ANALYSIS OF ECONOMIC CYCLES USING UNOBSERVED COMPONENTS MODELS

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1 ANALYSIS OF ECONOMIC CYCLES USING UNOBSERVED COMPONENTS MODELS Diego J. Pedregal Escuela Técnica Superior de Ingenieros Indusriales Universidad de Casilla-La Mancha Avda. Camilo José Cela, Ciudad Real Spain Tlf: Ex FAX: pedregal@ind-cr.uclm.es ABSTRACT The Unobserved Componens Models represen a framework in which phenomena like any periodic behaviour, economic cycles in paricular, may be modelled and forecas naurally. The main disinc feaure of he mehodology used in his paper is he use of a Dynamic Harmonic Regression model, characerised by ime variable parameers ha may vary following a rich family of models. This class of models are se up in a Sae Space conex ha akes advanage of he exraordinary flexibiliy of he recursive algorihms known as he Kalman Filer and Fixed Inerval Smooher. Differen versions of he models have o be applied o ime series, depending on heir ime properies. In paricular, ime series wih a non-consan period cycle has o be analysed in a oally differen way o oher series ha exhibi a consan period cycle. A simple mehod o exrac he cyclical informaion from he series in he case of non-consan period cycles is presened in he paper. The mehodology is compared wih ohers and shown working in pracice wih several examples. Key words: Hodrick-Presco filer, Inegraed Random Walk, Kalman Filer, Fixed Inerval Smooher, Unobserved Componens Models. 1

2 1. INTRODUCTION I was long ime ago when he necessiy of exracing basic unobserved signals from observed ime series appeared. One ineresing aspec of such an special analysis of ime series was he possibiliy o follow he underline evoluion of variables independenly of all kind of shor run variaions, such as seasonal paerns or oulier observaions. In his regard, he sudy of rends, cycles and seasonal componens (or alernaively seasonaliy-adjusmen) have been he goals of many researchers worldwide. Regarding he analysis of he economic cycles, a grea deal of confusion has appeared in he lieraure since he iniial sudies abou he opic. The imporan work of Burns and Michell (1946) Measuring he Business Cycles may be considered an iniial sep in he direcion of he empirical evaluaion of his maer. However, i was criicised by imporan researchers, like Koopmans (1947), mainly because i presened measuremen wihou heory. Neverheless, Kydland and Presco (1990) argued ha while he criicism is accepable, hypohesis abou he saisical disribuions of variables should no be done based on he Economic Theory. These are examples of he widely acceped inheren subjeciviy of his opic. While he disincion beween rend and cycle is somewha problemaic from a heoreical poin of view, however, he empirical definiion is quie ransparen if i is provided in he frequency domain and using Unobserved Componens (UC) models. When dealing wih obaining empirical measures of he economic cycle, he confusion is complex, because he lack of consensus abou he definiion of he economic cycle has moivaed he appearance in he lieraure of a variey of very differen mehods. Some of he mos ypical are linear or polynomial rends; differeniaion; applicaion of moving average filers; Hodrick-Presco (HP) filer; nonparameric esimaes; ec. Due o he raher differen naure of each of hese procedures he esimaed cyclical componens for he same series have very differen properies. Burns and Michell made very imporan conribuions o he problem ha have been blurred by poserior discussions beween heoriss and praciioners. These basic conribuions are he definiion of he differen phases of he cycle and is characerisics, bu above all and from he poin of view of he presen paper, he explici delimiaion of he bandwidh of ineres in order o provide a clear definiion of he cycle (such bandwidh in heir original work was beween 18 and 10 o 144 monhs). This paper presens a new UC mehodology for he analysis of he economic cycle very successful in he pas when applied o a wide range of ime series. In addiion, a general mehod for obaining an esimaion of he cycle based on a frequency domain approach is proposed. This mehod consiss of he applicaion of a band-pass filer o he series in which he bandwidh is defined by he user depending on his/her paricular definiion of he cycle. This filer is effecively buil as a simple UC model wihin his new class applied wice o he series and is closely relaed o he HP filer.

3 . THE DIVERSITY OF UNOBSERVED COMPONENTS MODELS The UCM are models in which he ime series are decomposed as he sum or a produc of a number of oher simple ime series wih economic or physical meaning. One widely acceped univariae version of UCM is y = T + C + S + e (1) where y is he observed ime series; T is a rend or low frequency componen; C is a susained cyclical or quasi-cyclical componen (e.g. an economic cycle) wih period differen from ha of any seasonaliy in he daa; S is a seasonal componen (e.g. annual seasonaliy); and e is an irregular componen, normally defined for analyical convenience as a normally disribued Gaussian sequence wih zero mean value and variance σ. In order o allow for nonsaionariy in he ime series y, he various componens in he model, including he rend T, can be characerised by sochasic, Time Variable Parameers (TVP s), wih each TVP defined as a nonsaionary sochasic variable, as discussed below. While mos researchers in he area would agree basically wih he general formulaion of he problem given by equaion (1), he agreemen ends jus here. There are an endless amoun of philosophical and pracical deails in such a formulaion ha produces a big amoun of alernaive mehodologies. From all he mehods available nowadays, one of he oldes and bes known echniques, in essence beer known as a seasonal adjusmen mehod, is he X-11 and exensions i.e. X-11 ARIMA and X-1 ARIMA (see e.g. Findley e al., 1996). Due o he fac ha i has been he main mehod of rend esimaion and seasonal adjusmen used by Governmen Agencies all over he World, i has been applied mainly o macroeconomic monhly, quarerly and annual daa. This is a raher ad-hoc mehod of seasonal adjusmen in which smoohing procedures are used o exrac rend and seasonal componens from he ime series. The design of hese filers is based mainly on previous experience and hey have been coninuously refined as he mehod has been applied o more and more series. One of he mos imporan limiaions is is ad-hoc naure, forcing a coninuos refinemen as he mehod is applied o more ime series. Some deerminisic opimisaion mehods have been proposed for signal exracion. These are considered from a variey of differen sandpoins such as regularisaion (e.g., Hodrick and Presco, 1980, 1997; Akaike, 1980) where consrains are imposed on he sae esimaes via a Lagrange Muliplier erm wihin he cos funcion, in order o ensure ha hey possess he required characerisics. More specifically, he variance of residuals is usually minimised subjec o a given degree of smoohness imposed on he componens, as defined by specified weighing marices in he Lagrange Muliplier erm of he cos funcion. The values of Lagrange Mulipliers are esimaed in differen ways, such as cross validaion, bu Akaike esimaes hem wihin a Bayesian framework. Oher alernaive sandpoins for deerminisic opimisaion mehods are smoohing splines ; smoohing kernels ; pseudosplines and wavele mehods. 3

4 There are wo addiional families of mehods ha have received a lo of aenion, especially among economiss. These are he Srucural form and he Reduced form mehods, following he analogy wih he Simulaneous Equaions models ypical of Economerics. If each of he componens in equaion (1) are assumed o follow ARIMA models (hese are he srucural form of he componens), hen he overall model, or reduced form, has o be anoher ARIMA of cerain orders depending on he individual, srucural, models. This wo families have been viewed usually as compeiors, given he radically differen saring poin of view. In Srucural form mehods he UC model is considered as he observaion equaion of a discree ime sochasic Sae Space (SS) model and he associaed sae equaions are used o model each of he componens in Gauss-Markov form. This formulaion has is origin in he he 1960 s when conrol engineers realised ha recursive esimaion and, in paricular, he Kalman Filer, could be applied o he problem of esimaing ime variable parameers in regression models, usually wihin a dynamic sysems conex. More recen developmens have shown how his approach can be exended in various ways o problems of forecasing, backcasing, smoohing (by means of he recursive Fixed Inerval Smoohing) and signal exracion. Here, he mos influenial conribuions are probably hose of Harvey, whose Srucural Model approach is now widely available in he successful STAMP compuer programme (Srucural Time Series Analyser, Modeller and Predicor: see Koopman e al, 1995; Harvey, 1989); and he Bayesian approach by Harrison and Wes, implemened in he Bayesian Analysis of Time Series programme (BATS: see Pole e al., 1995; Wes and Harrison, 1989). The Reduced Form approach o UC model idenificaion and esimaion follows from he success of Box-Jenkins mehods of ime series analysis and forecasing (Box and Jenkins, 1970; Box e al., 1994). I is assumed ha he series can be modelled as an ARIMA model (or reduced form) and find he srucural models for he n individual componens hrough a process of idenificaion, i.e. y ϑ = φ ( B) ( B) e = ϑ n i i = 1 φ i ( B) ( B) where B is he backward-shif operaor; ϑ ( B) and ( B) φ are general MA and AR polynomials of order q and p idenified and esimaed a la Box-Jenkins 1 ; each of he e i 1 The erm general is used here in he sense ha MA and AR polynomials may, and in mos cases will, be he resul of a produc of regular and seasonal polynomials. Also, he AR polynomial here conains all he uni roos in he model included by differencing in order ha he process a hand may be reaed as mean saionary. 4

5 ϑ ( B) and ( B) i φ i are unknown polynomials which orders would depend on he naure of each of he componens and he q and p orders of he overall ARIMA model. From he previous formulae, i is obvious o see ha here exis an infinie number of srucural forms compaible wih each reduced form. Therefore, an idenificaion process is necessary in order o find a saisfacory soluion o he problem. Such process relies on he imposiion of a number of arbirary consrains o ensure exisence and uniqueness of he decomposiion. Many differen ways o achieve his goal have been proposed in he lieraure, being he mos popular he canonical decomposiion by which he variance of he irregular componen is maximised in order o obain componens as free from noise as possible (Hillmer e al., 1983). However, his is no he unique mehod, oher less popular are e.g. he formal decomposiion and he saisical decomposiion (see Piccolo, 198). A full mehodology ha akes advanage of he canonical decomposiion is implemened in he sofware package known as Signal Exracion in ARIMA Time Series (SEATS, Gómez and Maravall, 1998). The new mehodology inroduced in his paper falls ino he Srucural class of models, bu wih novel aspecs discussed below, like new mehods of esimaion ha akes advanage of he excepional specral properies of he Dynamic Harmonic Regression (DHR) model (see e.g. Ng and Young, 1990; Young e al., 1999). Such a mehodology has been implemened in a sofware package, he muli-plaform, CAPTAIN Time Series Analysis and Forecasing Toolbox in Malab 3 (and is predecessor he MS-DOS based microcaptain program). 3. THE STRUCTURAL UNOBSERVED COMPONENTS TIME SERIES MODEL In he Srucural framework, (1) is considered as he observaion equaion in discreeime (possible coninuous-ime of discree differenial) Non Minimal Sae Space model which describes he sochasic evoluion of sae variables associaed wih he UC s in (1). The SS form is an exraordinarily powerful and flexible ool for ime series analysis, as i will be clear in he nex pages. I is also a framework in which he UC ype of models fis naurally. The SS formulaion is described by he following sae and observaion equaions: Sae Equaions : Observaion Equaion : x y = Fx + -1 = H x + e η For example he rend model would incorporae he uni roos in he overall model; he seasonal componen would include he seasonal roos; and so on. 3 Informaion abou his sofware is available in hp://cres1.lancs.ac.uk/capain/ and a bea-es version is available from he auhor. 5

6 where x is an n dimensional sae vecor; y is he scalar observed variable; η is an n dimensional vecor of zero mean, whie noise inpus (sysem disurbances) wih diagonal covariance marix Q and e is he whie noise variable in equaion (1), which is assumed o be independen of η. F and H are, respecively, he nxn sae ransiion marix and he 1xn observaion vecor, which relaes he sae vecor x o he scalar observaion y. Given his form for he overall model, he well-known Kalman Filer and he associae recursive Fixed Inerval Smoohing algorihm provide he basis for forecasing, inerpolaing and smoohing (ha is, esimae he componens). For a daa se of N samples, he former algorihm yields a filered esimae of he sae vecor a every sample, based on he ime series daa up o sample. The laer produces a smoohed esimae of he saes which, a every sample, is based on all N samples of he daa. This means ha, as more informaion is used in he laer esimae, is Mean Square Error canno be greaer han he former. As hese algorihms are discussed in deail in he previous references, we will no pursue he opic furher. These algorihms allow inherenly for missing daa and provide auomaic forecasing, inerpolaion and backcasing. The off-line FIS algorihm is paricularly useful for inerpolaion, signal exracion, seasonal adjusmen and lag-free TVP esimaion. Also i is useful in he so call variance inervenion for handling sudden changes in he rend level. Previous o he applicaion of hese algorihms, variances of all he noises η and e (ofen called hyper-parameers) presen in he model mus be known or esimaed in some way. A usual way o deal wih he problem is o formulae i in Maximum Likelihood (ML) erms (e.g. Harvey, 1989). Assuming ha all he disurbances in he sae space form are normally disribued, he ML funcion can be compued using he Kalman Filer via predicion error decomposiion. This is he generally acceped mehod, because of his well-known heoreical basis. However, he opimisaion can be very complex even for relaive simple models due o he flaness around he opimum (Young e al., 1999). There are several alernaives o ML, from which he one preferred here is buil in he frequency domain (Young e al., 1999). Basically, he parameers are esimaed so ha he logarihm of he model specrum fis he logarihm of he empirical pseudospecrum (eiher an AR-specrum or periodogram) in a leas squares sense. Such mehod provides he esimaion of a noise variance raio marix (NVR), insead of he variances in he SS model. This NVR marix is defined as he raio of he sae noise covariance marix o he variance of he observaion equaion noise i.e. NVR = Q / σ. Since Q is assumed diagonal, i is common o speak abou he NVR parameers, referring o he diagonal elemens of he NVR marix. In he UC conex all he componens in (1) has o be represened in SS form, and hen an overall SS form for he whole model is buil by he assembly of he individual models. Very ofen, he Generalised Random Walk (GRW) is he preferred model for he rend, given by 6

7 x x 1 x = α β 0 γ x η η 1 T = ( 1 0) x1 x () Here, α, β, and γ are consan parameers; T is he rend componen (i.e. he firs sae, x 1 ); x is a second sae variable (generally known as he slope ); η1 and η are zero mean, serially uncorrelaed whie noises wih consan block diagonal covariance marix Q. This model includes, as special cases, he Random Walk (RW: α = 1; β = γ = 0 ; η = 0 ); Smoohed Random Walk (0 < α < 1; β = γ = 1; η 1 = 0 ); he Inegraed Random Walk (IRW: α = β = γ = 1; η 1 = 0 ); he Local Linear Trend (LLT: α = β = γ = 1); and he Damped Trend (α = β = 1; 0 < γ < 1). In he case of he IRW, x 1 and x can be inerpreed as level and slope variables associaed wih he variaions of he rend, wih he random disurbance enering only via he x equaion. Wih respec o periodic componens (eiher cyclical or seasonal) here are many differen alernaives quoed in he lieraure. The main ones are he Dummy Seasonaliy; he Trigonomeric Cycle or Seasonaliy (Wes and Harrison, 1989; Harvey, 1989); he Dynamic Harmonic Regression (DHR: Ng and Young, 1990; Young e al., 1999); he Modulaed periodic componens (Young e al., 1999); and he General Transfer Funcion model (Ng and Young, 1990). From all he above, he DHR opion is he one prefered in his paper. The DHR model has a linear regression form wih deerminisic periodic funcions of ime as inpus wih TVP ha follow GRW models. The SS form for his opion is a a f f a = α β 0 γ f a + η f f f * * * f f η f 1 { cos( ω ) sin( ω )} R f = 1 f f f f S = a + b where b f are parameers defined in he same way as a f. The rigidiy inroduced by he deerminisic funcions is compensaed by he TVP giving he model a grea flexibiliy, capable of represen many ypes of differen seasonal paerns. This is exremely useful for series which exhibi he kind of non-saionariy behaviour in he seasonal componen so commonly observed in economic ime series. 4. THE ANALYSIS OF CYCLES USING STRUCTURAL UCM Two paricular versions of he models shown above are especially relevan for he research of cycles in ime series. These are (i) a single IRW rend and (ii) a model ha includes an IRW rend and a DHR model for he cyclical and/or seasonal par. The selecion beween hese wo models and he advanages of each of hem depend on he properies of he ime series and he objecives of he analysis, as illusraed below. I is obvious ha he objecive of producing forecass of he ime series (or he cyclical componen) ino he fuure is much more complicaed han analysing he peaks 7

8 and roughs of he economic cycle ino he pas. For he presen auhor, analysing and forecasing are wo disinc objecives ha may be reaed separaely in his conex in he sense ha a valid model for he analysis of he periodiciies of he daa in he pas do no necessarily has o be a good model for forecasing he series ino he fuure. However, a valid model in forecasing erms may be also valid for he analysis of he economic cycle inside he esimaion sample. In addiion o his, forecasing is also considerably more complicaed for series ha exhibi a non-consan period cycle, han for hose series wih a consan period cycle, and models used should reflec hese feaures of he daa. Based on he previous consideraions, he res of his secion is devoed o he analysis of wo differen scenarios, illusraed wih ime series ha show he power of he UC models. The firs scenario is a ime series wih a consan period for which a model may be buil and i, in general, will be useful boh for he srucural analysis and for forecasing objecives. The second scenario is one in which he series is somewha more complicaed, because he period of he cycle is ime varying, in his case he srucural analysis of he cycle is relaively simple, bu forecasing is far more complicaed Cyclical componens wih a consan period. The monhly aggregaed elecriciy consumpion in Spain from January 1971 o April 000 in KW per hour is shown in figure 1 and is used here in order o exemplify he resuls obained wih a number of mehodologies. These models are DHR; he Basic Srucural Model (BSM) of Harvey; and an ARIMA model (in which he esimaion of componens is carried ou using he canonical decomposiion given in Gómez and Maravall, 1998). The series reveals he nonsaionariy behaviour in he mean and variance, he ampliude of which seems more imporan in he middle of he series. Pre-processing of he daa in BSM and ARIMA mehodologies are compulsory in hese circumsances. In order o implemen ML esimaion in he BSM model, he logarihmical ransformaion was done. Apar from his, a regular and seasonal difference has o be done previous o he applicaion of he ARIMA mehodology. However, DHR models are able o avoid any pre-judgemen of his kind by modelling and forecasing he basic daa wihou any ransform a all. In his way he DHR model explains he nonsaionariy in boh he mean and he variance of he original daa, insead of removing such problems. However, in order o faciliae comparisons among he mehods considered, he series is also logarihmically ransformed for DHR modelling. (INSERT FIGURE 1) The sandard idenificaion ools sugges an ARIMA airline model for he logarihmically ransformed series (i.e. ARIMA( 011,, ) ( 0, 11, ) 1 ). On he oher hand, he idenificaion in he frequency domain, necessary for he DHR analysis, reveals he exisence of he rend and a clear seasonal paern in he series confirmed by he AR specral esimaes, wih visually significan peaks a periods of 1, 6, 4, 3,.4 and monhs. 8

9 DHR models were esimaed in he frequency domain using he CAPTAIN Malab version (Young e al., 1999); BSM models were esimaed by ML implemened in STAMP (Koopman e al, 1995); and ARIMA models were esimaed by Exac ML using SEATS (Gómez and Maravall, 1998). Table 1 exhibis some relevan comparisons among he innovaions obained by he models. In paricular, i shows he esimaed models for he logarihmically ransformed daa in erms of he variance of he innovaions; he Ljung-Box auocorrelaion es for 1 and 4 lags; and he Jarque-Bera normaliy es 4. The immediae conclusion exraced from able 1 is ha he DHR model ouperforms in saisical sense he oher wo mehodologies. This is specially ineresing, because he DHR model is opimised in he frequency domain, while BSM and ARIMA are boh esimaed in he ime domain. I is surprising in he sense ha ML crierion is defined as an explici, ime domain funcion of he normalised innovaions, whereas opimisaion in he frequency domain, is primarily concerned wih ensuring ha he specral properies of he esimaed componens mach he empirical specrum of he daa. σ ˆ a DHR Model Basic Srucural Model ARIMA 0.58e e e-03 Q(1) Q(4) Jarque-Bera 0.06 (0.96) 1.14 (0.56) 0.53 (0.76) Table 1: Innovaions variance, Ljung-Box and Jarque-Bera normaliy ess of some models for he elecriciy consumpion series in Spain. Due o convergence problems arising in ML esimaion for BSM models when individual hyper-parameers are esimaed for each harmonic in he seasonal componen, i is a common pracice o consrain all of hem o a single parameer. I is he case of he STAMP and oher similar compuer programs (see e.g. Koopman e al., 1995; Pole e al, 1995). This may be one reason explaining he superioriy of he DHR in frequency domain seen in able 1. The DHR model effecively has a higher number of parameers. Neverheless, he compuaional burden of such esimaion is less imporan han in ime domain for BSM and ARIMA models, provided he specral peaks are well defined, as i is he case of mos seasonal economic daa. There are also advanages in forecasing erms, bu hese are discussed laer 5. 4 These saisics are esimaed consisenly by he auhor based on he innovaions exraced from he respecive sofware packages, insead of using he saisical oupu of each package, in order o ensure he comparabiliy of resuls. 5 Resuls similar o he ones presened here can be found in Young e al. (1999) for he well-known airline passengers ime series in he US in Box and Jenkins (1970). 9

10 The signal exracion exercise reveals raher nicely anoher feaure in he series ha is hardly disinguishable in he original raw daa; namely he fac ha he rend appears o include a long period cycle of jus over en years which is, presumably, relaed o he economic cycle (see he esimaed rend in figure 1). Such cyclical behaviour is also clear by means of he rend derivaives, a variable ha has been used commonly for he analysis of he so called underlined growh. Figure shows he derivaives of IRW, LLT and ARIMA rends obained by he hree mehodologies discussed above. I seems clear ha he rends oher han IRW are very dangerous in he analysis of he underlined growh because of he high level of noise. On he oher hand, he smoohness of he IRW rend suppors he exisence of an approximaely consan period cycle of abou four years in he AP series. (INSERT FIGURE ) The cyclical componen deeced may be included ino he model in several ways, bu he opion chosen here is he one already exploied in Young e al. (1999). I consiss of obaining a beer esimae of he specrum a he low frequencies relaing o he cycle by means of a higher order AR specrum. An AR(10) specrum shows a clearly defined peak in he specrum a a fundamenal period of abou 16 monhs, wih wo oher peaks a 60 and 37 monhs, which will be recognised as approximae harmonics of his fundamenal period. Simply by concaenaing he original AR(18) and he new AR(10) specra, using he higher order AR specrum o define he lower frequency cyclical band of he specrum, and he lower order AR specrum o specify he higher frequency seasonal behaviour, he model including he cycle can be esimaed in he frequency domain. The resuling AR-specrum is shown in figure 3. (INSERT FIGURE 3) In order o compare he forecasing resuls, four models are used. These are he hree models shown in able 1 and a DHR model ha incorporaes periodic componens a he frequencies of he long erm cycle. A rolling experimen was performed in order o make forecasing comparisons. The las six years of daa were reserved for he comparaive sudy. The four models were esimaed and used o obain up o 4 sep-ahead forecass. The process is repeaed, expanding he sample by one observaion a each sep. The resuls of his exercise are shown in figure 4, where he mean, minimum, maximum and sandard deviaions of he Mean Absolue Percenage Error (MAPE) are ploed as a funcion of he forecas lead ime. The hick solid lines show he resuls for he DHR model including he economic cycle; he hin solid line represen he simpler DHR model; he doed lines represen he resuls for he BSM model; and he dashed lines show he ARIMA resuls. (INSERT FIGURE 4) I is clear ha he wo DHR model forecass, wih and wihou he cyclical componen, are boh significanly beer han hose obained using BSM and ARIMA over he range of forecas lead imes. Moreover, he superioriy of he DHR model resuls increases subsanially as he forecas horizon increases, especially in he case of he improved DHR model. 10

11 4.. Cyclical componens wih a non-consan period. All he models discussed in he previous subsecion are se up for ime series which exhibi a consan period cycle. I is obvious ha hose models are sricly incorrec when such assumpion is no fulfilled and he specificaion error would be more serious when he cyclical componen is relaively more imporan wih respec o he res of componens. I seems surprising ha he ype of models in he previous secions have been used by qualified analyss for ime series which irregulariy of he periodic behaviour is widely acknowledged. From he poin of view of he curren auhors, such analysis may be dangerous, especially when used for forecasing purposes, bu no necessarily for he srucural analysis of he cycles. Typical examples of he above problem are he US Naional Accouns quarerly ime series, ha have been used sysemaically for he analysis of he economic cycle from very differen heoreical and pracical poins of view, boh for he analysis of he economic cycle iself or for he deecion of co-movemens among variables coheren wih some heoreical dynamic macroeconomic models. In a significan amoun of hese sudies, he Hodrick-Presco filer (HP; see Hodrick and Presco, 1980, 1997) plays a cenral role as a mean o obain he periodic componens of he series. Two mos ineresing procedures for he esimaion of cyclical componens for he US GNP series aken from he lieraure are: a) Definiion of he economic cycle as he perurbaion around a HP rend, ha is effecively a raher simple UCM (see below). b) UCM ha includes explicily a cyclical componen, in addiion o he rend (Koopman e al., 1995). Alhough opion b) seems a more comprehensive model because of he specific inclusion of he cycle in he model, he cyclical componen is effecively esimaed once more as he difference beween he series and a HP rend. This is a consequence of he blind Maximum Likelihood esimaion process in which he period of he cyclical componen and he res of hyper-parameers are esimaed joinly. To be exac, when Harvey s BSM was esimaed for he logarihmically ransformed US GNP series (from he second quarer of 1948 o he second quarer of 1998) he opimum was found wih very srong convergence a he following soluion y = T + C Irregular : Trend : Cycle : + e σ σ σ e Level = 0 = 0 Cycle σ = Slope = ρ Cycle = 0.90 Period = 4.37 years In words, his model implies ha he rend is effecively a HP ype (bu wih a differen smoohing consan han he HP filer, see below) wih no irregular componen 11

12 and wih a cycle ha has a period somewha longer han four years. In pracice, his model is raher similar o he previous one, in which we were assuming ha he series is jus a HP rend plus a perurbaional componen abou i, wih he only difference ha he smoohing consan for he HP filer is differen. One may wonder wheher he BSM model above is a sensible opion or no in general, mainly because he model is iniially specified including an irregular componen, bu he oupu is a model wihou i, meaning ha a model has no sochasic noise added o i. The proposal of his paper is a mixure of he wo models seen above, in which he cycle and an irregular componens are explicily inroduced in he model. A he same ime, he non-consan feaures of he cycle are direcly addressed by means of he exac definiion of he cycle in erms of a frequency band. The HP rend is se up in a regularisaion framework in which smooh signal, he rend, is exraced from he daa in a way such ha he sum of squares of residuals are minimised subjec o he consrain ha he rend mus have a cerain level of smoohness. Such smoohness is conrolled by a lagrange muliplier ha is mainained a a value of 1600 for quarerly series. I has been demonsraed (e.g. see Pedregal, 1995 and Young and Pedregal, 1996) ha he HP filer is exacly equivalen o a simple UC model composed of an IRW rend plus an irregular componen. In addiion, i has been shown ha he IRW filer is effecively a low pass filer in which he bandwidh is conrolled by he NVR parameer, ha is exacly he inverse of he smoohing consan in he HP filer. In his way, he filered series would conain all he informaion of he series in he defined frequency band, independenly of he sysemaic or asysemaic properies of he series. The exac formulae is y σ η NVR = σ e η = T + e T = (3) Pedregal (1995) gives he exac formulae o calculae he bandwidh of he RW family filers as a funcion of he NVR parameer. Table shows some examples for an IRW filer. NVR Period (Time samples) Years (quarerly series) Years (monhly series) / Table : Relaion beween he NVR parameer and he associaed bandwidh of he IRW filer, i.e. minimum period of cycles include in filered series. Based on able, if a filered series has o be esimaed in a way ha conains all he informaion of he ime series abou 5 years and above, a NVR=0.01 would be he opion for a quarerly series, while a NVR= should be chosen for a monhly series. 1

13 Now, if an exac definiion of he cycle is provided in erms of a frequency band, an esimaion of such cycle may be obained by he applicaion of he IRW wice, in order o build a band pass filer. In he firs pass a rend-cycle componen would be obained by applying he IRW filer wih an appropriae NVR. In he second pass, he rend-cycle (or he original series) is filered again in order o remove he low frequency informaion relaed o he rend and o ge he cyclical informaion of he series. The resul of such a process is a decomposiion of he series in a low frequency rend; a cycle wih consan or non-consan period depending on he properies of he series; and an irregular componen ha will no be whie noise necessarily. For example, applying Burns and Michell (1946) definiion of he cycle o he US GNP quarerly series, such componen would conain he informaion of he series beween 18 monhs (6 quarers) and 10 o 144 monhs (40 o 48 quarers). The proposed esimaion of he cyclical componen could be obained in wo seps, i.e. (1) Apply model (3) wih NVR= 1. The filered series would conain all he informaion of he series abou 6 quarers and above (rend-cycle componen). () Apply model (3) o he rend-cycle componen in (1) or o he original 4 series wih NVR=.9 *10. The filered series would conain he informaion abou 48 quarers (1 years) and above. The esimaed cycle is he perurbaion in (), if () is applied o he rend-cycle componen in (1), or he difference beween he rends in (1) and (), if () is applied o he original ime series. The acual differences beween applying () o he original series or o he rend-cycle in (1) are negligible. This mehod exhibis a number of advanages: I relies on an explici definiion of he cycle based on a frequency band, insead of defining i as a residual (HP) or jus one frequency or a narrow band around i (BSM). This may be seen as a maer of ase, and i is indeed so for he HP approach, bu may be dangerous he definiion of he cycle given by BSM for series wih a non-consan period cycle. I does no require any preprocessing of he daa (e.g. variance sabilisaion) ha would be necessary in he BSM approach. Exacly he same procedure, as i is wihou any addiion, may be applied o seasonal daa. The BSM model may be applied o such daa wih he (simple) addiion of a seasonal componen, bu he cyclical componen esimaed by HP for seasonal daa would incorporae he seasonal componen, and he correcion of his problem do no seem easy wihin he HP conex a firs sigh. An irregular componen is always esimaed from he daa, avoiding his rare peculiariy of some UC mehodologies. (INSERT FIGURE 5) 13

14 Figure 5 shows he applicaion of he hree alernaives (BSM, HP and IRW) menioned in his subsecion o he GNP series. I is clear from he plo ha all are very similar, hough he heoreical definiion is quie differen in each model: he BSM model produces a cycle ha is effecively he informaion of he series of period abou 3.47 years; he HP filer assumes ha he cyclical componen incorporaes he informaion of period less han 9.9 years (see able ); while he IRW cyclical componen is defined as he informaion of he series beween 1.5 and 1 years. The logical consequence is ha mos of he informaion of his series is precisely conained in he band beween 1.5 and 10 years (approximaely), while he res of he frequencies are much less relevan. Because of his, differences among esimaed cycles are expeced if we play wihin his frequency band. The problem of figure 5 is he lack of smoohness ha may induce o a number of oscillaions superior o wha one would like for he analysis of he cycle: some of hem are indeed very shor, some of hem of very low ampliude, making difficul he deecion of recessions and expansions. One possibiliy o solve such a problem is selecing a differen bandwidh for he cycle componens. For example, figure 6 shows some alernaives for differen bandwidhs, namely beween 1.5 and 1 years (as in figure 5); 4 o 1 years and 6 o 1 years. The final choice of he frequency band is subjecive because he conclusions abou he cyclical oscillaions are differen. For example, given he definiions of he cycles in figure 6, he smooher esimae canno see he cycles of period inferior o six years. This is he reason why his esimae considers one single cycle beween years 1950 and 1958; 1965 and 1971; 1984 and 1993; while he oher esimaes would consider he exisence of wo cycles wihin each of hose daes. On he oher hand, he leas smooh cycle esimae consider a very shor cycle beween 1959 and In order o define a sensible bandwidh, oher kind of analysis, ypically recessions and expansions records made by expers in he field, become very relevan. (INSERT FIGURE 6) 5. CONCLUSIONS This paper proposes new alernaives for he analysis of he economic cycles relaed o well-known generally available mehods, bu wih a number of novel feaures. The analysis of he cyclical behaviour of ime series fis naurally in he Unobserved Componens (UC) models framework. From all he complex diversiy of such mehods, he one proposed in he main ex is of he Srucural form ype. Two versions are proposed, based on he specific properies of he daa a hand. Firsly, a Dynamic Harmonic Regression for hose series ha exhibi a consan period cycle. The relevance of such model is demonsraed wih he monhly aggregaed elecriciy consumpion daa in Spain. In his series, he forecasing performance of a model ha includes a cycle (esimaed objecively from he daa) is considerable superior o oher simpler univariae UC and ARIMA models. 14

15 Secondly, a double process of filering for series wih non-consan period cycles is proposed. I is based on a very simple UC model (jus an Inegraed Random Walk rend) ha provides he esimaion of he informaion conen of he series wihin a frequency band ha should be provided subjecively by he user. The resuls of such procedure compare very favourably wih he Basic Srucural Model and Hodrick- Presco rend modelling. 6. REFERENCES Akaike, H. (1980). Seasonal adjusmen by a bayesian modelling. Journal of Time Series Analysis, 1, pp Box, G. E. P., and Jenkins, G. M., (1970, 1976). Time Series Analysis, Forecasing and Conrol. Holden-Day: San Francisco. Box, G.E.P., Jenkins, G.M., Reinsel, G.C. (1994). Time Series Analysis, Forecasing and Conrol. Prenice Hall Inernaional, Englewood Cliffs, New Jersey. Burns, A.M. and Michell, W.C. (1946). Measuring Business Cycles. NBER, New York. Findley, D. F., Monsell, B. C., Bell, W. R., Oo, M. C. and Chen, B.C. (1996). New capabiliies and mehods of he X-1 ARIMA seasonal adjusmen program. U.S. Bureau of he Census. Mimeo, May 16. Gómez, V. and Maravall, A. (1998). Programs TRAMO and SEATS, Insrucions for he User (Bea Version: June 1998). Bank of Spain, Madrid. Harvey, A. C. (1989). Forecasing Srucural Time Series Models and he Kalman Filer. Cambridge Universiy Press, Cambridge. Hillmer, S. C., Bell, W. R. and Tiao, G. C. (1983). Modelling Consideraions in he Seasonal Adjusmen of Economic Time Series. In Zellner, A. (ed), Applied Time Series Analysis of Economic Daa, US Dep. of Commerce-Bureau of he Census, Washingon D. C., pp Hodrick, T. and Presco, E. (1980). Pos-war US business cycles: an empirical invesigaion. Carnegie Mellon Universiy, manuscrip. (1997). Journal of Money, Credi and Banking, 9, pp Koopman, S. J., A. C. Harvey, J. A. Doornik y Shephard, N. (1995). STAMP 5.0: Srucural Time Series Analyser, Modeller and Predicor. Chapman & Hall, London. Koopmans, T. (1947). Measuremen wihou heory. Review of Economic Saisics, 9, pp Kydland, F.E. and Presco, E.C. (1990). Business cycles: real facs and a moneary mih. Federal Reserve Bank of Minneapolis Quarerly Review, Spring, pp Ng, C. N. and Young, P. C. (1990). Recursive Esimaion and Forecasing of Nonsaionary Time Series. Journal of Forecasing, 9, pp

16 Pedregal, D.J. (1995). Comparación Teórica, Esrucural y Prediciva de Modelos de Componenes no Observables y Exensiones del Modelo de Young. PhD hesis. Universidad Auónoma, Madrid. Piccolo, D. (198), A Comparison of Some Alernaive Decomposiion Mehods for ARMA Models, in O. D. Anderson (ed), Time Series Analysis: Theory and Pracice 1, Norh-Holland Publishing Company. Pole, A., Wes, M. y Harrison, J. (1995). Applied Bayesian Forecasing and Time Series Analysis. Chapman & Hall, New York. Wes, M. and Harrison, J. (1989). Bayesian Forecasing and Dynamic Models. Springer- Verlag, New York. Young, P.C. and Pedregal, D.J. (1999). Recursive and En-Block Approaches o Signal Exracion. Journal of Applied Saisics, 6, pp Young, P.C., Pedregal, D.J. and Tych, W. (1999). Dynamic Harmonic Regression. Journal of Forecasing, 18, pp

17 FIGURES Series and Trend Thousand KW/h Figure 1: The Elecriciy consumpion in Spain from January 1971 o April 000, and esimaed rend Figure : Trend derivaives of alernaive models: IRW rend in DHR model (wide solid); canonical decomposiion rend in ARIMA model (hin solid); and LLT in BSM model (doed). 17

18 Log(Power) AR Specrum of Elecriciy Consumpion Figure 3: AR specrum of elecriciy consumpion series Monhs Mean value Minimum value % % Maximum value Seps ahead % % Sandard Deviaion Seps ahead Figure 4: Relaive forecasing performance for he elecriciy consumpion series in Spain, based on MAPE measures. DHR seasonal plus cyclical model resuls (hick 18

19 solid); sandard seasonal DHR model resuls (hin solid); BSM resuls (doed); and ARIMA resuls (dashed). 00 Esimaed cycles for he US GNP Cycles Figure 5: Three possible esimaions of he cyclical componen for he US GNP series: IRW filer (solid); HP (dashed) and BSM (doed). Esimaed cycles for he US GNP Cycles Figure 6: Esimaed cycles for he US GNP using he IRW approach for differen bandwidhs: from 1.5 years o 1 years (dashed); 4 o 1 years (doed) and 6 o 1 years (solid). 19