Garratt, A.; Lee, K.; Mise, E. and Shields, K. Real-time representations of the output gap

Transcription

1 Birkbeck eprins: an open access reposiory of he research oupu of Birkbeck College hp://eprins.bbk.ac.uk Garra, A.; Lee, K.; Mise, E. and Shields, K. Real-ime represenaions of he oupu gap Review of Economics & Saisics - 90(4), pp (2008) This is an exac copy of an aricle published in Review of Economics & Saisics (ISSN: ) made available here wih kind permission of: 2008 Elsevier. All righs reserved. All aricles available hrough Birkbeck eprins are proeced by inellecual propery law, including copyrigh law. Any use made of he conens should comply wih he relevan law. Ciaion for his version: Garra, A.; Lee, K.; Mise, E. and Shields, K. Real-ime represenaions of he oupu gap London: Birkbeck eprins. Available a: hp://eprins.bbk.ac.uk/1939 Ciaion for publisher s version: Garra, A.; Lee, K.; Mise, E. and Shields, K. Real-ime represenaions of he oupu gap Review of Economics & Saisics - 90(4), pp (2008) hp://eprins.bbk.ac.uk Conac Birkbeck eprins a lib-eprins@bbk.ac.uk

2 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 encounered 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, probabiliy forecass. JEL Classificaion: E52, E58. Birkbeck College, London, UK, Universiy of Leiceser, UK, Universiy of Melbourne, Ausralia. We have received helpful commens from wo referees, Simon van Norden, Adrian Pagan and paricipans a he CIRANO and Bank of Canada s Workshop on Macroeconomic Forecasing, Analysis and Policy wih Daa Revisions. Corresponding auhor: Kalvinder K. Shields, Deparmen of Economics, Universiy of Melbourne, Vicoria, 3010, Ausralia. k.shields@unimelb.edu.au, el: , fax:

3 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. (Chaper 4, 2006) for reviews. [1]

4 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. 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. For example, as explained in Mise, Kim and Newbold (2005a,b) [denoed MKN], his laer poin helps o address he end-of-sample problems associaed wihhewidely-usedhodrick-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 poininime,nojushemosrecenvinageavailable. 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 any end-of-sample problems associaed wih he measure of he rend. Second, by linking he rend measure o forecass of fuure oupu levels, i can readily inerpreed in erms of economically-meaningful conceps such as poenial oupu. And hird, as well as 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 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 hem, when he model is used o supplemen he poin forecass [2]

5 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 end-of-sample problems encounered when measuring rends in real ime and hrough a commen on he calculaion of probabiliy forecass relaing o he oupu gap. Secion 3 describes he applicaion of he proposed mehods o obain oupu gap measures for he US and compares hese wih measures obained following alernaive procedures. 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 1andbefore. Noehaiisassumedhahefirs release of oupu daa for any period akes place afer a one-period delay; his corresponds o pracice in he US, for example. 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 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 ),...}. 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 [3]

6 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. 3 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 arising from he revision of he rend in he ligh of subsequen daa. OvN find his elemen o be significanburelaivelysmall,andiisarguedhaiis 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 poenially misleading. For example, 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, since hese revisions migh have been anicipaed. Indeed, even if only vinage- daa is used, predicions of fuure oupu levels will be helpful in measuring he rend a he end of he sample whenever he ime- value of he rend is relaed o is value in adjacen periods. 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 paid o forecass of fuure values of he oupu series. The appropriae modelling framework for accommodaing all informaion 3 For exposiional purposes, we iniially focus on he HP filer, bu OvN also illusrae he uncerainy in he gap measures arising from he choice of derending echnique: deermininsic rends, HP filer, unobserved componens, and so on. See also Canova (1998). [4]

7 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 measures of he rend. 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. 4 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 ha if revisions coninue up o 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, 5 and he laer is eminenly reasonable. Under hese assumpions, any linear combinaion of hese wo series can be modelled in a bivariae VAR. 6 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) 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, 4 See also Howery (1978) and Diebold and Rudebusch (1991). 5 See, for example, Pappell and Prodan (2004). 6 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 ). [5]

8 assumed o be absoluely summable, and L is he lag-operaor. Also, 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 rewrien o obain he AR represenaion y 1 1 y 2 y 2 1 y 2 and hence y 1 y 2 = a B 1 = a + Φ 1 where a = A 1 (1)α, 1 y 2 2 y 3 1y 3 2 y 3 1 y 2 1y 3 + Φ 2 2 y 3 2y 4 B p Φ p p+1 y p p y p 1 p+1y p 1 p y p 1 p y p 1 + 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 (,ξ ) 0 has a clear inerpreaion: 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 [6] + ξ (2.2)

9 acual oupu levels are coinegraed by assumpion. 7 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. 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 up o 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 ( ) 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 + 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 7 The VECM represenaion also has implicaions for he corresponding MA represenaion in firs differences; see Appendix for deails. [7]

10 hese can be usefully applied in he measuremen of he oupu rend whenever his is relaed o he rend in adjacen periods (i.e. boh backwards and forwards in ime). To moivae his procedure, we focus on he HP filer which 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 generaing processes of he form (1 L) 2 ey = A(L)ε ; x = A(L)u (2.5) A(L) = X j=0 a j L j ; X 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 = h i 2 σ u σε,wih λ being he value of he smoohness parameer. 8 Moreover, alhough he opimaliy condiions are expressed in erms of unobserved componens, MKN show ha all ARIMA(p, 2,q) models ha can be fied o he observed series y canbeexpressedinhisframework. In paricular, his holds rue for all possible ARIMA(p, 1,q)models,wihA(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)andx 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 he filer coninues o provide an unbiased esimae of he quaniy x a he endpoins of a finie series bu ha 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, finding ha he esimaion variance 8 This parameer is convenionally se o 1600 for quarerly daa, following a suggesion by Hodrick and Presco (1997). [8]

11 of he rend is up o 40 imes ha of he error inheren in he series in some circumsances (see also Baxer and King, 1999, and S-Aman and van Norden, 1998). To address he inefficiency issue, MKN noe Burman s (1980) suggesion o augmen 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 forecas, he sandard deviaion of he esimaion error for he cyclical componen is reduced by up o half (relaive o he sandard applicaion of he HP filer) in heir various simulaions. 9 The clear implicaion of hese resuls is ha he oupu gap should be calculaed using a rend obained by applying he 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. We shall denoe he end-of-sample 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 uh and he corresponding measure obained using he bivariae model of (2.1) by ey 1 Ω mh. In he empirical secion, we shall also consider gap measures obained by applying an exponenial smoohing filer and Wason s (1986) unobserved componens model o he forecas-augmened daa for he purpose of comparison; hese are denoed wih a e and w superscrip so ha he mulivariae versions of he series are ey 1 Ω me and ey 1 Ω mw respecively. The applicaion of he HP filer o he forecas-augmened series no only improves he saisical properies of he derived series bu i also jusifies an economically-meaningful inerpreaion of he rend. Specifically, forecass of fuure oupu levels show he expeced 9 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. [9]

12 evoluion of he series in he absence of furher shocks, so ha he infinie-horizon oucome can be readily inerpreed as he economy s poenial oupu level. 10 A rend measure based on a forecas-augmened series will coincide wih his poenial oupu series a long horizons by consrucion. As discussed above, he opimaliy of a paricular filer in idenifying he rend a shorer horizons depends on he underlying daa generaing process. Unless economic heory can provide sufficien deail on he naure of he shor run dynamics, an invesigaor migh wan o consider a number of alernaive rend measures. Bu focusing aenion on rends using forecas-augmened series ensures he rend is consisen wih expeced fuure oupu levels and maches he poenial oupu concep in he long run. 2.3 Conveying he Uncerainy Surrounding he Oupu Gap Measures In pracice, decision-makers faced wih he complee se of vinages of daa up o 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 fk T = T +2 y T ey T k Ω T +N for a rend measure k and for large N. Hence, he relevan 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 + N) 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 +2 y T is obained sraighforwardly from he bivariae model of (2.2) based on Ω T ; 10 The Beveridge-Nelson (1981) rend highlighs precisely his infinie-horizon oucome, absracing from he dynamic pah ha will be involved in reaching he poenial oupu level. [10]

13 and he forecas of ey k T Ω T +N, based on Ω T,issimplyheperiod-T observaion of ey k T Ω T. 11 Bu 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 for example, here will be uncerainy associaed wih he oupu gap measure a ime T 2 because of he need o forecas he values of oupu beyond T and he consequen imprecision in he measure of he rend. (Of course, he esimaion variance due o he end-of-sample problem 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, i is someimes argued ha decision-makers objecive funcions are concerned wih booms and recessions (i.e. wheher he oupu gap is posiive or negaive, irrespecive of is 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. (See Cukierman and Gerlach, 2003, and references herein, for example). Similarly, here is an argumen ha policy-makers are concerned wih wheher condiions are improving or deerioraing, wih he gap rising or falling (see Walsh 2003, for example). In hese circumsances, he decision-maker requires he enire probabiliy densiy funcion (pdf) of he esimaed oupu gap measure raher han is poin forecas or, a leas, explici forecass of he probabiliy of he even of ineres (i.e. he probabiliy ha he oupu gap will exceed or fall below zero, or he probabiliy of a urning poin) This follows because he measure ey T k Ω T is iself based on forecas values of he fuure unrevised and revised series and in he absence of any addiional informaion, he value of he updaed series expeced o be observed in T +N is unchanged from ha measured in period T (cf. he Law of Ieraed Expecaions). 12 Poin forecass will provide sufficien informaion for decisions only in he special case of he LQ problem involving a single decision variable (where he objecive funcion is quadraic and consrains, if hey exis, are linear); see Pesaran and Skouras (2002). For oupu gap measures, i is widely recognised [11]

14 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 fk T = T +2 y T ey T k Ω T +N, one would use he esimaed model of (2.2) o generae R replicaions of he fuure vinages of daa, denoed Y b (r) T +n for n =1,..., N and r =1,..., R. These include values of T +2+n by(r) T +n, n =0, 1, 2,..., N 2, on which he rend measure ey k(r) T The simulaed disribuion of bx fk(r) T = T +2 by (r) T Ω (r) T +N can be based. ey k(r) T Ω (r) T +N obained in his way provides he pdf of he oupu gap measure direcly. Equally, 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 fk(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) and he mehods can also readily accommodae he use of alernaive rend measures. 13 Hence, a complee characerisaion of he uncerainy surrounding he oupu gap measure can be obained, accommodaing sochasic uncerainy, parameer uncerainy, and he uncerainies associaed wih he appropriae measure of rend. 3 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 up o one period prior o he ha he design of opimal moneary policy requires a more sophisicaed reamen of uncerainy han he LQ framework; see, for example, Svensson (2001, 2002). 13 Specifically, he alernaive measures of he rend can be calculaed in each of he simulaion exercises o provide alernaive gap measures. Assigning appropriae weighs o he alernaive rend measures, he simulaions for each rend can hen be pooled o provide densiy funcions for he gap measures and associaed even probabiliy forecass. [12]

15 release dae; i.e. Y = { y 1947q1,..., y 1 },=1965q4, q4. The US Naional Income and Produc Accoun (NIPA) figures ha include an observaion of oupu in quarer for he firs ime are released a he end of he firs monh of quarer +1. This is he vinage ha is idenified in he Philadelphia daabase as being he daa ha exiss a he mid-poin of he quarer ( +1) and which we erm Y +1. The effecs of wo subsequen revisions o he NIPA daa, aking place a he end of he second and hird monhs of quarer ( + 1), are capured in he Philadelphia daabase when i repors he available daa a he mid-poin of he following quarer ( + 2), ermed Y +2 in his paper. 14 The firs exercise underaken on his daa aims o invesigae he gains from using he forecas augmened approach o defining he rend, focusing on he case where he rend is obained using he HP filer. 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 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. 15 Table 1 repors saisics relaing o he oupu gaps considered by OvN, namely x ro = ( +1 y ey o Y +1 ), x qo =( 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-of-sample effecs on he underlying rends. The Table 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. 14 The NIPA daa is also usually revised each July for he prior hree years. These July revisions are differen in naure o hose occurring a oher imes and his could mean ha heir differenial impac, and he consequen seasonal effecs, should be aken ino accoun in he model. However, exensions made o accommodae hese July effecs in he mulivariae models discussed below failed o add significanly o he fi of he models. (Deails are available from he auhors on reques.) 15 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. [13]

16 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 in reflecing 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 approach o calculaing he rends on he hree gap measures, x ruh, x quh and x fuh where he uh superscrip indicaes ha he underlying rend is based on he HP filer applied o a forecas-augmened series obained using he univariae model. 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. 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 fuh 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 forecasaugmenaion mehod is pronounced and shows he imporance of he augmenaion in calculaing oupu gap measures. 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 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 = 4 and he 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 [14]

17 heeniredaaupoandincludingy 2004q4. 16 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. Table 2 herefore confirms 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 ). 17 The regression analysis shows ha only he firs wo revisions ( y 2 1 y 2 )and ( y 3 1 y 3 ) conain sysemaic conen, and his suggess ha i migh be reasonable o work wih an adjused daase in which he subsequen revisions are assumed o be precisely zero (so ha k+s y k = y k, k =4, 5,...and for s =3, 4,..., k 1). The reamen of he unsysemaic revisions in he regression analysis, and he choice beween using he adjused or unadjused daase, deermines he way in which measuremen error eners he sysem and could poenially inroduce biases in he esimaed parameers. Thechoicebeweenhewodaasesdepends on he naure of he (unobservable) daa generaing process for he revisions and oupu daa. The use of he adjused daa is appropriae if he revision process is a funcion of he rue oupu whose hisorical values are accuraely measured by he mos recen vinage of daa. The use of he unadjused daa would be more reasonable if he revisions arefuncionsofgrowhasmeasureda he ime (cf. Koenig e al., 2003). In he even, he correlaion beween he real ime and final vinage measures of he gaps based on he adjused and unadjused daases are 0.95 and 0.93 respecively (wih agreemen on booms and slumps in 91% and 93% 16 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. 17 Similar sysemmaic elemens are found in Swanson e al. (1999). [15]

18 of he sample). This confirms ha he adjusmen and he choice of he daase has a relaively minor impac in his case. Furher, employing he adjused daase ensures ha he mos up-o-dae informaion on hisorical oupu levels is used in consrucing he oupu gap measures a any ime. Hence, our suggesed measure of he oupu gap based on he HP filer is ha obained by applying he forecas-augmened echnique based on he mulivariae model esimaed using he adjused daase; his is denoed x fmh = T y ey mh Ω T, =1,..., T Table 3 provides summary saisics relaing o his series, and he corresponding real ime measure obained applying he procedure recursively over ime, x rmh, for our daa up o 2004q1 (i.e. for T =2004q4). These figures show ha he advanages of he forecasaugmenaion remain, wih a correlaion beween he real ime measure and he final measure of 0.75 and agreemen on booms and recessions in 84% of he sample. Table 3 also provides saisics relaing o he oupu gap measures obained using wo alernaive mehods for measuring he rend in place of he HP filer. The measure denoed x rme refers o he gap obained in real ime and based on an exponenial smoohing (ES) filer. The filer is applied o he pos-revision series augmened wih forecass based on our mulivariae model. 19 The measure x rmw applies Wason s (1986) unobserved componens (UC) model o he same series. 20 AsdiscussedinKingandRebelo(1993), he ES smoohing can be considered as a resriced version of he HP filer and can be moivaed as providing he filer ha minimises rend growh (as opposed o he HP filer which minimises he change in rend growh). The UC model permis more complex dynamics and is consisen wih a more volaile rend measure, han HP characerising he rend as a random walk wih drif Resricing aenion o =1,..., T 3 implies ha only pos-revision measures of acual oupu are involved and forecass are used only in measuring rends. 19 The smoohing parameer was se equal o 10. This means ha 85% of he weigh is on observaions one year eiher side of he observaion of ineres and 95% on wo years eiher side. 20 The forecas augmenaion here refers o he forecas of he pos-revision oupu daa for he duraion of he revision process only. The sample period on which he UC measure is based runs from 1975q1, using he earlier observaions o obain iniial values for he Kalman filer esimaion. 21 The variabiliy of growh in he ES and UC rends is 70% and 67% of ha of oupu growh compared o 15% for he HP rend. [16]

19 The resuls in Table 3 show ha he (relaively) reassuring resuls obained for he HP are also found wih he oher wo smoohers. Hence, he correlaion beween he real ime and final vinage gap measures are relaively high, a 0.89 and 0.78 for he UC and he ES model respecively (and agreemen on booms and recessions are also high a 88% and 84%). Perhaps more surprisingly, he able also shows reasonably high correlaions beween he gap measures obained using he hree alernaive rends. The (pairwise) correlaion beween he hree final vinage measures are in he range [0.86, 0.96] and agreemen on booms and recessions is in he range [0.80, 0.93] despie he differences in he form and moivaion of he alernaive rend measures. Table 3 herefore no only confirms ha he advanages of applying our modelling approach carries over o oher mehods of derending, bu also shows ha he alernaive gaps obained in real ime provide a reasonably consensual picure of he macroeconomy, a leas as far as he size of he gap is concerned. Before discussing he reamen of uncerainy in hese measures, i is worh commening on he conribuion of our modelling framework, and is use of revisions daa, o hese resuls. This conribuion can be judged by comparing he forecasing performance of he mulivariae model wih ha of he univariae model and by comparing he in-sample fi of he associaed gap measures. In erms of forecasing, he univariae and mulivariae models can be esimaed recursively over he period 1970q1-2004q1 and he models forecass of oupu can be compared wih eiher of wo oupu oucomes: namely, he firs release observaion of oupu a ime, +1 y,orhefinal vinage measure T y. Using he firs release series as he appropriae measure of he oupu oucome, he roo mean square forecas error (RMSFE) defined by q 1 P T 3 T 3 =1 ( +1 y +1 [ y ) 2 akes he value for he univariae model and for he mulivariae model, represening a 51% improvemen over he univariae model. Using he final vinage daa as he measure of oupu oucome, calculaing q 1 P T 3 T 3 =1 ( T y T dy ) 2, also suppors he use of he mulivariae model: he RMSFE for he univariae model, for which T dy = E[[ +1 y Y ], akes he value , while he RMSFE for he mulivariae model is equal o where d T y is measured by d T y = E[[ +1 y Ω ] and equal o where i is measured by E[[ +3 y Ω ]; i.e. depending on wheher he mulivariae model is used o forecas he [17]

20 firs-release series +1 y or he pos-revision series +3 y. 22 In eiher case, he mulivariae model represens an improvemen over he univariae model, of around 15% for he firs release series and 73% for he full pos-revision case. The advanages of using he mulivariae model are confirmed also by he in-sample roo mean squared error (RMSE), defined as he gap beween he real ime gap measures and he final vinage gap measures, obained using he wo models. Values of he RMSE for x rmh,x rme and x rmw are , and , respecively, represening improvemens of 7%, 26% and 57% over heir univariae counerpars. The saisical significance of he explanaory variables in he model of he regressions explaining he revisions, he gains in forecasing of oupu levels and he gains in he fi of he real ime gap measures all confirm ha he mulivariae model of growh and revisions is appropriae and will provide a firm basis on which o calculae rends and oupu gap measures. 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, some of he unreliabiliy of he measures highlighed by OvN remains and so i is imporan ha he uncerainies surrounding he measures are properly represened for decision-making purposes. Figure 1 illusraes he order of magniude of he uncerainies involved based on x fmh using he informaion available a 2002q2 (leaving en periods, o 2004q4, for he purpose of ou-of-sample forecas evaluaion). According o he analysis based in real ime, he plo shows a period of expansion, in which he economy moves from recession (where x fmh < 0) o boom (where x fmh > 0) up o 2000q2, followed by a conracion ha 22 For he univariae model, no revisions are expeced o ake place afer he firs-release daa and he forecas of +1 y is he measure of he ineres. If i is known ha wo sysemaic revisions will occur (as shown in he regression analysis), he forecas of he rue pos-revision series is he expeced value of +3y and i is his measure, which fully akes ino accoun he role of prediced revisions, ha is mos appropriae for he mulivariae model. [18]

21 ends in 2001q4. These measures are conveyed wih a good degree of precision a firs (wih 95% confidence inervals no more han ±0.7% prior o 2000q2) when uncerainy is derived solely from he esimaion error in he underlying rend. 23 Bu he uncerainy rises considerably when he daa uncerainies are accommodaed owards he end of he sample and when forecasing ou-of-sample. During he laer sages, i is difficul o inerpre he informaion conen of he gap measures eiher in erms of he size of he gap, he likelihood of urning poins or he occurrence of boom or recession when judged simply according o he size of he confidence inerval. Theinformaiononhesizeandheprecisionofhegapcanbeconveyedmoreusefully and more direcly hrough he corresponding probabiliy densiy funcions showing prob(x fm 2002q2+n Ω 2002q2 <c) for a range of criical values c a a various esimaed horizons, n. Figure 2 shows such densiy funcions for n = 3, 0and4, generaed using he simulaion mehods of Secion 2.3 and 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 become progressively flaer reflecing he accumulaing uncerainy a he end-of-sample and ino he forecasing horizons. This sequence of densiies illusraes well he form in which he oupu gap can be usefully presened for he purpose of decision-making. Furher, he analysis underlying he densiies can also convey insigh ino paricular evens involving he gap. Hence, he probabiliy of a negaive oupu gap can be seen direcly from he densiies, falling from 0.93 in 2001q3 o 0.72 in 2002q2 and 0.55 in 2003q2. Bu he underlying simulaions also show, for example, ha he probabiliy of a urning poin rises from 0.18 in 2001q3 o 0.64 in 2001q4 (where he poin forecas urns) and o 0.80 by 2003q1 (when he upurn acually occurred). 24 These probabiliies convey far more precisely he srengh of convicion wih which hese evens are perceived o ake place. Given he uncerainies associaed wih he gaps discussed in 23 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. 24 A urning poin is defined here as wo consecuive periods of posiive oupu growh following wo periodsofnegaivegrowh. [19]

22 he previous secion, i is clear ha he densiy funcions of Figure 2 and he associaed probabiliy forecass provide a more useful form for represening he oupu gap han he poin forecass and confidence inervals given in Figure 1. A final illusraion of he usefulness of probabiliy forecass in conveying he uncerainies associaed wih he gap measures is provided in Figure 3. The figure plos he probabiliy of a posiive gap occurring one-sep ahead, as measured in real ime hrough he sample 1978q1 2004q4. In he figure, x rmh +1 and x ruh +1 represen he real ime measures of he gap obained by applying he HP filer o daa augmened by daa from he mulivariaeandheunivariaemodels. Themeasurex rm +1 is he gap in ime + 1 based on he densiy derived by aggregaing he hree densiies obained for he alernaive rend measures HP, ES and UC (using he mulivariae model in each case and wih equal weigh given o each rend measure). The aggregaed densiy accommodaes he uncerainies associaed wih he choice of he rend measure as well as he sochasic and parameer uncerainy underlying he individual gaps in a sraighforward way. In Figure 3, he informaion in he aggregaed densiy is ranslaed ino a form ha is direcly usable by a decision-maker who is concerned wih booms and recessions. As i urns ou, he probabiliy series based on he aggregaed densiy rarely differs from ha based on x rmh +1 by more han 10%, while here are some subsanial differences beween he gaps based on x rmh +1 and x ruh +1 for example. Hence, in his case a leas, he rend uncerainy appears less imporan han he choice of he model used o implemen he forecas-augmened approach. 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. This is because of he imprecision of he oupu daa available a he ime decisions have o be made and because of he difficulies in esablishing a precise measure of rend oupu. We have shown ha hese uncerainies can be miigaed by modelling he oupu process alongside he revision process, making use of forecass of curren and fuure posrevision oupu levels, o obain more precisely esimaed measures of he gap for use [20]

23 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 shown, 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 densiy funcions ha we have discussed, along wih he esimaed probabiliies of paricular evens of ineres, provide a very informaive and helpful means of represening he oupu gap daa for use by decision-makers. [21]

24 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 + ξ (6.6) 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 (6.7) 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 ξ (6.8) 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 (6.9) k 3 k 4 [22]

25 References Amao, J. D., and N. R. Swanson (2001) The Real-Time Predicive Conen of Money for Oupu, Journal of Moneary Economics, 48, Beveridge S. and C.R. Nelson (1981), A New Approach o Decomposiion of EconomicTimeSeriesinoPermanenandTransiory Componens wih Paricular Aenion o Measuremen of he Business Cycle, Journal of Moneary Economics, 7, 2, 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, M. and King, R.G. (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?, Ch. 2 in J.B Taylor and M. Woodford (eds.), Handbook of Macroeconomics, Vol.1A. Norh-Holland, Elsevier: Amserdam. Cukierman, A. and S. Gerlach (2003), The Inflaion Bias Revisied: Theory and Some Inernaional Evidence, Mancheser School, 71, 3, Diebold, F. X., and G. D. Rudebusch (1991) ForecasingOupuwihheComposie Leading Index: A Real-Time Analysis, Journal of he American Saisical Associaion, 86, [23]