls (ENAP), which is subsiued wih hours of employees on nonagriculural payrolls (HENAP). Model Selecion and Specificaion Tess Several differen specificaions of he various models were esimaed, including AR(), AR(2) and AR(3) processes for he facor in he ransiion and measuremen equaions, AR() and AR(2) for he idiosyncraic erms, and combinaions of hese using differen coinciden variables. More highly parameerized models were also esimaed, bu he coefficiens of 2 The daa, kindly supplied by Frank Diebold and Glenn Rudebusch, were obained from he Federal Reserve Board s daabank, released in June We also esimaed he model using linear derended series as well as by applying he HP filer derending echnique. Under he linear derending mehod, if a srucural break in he 7s is no aken ino accoun, he sample daa idenify he swiching as permanen changes raher han cycling back and forh. On he oher side, he HP filer is designed o remove aspecs of he daa, such as low frequency cycles. Thus, since our goal is o uncover dynamics underlying he daa wihou imposing any a priori informaion on i, hese derending mehods urn ou o be inappropriae for his paper. 3
15 higher dynamic orders were no significan a he 5% saisical level. 4 Akaike Informaion Crierion, Schwarz Crierion and he likelihood raio es were used o choose among alernaive specificaions of he model. In order o check he adequacy of he model specificaion, we analyze he disurbances in he observable variables. If he model is correcly specified, he esimaed residuals for each observable variable are serially uncorrelaed and nearly uncorrelaed wih each oher. Thus, he residuals' sample auocorrelaion should be close o zero for observaions more han one period apar and ε should be a whie noise. We also use Brock, Decher, and Scheinkman s (987) BDS es for nonlinear models o check he i.i.d. assumpion for he disurbances. 5 The diagnosic ess for boh quarerly and monhly daa indicae ha he specificaions seleced are adequae for all equaions. The BDS es fails o rejec he i.i.d. hypohesis for he residuals. In addiion, he auocorrelaion funcions for he disurbances are wihin he limi of wo imes heir asympoic sandard deviaion, and he pairwise covariance beween he disurbances is nearly zero. We also es for he number of saes, as was done in Diebold and Rudebusch (996). In paricular, we employ he approach described in Garcia (992). Garcia shows ha if he ransiion probabiliies are reaed as nuisance parameers, resuls from Hansen (993, 996) 4 The exraced swiching facors are almos idenical for all specificaions when using he same variables. 5 For a vecor ε m = ε, ε+, ε+2,..., ε+m, we use m=2,..., 5 and λ=sandard deviaion of ε, where λ is he disance beween any wo vecors, ε m and εs m. The es amouns o esimaing he probabiliy ha hese vecors are wihin he disance λ. 4
16 can be applied o es regime swiching models. 6 We consruc Garcia s es saisic and use criical values ha are repored in his paper. Alhough hese criical values are designed for an AR() regime swiching model and he es is parameer dependen, he highes value in Garcia s able for he % significance level is abou 2.5 imes smaller han he likelihood raio es for he dynamic facor wih regime swiching. 7 Hence, his es provides some evidence rejecing he one sae null hypohesis. We seleced he bes performing model for each daa frequency condiional on he resuls of he diagnosic ess. The quarerly specificaion, henceforh Model, is composed of he coinciden variables MTS, PILTP, ENAP and GDP. Boh he disurbances, v, and he facor, F, follow a second order auoregressive process in he ransiion equaions (b=2,f=2). For monhly ENAP, i is necessary o inroduce a high order auoregressive process o eliminae he misspecificaion in is equaion. Since his would amoun o sudying a lagging indicaor, we also examined oher alernaive measures of employmen such as NACE, TCE or HENAP. Parsimonious versions of he swiching dynamic facor model pass specificaion ess when we use hese coinciden variables. Thus, for he monhly analysis (Model 2), he series used are 6 Markov swiching models require nonsandard esing mehods since several of he classical assumpions of asympoic disribuion heory do no hold. For example, he ransiion probabiliies are no idenified under he null hypohesis, which implies ha he likelihood funcion wih respec o hem is fla a he opimum poin. Hansen (993, 996) proposes simulaion mehods o approximae he asympoic null disribuion of a sandardized likelihood es under nonsandard condiions. If he ransiion probabiliies are reaed as nuisance parameers, he asympoic one sae null disribuion is he supremum over all admissible values in he space of ransiion probabiliies. 7 We specify a grid in he space of he Markov parameers p and p22, where he range is from.2 o.95 in seps of.5. In order o overcome he problem of local maxima, we also esimaed he likelihood funcion under he alernaive using many differen ses of saring values. 5
17 MTS, PILTP, NACE and IP. A firs order auoregressive process was seleced for boh he disurbances and for he facor (b=,f=). Esimaion Resuls The quarerly daa are obained as simple averages of he monhly daa, which ends o smooh he series somewha. For monhly analysis, empirical models have no been as successful in exracing informaion abou flucuaions in economic aciviy, possibly because of noise inheren o he daa. For example, a univariae Markov swiching model, such as he one sudied in Hamilon (989), fails o accoun for several of he hisorical recessions when applied o some monhly coinciden variables. In addiion, according o Sock and Wason (993), heir monhly experimenal index gives only mild and delayed signals of he las recession and heir ouofsample esimaed probabiliies fail o forecas i. In conras wih he exising lieraure, he inferred probabiliies for monhly daa esimaed from he swiching dynamic facor model are highly correlaed wih NBER business cycle daing. In paricular, in an ouofsample exercise, he inferred probabiliies predic he las recession. The esimaes obained hrough numerical maximizaion of he condiional log likelihood funcion (2) are presened in Tables and 2, for quarerly and monhly daa, respecively. 8 The empirical resuls provide suppor o he Markov swiching framework. Boh monhly and quarerly samples are well characerized by he wosae specificaion. There is a significanly posiive growh rae in sae and a significanly negaive growh rae in sae 2. The asymmeries in he phases of he business cycle are also well characerized by he swiching dynamic facor. The esimaed ransiion probabiliies are subsanially 8 We also summarize he resuls of a more highly parameerized specificaion for he monhly frequency, alhough he specificaion ess favor more parsimonious versions. 6
18 significan and he probabiliy of saying in expansion, p, is higher han he probabiliy of saying in a conracion, p 22. This confirms previous findings ha he average duraion of recessions is smaller han he duraion of expansions. The expeced duraion for recession and expansion implied by he quarerly swiching models is, respecively, 5 and 3.7 quarers, compared o 4.4 and 4.3 quarers implied by he NBER daing echnique. 9 INSERT TABLES AND 2 Wih respec o he facor loadings, sales (MTS) and indusrial producion (IP) have he highes coefficiens and variances in boh models, supporing he observaion ha hey are he mos sensiive coinciden variables o business cycles. In fac, a hisorical examinaion of U.S. business cycles indicaes ha sales and producion respond immediaely and more inensively o changes in economic condiions close o urning poins han he oher variables analyzed. We also esimaed he model allowing he facor variance o be sae dependen. When he variance and he mean follow a Markov process, he asymmery in he daa is mosly absorbed by he variances. There is a low variance sae wih a high posiive mean, corresponding o he long and gradual expansions, and a high variance sae wih a posiive low mean, associaed wih seep and shor conracions. Probabiliies of Turning Poins The NBER s business cycle daing is generally recognized as he official chronology of urning poins. The deerminaion of peaks and roughs is he resul of a consensus among he Daing Commiee members, who each use differen procedures o examine business cycle phases. Various coinciden macroeconomic variables are examined and urning poins are chosen based on quasisimulaneous accumulaion of inflecion poins. The subjecive aspec of k / ( ). k = The expeced duraion of recession is deermined by he formula: kp ( p ) = p 7
19 he NBER decision process may involve he use of differen crieria over ime, which suggess ha conclusions abou some feaures of expansions and conracions should be confroned wih alernaive daing mehodologies. 2 Anoher drawback of he NBER daing is ha decisions abou urning poins are released wih a long delay. In our case, inferences of he filered and smoohed probabiliies can be used o idenify peaks and roughs. Hence, our mehod allows for real ime assessmen of he economy and resuls can be consisenly reproduced. Figures and 3 graph he esimaed probabiliy ha he economy is in he recession sae a ime, based on informaion up o ime, Prob(S =2 I ), for Models and 2, respecively. Figures 2 and 4 repor he corresponding full sample smooher, Prob(S =2 I T ), for Models and 2. 2 The plos show ha he probabiliies of he recession sae are remarkably similar o he NBER daing of business cycles. All recessions are well characerized by boh filered and smoohed probabiliies, including he las recession of 99. For comparison, we also plo he smoohed probabiliies of a recession obained by fiing Hamilon s univariae model o monhly growh raes in indusrial producion, using an AR(8) process (Figure 5). 22 The inferred probabiliies are no srongly correlaed wih he NBER 2 Boldin (994) presens an exensive review and analysis of exising mehods for daing business cycle urning poins. 2 The bars in he figures represen hisorical recessions as deermined by he NBER. This informaion was no used in esimaing he model and is shown only for comparison wih he inferred probabiliies. The NBER business cycle daing is obained from Business Condiions Diges, June Alhough i may be appropriae o use 2 lags, o correspond o he four quarers in Hamilon s specificaion, he algorihm becomes compuaionally prohibiive for higher auoregressive processes. 8
20 daing of business cycle urning poins and hey fail o accoun for he 97, 982 and 99 recessions. 23 INSERT FIGURES TO 5 Table 3 repors recession urning poins derived from our smoohed probabiliies, using Hamilon s (989) crierion o characerize peaks and roughs. 24 Changes in he probabiliies rack very closely hisorical urning poins, and discrepancies wih he NBER daing are very small (no more han 2 periods), wih he excepion of he 957 and 99 recessions. For he 957 recession, he esimaed smoohed probabiliies for boh models indicae a peak before he official NBER dae. The probabiliies sugges, as also observed by Hamilon (989), ha he recession came as an immediae response o he oil price shock in he firs quarer of 957. In general, daing differences beween he proposed mehod and he NBER daing are concenraed in he deerminaion of peaks more han in roughs. In paricular, he roughs from Models and 2 almos all coincide wih he NBER s, wih he excepion of he las recession. INSERT TABLE 3 23 Hamilon s univariae Markov swiching model does no succeed in yielding resuls ha are srongly correlaed wih he NBER recession daing when applied o some coinciden macroeconomic variables, including monhly indusrial producion (IP), sales (MTS), personal income (PILTP), DOC coinciden indicaor (CCI) and employmen (ENAP). I is successful when applied o employmen (HENAP, NACE, TCE). The order of he assumed auoregressive process also affecs he resuls. 24 Under his mehod, he economy is in a recession if he fullsample smoohed probabiliy of recession is greaer han.5, Prob(S=2 IT)>.5. This meric is no necessarily opimal and flexibiliy should be considered in dubious cases, such as when he recession probabiliies are clusered in he inerval (.3,.7). We follow he NBER rule in considering he minimum duraion of a recession o be six monhs. 9
21 The economic recession in 99 exhibied some unusual feaures no observed in previous ones, in paricular during he final sage as he economy began o recover. Alhough, as in prior conracions, producion and sales had a seep upurn near he rough, real personal income recovered very slowly, and employmen remained low for a long ime afer he NBER declared he recession o be over in March Generally, a faser rise in employmen and income are observed during he recovery phase. This feaure is capured by he swiching dynamic facor srucure and is especially accenuaed in he model using quarerly daa, in which he probabiliies of recession decrease very slowly in he end of he 99 recession. In fac, for Model, he probabiliies indicae ha he recession did no end unil he firs quarer of According o he Bureau of Economic Analysis (BEA) official daing, which is based on he DOC coinciden indicaor, he rough occurred in January This is in close agreemen wih he findings from our exraced coinciden indicaor using quarerly daa. The inferred probabiliies are no only useful in idenifying he beginning and end of recessions as hey occur, bu hey also reveal momens of grea uncerainy in he economy. Almos all recessions were preceded by an increase in he recession probabiliies immediaely before i. Also, a mild rise in he probabiliies 5 o 2 monhs before a recession forewarns a subsequen downurn, wih he excepion of he 97 and 975 recessions. While i is he case ha no all recessions had hese prior indicaive spikes in he probabiliies, every ime here 25 The ensuing uncerainy in he economy during caused he NBER o delay is decision abou he rough of he recession for over a year. The rough in March 99 was chosen by he NBER based primarily on he recovery of indusrial producion, alhough a general rebound happened almos wo years laer. 26 Boldin (994), using a Markov swiching model for unemploymen o dae business cycle urning poins, finds ha an expansion did no begin unil mid Since his paper was compleed, he BEA, hrough exensive revisions of he series, changed he daing of he rough so ha i coincides wih he NBER daing. 2
22 was an abnormal change in he average expansion probabiliies, a recession followed beween half a year and a year laer. The mos noiceable spikes occurred in mid 956 and mid 959. These evens are no considered recessions due o heir very shor duraion of a mos a quarer. Thus, he swiching dynamic facor model migh be useful in capuring signals of an imminen recession ha are implici in coinciden macroeconomic variables. 28 Comparison wih he Deparmen of Commerce Indicaor The similariies beween he growh raes of he exraced facor and he Deparmen of Commerce coinciden indexes are sriking. As seen in Table 4, he growh raes of he exraced coinciden facors and he DOC indexes are highly correlaed for boh models. In paricular, he exraced facor and CCI growh raes for Model exhibi a correlaion of.96. The sandard deviaions of hese indexes are very close, differing by only percen. For Model 2, he correlaion of SFC wih CCI is.94. The SFC growh raes for boh models are ploed on Figures 6 and 7. INSERT TABLE 4 INSERT FIGURES 6 AND 7 28 The purpose of he coinciden indicaor is no o forecas business cycles, bu o obain real ime predicion of he sae of he economy and o dae urning poins. A leading indicaor would be a more appropriae ool for forecasing business condiions. 2
23 For graphical analysis i is easier o examine he indicaors in levels. 29 The level of he exraced index (SFC) for Models and 2, as well as he Composie Coinciden Index from he Deparmen of Commerce are ploed in Figures 8 and 9, respecively. For quarerly daa, he esimaed coinciden indicaor racks very closely he CCI index and, for almos he whole sample sudied, he wo series show he same paern regarding ampliude, iming and duraion of flucuaions. The ime series for he monhly SFC and CCI in levels are also very similar wih respec o he iming and duraion of he cycles, alhough he esimaed coinciden index exhibis more accenuaed oscillaions. INSERT FIGURES 8 AND 9 For he 99 recession, he exraced SFC indicaors for boh models show a deeper decline han he CCI. A possible reason for his is relaed o he weigh of each coinciden variable in he index consrucion. The Deparmen of Commerce calculaes is coinciden index as a weighed average of individual componens, where he weighs are inversely relaed o he series esimaed volailiy. This implies, for example, ha employmen and personal income receive he highes weigh in he consrucion of CCI. In our model, we do no place any a priori resricions on he weighs assigned o each variable ha eners our coinciden indicaor. As i urns ou, he mos volaile variables, sales and producion, are he ones more highly correlaed o he exraced coinciden index, alhough employmen and personal income also 29 We use he ideniy F  = F  +F 2 in he nonlinear filer o obain he facor level, FL, where F  =*log(fl  ). From he esimaed resuls, he exraced facor in levels is obained by exponeniaing.*f. The Deparmen of Commerce calculaes he coinciden index as a weighed average of is individual componens, CCI, ha is: n CCI = i bi Yi, +, where, bi are he weighs, Yi are he growh rae of he coinciden variables and (=3% a = monh) is an adjusmen so ha he index has he same rend as GDP. For graphical comparison, we also accoun for he rend adjusmen imposed on he calculaion of he DOC coinciden index. 22
24 play an imporan role. The more volaile series are he ones ha displayed a seeper decline during he las recession, which migh accoun for he difference beween our coinciden indicaor and he DOC index. The slow recovery following he 99 recession is capured by our SFC index and he Deparmen of Commerce indicaor, boh of which indicae an economic conracion lasing unil he beginning of 992. The monhly SFC index shows a shorer recession in 99 han he quarerly exraced index. This migh be caused by he faser recovery of he monhly IP in he firs quarer of 99. This is he variable wih he highes esimaed weigh in our monhly coinciden indicaor. Figure shows he CCI and he monhly index obained by Sock and Wason (99). Sock and Wason s index has a sandard deviaion 8% smaller han he CCI index. Thus, any comparison of hese wo measures involves an exra correcion. 3 The SW index is more jagged, has a smaller mean han he CCI, and underpredics i for he whole sample period. Wih respec o he 99 recession, he SW index provides weak and lae signals of a recession exhibiing a fas revival a a ime when he economy was sill in a slow recovery as indicaed by he CCI index. Our exraced swiching facor series, as seen in Figures 8 and 9, capures in a imely manner he las recession as well as he slow recovery. INSERT FIGURE 5. OUTOFSAMPLE PERFORMANCE  THE 99 RECESSION We examine he performance of inferred probabiliies in predicing urning poins in an ouofsample exercise. The probabiliy forecass are evaluaed wih respec o heir accuracy 3 Sock and Wason (99) use a modified facor growh rae o exrac he level, whose sandard deviaion is se o he same value as he DOC coinciden index and is mean is kep he same. The series is hen scaled o equal in July
25 and calibraion in predicing observed realizaions by using he quadraic probabiliy score (QPS), he raw correlaion, and he global square bias (GSB), for boh he NBER and he BEA daing. 3 Two ses of possample daa are examined: hisorically revised daa as well as parially revised or real ime daa. The idea is o compare and evaluae no only he model performance of expos forecass, bu also real ime exane forecass using only daa available a he ime of forecasing. As discussed in Diebold and Rudebusch (99), since macroeconomic series undergo several revisions and definiional changes over ime, he use of real ime daa in an ouofsample exercise provides a more rigorous es of model performance for exane forecass. The parameers were esimaed using daa up o 989.4, for quarerly daa, and up o 989.2, for monhly daa. The insample esimaes were hen used o generae ouofsample forecass of he filered probabiliies. For quarerly analysis, ouofsample performance is analyzed from 99. hrough 993. and for monhly, from 99. hrough Given he sample selecion, he ouofsample exercise amouns o esing he model for a very unusual period corresponding o he economic downurn in 99 and he sluggish recovery of he economy in The dynamic facor models wih regime swiching successfully characerize he las recession using boh filered and smoohed probabiliies and for boh quarerly and monhly daa. The monhly model yields daing of urning poins closer o he NBER while he quarerly model is more in accord wih he BEA daing. ExPos Performance  Revised Daa 2 3 The quadraic probabiliy score and he global square bias are, respecively: QPS = T T and GSB = 2{ {Prob[S = 2 I ]  T T = = N } Prob[S=2 I] are he filered probabiliies of recession. T 2 T = {Prob[S = 2 I ]  N }, where N is he / dummy for he NBER or BEA recessions and τ 2, 24
26 Table 5 repors he ouofsample performance of he probabiliies in forecasing urning poins. The dynamic facor model displays a slighly beer pos sample performance in erms of he QPS for regime forecass han he insample resuls. Also, he proposed model shows beer ouofsample performance in erms of he QPS for he NBER regime forecass when compared o alernaive models. The QPS obained for Models and 2 are, respecively,.29 and.. These values are smaller, for example, han he QPS=.34 obained by Hamilon and PerezQuiros (996) for a bivariae Markov Swiching VAR of he DOC leading indicaor and GNP. In fac, he inferred probabiliies from heir model miss he 99 recession. In he middle of he recession, heir inferred probabiliies of a recession were no greaer han 25%. INSERT TABLE 5 The uncerainy in he pah of he economy during is also capured by he ouofsample filered probabiliies obained from boh Models and 2, bu i is paricularly accenuaed for he quarerly frequency (Figure ). 32 A graphical comparison of he ouofsample filered probabiliies for Model and he DOC Coinciden Indicaor (Figure ) shows ha he probabiliies of recession characerize he economy in a very similar way as he CCI. The rough deermined by he NBER o be in he firs quarer of 99 corresponds o a period of flaness in he CCI. I also coincides wih he momen in which he probabiliies of recession sar decreasing. However, hese probabiliies decrease very slowly during he firs and second quarer of 99, and increase again in he hird quarer when he CCI also shows a decline. Only in he firs quarer of 992 did he probabiliies fall below 5%, indicaing he end of he recession. This is in closer agreemen wih he rough decided by he BEA. The QPS 32 This difference migh be explained by he fac ha he monhly model uses he variable IP, which showed a seep upurn in March 99. The variable GDP, used in he quarerly model, shows a mild increase in he firs hree quarers of 99 and a seeper rebound only in he las quarer. 25
27 measuring he closeness of he filered probabiliies o he BEA daing is only.78, and he GSB=.39. INSERT FIGURE Real Time Analysis Ouofsample performance is examined wih real ime monhly daa available a he dae of each forecas, obained from he Survey of Curren Business. 33 Table 5 repors he ouofsample performance for monhly daa, using our real ime daa se. Model 2 performs well in erms of forecasing he NBER regimes, achieving a QPS=.6, a GSB=.3 and a raw correlaion beween he filered probabiliies and he NBER business cycle daes of.6. Figures 2 and 3 show he filered probabiliies of recession for revised and real ime monhly daa. The probabiliies using real ime daa are more volaile, reflecing he uncerainy of he economy during he period. In he beginning of 99 he probabiliy of a recession had a shor and pronounced increase o.8, indicaing he subsequen economic conracion. The probabiliies were above 5% again in May 99. Tha is, he swiching facor model indicaes an economic downurn a he same ime he economy was signaling a recession, using only daa available hen. The model also forecass he beginning of he recession before i occurred. The onesep ahead probabiliy of a recession, based on real ime daa up o May, signals a 5.4% chance of a recession in June (Figure 4). The zerosep ahead probabiliy forecass idenify he end of he recession in March 99. These probabiliies also indicae uncerainy in he economy a he end of 99, increasing close o he 5% level a he 33 Real ime dae corresponds o daa released a +2. For example, daa for January 99 was obained from he March 99 issue of he Survey of Curren Business. This daa selecion is based on Diebold and Rudebusch s (99) evidence ha using preliminary and incomplee daa, as released a +, leads o a poor forecasing performance. 26
28 ime sales and indusrial producion showed a modes decline. INSERT FIGURES 2 AND 3 According o Sock and Wason (993), heir experimenal index fails o characerize and forecas he las recession. Their ouofsample esimaed probabiliies signal a recession only in November of 99. In Ocober, a quarer afer he beginning of he recession, he zerosep ahead recession probabiliy was only.28. The hreemonh ahead probabiliy forecas was.23 in November, based on informaion up o Augus, and he onesep ahead probabiliy of a recession was.5 in boh May and in June, failing o forecas he las recession. 6. SUMMARY AND CONCLUDING REMARKS This paper proposes a model in which business cycles are empirically characerized by a dynamic facor wih regime swiches. The approach capures boh he idea of business cycles as comovemens in several macroeconomic variables and he asymmeric naure of business cycle phases. The opimally inferred daes of business cycle urning poins display a srong correlaion wih he NBER daing of business cycles and he exraced dynamic facor is remarkably similar o he Deparmen of Commerce coinciden indicaor. In paricular, he resuls highligh he imporance of nonlineariies in business cycles. The model provides a more rigorous and imely approach for daing business cycle urning poins han radiional mehods. In paricular, our approach is based on a probabilisic framework ha can be used in real ime o assess he sae of he economy and ha can be replicaed consisenly a any ime. The resuls sugges ha a very saisfacory represenaion of he sample daa is obained by modeling business cycles as he common elemen underlying a se of coinciden variables subjec o sporadic regime shifs. The proposed framework is also a useful ool for exane predicion of business cycle urning poins. Invesigaion using boh revised daa and 27
29 informaion available o agens in real ime indicaes ha he exraced coinciden index and esimaed probabiliies perform very well in heir abiliy o characerize hese urning poins. In he fuure, i migh be worhwhile o invesigae wheher exending he approach in his paper o include leading macroeconomic variables migh yield a leading indicaor ha could be successfully used o forecas urning poins. Also, including addiional saes in he Markov sochasic process migh improve he performance of he model. For example, Burns and Michell (946) conceive business cycles as composed of four disinc periods: prosperiy, crisis, depression and revivals. I migh be ineresing o invesigae his noion using he framework sudied in his paper. In closing, le us address he relaionship of his paper o he conemporaneous and independen work of Kim and Yoo (995). Alhough he KimYoo approach is similar o ours, here are imporan differences, relaed o choice of sample period, model specificaion, and he variables used. As regards sample period, he KimYoo sample sars in 96, which excludes wo recessions compared o our sample daa. Thus, our esimaion uses informaion obained from 8 recessions, while heirs uses informaion from 6 recessions. As regards model specificaion and he variables used, Kim and Yoo use he ENAP employmen variable, which according o Sock and Wason (99) requires exra lags in is equaion o avoid model misspecificaion. This resuls in a mixed coinciden/lagging index specificaion, which is unforunae given ha he objecive is o consruc a coinciden index model. As was discussed in Secion 4, we obain a beerspecified version of he swiching dynamic facor model by using he employmen series NACE insead of ENAP. The upsho is simply ha, because of differences in sample period, model specificaion, and he variables used, he KimYoo coinciden indicaor behaves very differenly from ours. The KimYoo coinciden indicaor is highly correlaed wih Sock and Wason's (he correlaion 28
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