Real Time Representations of the Output Gap

Transcription

1 Real Time Represenaions of he Oupu Gap by Anhony Garra, Kevin Lee, Emi Mise and Kalvinder Shields Absrac Mehods are described for he appropriae use of daa obained and analysed in real ime o represen he oupu gap. The mehods employ coinegraing VAR echniques o model real ime measures and realisaions of oupu series joinly. The model is used o miigae he impac of daa revisions; o generae appropriae forecass ha can deliver economically-meaningful oupu rends and ha can ake ino accoun he end-of-sample problems associaed wih he use of he Hodrick-Presco filer in measuring hese rends; and o calculae probabiliy forecass ha convey in a clear way he uncerainies associaed wih he gap measures. The mehods are applied o daa for he US 1965q4-2004q4 and he improvemens over sandard mehods are illusraed. Keywords: Oupu gap measuremen, real ime daa, daa revision, HP end-poins, probabiliy forecass. JEL Classificaion: E52, E58. Birkbeck College, London, UK, Universiy of Leiceser, UK, Universiy of Melbourne, Ausralia. Corresponding auhor: Kevin Lee, Deparmen of Economics, Universiy of Leiceser, Leiceser, LE1 7RH, UK; web Version daed April

2 1 Inroducion The measuremen of he oupu gap, i.e. he difference beween he economy s acual oupu and is poenial or rend level, is cenral o much applied macroeconomeric work and paricularly he analysis of moneary policy. However, i is widely recognised ha he oupu gap is measured wih considerable uncerainy, and his is especially rue for he measures considered in real-ime decision-making. 1 For example, Orphanides and van Norden (2002) [OvN] show, using US daa, ha he sandard measures of his cenral concep are exremely unreliable, wih ex pos revisions of he gap in he US of he same order of magniude as he esimaed gap iself. Much of he unreliabiliy arises because he gap measures are based on oupu daa which is subsequenly revised and on measures of he rend oupu level which are subjec o esimaion error. OvN decompose he revisions observed in heir oupu gap measures ino wo pars reflecing hese wo sources of change. They show ha, for heir daa, he effec of changes in he measuremen of he rend exceed he effecs of changes in he published daa bu ha boh effecs are significan. 2 The OvN analysis highlighs he problems involved in real ime decision-making by illusraing how heir gap measure changes as new informaion on he acual and rend oupu levels becomes available wih he release of each new vinage of daa. However, he OvN decomposiion is based on a recursive analysis of each successive vinage of daa aken in urn. This ignores he possibiliy ha he sequence of vinages released over ime may in iself conain useful informaion wih which o inerpre he mos recen vinage of daa and o anicipae fuure oucomes (as discussed in Howery, 1978). Hence, for example, here migh be sysemaic paerns in he daa revisions ha can be used, in 1 I is also acknowledged ha he use of ex pos revised daa can yield misleading descripions of hisorical policy and ha he use of real-ime daa generaes differen real-ime policy recommendaions o hose obained on he basis of ex pos revised daa (see, for example, Roemberg and Woodford (1999), Brunner (2000), Orphanides e al. (2000), Orphanides (2001),and Amao and Swanson (2001)). 2 These differences are poenially exremely imporan given he reliance of recen empirical work on he idenificaion of moneary policy shocks and impulse responses on assumpions on he ordering of decisions and he iming of he release of informaion. See Chrisiano, Eichenbaum and Evans (1999) or Garra e al. (2005) for reviews. [1]

3 conjuncion wih he real ime daa, boh o moderae he direc impac of he revisions obained in successive vinages of daa on he perceived curren oupu level and o look forward o offse heir impac on he oupu rend measure. 3 In his paper, we exploi he informaion conained in he sequence of vinages more fully han OvN hrough a coinegraing VAR model which, under reasonable assumpions on he naure of he oupu series and measuremen errors, explains boh he changes in he real ime daa and is revisions. The model is used o generae forecass of conemporaneous and fuure values of oupu. The forecass improve he accuracy wih which he rue level of aciviy is measured and hey can be used o supplemen he hisoricallyobserved series o obain improved measures of he underlying rends also. As explained in Mise e al. (2005a,b) [denoed MKN], his laer poin helps o address he end-of-sample problems associaed wih he widely-used Hodrick-Presco (1997) [HP] filer in he measuremen of he rend (his being he source of considerable esimaion error variance). The model can be esimaed recursively, aking ino accoun successive vinages of daa. Bu, because i describes he revision process as well as he underlying oupu process, he model makes use of all he informaion available a each poin in ime, no jus he mos recen vinage available. The proposed approach o measuring he oupu gap has a leas hree very useful properies. Firs, he oupu gap is measured relaively precisely because modelling he revision process moderaes he effec of changes in published daa, while he use of he forecass miigaes he end-of-sample problem associaed wih he use of he HP filer. Second, by linking he rend measure o forecass of fuure oupu levels, i can be relaed o he frequenly-used Beveridge-Nelson rend and can be readily inerpreed in erms of economically-meaningful conceps such as poenial oupu. And hird, aswellas producing poin esimaes of he oupu gap ha are measured relaively precisely, he underlying model can be used o describe clearly he uncerainies associaed wih he 3 OvN also illusrae he uncerainy in he gap measure arising hrough he choice of derending echniques. (See also Canova, 1998.) We do no consider his aspec of he uncerainy in his paper, alhough he mehods described for conveying he exen of he uncerainy on he gap could be exended o accommodae alernaive derending echniques. [2]

4 measure of he gap. This is exremely useful because, while i is imporan o recognise he unreliabiliy of he oupu gap measures, he esimaed values of he gap a differen horizons are neverheless an essenial requiremen in many decision-making conexs. The oupu gap measures can be used appropriaely, aking ino accoun he uncerainies surrounding hese, when he model is used o supplemen he poin forecass wih forecass of he probabiliy of he occurrence of paricular evens involving he gap. The remainder of he paper is organised as follows. In Secion 2, he proposed mehod for measuring he oupu gap is elaboraed hrough a descripion of he coinegraing VAR model, hrough a discussion of he HP filer and is limiaions when he focus of aenion is he end-of-sample, and hrough a commen on he calculaion of probabiliy forecass relaing o he oupu gap. Secion 3 describes he applicaion of he proposed mehodsoobainoupugapmeasuresforheusandcompareshesewihmeasures obained following he procedures of OvN. Secion 4 presens some probabiliy forecass obained using our modelling framework, and Secion 5 concludes. 2 Measuring he Oupu Gap wih Real Time Daa To describe our proposed mehod of measuring he oupu gap, we need o inroduce some noaion and erminology. We wrie (he logarihm of) he oupu level a ime j by y j, and denoe he measure of oupu a ime j ha is released in ime by y j, j =0, 1, 2,,... Throughou he paper, he vinage- daase is defined by Y = { y 1, y 2, y 3,...} so ha i includes he ime- measure of oupu a ime 1 and before. Noe ha i is assumed ha he firs release of oupu daa for any period akes place afer a one-period delay; his corresponds o pracice in boh he US and UK. 4 The full informaion se available a ime, denoedω, conains he daases of all vinages daed a and earlier; i.e. Ω = {Y,Y 1,Y 2,...}. I is worh noing ha he ime-( + 1) measure of a variable is simply he ime- measure plus he revision; i.e. +1y 1 = y 1 +( +1 y 1 y 1 ). Hence, he full informaion se grows wih he 4 In he US, for example, he firs release of oupu daa provides an indicaion of oupu levels in he previous quarer. The subsequen releases provide progressively more informaion, and he July release in each year provides he mos definiive saemen of oupu over he previous year. [3]

5 addiion of successive vinages of daases by including he news on he oupu level in he previous period (he firs release of informaion on he oupu level in ha period) and he revisions on he oupu series in previous periods; i.e. Ω +1 = Ω { +1 y, ( +1 y 1 y 1 ), ( +1 y 2 y 2 ),...}. 5 Finally, urning o he oupu rend, we noe ha here are a variey of mehods employed in he lieraure o obain measures of he oupu rend a ime. Some of hese make use of daa ha becomes available boh before and afer ime, so ha care also needs o be exercised in describing he informaion se on which he rend measure is based. Specifically, wriing he rend oupu level a ime j by ey j,we denoe he measure of rend oupu a ime j ha is calculaed using mehod k on he basis of an informaion se available a ime, sayω,byey k j Ω. In OvN, aenion is focused on he differences beween real ime measures of he oupu gap based on successive vinages of oupu daa and final measures obained from he las available vinage of daa. Hence he comparison is beween he real ime measure of he gap x ro = +1 y ey o Y +1 and he final measure x fo = T y ey o Y T,where =1,..., T 1, and he o superscrip denoes he HP filer mehod used by OvN. OvN also consider a quasi-real esimae, x qo = T y ey o Y T,,inwhichheime-T measure of oupu a ime is compared o a rend measure obained on he basis of a subse of Y T ;namelyy T, = { T y, T y 1, T y 2,...}, <T. Evaluaing he differences beween he quasi-real measure of he oupu gap x qo and he real ime measure x ro isolaes he changes in he gap measures arising from he revision of he rend measure in he ligh of subsequen daa. OvN find his elemen o be significan bu relaively small, and i is argued ha i is he addiion of new poins o he sample, which causes ey o Y T o deviae from ey o Y T,, ha explains much of he difference beween he real ime and final measures of he oupu gap. OvN s hree measures of he oupu gap highligh he differen effecs of revisions in published daa and of differences in he use of informaion. Bu heir decomposiion is 5 Realisically, he measuremen of oupu a a paricular ime will be revised for some ime, say p periods, afer which no news becomes available on his oupu level and measured oupu will remain unchanged. Full informaion herefore consiss of he firs release of he oupu level measuremen plus he p subsequen revisions. [4]

6 poenially misleading. For example, MKN show ha he HP filer has very poor properies in esimaing he rend componen of he end-poin of a series, wih he esimaion variance upo 40 imes ha of he error inheren in he series in some circumsances (see also Baxer and King (1999), or S-Aman and van Norden (1998)). So, even if is decided o focus only on he vinage- daase in deermining he rend oupu measure, i is imporan o ake appropriae accoun of all he informaion available in ha daase, including is implicaions for expeced fuure values of he series, o ry o miigae he end-of-sample effec. OvN s chosen mehod of calculaing he HP filer does no do his so ha comparison of he quasi real and final measures do no reflec accuraely he impac of he effec of changes in he rend measure over ime. 6 Moreover, focusing on vinage- daa wihou reference o he revisions ha have aken place in previous periods daa poenially oversaes he effecs of changes in he published daa in ime, sincehese revisions migh have been anicipaed. The conclusion, hen, is ha all informaion available a ime should be employed in consrucing an oupu gap measure in real ime, wih paricular aenion payed o forecass of fuure values of he oupu series. The appropriae modelling framework for accommodaing all informaion is described in he secion below, and his is hen used o explain how forecass can be used o eliminae he end-of-sample problems associaed wih he HP filer. 2.1 A Join Model of Acual and Revised Oupu Series In order o make use of he full informaion available, he real ime measures of oupu should be modelled alongside he acual, realised value of oupu, aking ino accoun he revision process as well as he underlying oupu process. 7 In mos of his secion, we assume for illusraive purposes ha daa is revised jus once afer is iniial release, so ha we can model he wo processes joinly in a bivariae VAR. However, we noe also 6 As we discuss below, forecass of fuure values of he series (obained in real ime), can be used o supplemen he vinage- daa and exend he series beyond he end-of-sample. The mid-sample rend esimae obained for ime in his way is clearly more comparable o OvN s final measure han he rend underlying he quasi-real oupu gap. 7 See also Howery (1978) and Diebold and Rudebusch (1991). [5]

7 ha if revisions coninue upo q periods afer he firs release of daa, hen a VAR of size q + 1 would be required o model he processes adequaely, and we illusrae his more general case oo. Our modelling approach assumes firs ha acual oupu is firs-difference saionary. This means ha, if daa on oupu is released wih a one period delay and he acual oupu is observed wih he revision afer one furher period, ( y 2 1 y 3 ) is saionary. The approach also assumes ha measuremen errors (i.e. revisions) are also saionary. The firs of hese assumpions is suppored by considerable empirical evidence, 8 and he laer is eminenly reasonable. Under hese assumpions, any linear combinaion of hese wo series can be modelled in a bivariae VAR. 9 Hence, he oupu growh measure ( y 1 1 y 2 ) and he daa revision series have he following join fundamenal Wold represenaion: y 1 1 y 2 y 2 1 y 2 = α 1 α 2 + A(L) ² ξ (2.1) Here, α 1 is mean oupu growh (measured by firs-release daa), α 2 is he mean value of he revisions, A(L) = P j=0 A j (L), where he {A j } are 2 2 marices of parameers, assumed o be absoluely summable, and L is he lag-operaor. Also, ² and ξ are mean zero, saionary innovaions, wih non-singular covariance marix Ψ = ψ jk, j, k =1, 2. The model in (2.1) emphasises he poin ha he chosen measure of oupu growh a ime 1 and he revision of he measure of oupu a ime 2 beween 1and are boh revealed a ime. For noaional convenience, in wha follows we wrie α =(α 1, α 2 ) 0, where α 2 = 0 if here is no bias in he measuremen error. The general model in (2.1) can be expressed in various differen ways. For example, assume ha A 1 (L) can be approximaed by he lag polynomial A 1 (L) =B 0 + B 1 L B p 1 L p 1,whereB 0 = I 2 wihou loss of generaliy. In his case, (2.1) can be 8 See, for example, Murray and Nelson (2000) and Pappell and Prodan (2004). 9 For example, oupu growh measured by he change in he firs-release oupu level, ( y 1 1y 2 ), can be wrien in erms of acual growh and he relevan revisions and so is iself saionary; i.e. ( y 1 1 y 2 )=( +1 y 1 y 2 )+( y 1 +1 y 1 ) ( 1 y 2 y 2 ). [6]

8 rewrien o obain he AR represenaion y 1 1 y 2 y 2 1 y 2 = a B 1 1 y 2 2 y 3 1y 3 2 y 3 B p 1 p+1 y p p y p 1 p+1y p 1 p y p 1 + ² ξ (2.2) and hence y 1 y 2 = a + Φ 1 where a = A 1 (1)α, 1 y 2 1y 3 + Φ 2 2 y 3 2y Φ p p y p 1 + ² py p 2 ξ Φ j = B 1 0 j 1 B j for j =1,...,p 1, and Φ p = B 1 0 p (2.3) Seen in he conex of (2.3), he vecor of errors (², ξ ) 0 has a clear inerpreaion: ² is he news on oupu level in ime 1 conained in he firs-release daa becoming available a ime ; and ξ is he news on he level of oupu in ime 2 conained in he revised daa becoming available a ime. Alernaively, manipulaion of (2.3) also provides he VECM represenaion explaining he changes in he firs release measures and he change in oupu realisaions, [ y 1, y 2 ]where =(1 L) is he difference operaor. As shown in he Appendix, he VECM represenaion includes he lagged value of ( y 1 y 2 )asaregressorsince hese wo series are coinegraed, wih coinegraing vecor β 0 =[1, 1]. This propery holds because revisions are aken o be saionary in his model, so ha firs-release and acual oupu levels are coinegraed by assumpion. 10 Noe ha he model a (2.1), and is equivalen forms, are quie general and have no implicaions for he naure of he measuremen error oher han i is saionary. However, he assumpion ha real ime measures are unbiased (in he sense ha measuremen errors have no sysemaic conen) can be accommodaed in he model hrough he imposiion of resricions. If firs-release measures are unbiased, we would have y 2 = 1 y 2 + ξ so ha, in (2.3), he second µ µ row of Φ 1 = 1 0,andhesecondrowofΦ j = 0 0,j=2,..., p. 10 The VECM represenaion also has implicaions for he corresponding MA represenaion in firs differences; see Appendix for deails. [7]

9 Finally here, we noe ha he above models can be readily exended when he revision process exends beyond jus one period. Hence, for example, if quarerly daa coninues o be revised for upo a year, hen he daa requires a four-variable VAR o capure he join deerminaion of he firs-release oupu series and he hree successive revisions. Hence, he model ha will accommodae he news on oupu levels conained in he firs-release daa (² ) and in all he revised daa becoming available a ime on he previous periods (ξ 1, ξ 2, ξ 3 ) can be wrien in a form corresponding o (2.2), y 1 1 y 2 y 2 1 y 2 y 3 1 y 3 y 4 1 y 4 = a B 1 1y 2 2 y 3 1y 3 2 y 3 1y 4 2 y 4 1y 5 2 y 5 B p 1 p+1y p p y p 1 p+1y p 1 p y p 1 p+1y p 2 p y p 2 p+1y p 3 p y p 3 + ² ξ 1 ξ 2 ξ 3.(2.4) This can be rewrien in levels form, in VECM form and in MA form exacly as in (2.3) and he models of he Appendix. 2.2 Measuring Trend Oupu and he Oupu Gap Esimaes of he bivariae or mulivariae models derived above can be used o generae forecass of he oupu series infiniely ino he fuure and, in his secion, we argue ha hese can be usefully applied in he measuremen of he oupu rend using he HP filer. To moivae his procedure, recall ha he HP filer is an addiive decomposiion y = ey + x where ey is idenified as a growh (rend) componen and x as a cyclical componen. The HP filer is an exponenially weighed moving average filer, and is wosided symmeric in he sense ha i uses boh pas and fuure observaions wih equal imporance in order o decompose any one observaion in a series. The HP filer has he desirable propery ha i is opimal, in he expeced squared error sense, for daa [8]

10 generaing processes of he form (1 L) 2 ey = A(L)ε ; x = A(L)u (2.5) A(L) = X a j L j ; X j=0 j=0 a 2 j < where ε and u are muually sochasically uncorrelaed whie noise processes (i.e. E(ε u s )= 0, s), and where heir variance raio is λ = σu σ ε 2, (2.6) wih λ being he value of he smoohness parameer. 11. Moreover, alhough he opimaliy condiions (2.5) o (2.6) are expressed in erms of unobserved componens, MKN show ha all ARIMA(p, 2,q)modelshacanbefied o he observed series y can be expressed in his framework. In paricular, his holds rue for all possible ARIMA(p, 1,q) models, wih A(L) in (2.5) involving a uni moving average roo, so ha he series and is rend componen are I(1). Here, if y is an ARIMA(p, 1,q), hen ey is ARIMA(p +2, 1,q) and x is ARMA(p +2,q+1). However, an imporan feaure of he HP filer is ha, when we have a finie series, he opimaliy properies only hold for he mid-poin of he series. As we move owards he end of he series, he HP filer becomes increasingly one-sided, and for he las observaion of he series, he filer is compleely one-sided. MKN noe ha, while he filer coninues o provide an unbiased esimae of he quaniy x a he endpoins of a finie series, he esimaes are inefficien. They illusrae he exen of he inefficiency by comparing he esimaed HP rend measures wih he acual rends presen in a variey of simulaed series obained using differen rend and cycle specificaions. In paricular, MKN noe Burman s (1980) suggesion for addressing he inefficiency issue by augmening he observed series wih opimal linear forecass and demonsrae, hrough heir simulaion exercises, ha he applicaion of he HP filer o he augmened series provides an esimae of he endof-sample observaion which is opimal. Indeed, by augmening a series by is univariae forecas, he sandard deviaion of he esimaion error for he cyclical componen is 11 This parameer is convenionally se o 1600 for quarerly daa, following a suggesion by Hodrick and Presco (1997), made on somewha arbirary grounds. [9]

11 reduced by upo half (relaive o he sandard applicaion of he HP filer) in heir various simulaions. 12 The clear implicaion of hese resuls is ha he oupu gap should be calculaed using a rend obained by applying he HP filer o he forecas-augmened oupu series. For he series described in he previous secion, he model a (2.1), or is equivalen forms in (2.2) or (2.3), can provide he vehicle for generaing hese forecass. Forecass of he oupu series +1 y, +2 y +1, +3 y +2,... could be generaed using a univariae model of he vinage- daa, bu his will generally be less efficien han ha provided by he bivariae model of (2.1) which uses all he informaion available. 13 We shall denoe he end-ofsample rend measure obained by applying he HP filer o he oupu series augmened by forecass from he univariae model obained using vinage- daa by ey 1 Y u and he corresponding measure obained using he bivariae model of (2.1) by ey 1 Ω m, where he u and m superscrips indicaes he use of forecas-augmenaions, suggesed in MKN, based on univariae and mulivariae models respecively. The applicaion of he HP filer o forecas-augmened series no only improves he saisical properies of he derived series, bu i also provides an inerpreaion of he rend ha he radiional HP rend does no have and helps reconcile he use of he HP rend wih hose who prefer o use Beveridge-Nelson (BN) rends. Specifically, he BN rend obained from a ime series analysis of oupu measures he infinie-horizon effec of shocks on he series. Since his rend measure shows he permanen long-run effec of a shock o oupu, he BN rend is ofen inerpreed as poenial oupu since his is he level o which he economy will converge in he absence of any furher shocks. However, while his measure has an inuiively reasonable inerpreaion, i has he disadvanage ha i does no pay aenion o he dynamic pah ha is aken by he oupu series 12 MKN also noe ha he HP filer is ofen used in conexs where here is no assumed underlying rue rend and cycle measures of he form (2.5) or indeed any oher form. They commen ha he reliabiliy of a rend measure can be assessed in hese circumsances if a measure based on a sample of daa 1,.., T is revised as lile as possible in he ligh of subsequen observaions; his maches he discussion of OvN on he comparison of heir quasi real and final rend esimaes. MKN confirm hrough heir simulaions ha hese revisions are indeed minimised when he HP filer is applied o he forecas-augmened series. 13 The excepion is when he real ime measures are unbiased as discussed in he earlier foonoe. [10]

12 as i moves o is infinie-horizon forecas. This means ha he BN rend may be even more volaile han he acual oupu series iself (especially if i is based on a simple auoregressive model of he oupu series) and i is ofen argued ha his reduces he usefulness of he rend in he conex of measures of he oupu gap for moneary policy or business cycle analysis. This is clearly no he case for he rend obained from he forecas-augmened HP filer, ey 1 Ω m, which will display he smoohness ypically desired of a rend measure o be used as a measure of he business cycle or oupu gap. Furher he forecas-augmened HP filer series is reconciled wih he BN rend since forecass of fuure values of ey 1 Ω m will, by consrucion, hemselves converge owards he oupu level forecas a he infinie-horizon. The proposed measure herefore has he smoohness properies and, a he long horizon, i can be inerpreed as a poenial oupu series. 2.3 Conveying he Uncerainy Surrounding he Oupu Gap Measures In pracice, decision-makers faced wih he complee se of vinages of daa upo and including ha a ime T are concerned wih obaining a measure of he oupu gap for he end-of-sample period (and possibly ino he fuure). In some cases, aenion focuses simply on wheher he gap is posiive or negaive, bu in any case i is he ime-t (and fuure) magniudes ha maer in real ime decision-making. Here, assuming again ha daa is released wih a one period delay and here is a single revision made, his means decision-makers are ineresed in forecass of x fm T = T +2 y T ey T m Ω T +H.Hence,herelevan oupu level o be forecas is T +2 y T,heime-T oupu level ha will be observed in T +2, aking ino accoun he one period delay in he release of daa and afer any revisions in he daa have been fully aken ino accoun. And he relevan rend measure o be forecas is ha obained on he basis of an informaion se ha is available a some forecas horizon well ino he fuure (a T +H) so ha here are no end-of sample problems for he measure a T. We can obain poin forecass of his magniude relaively easily: he poin forecas of T +2y T is obained sraighforwardly from he bivariae model of (2.2) based on Ω T ;and he forecas of ey T m Ω T +H, based on Ω T,issimplyheperiod-T observaion of ey T m Ω T. 14 Bu 14 This follows because he measure ey m T Ω T is iself based on forecas values of he fuure unrevised and [11]

13 he poin forecas of he gap obviously does no convey he uncerainy associaed wih he oupu gap measure, and his is poenially significan here given ha forecass of he revised and unrevised series are used in various differen ways in he consrucion of he measure. So, using he informaion se Ω T a ime T for example, here will be uncerainy associaed wih he oupu gap measure a ime T 2 because of he uncerainy over he values of oupu beyond T and, hence, over he measure of he rend (his is due o he end-of-sample problem which is reduced by he forecas augmenaion bu no eliminaed). This uncerainy is compounded in he measure daed a T 1 by he forecas revisions ha will be made o he firs-release daa on T y T 1 and hen furher compounded a T and beyond as he unrevised oupu series and revisions are subsequenly forecased. I is imporan, herefore, ha any oupu gap measure is supplemened wih informaion on he uncerainies associaed wih he measure. Indeed, as noed above, i is frequenly he case ha ineres focuses no on he size of he oupu gap bu raher on is sign (i.e. wheher i is posiive or negaive). This reflecs he fac ha decision-makers objecive funcions are ofen concerned wih booms and recessions (irrespecive of heir size) and ha hese episodes are no valued symmerically (so ha he coss incurred during a recession migh ouweigh he benefis experienced in boom, say). In hese circumsances, he decision-maker requires he enire probabiliy disribuion funcion (pdf) of he esimaed oupu gap measure, raher han is poin forecas, or a leas an explici forecas of he probabiliy ha he oupu gap will exceed or fall below zero. 15 The calculaion of probabiliy forecass and pdf s of his sor is relaively unusual in economics (where uncerainy is ypically conveyed, if a all, by he reporing of confidence inervals). Bu he mehods are relaively sraighforward o implemen and are described in Garra e al. (2003). For example, absracing from parameer uncerainy for he ime being, o calculae he pdf associaed wih he forecas of x fm T = T +2 y T ey T m Ω T +H,one revised series and in he absense of any addiional informaion, he value of he updaed series expeced o be observed in T +H isunchangedfromhameasuredinperiodt (cf. he Law of Ieraed Expecaions) 15 There is widespread recogniion ha he design of opimal moneary policy mus ake ino accoun he various forms of uncerainies faced by he moneary auhoriies, including hose involving imperfec informaion abou he curren sae of he economy as well as fuure developmens. See, for example, Svensson (2001, 2002). [12]

14 would use he esimaed model of (2.2) o generae R replicaions of he fuure vinages of daa, denoed b Y (r) T +h for h =1,..., H and r =1,..., R. These include values of T +2+h by (r) T +h, h =0, 1, 2,..., H 2, on which he rend measure ey m(r) T Ω (r) T +H can be based. The simulaed disribuion of bx fm(r) T = T +2 by (r) T ey m(r) T Ω (r) T +H obained in his way provides he pdf of he oupu gap measure direcly, while couning he number of imes an even occurs in hese simulaions provides a forecas of he probabiliy ha he even will occur; he fracion of he simulaions in which bx fm(r) T > 0 provides an esimae of he forecas probabiliy ha he ime-t oupu gap is posiive, for example. Exending he simulaion exercise o accommodae parameer uncerainy is relaively sraighforward (see Garra e al. (2003) for more deails), so ha a complee characerisaion of he uncerainy surrounding he oupu gap measure can be obained Oupu Gaps in he US The mehods described above are applied o he vinages of US oupu daa provided by he Federal Reserve Bank of Philadelphia a hp// This daase includes 157 vinages of daa; he firs vinage is daed 1965q4 and he final vinage is daed 2004q4. All vinages of daa run from 1947q1 upo one period prior o he release dae; i.e. Y = { y 1947q1,..., y 1 },=1965q4, q4. The firs exercise underaken on his daa aims o invesigae he gains from using he forecas augmened HP filer approach o defining he rend. In he firs insance, we follow OvN and consider he successive vinages of daa, applying he HP filer, o derive he real-ime measure ey o Y +1,=1965q4, q3 as he end-of-sample observaion of he rend in each recursion. We compare his wih he quasi real measure ey o Y T,, also derived recursively, and he final measure ey o Y T. We also derive he corresponding rends 16 In his exercise, we resric aenion o he forecas-augmened HP filer as a measure of he rend. Bu alernaive measures of he rend exis and, in principle, hese could be calculaed in each of he simulaion exercises also o provide alernaive gap measures. If he reasonableness of he alernaive approaches o measuring he rend could be capured by appropriae weighs, hese various ses of simulaions could be pooled o provide densiy funcions for he gap measures, and even probabiliy forecass, ha accommodae sochasic uncerainy, parameer uncerainy, and he uncerainies associaed wih he appropriae measure of rend. [13]

15 based on daa augmened by forecass. The forecass are based on eighh-order univariae auoregressions explaining ( y 1 y 2 ); an eighh-order auoregression is applied o ensure here is no serial correlaion in he residuals. 17 x qo Figure 1 shows he oupu gaps considered by OvN, namely x ro =( +1 y ey o Y +1 ), =( T y ey o Y T, )andx fo =( T y ey o Y T ), for =1965q3,...,2004q3, and T =2004q4, and illusraes he considerable differences arising ou of daa revisions and he end-ofsample effecs on he underlying rends. Table 1 shows ha he correlaion beween he real ime and final measures of OvN is jus 0.526, and he wo measures agree on wheher oupu growh is above or below rend in only 63% of he sample period. These figures rise o and 69% respecively when he comparison is beween he quasi real ime and final measures (absracing from effecs of daa revision) bu he figures are clearly sill no high. Taking he final measure x fo as he bes indicaor of he rue oupu gap available o OvN, i is he poor performance of he x ro and x qo measures o reflec he rue oupu gap ha is he basis of OvN s conclusion ha real ime measures of he gap are unreliable. Table 1 also describes he effec of employing he forecas-augmenaion mehod of calculaing he rends on he hree gap measures. This has a subsanial impac on he variabiliy of he oupu gap series, cuing he sandard deviaion and range of values for he real ime measure by around 30% and by nearer 40% in he case of he quasi-real measure (his reducion in variabiliy is clearly illusraed in Figure 2). This illusraes ha he forecas-augmenaion is having a considerable impac on he rend measure as he esimaion error variance associaed wih he applicaion of he HP filer a he end-of-sample is reduced. The effec is o raise he correlaion beween he final measure x fu and he real and quasi real measures o 0.77 and 0.78 respecively, and agreemen on he occurrence of booms and recessions rise o 83% and 81% respecively also. The improvemen in reliabiliy using he forecas-augmenaion mehod is pronounced and shows ha he augmenaion should be applied in oupu gap measures. However, hey remain far from perfec, and i is clear ha oupu gap measures obained in real ime 17 Deails of regressions, and diagnosic ess relaing o he order of inegraion of he oupu and revision series, are no presened for space consideraions bu are available from he auhors on reques. [14]

16 need o be reaed wih cauion and he uncerainies on heir measuremen appropriaely aken ino accoun. Nex, we urn o he mulivariae analysis of he oupu growh and revision processes ogeher, considering wheher here are sysemaic paerns in he daa revisions ha underlie he successive vinages of daa ha are released and he exen o which a model of he oupu growh daa is enhanced by modelling he measured oupu growh and revisions daa joinly. To do his, we need o choose he lag lengh p in he mulivariae model in (2.4) and he lengh of he revision horizon (afer which revisions are unsysemaic and insignifican). The maximum lag lengh we consider is p = 4andhe maximum lengh of he revision horizon we consider is 3, as in (2.4). I urns ou ha he daa is described adequaely if we allow for a revision horizon of wo quarers and lags in he VAR of order 2. To demonsrae his, Table 2 provides esimaes of (2.4) obained using he enire daa upo and including Y 2004q4. 18 The Table shows ha a revision horizon of 2 is sufficien o capure sysemaic elemens in he revision process, since none of he variables in he fourh column, explaining ime- revisions of daa a 4 are individually or joinly saisically significan. And he Table also provides variable exclusion ess, denoed χ 2 LM(10), showing ha he hird and fourh lags of he firs hree variables in our sysem and all four lags of he fourh can be safely dropped from he regressions wihou violaing he daa. Moreover, i is apparen ha he join modelling of he growh series and he revisions is a useful approach. Boh he lagged growh series and he lagged revisions conribue significanly o he explanaion of he ime- growh ( y 1 1 y 2 ), meaning ha he univariae model is misspecified, and here are very significan sysemaic elemens in he revisions ( y 2 1 y 2 )and( y 3 1 y 3 ). 19 We are confiden, herefore, ha he mulivariae model of growh and revision is appropriae and will provide more accuracy in he forecass of fuure oupu on which o base he rends and oupu gap measures. 18 In fac, we conduced his exercise recursively on he full informaion se, Ω, consising of all of he vinages of daa upo and including Y,for = 1965q4,.., 2004q4. Alhough we only repor he resuls of he 2004q4 analysis, qualiaively similar resuls were obained hroughou. 19 Similar sysemmaic elemens are found in Swanson e al. (1999). [15]

17 The regression analysis shows ha he firs wo revisions ( y 2 1 y 2 )and( y 3 1 y 3 ) conain sysemaic elemens bu ha ( y 4 1 y 4 )doesno. Inhissense, y 3 is he firs measure of he rue oupu level in ime 3 and in pracice he measure of he oupu gap based on rue daa ha is available a ime T, using he forecas-augmened echnique based on our preferred mulivariae model, is given by x fm = T y ey m Ω T, =1,..., T 3. Table 3 provides summary saisics relaing o his series, and he corresponding real ime measure obained applying he procedure recursively over ime, x rm, for our daa upo 2004q1 (i.e. for T = 2004q4). These figures show ha he advanages of he forecas-augmenaion remain, wih a correlaion beween he real ime measure and he final measure of 0.80 and agreemen on booms and recessions in 83% of he sample. The Table also presens corresponding saisics based on an adjused daase in which revisions relaing o daa one year earlier (i.e. afer wo revisions) are assumed o be zero; i.e. y 4 k = s y 4 k, k =0, 1, 2,...for all s =1, 2,... Of course, his is a more severe assumpion han ha hese revisions are unsysemaic, as shown in he daa. Bu he assumpion effecively means ha, in order o calculae he oupu gap a any poin in ime, one requires jus he hree mos recen vinages of daa (no every vinage of daa) 20 which will be mos useful for he pracical implemenaion of he mehods. The correlaion coefficiens in he Table show ha his simplificaion has a relaively minor impac on he series: correlaions beween he unadjused and adjused series are 0.95 and 0.93 for x rm and x fm respecively (wih agreemen on booms and recessions in 91% and 93% of he sample). Given he pracical advanages of he adjused series in erms of heir daa requiremens, hese represen our preferred measures of he oupu gap and hese series are presened in Figure 3 (x fm represening he bes measure of he oupu gaphawehaveavailableous,given he informaion se in 2004q4, and x rm represens he corresponding measures ha would have been produced in real ime). 21 This figure illusraes clearly he advanages of applying our mehods for esimaing he oupu gap 20 Since he older vinages are assumed o be runcaed versions of he mos recen daa; i.e. Y T 3 = Y T,T 4, Y T 4 = Y T,T 5,andsoon. 21 I is also he adjused measures on which we concenrae in he secion below. [16]

18 which show a relaively close correspondence beween he real ime and final measures over he whole period. 4 Represening he Oupu Gap under Uncerainy The analysis above shows ha he uncerainy surrounding he oupu gap measure can be reduced hrough he appropriae use of forecas-augmened daa, and ha he forecass are bes calculaed using a mulivariae model ha describes he measured oupu growh series and daa revisions joinly. Noneheless, i shows ha he unreliabiliy of he measures highlighed by OvN remains and, as argued earlier, i is herefore imporan ha he uncerainies surrounding he measure are properly represened for decision-making purposes. Figure 4 illusraes he order of magniude of he uncerainies involved using he informaion se available a 2002q2. The end period was chosen o be 2002q2 o leave en periods, o 2004q4, for he purpose of ou-of-sample forecas evaluaion. As we see from he solid line, analysis underaken in 2002q2 would show he measured oupu gap rising above zero in early 1997, peaking in 2000q2 a close o 2%, before falling o -1.3% in 2001q3. These gap measures relae o periods when rue oupu daa is available (i.e. daa ha will no be subsequenly revised given our finding ha revisions coninue for only wo periods) and so he uncerainy surrounding hese figures arises from he esimaion error in he underlying rend measure only. The 95% confidence inervals ploed in he Figure lie approximaely ±0.3% around he poin forecas in 1997, rising o ±0.7% in 2000q2 and ±1.9% in 2001q3 reflecing he uncerainies associaed wih he rend measure as we move owards he end of he sample. 22 As i urns ou, he measured oupu gap falls furher in 2001q4 before reversing direcion in 2002q1 (o 1.7% and 1.1%). Alhough hese measures are informed by daa on y 2001q4 and y 2002q1 published in 2002q2, i is recognised ha hese figures will be revised in he coming 22 These inervals are generaed by he simulaion mehods discussed in Secion 2.3 and relae o he sochasic and parameer uncerainy surrounding he measures. Absracing from parameer uncerainy, by underaking simulaions aking ino accoun sochasic uncerainy only, generaes slighly igher bu very similar confidence inervals. [17]

19 periods, and he gap measures are subjec o he addiional uncerainies associaed wih he forecass of hese revisions. The 95% confidence inervals widen o ±2.4% a his ime herefore, and hen widen sill furher o around ±3% over he following wo years as he gap esimaes rely more comprehensively on forecass of ou-of-sample oupu levels and rend levels. The (considerable) uncerainy associaed wih he forecas of he gap a his horizon reflecs he uncondiional variabiliy in he oupu gap observed over he sample (as illusraed in Figure 3). Figure 4 shows ha he oupu gap measures based on informaion a 2002q2 remain negaive bu, having reached is minimum level in 2001q4, he gap measure ends o zero by 2003q4. As noed previously, he gap ends o zero by consrucion, bu he speed of adjusmen reflecs he dynamics of he underlying model of Table 2 and he smoohing properies of he HP filer. The figure indicaes ha, saring from a posiion of around -1.5%, an invesigaor would have expeced he impac of shocks o oupu o be fully worked hrough, and acual and rend oupu o reach heir poenial level, wihin wo years. As i urned ou, however, he negaive oupu gap persised for some ime. Figure 4 also plos, wih he dashed line, he oupu gap measures obained on he basis of he informaion available in 2004q4, x fm Ω 2004q4. These show ha 2001q4 was no in fac a urning poin for he oupu gap, defining a urning poin o occur when wo periods of negaive growh is followed by wo periods of posiive growh (or vice versa), and ha he recovery began raher laer, in 2003q1. Hence, he model performs relaively well, boh in erms of he poin forecas (he x fm Ω 2004q4 all lie well wihin he confidence inervals obained on he analysis of he 2002q2 daa) and in erms of forecasing recession. Bu policy prescripion based on he poin forecass alone would have misjudged he exen and duraion of he recession over he following wo year period. The resuls of Figure 4, and associaed confidence inervals, give a good indicaion of he likely size of he oupu gap wihin he sample, a he end of he sample and a various forecas horizons, bu he informaion is no presened in he mos useful way from a decision-maker s perspecive. This informaion can be conveyed more usefully and more direcly hrough he corresponding probabiliy disribuion funcions showing prob(x fm 2002q2+h Ω 2002q2 <c) for a range of criical values c a a various esimaed horizons, [18]

20 h. Figure 5 shows such densiy funcions for h = 3, 0and4, again aking ino accoun sochasic and parameer uncerainy. The funcions shif o he righ over ime, reflecing he rising value of he poin forecas, and show, for example, ha he probabiliy of an oupu gap less han zero is 0.93 in 2001q3, 0.72 in 2002q2 and0.55 in 2003q2. The densiy funcions are relaively seep in 2001q3, showing ha here is relaively lile uncerainy on he measure a ha ime, bu become progressively flaer a 2002q2 and 2003q2 reflecing he accumulaing uncerainy a he end-of-sample and ino he forecasing horizons. The densiy funcions of Figure 5 convey informaion on he oupu gap in precisely he form ha is required by mos decision-makers whose objecives are influenced by he expeced oupu gap. Only in he special case where decision-makers face consrains ha are linear in he oupu gap and pursue objecives ha are quadraic in he oupu gap (he LQ case) will aenion focus on he poin forecass. More generally, decision-makers will require o maximise complex objecive funcions and he soluion will require he enire densiy funcion describing he likely oupu gap oucomes. The unreliabiliy of he oupu gap measures noed by OvN and characerised in he previous secion does no mean ha measures of he oupu gap canno be used. Raher, i means ha he decision problem needs o be fully-specified, possibly including saemens on he risks involved in decisions as well as possibly non-linear objecive funcions, and evaluaed in he ligh of he various saes of naure ha migh be faced. The densiy funcions of he ype presened in Figure 5 provide precisely his informaion. One imporan example of nonlineariies in objecives arises when decision-makers are concerned wih he sign or rae of change of he oupu gap raher han he size. The emphasis in he media on booms and recessions reflecs hese ideas, indicaing ha may be a sharp difference in he consequences of a posiive oupu gap compared o a negaive one, irrespecive of he size of he gap, or ha aenion should focus on he rae of change of he oupu gap, irrespecive of is size or sign. Cerainly here is a view ha moneary auhoriies do no rea posiive and negaive oupu gaps symmerically, reacing more srongly o he inflaionary pressures associaed wih a posiive oupu gap han hey do o he recessionary pressures associaed wih a negaive oupu gap. And here is an argumen ha policy-makers are concerned wih wheher condiions are improving or [19]

21 deerioraing, wih he gap rising or falling (see Walsh (2003), for example). If hese argumens are imporan, hen here are paricular evens ha are relevan o decision-makers and direc saemens on he likely occurrence of hese evens will be more helpful o decision-makers han he poin forecass of he oupu gap and associaed confidence inervals of Figure 4. Specifically, if ineres focuses on he sign of he gap, hen prob(x fm Ω T > 0) is a key saisic for decision-makers, and he evoluion of his saisic over ime migh provide a useful indicaor of inflaionary pressure. This indicaor is provided in Figure 6a, again based on he daa available in 2002q2, and ploing prob(x fm 2002q2+h Ω 2002q2 > 0) for h = 3, 2, 1, 0, 1, 2,... The probabiliy sars a less han 0.1 in 2001q3, reflecing he fac ha he poin forecas of he gap sars low a -1.3% and ha here is relaively lile uncerainy because his figure relaes o oupu daa ha is no going o be revised. Neverheless, he probabiliy of a posiive oupu gap is sill non-zero as here remains uncerainy abou he underlying rend even a his sage. The probabiliy remains low hrough o 2002q2, bu rises o 0.5 as he oupu level is forecas o converge on is rend. Equally, if ineres focuses on wheher he gap is rising or falling, hen decision-makers will focus on he likelihood of a urning poin in he daa. Taking i as given ha he oupu gap peaked in 2000q2, and defining a urning poin as wo consecuive periods of posiive growh following wo consecuive periods of negaive growh (or vice versa), decision-makers would be ineresed in esablishing he likelihood of an upurn in ime given by prob(a) wherea = { h x fm 2 Ω T >x fm i 1 Ω T [x fm 1 Ω T >x fm Ω T ] [x fm Ω T < x fm +1 Ω T ] [x fm +1 Ω T <x fm +2 Ω T 0]}. This probabiliy, based again on informaion available in T =2002q2, is presened in Figure 6b. We noed earlier in he discussion of Figure 4 ha, based on informaion available in 2002q2, he poin forecass of he gap indicaed ha a urning poin had been experienced in 2001q4 alhough in fac, based on informaion available in 2004q4, he recovery sared laer in 2003q1. Figure 6b reflecs he uncerainies surrounding he saemens on urning poins much more precisely and informaively, showing ha he probabiliy ha an upurn was experienced in 2001q4 was esimaed o be Given ha he probabiliy exceeds 0.5, he invesigaor s bes 23 The probabiliy was zero unil 2001q2 and0.18 in 2001q3. [20]

22 guess would be ha he upurn had occurred, herefore, bu here is considerable uncerainy associaed wih his view. The probabiliy ha an upurn will have happened by 2003q1 (when he upurn did occur) is esimaed o be Obviously hese probabiliy measures correspond o he poin forecass described above in Figure 4, and which urned ou o be incorrec in erms of he size and duraion of he recession. Bu, because hey are expressed in probabilisic erms, raher han as poin forecass, hey convey much more accuraely he srengh of convicion wih which he forecass are held and will be much more direcly useful for hose whose ineres is in he occurrence of booms and recessions. 5 Conclusions The analysis of his paper sars from he poin ha oupu gap measures are an essenial elemen of many decisions bu ha hey are measured wih considerable uncerainy boh because of he imprecision of he oupu daa ha hey face a he ime decisions have o be made and because of he difficulies in esablishing a precise measure of rend oupu. We have argued ha hese uncerainies can be miigaed by modelling he oupu process alongside he revision process, making use of forecass of fuure oupu levels, o obain more precisely esimaed measures of he gap for use in real ime decision-making. Bu he uncerainies surrounding he measures, correcly idenified as imporan by OvN and ohers, remain and are subsanial. We have also argued, herefore, ha he producion of forecass of probabiliies of evens involving he gap convey he informaion on he level of he gap and he uncerainies associaed wih his measure more precisely han he poin forecass and confidence inervals ypically delivered by analyss. The cumulaive disribuion funcions ha we have presened, along wih he esimaed probabiliies of posiive gap measures and of urning poins in he gap, provide a very informaive and helpful means of represening he oupu gap daa for use by decision-makers. [21]

23 6 Appendix Manipulaion of (2.3) in he ex provides he VECM represenaion where y 1 y 2 = a + Γ 0 1 y 2 px + Γ j 1y 3 j=1 Γ 0 = (I 2 Φ 1 Φ 2 Φ p )= Φ(1) j+1y j j+1 y j 1 Γ j = (Φ j+1 + Φ j Φ p ) j =1, 2,...,p 1 + ² ξ (6.7) Given he form of he Φ i described in (2.3), i is easily shown ha Γ 0 akes he form Γ 0 = k 1 k 1 = k (6.8) k 2 k 2 k 2 where k 1 and k 2 are funcions of he elemens of he B j, j =0, 1,.., p 1. Hence, he model a (2.1) can be wrien in a VECM form where Γ 0 = αβ 0 and α 0 =[ k 1, k 2 ]conains he parameers deermining he speed of adjusmen o equilibrium and β 0 =[1, 1] is he coinegraing vecor. The form of he coinegraing vecor capures he assumpion ha revision errors are saionary hrough he inclusion of he error correcion erm β 0 [ 1 y 2, 1 y 3 ] 0 = 1 y 2 1 y 3. A final alernaive for describing he model is he MA represenaion obained hrough recursive subsiuion of (2.3): y 1 = b + C(L) y 2 ² ξ (6.9) where b = C(1)a, C(L) = P j=0 C j (L), C 0 = I 2, C 1 = Φ 1 I 2 and C i = P p j=0 C i j Φ j, i>1, C i =0,i<0. As is well known, following Engle and Granger (1987), he presence of a coinegraing relaionship beween he y 1 and 1 y 2 imposes resricions on he parameers of C(L); namely, β 0 C(1) = 0. Furher, given ha β 0 =[1, 1], his ensures ha C(1) akes he form for scalars k 3 and k 4. C(1) = k 3 k 4 (6.10) k 3 k 4 [22]

24 References Amao, J. D., and N. R. Swanson (2001) The Real-Time Predicive Conen of Money for Oupu, Journal of Moneary Economics, 48, Brunner, A.D. (2000), On he Derivaion of Moneary Policy Shocks: Should we Throw he VAR Ou Wih he Bah Waer, Journal of Money Credi and Banking, 32, Baxer and King (1999), Measuring Business Cycles: Approximae Band-Pass Filers for Economic Time Series, Review of Economics and Saisics, 81(4) Burman, J.P. (1980), Seasonal Adjusmen by Signal Exracion, Journal of Royal Saisical Sociey A, 143, Canova, F. (1998), Derending and Business Cycle Facs, Journal of Moneary Economics, 41, 3, Chrisiano, L.J., M. Eichenbaum and C.L. Evans (1999), Moneary Policy Shocks: Wha Have We Learned and o Wha End?, Chaper 2 in J.B Taylor and M. Woodford (eds.), Handbook of Macroeconomics, Volume 1A. Norh-Holland, Elsevier: Amserdam. Diebold, F. X., and G. D. Rudebusch (1991) ForecasingOupuwihheComposie Leading Index: A Real-Time Analysis, Journal of he American Saisical Associaion, 86, Garra, A., K. Lee, M.H. Pesaran, and Y. Shin, Y. (2003) Forecas Uncerainies in Macroeconomeric Modelling: An Applicaion o he UK Economy, Journal of American Saisical Associaion, 98, 464, Garra, A., K.C. Lee, M.H. Pesaran and Y. Shin, (2005), A Srucural Coinegraing Macroeconomic Model of he UK, monograph in preparaion, Universiy of Cambridge. [23]