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1 constor Mak Your Publications Visibl. A Srvic of Wirtschaft Cntr zbwlibniz-informationszntrum Economics Rich, Robrt; Tracy, Josph Working Papr Th rlationship btwn xpctd inflation, disagrmnt, and uncrtainty: Evidnc from matchd point and dnsity forcasts Staff Rport, Fdral Rsrv Bank of Nw York, No. 53 Providd in Coopration with: Fdral Rsrv Bank of Nw York Suggstd Citation: Rich, Robrt; Tracy, Josph (006) : Th rlationship btwn xpctd inflation, disagrmnt, and uncrtainty: Evidnc from matchd point and dnsity forcasts, Staff Rport, Fdral Rsrv Bank of Nw York, No. 53, Fdral Rsrv Bank of Nw York, Nw York, NY This Vrsion is availabl at: Standard-Nutzungsbdingungn: Di Dokumnt auf EconStor dürfn zu ignn wissnschaftlichn Zwckn und zum Privatgbrauch gspichrt und kopirt wrdn. Si dürfn di Dokumnt nicht für öffntlich odr kommrzill Zwck vrvilfältign, öffntlich ausstlln, öffntlich zugänglich machn, vrtribn odr andrwitig nutzn. Sofrn di Vrfassr di Dokumnt untr Opn-Contnt-Liznzn (insbsondr CC-Liznzn) zur Vrfügung gstllt habn solltn, gltn abwichnd von disn Nutzungsbdingungn di in dr dort gnanntn Liznz gwährtn Nutzungsrcht. Trms of us: Documnts in EconStor may b savd and copid for your prsonal and scholarly purposs. You ar not to copy documnts for public or commrcial purposs, to xhibit th documnts publicly, to mak thm publicly availabl on th intrnt, or to distribut or othrwis us th documnts in public. If th documnts hav bn mad availabl undr an Opn Contnt Licnc (spcially Crativ Commons Licncs), you may xrcis furthr usag rights as spcifid in th indicatd licnc.

2 Fdral Rsrv Bank of Nw York Staff Rports Th Rlationship btwn Expctd Inflation, Disagrmnt, and Uncrtainty: Evidnc from Matchd Point and Dnsity Forcasts Robrt Rich Josph Tracy Staff Rport no. 53 July 006 This papr prsnts prliminary findings and is bing distributd to conomists and othr intrstd radrs solly to stimulat discussion and licit commnts. Th viws xprssd in th papr ar thos of th authors and ar not ncssarily rflctiv of viws at th Fdral Rsrv Bank of Nw York or th Fdral Rsrv Systm. Any rrors or omissions ar th rsponsibility of th authors.

3 Th Rlationship btwn Expctd Inflation, Disagrmnt, and Uncrtainty: Evidnc from Matchd Point and Dnsity Forcasts Robrt Rich and Josph Tracy Fdral Rsrv Bank of Nw York Staff Rports, no. 53 July 006 JEL classification: C1, C, E37 Abstract This papr xamins matchd point and dnsity forcasts of inflation from th Survy of Profssional Forcastrs to analyz th rlationship btwn xpctd inflation, disagrmnt, and uncrtainty. W xtnd prvious studis through our data construction and stimation mthodology. Spcifically, w driv masurs of disagrmnt and uncrtainty by using a dcomposition proposd in arlir rsarch by Wallis and by applying th concpt of ntropy from information thory. W also undrtak th mpirical analysis within a smingly unrlatd rgrssion framwork. Our rsults offr mixd support for th propositions that disagrmnt is a usful proxy for uncrtainty and that incrass in xpctd inflation ar accompanid by hightnd inflation uncrtainty. Howvr, w documnt a robust, quantitativly and statistically significant positiv association btwn disagrmnt and xpctd inflation. Ky words: Survy of Profssional Forcastrs, dnsity forcasts, point forcasts, inflation prdictions, smingly rlatd rgrssion Rich: Fdral Rsrv Bank of Nw York (-mail: robrt.rich@ny.frb.org). Tracy: Fdral Rsrv Bank of Nw York (-mail: josph.tracy@ny.frb.org). Th authors thank J. S. Butlr, Timothy Cogly, John Ham, Bart Hobijn, John Lahy, Jos Lopz, Simon Pottr, Til Schurmann, Tom Stark, and Giorgio Topa for hlpful commnts and suggstions. Bss Rabin and Aril Ztlin-Jons providd xcllnt rsarch assistanc. Th viws xprssd in this papr ar thos of th authors and do not ncssarily rflct th position of th Fdral Rsrv Bank of Nw York or th Fdral Rsrv Systm.

4 1. Introduction Thr is widsprad agrmnt that inflation xpctations ar important for undrstanding th bhavior of individuals and obsrvd macroconomic outcoms. Whil a grat dal of rsarch continus to focus on how popl form xpctations, thr is also intrst in xamining othr aspcts of prdictiv bhavior and charactrizing thir rlationships. For xampl, Zarnowitz and Lambros (1987) and Giordani and Södrlind (003) invstigat th linkag btwn th disprsion of individual man forcasts of inflation (a masur of disagrmnt ovr inflation forcasts) and th avrag disprsion of corrsponding dnsity forcast distributions (a masur of uncrtainty ovr inflation forcasts). This issu bars upon th validity of using disagrmnt as a proxy for inflation uncrtainty in mpirical invstigations. Othr studis sk to dtrmin if changs in anticipatd inflation ar associatd with paralll changs in uncrtainty about inflation. If this rlationship holds, thn an additional cost of rising inflation is th advrs ral ffcts associatd with incrasd uncrtainty. Mor rcntly, Mankiw, Ris and Wolfrs (003) xplor th rlationship btwn th disprsion of individual man forcasts and xpctd inflation to tst prdictions of th sticky-information modl of Mankiw and Ris (00). This papr xamins matchd point and dnsity forcasts of inflation from th Survy of Profssional Forcastrs (SPF) to analyz th rlationship btwn (aggrgat) xpctd inflation, disagrmnt and uncrtainty. Our study improvs upon prvious studis in trms of data construction and stimation mthodology. With rgard to data construction, w driv mpirical masurs of disagrmnt and uncrtainty using two altrnativ approachs. On approach draws upon th work of Wallis (004, 005) and 1

5 uss a dcomposition of th varianc of th aggrgat dnsity forcast distribution. Th scond approach applis th concpt of ntropy from information thory. Whil w argu that ach approach has its own mrits, th us of both approachs has th addd bnfit of allowing us to assss th snsitivity of th rsults to diffrnt data constructs. With rgard to stimation mthodology, th matchd point and dnsity inflation forcasts from th SPF involv four forcast horizons. Prvious studis hav ithr slctd a singl horizon for analysis or xamind th horizons sparatly. W adopt a smingly unrlatd rgrssion (SUR) approach in which w group th quations for ach horizon. This choic of stimation stratgy not only stms from thortical considrations suggsting th rgrssion rsiduals should b corrlatd across horizons, but also from formal statistical tsts that confirm this fatur of th data. Th SUR framwork provids fficincy gains rlativ to convntional stimation mthods and also allows us to assss th robustnss of th rsults across diffrnt forcast horizons. Our findings offr mixd vidnc concrning th natur of th rlationships btwn disagrmnt (across inflation forcasts) and inflation uncrtainty as wll as btwn xpctd inflation and inflation uncrtainty. Spcifically, whn w mploy th Wallis-basd masurs of disagrmnt and uncrtainty, th rlationships btwn disagrmnt and uncrtainty as wll as btwn xpctd inflation and uncrtainty display littl conomic importanc. On th othr hand, th ntropy-basd masurs of disagrmnt and uncrtainty rval a positiv association btwn th variabls in ths two rlationships that is conomically and statistically significant. Whil w ar unabl to offr a complling argumnt that would favor on st of findings ovr th othr, w can nvrthlss draw som conclusions concrning th us of disagrmnt as a proxy for

6 inflation uncrtainty. Th analysis not only raiss qustions about th validity of this practic, but also suggsts that th masurs of disagrmnt commonly adoptd within this practic (i.., Wallis-typ masurs) may b particularly problmatic. 1 In contrast, th natur of th rlationship btwn disagrmnt and xpctd inflation is robust across both data constructs. Spcifically, w find strong vidnc that mor divrsity among rspondnts point prdictions of inflation coincids with incrass in xpctd inflation, with th linkag btwn th variabls displaying both conomic and statistical significanc. Whil w ar cautious about th intrprtation and implications of ths findings at th aggrgat lvl for spcific modls of xpctations formation, w acknowldg that th positiv co-movmnt btwn disagrmnt and xpctd inflation appars to b an important fatur of prdictiv bhavior and an issu warranting gratr attntion on th part of rsarchrs. In th nxt sction of th papr, w provid an ovrviw of th SPF inflation data. Sction 3 dscribs our conomtric mthodology. W prsnt th mpirical rsults in Sction 4. W thn conclud with a short summary of our findings.. Data This sction bgins with a dscription of th statistical framworks that undrli our masurs of xpctd inflation, disagrmnt and uncrtainty for th SPF inflation 1 Thr is an xtnsiv litratur that has usd forcast disprsion masurs from survys of inflation xpctations as a proxy for inflation uncrtainty. Zarnowitz and Lambros (1987) and Giordani and Södrlind (003) contain rfrncs to various studis that hav sought to dtrmin th ffct of inflation uncrtainty on macroconomic and financial variabls such as output growth, unmploymnt, nominal intrst rats, and labor contract durations. This finding initially might b viwd as corroborating vidnc in support of th sticky-information modl of Mankiw and Ris (00). In a rlatd papr, howvr, Rich and Tracy (004) argu that anothr implication of th sticky-information modl is that thr should b no prsistnt diffrncs across SPF rspondnts in thir forcast bhavior. Whn w xamin th SPF inflation data at th individual lvl, w strongly rjct th modl s prdiction that thr ar no significant fixd ffcts associatd with ithr th rspondnts x ant forcast uncrtainty or thir x post forcast accuracy. 3

7 data. W thn provid dtails on th construction of th variabls for th mpirical analysis and also discuss particular faturs of th SPF inflation data that bar upon stimation of th rlationships of intrst. W conclud by comparing our approach to that in Zarnowitz and Lambros (1987) and Giordani and Södrlind (003). A. Variabl Dfinitions Th SPF has undrgon significant changs throughout its history. Th survy was jointly initiatd in lat 1968 by th National Burau of Economic Rsarch (NBER) and th Amrican Statistical Association (ASA), and was first known as th NBER-ASA Economic Outlook Survy. Th survy is maild four tims a yar, on th day aftr th first rlas of th National Incom and Product Accounts data for th prcding quartr. Ovr tim, th numbr of rspondnts dclind, and in arly 1990 th NBER-ASA Economic Outlook Survy was discontinud. Howvr, latr that yar th Fdral Rsrv Bank of Philadlphia rvivd th survy and rnamd it th SPF. Th survy originally askd rspondnts to provid point forcasts for 10 variabls ovr a rang of forcast horizons. Unlik othr survys, th qustionnair also solicits dnsity forcasts for aggrgat output and inflation in th form of histograms. That is, rspondnts ar askd to attach a probability to ach of a numbr of pr-assignd intrvals, or bins, in which output growth and inflation might fall. Bcaus ths forcasts rlat to th sprad of a probability distribution of possibl outcoms, thy provid a uniqu basis from which to driv mpirical masurs of uncrtainty. W will rstrict our attntion to data on th inflation forcasts du to th lack of a homognous sampl for th output forcasts. 3 With rgard to th dnsity forcasts of inflation, in th fourth quartr th survy asks rspondnts about th annual avrag 4

8 prcntag chang in prics btwn th currnt yar and th following yar. In th first, scond and third quartrs, howvr, th survy asks rspondnts about th annual avrag prcntag chang in prics btwn th currnt yar and th prvious yar. Consquntly, th targt variabl for th dnsity forcasts rmains fixd for four conscutiv survys (from th fourth quartr of yar t through th third quartr of yar t+1), with a corrsponding forcast horizon (h) that dclins from approximatly 4 1 quartrs to1 1 quartrs. 4 For convninc, w rfr to ths horizons as h = 4, K,1. Dfining notation, lt φ, ( π ) dnot rspondnt j s h-quartr-ahad dnsity j h t forcast of inflation ( π ) in yar t. Thrfor, φ4, ( π ) will dnot rspondnt j s dnsity forcast in th fourth quartr (h=4) of yar t, whil ( ) jφ3, t+ 1 π will dnot th subsqunt dnsity forcast in th first quartr (h=3) of yar t+1. W will thn lt φ, ( π ) and j t j h t jσ h, t( π ) dnot, rspctivly, th man and varianc of th corrsponding dnsity forcasts. With rgard to th point forcasts, th SPF asks rspondnts for prdictions of th pric lvl for th currnt quartr and th nxt four quartrs. Bcaus data is availabl on th pric indx in prcding quartrs, a point forcast, j f h, t, can b constructd that matchs ach dnsity forcast. Thrfor, w will lt j f 4, t dnot rspondnt j s point forcast of th annual avrag prcntag chang in prics in th fourth quartr (h=4) of yar t. Th subsqunt point forcast of th annual avrag prcntag chang in prics in th first quartr (h=3) of yar t+1 will b dnotd by j f. 3, t+ 1 3 Spcifically, rspondnts switchd from forcasting nominal output to ral output in th arly 1980s. 5

9 Our study considrs two altrnativ approachs to driv masurs of disagrmnt and uncrtainty. Th first is basd on th statistical framwork of Wallis (004, 005) that yilds a formal rlationship among masurs of disagrmnt and uncrtainty. Spcifically, lt φ, ( π ) dnot th h-quartr-ahad aggrgat dnsity forcast of inflation in yar t dfind as: ht N ht, φ ( π) = (1/ N ) φ ( π), (1) ht, ht, j ht, j= 1 which avrags th dnsity forcasts across all N ht, rspondnts. As Wallis nots, th combind dnsity forcast in quation (1) is an xampl of a finit mixtur distribution. If w assum that th individual point forcasts ( j f h, t) ar th mans of th individual forcast dnsitis ( jφh, t( π )), thn th first two momnts of th aggrgat dnsity forcast about th origin ar givn, rspctivly, by: 5 N μ 1 = (1/ N) j f = f () j= 1 and N (1/ N) ( j f ) j ( ), j= 1 μ = + σ π (3) whr for convninc w tmporarily supprss th subscripts dnoting th spcific forcast horizon and yar. Consquntly, th varianc of φ ( π ) is givn by: 4 Zarnowitz and Lambros slct ths valus for th distancs btwn th dats of th survys and th nd of th targt yar. As w dmonstrat shortly, th horizons also rflct publication lags in th pric indx. 5 Englbrg, Manski and Williams (006) provid vidnc that most SPF forcastrs giv point prdictions that ar consistnt with th mans/mdians/mods of thir dnsity forcast distributions. 6

10 N N [ φ ( π)] = μ ( μ 1 ) = (1/ ) ( j ) + (1/ ) jσ ( π). j= 1 j= 1 Var N f f N = s + σ f φπ ( ) (4) Th rsulting dcomposition of th varianc of th aggrgat distribution undrlis our choic of this stratgy to obtain masurs of disagrmnt and uncrtainty. Th first trm on th right-hand sid of (4) is th cross-sctional varianc of th point forcasts ( s ) f and provids th corrsponding masur of disagrmnt. Th scond-trm is th avrag individual varianc ( σ ) and provids a natural masur of aggrgat uncrtainty. φπ ( ) Our scond approach to driv masurs of disagrmnt and uncrtainty draws upon information thory and th concpt of ntropy. 6 To bttr undrstand th motivation for th ntropy-basd masurs, on can think about trying to assss th information in a mssag confirming th occurrnc of a particular vnt. If th vnt was xpctd to occur with almost complt crtainty, thn th mssag causs littl surpris and contains littl information. On th othr hand, if thr was vry littl rason to bliv th vnt would occur, thn th mssag causs considrabl surpris and contains a grat dal of information. Thus, th informational contnt of th mssag is invrsly rlatd to th liklihood of th vnt. Th concpt of ntropy xtnds th prvious illustration by computing th xpctd informational contnt of th mssag basd on all possibl vnts and thir associatd probabilitis. As such, thr is a dirct connction btwn th xpctd information of th mssag and th notion of uncrtainty. If thr is littl uncrtainty prior to th mssag, du to th numbr of vnts bing small or th xistnc of on 6 Intrstd radrs can consult Th Nw Palgrav Dictionary of Economics for a usful summary of th history and dvlopmnt of information thory. 7

11 highly anticipatd vnt, thn its arrival is xpctd to convy littl information. Howvr, if thr is gratr uncrtainty arising from an incras in th numbr of vnts and/or a gratr uniformity of probabilitis across vnts, thn mor information is xpctd from th mssag. Whil w hav originally introducd th ntropy as th xpctd information of th mssag, it is clar that it can also b rgardd as a masur of th uncrtainty associatd with an mpirical distribution, and hnc with th SPF histograms. Following convntion in th information litratur and continuing to supprss subscripts dnoting forcast horizon and yar, w calculat th ntropy of an individual SPF histogram as: 1 = pk ( ) log, (5) j pk ( ) n jσ H j k = 1 whr j p( k ) dnots th probability that individual j attachs to intrval k. Th ntropy is nonngativ, and can attain a valu of zro whn pk ( ) = 1 for on of th n bins. If w hold th numbr of bins fixd at n, thn th ntropy is maximizd whn p( k) = (1 n). Howvr, this maximum incrass whn th numbr of possibl outcoms (n) incrass. Our ntropy-basd masur of aggrgat uncrtainty is thn obtaind by avraging th individual valus of (5) across th N rspondnts: σ N H (1/ N) jσ H j= 1 = (6) Whil our prvious discussion of ntropy has bn cast in trms of uncrtainty, its clos association with th notion of divrgnc suggsts that it can also b usd to masur disagrmnt. Consquntly, w can driv an ntropy-basd masur of 8

12 disagrmnt that parallls that for uncrtainty. Using th sam pr-assignd bins as thos for th SPF dnsity forcasts, an aggrgat histogram for th individual point forcasts can b constructd. W can thn rcast th formula in (5) in trms of aggrgat probabilitis to obtain an ntropy masur of disagrmnt which w dnot by B. Variabl Construction 7 s H. Whil th xprssions for xpctd inflation, disagrmnt and uncrtainty in th prvious sction srv as usful dfinitions, thy nd to b mad oprational for our mpirical analysis. W now provid dtails on th construction of ths masurs. For our purposs, it is rlativly straightforward to construct th individual point forcasts of inflation and th masurs of disagrmnt. Rcalling th structur of th targt variabl for th dnsity forcasts, th matching point forcast for rspondnt j of th annual avrag prcntag chang in prics in th fourth quartr of yar t is givn by: P + P + P + P j t+ 1,1 j t+ 1, j t+ 1,3 j t+ 1,4 j f4, t = 100 1, Pt,1 + Pt, + Pt,3 + jpt,4 (7) whr jp t, q is rspondnt j s prdictd valu of th pric lvl in quartr q of yar t and Ptq, is th actual valu of th pric lvl in quartr q of yar t. 8 Th subsqunt point forcast of th annual avrag prcntag chang in prics in th first quartr of yar t+1 is thn givn by: 7 W rcogniz that it would b usful if th ntropy-basd masurs of disagrmnt and uncrtainty could b constructd along th sam lins as in Wallis (004, 005). Howvr, whil th ntropy for th aggrgat dnsity forcast of inflation can b calculatd and dcomposd into two trms, thir intrprtation would not b idntical to thos in (4). On of th trms would corrspond to avrag uncrtainty, but th othr trm would corrspond to th disprsion in rspondnts forcast uncrtainty and not in thir inflation forcasts. This considration accounts for th disconnct btwn th construct of th ntropy-basd masurs of uncrtainty and disagrmnt rlativ to th Wallis-basd masurs. 8 Th trm actual valu includs rcntly rportd figurs that th SPF provids to assist rspondnts with thir forcasts. 9

13 j f 3, t+ 1 P + P + P + P = Pt,1 + Pt, + Pt,3 + Pt,4 100 j t+ 1,1 j t+ 1, j t+ 1,3 j t+ 1,4 1, (8) whr th P ' s and Ps ' rflct th nw quartrly pric lvl prdictions and ralizations, rspctivly. A similar updating would occur for j f and, t+ 1 j f. Th 1, t+ 1 availability of th individual point forcasts thn allows us to calculat th man point forcast ( f ), th cross-sctional varianc of th point forcasts ( s ) f, and th ntropybasd masur of disagrmnt s with littl ffort. ( H ) Turning to th dnsity forcast data, th construction of th ntropy-basd masur of avrag uncrtainty ( H ) σ is also rlativly straightforward. Howvr, th natur of th data dos not immdiatly lnd itslf to driving th rmaining variabls of intrst. Thrfor, w procd by making additional assumptions and calculating momnts of th aggrgat distribution of inflation. Th stimat of th man will provid a masur of xpctd inflation ( φ ( π )) from th dnsity forcast data. Givn an stimat of th corrsponding varianc, w can thn us th dcomposition in (4) and th calculatd valus of th sris s to back out th Wallis-basd masur of avrag f uncrtainty ( σ ): φ( π) σφπ ( ) = Var[ φ( π)] s f (9) Continuing th prvious discussion, thr ar two common approachs that hav bn usd to stimat th man and varianc of th SPF aggrgat histograms. Th first approach assums all th probability mass is locatd at th intrval midpoints. Th altrnativ approach assums th probability mass is distributd uniformly across ach 10

14 intrval. For th analysis, w adopt th first approach and apply th following formulas to comput th man and varianc of th aggrgat dnsity forcast, rspctivly: n Mid φ ( π) = pk ( ) π ( k) k = 1 n Mid Var[ φ ( π)] = p( k) π ( k) φ ( π) k = 1 (10) whr p( k ) dnots th aggrgat probability of intrval k, and π Mid ( k) dnots th midpoint of th corrsponding intrval. W omit th rsults for th uniform assumption bcaus thy ar similar, although slightly wakr. 9 C. Othr Faturs of th SPF Inflation Data Th discussion up to this point has abstractd from a numbr of othr important faturs of th SPF inflation data. For xampl, thr hav bn occasional rrors in th conduct of th survy whr th probability variabls hav bn subjct to a mismatch btwn th intndd and rqustd forcast horizon. As notd arlir, th matching of th point forcast and dnsity forcast sris is basd on dfinitions in which th probability variabls in th fourth quartr rfr to th following yar, whras th probability variabls in th first through third quartrs rfr to th currnt yar. Howvr, th survys conductd in 1974:Q4 and 1980:Q4 mistaknly askd rspondnts for dnsity forcasts of inflation btwn and , rspctivly. Convrsly, th survys conductd in 197:Q3, 1979:Q-Q3, 1985:Q1 and 1986:Q1 mistaknly askd survy rspondnts for dnsity forcasts of inflation btwn , , , and , rspctivly. Thus, ths data ar xcludd from th analysis du to thir forcast horizons not bing comparabl to thos in rlatd quartrs. 9 Ths rsults ar availabl from th authors upon rqust. 11

15 Thr hav also bn changs in th pric indx usd to dfin inflation in th survy as wll as priodic changs in th bas yar of th rlvant pric indxs. Thr is also a qustion of whthr to us ral-tim or final rvisd data. Anothr issu concrns th xclusion of rspondnts du to ithr thir failur to provid matching point and dnsity forcasts or du to discrpancis btwn thir point and dnsity forcasts that ar judgd to b xcssiv. 10 W rfr th radr to Appndix A for furthr dtails. D. Comparison To Othr Studis For our purposs, th statistical framwork of Wallis (004, 005) is xtrmly attractiv for analyzing th SPF inflation data. Th dcomposition of th varianc of th aggrgat dnsity forcast and th rsulting masurs of disagrmnt and uncrtainty corrspond closly to th notions undrlying prvious studis. Morovr, and in contrast to othr studis, thr is a formal drivation undrlying th masurs of uncrtainty and disagrmnt. For xampl, Zarnowitz and Lambros (1987) gnrat masurs of uncrtainty by calculating th avrag standard dviation from th individual dnsity forcasts. 11 With rgard to masurs of disagrmnt, thy calculat th cross-sctional standard dviation of th point forcasts. Whil ths masurs ar analogous to th two trms on th right-hand sid of (4), th us of standard dviations rathr than variancs braks th link to th dcomposition. For thir analysis, Zarnowitz and Lambros xamin th sam thr rlationships that ar of intrst to us. Thy find that disagrmnt and uncrtainty display a wak positiv rlationship, whil xpctd inflation contributs almost nothing to movmnts 10 This is similar to Englbrg, Manski and Williams (006) who also find thr ar som SPF forcastrs whos point prdictions appar to b inconsistnt with th mans/mdians/mods of thir dnsity forcasts. 11 Zarnowitz and Lambros mak thir calculations assuming th probability mass is distributd uniformly within bins. 1

16 in disagrmnt. Thy do, howvr, documnt an conomically and statistically significant association btwn xpctd inflation and uncrtainty. It should b notd that Zarnowitz and Lambros bas thir findings on a sampl that runs from 1968:Q4 1981:Q, rsulting in stimatd rgrssions for individual forcast horizons that only us obsrvations. Giordani and Södrlind (003) xtnd th work of Zarnowitz and Lambros by dvloping a statistical framwork that faturs individual forcastrs with privat information. Thir analysis yilds th following xprssion that is similar to (4): Var φ ( π) = Var ( jφ ( π)) + E( jσ ) (11) whr E dnots th xpctations oprator. Howvr, Giordani and Södrlind mak no subsqunt us of th varianc of th aggrgat distribution or th quality in (11). Rathr, thy lct to follow th approach of Zarnowitz and Lambros and calculat th masurs of disagrmnt and uncrtainty as standard dviations and not variancs. Unlik Zarnowitz and Lambros, howvr, th standard dviation calculations ar basd on normal approximations to th individual forcast histograms. Consquntly, Giordani and Södrlind xclud th individual point forcast data from thir analysis. 1 In contrast to Zarnowitz and Lambros, Giordani and Södrlind rstrict thir attntion to th qustion of whthr disagrmnt is a valid proxy for uncrtainty. Thy principally focus on first quartr (h=3) data and find a corrlation of 0.60 btwn thir 1 Th normal approximation provids th stimats of th man and standard dviation of ach individual forcast histogram. As prviously notd, it is straightforward to construct a masur of disagrmnt from th point forcast data. On th othr hand, it is much mor problmatic to driv a masur of disagrmnt from th dnsity forcast data. As w will discuss, th natur of th data may limit th ability to stimat a man for ach individual forcast histogram. Th us of an stimat may also introduc a sourc of masurmnt rror into th analysis. Abstracting from th prvious two considrations, thr is a mor gnral qustion of rlvanc in that Giordani and Södrlind s approach is not consistnt with th convntional practic of using disagrmnt across point forcast data as a proxy for uncrtainty. W rturn to this lattr issu in Sction 3 whr w discuss th spcification of th rgrssion quations. 13

17 masurs of disagrmnt and uncrtainty, although thy rport that corrlations for th othr quartrs ar similar and rang from 0.46 to Thy intrprt thir findings as showing that disagrmnt is a bttr proxy of inflation uncrtainty than prviously thought. With rgard to th mthodology of Giordani and Södrlind, thr ar two issus that mrit spcial discussion. Th first is thortical in natur and rlats to th statistical foundation of thir modl. As notd by Wallis (004, 005), th pooling of disparat information sts actually prsnts concptual difficultis and gratly complicats th issu of aggrgation. This considration may xplain why Giordani and Södrlind ar unabl to provid an intrprtation for th aggrgat dnsity forcast and may also undrli thir acknowldgmnt that th xprssion in (11) may b problmatic. In contrast, th finit mixtur distribution proposd by Wallis provids an appropriat rprsntation for combining th individual dnsitis of th SPF rspondnts. Morovr, as Wallis nots, th sampl avrag notation on th right-hand sid of (4) is statistically mor accurat than th us of E and Var on th right-hand sid of (11). Th scond issu is mpirical in natur and rlats to fitting distributions to th individual dnsity forcasts. Spcifically, an xamination of th histograms rvals that rspondnts typically assign probabilitis to only a fw bins. As w discuss in gratr dtail in Appndix B, this concntration of probabilitis raiss concrns about th fasibility and rliability of stimating mans and standard dviations basd on fittd normal distributions. 13 Whil our approach also involvs stimating momnts of a 13 Th ability to fit a uniqu normal distribution to a histogram is only possibl whn a rspondnt uss thr or mor bins. Th rlvanc of this condition is not trivial for th SPF inflation data, spcially as th forcast horizon dclins. Englbrg, Manski and Williams (006) and D Amico and Orphanids (006) 14

18 distribution, w ar much mor comfortabl working with aggrgat histograms du to th gratr diffusion of prdictiv probabilitis. Morovr, w fl that th assumption concrning th location of probability mass in (10) is lss tnuous than th maintnanc of a particular distributional assumption. 14 With rgard to th ntropy-basd masurs of disagrmnt and uncrtainty, thy ar lss formal than th Wallis-basd masurs. Howvr, th ntropy approach has th advantag of not rquiring any assumption for th location of probability mass for th dnsity forcasts and thrby circumvnts concrns rlatd to th accuracy of approximations to th undrlying distributions. Morovr, our ntropy-basd masur of avrag uncrtainty ar drivd using data on th individual dnsity forcasts and thrfor afford som comparability to th constructs in Zarnowitz and Lambros (1987) as wll as Giordani and Södrlind (003). 3. Empirical Framwork Th prvious discussion focusd on th construction of masurs of xpctd inflation, disagrmnt and uncrtainty. W now turn our attntion to valuating th conomic and statistical significanc of th various rlationships of intrst. Spcifically, w will considr th following modl to gaug whthr disagrmnt is a symptom of uncrtainty: σ α βs ε = + + (1) also qustion th appropriatnss of using a normal distribution to approximat ach rpondnt s probabilistic blifs. Consquntly, thy considr altrnativ distribution fitting mthods. 14 Giordani and Södrlind cit th visual normality of th aggrgat dnsity forcasts to motivat thir approach. W will also discuss th issu of fitting normal distributions to th SPF histograms at th aggrgat lvl in Appndix B. It is worth noting hr, howvr, that w will rport formal statistical tsts that ovrwhlmingly rjct th normality assumption for th aggrgat dnsity forcasts. 15

19 whr σ is avrag uncrtainty, s is th dgr of disagrmnt among forcasts, and ε is a man-zro, random disturbanc trm. W will considr both data approachs in th cours of stimating (1), although w will maintain a consistncy across th masurs of avrag uncrtainty and disagrmnt. That is, w will xamin th rlationship btwn σ and φ ( π ) s as wll as th rlationship btwn σ H and f s H. 15 With rgard to analyzing th contribution of xpctd inflation to movmnts in uncrtainty and disagrmnt, w diffrntiat btwn th us of th dnsity forcast data and th point forcast data. 16 Howvr, w allow for diffrncs in th construction of th masurs within ach of th rlationships w xamin. Spcifically, w adopt th following modl to invstigat th linkag btwn xpctd inflation and uncrtainty: σ α βφ ( π) ε = + + (13) whr w considr both σ and φ ( π ) σ H as masurs of inflation uncrtainty, and whr φ ( π ) again dnots th man of th aggrgat dnsity forcast. In th cas of th linkag btwn xpctd inflation and disagrmnt, w adopt th following modl: s = α + β f + ε (14) whr w considr both s and f s H as masurs of disagrmnt, and whr f again dnots th man of th point forcasts. 15 With th xcption of using variancs rathr than standard dviations, quation (1) is idntical to th modl usd in Zarnowitz and Lambros. This similarity also includs th us of th dnsity forcast data to construct th uncrtainty masur and th us of th point forcast data to construct th disagrmnt masur. Ths slctd masurs ar th appropriat choic to assss th validity of using masurs of disagrmnt across point forcasts as a proxy for uncrtainty. 16 Zarnowitz and Lambros adoptd this sam approach. 16

20 Whil w hav touchd on diffrncs btwn th analyss of Zarnowitz and Lambros (1987) and Giordani and Södrlind (003), it is worth noting ths studis shar on important fatur. Spcifically, thy almost xclusivly bas thir analysis on data for a singl horizon or for individual horizons. W will argu, howvr, that th natur of th data lnds itslf to applying th mthod of smingly unrlatd rgrssion (SUR). As prviously discussd, th forcasting horizon for th SPF inflation data is not constant and instad dclins from th fourth quartr of yar t through th third quartr of yar t+1. Bcaus of th variation in forcast horizons, it is mor rasonabl to trat th data as annual obsrvations on four diffrnt sris than as quartrly obsrvations on a homognous sris. By itslf, this considration would suggst stimation of th following rgrssion quations across th individual horizons: whr Yht, and X, ht Yht, = αh+ βhxht, + εht,, h= 4,3,,1 (15) dnot, rspctivly, th rlvant indpndnt and dpndnt variabls spcifid in (1), (13) and (14), and whr w allow th intrcpt and slop cofficints to vary across forcast horizons. Whil th diffrnt forcast horizons argu for sparat quations for th data, it dos not sm rasonabl to viw th quations as compltly unrlatd du to thir sharing a common inflation targt ovr four contiguous quartrs. This fatur of th survy suggsts that th corrsponding rror trms ( ε4, t, ε3, t+ 1, ε, t+ 1, ε1, t+ 1) ar likly corrlatd with ach othr. If this is th cas, thn it is possibl to xploit th corrlation structur of th rror trms and apply th gnralizd-last squars stimators proposd 17

21 by Zllnr (196) to gnrat mor fficint paramtr stimats than thos obtaind by th application of ordinary last squars (OLS) to ach quation individually. 17 Our smingly unrlatd rgrssion (SUR) stimation stratgy is standard xcpt for on minor modification. Spcifically, w group th quations basd on thir affiliation with th forcast horizon and targt rat of inflation. In particular, w stack th four tim sris rgrssions as follows: Y1, α1 + β1x1, + ε1, M M Y 1, T α1 + β1x1, T + ε 1, T M = M Y 4,1 α4 + β4x4,1 + ε 4,1 M M Y α + β X + ε 4, T , T 1 4, T 1 (16) whr w ordr th quations from horizon h=1 to horizon h=4. 18 W will follow convntion with rgard to th structur of th varianc-covarianc matrix Ω. Spcifically, w assum th disturbanc trm in any singl quation is conditionally homoscdastic and non-autocorrlatd, although allowanc is mad for th data to b conditionally htroskdastic across quations. Ths assumptions imply th following corrlation pattrn for th rrors: E[ ε ε ] = σ, i = j it, jt, j E[ ε ε ] = δ, i j, τ = 0 it, jt, + τ ij E[ ε ε ] = 0, othrwis it, jt, + τ (17) Consquntly, our stimat of Ω will hav th following form: 17 It should also b notd that th xplanatory variabls will not b idntical across th diffrnt forcast horizons. If this condition did not hold, thn no gains in fficincy could b ralizd from th SUR stimator ovr th OLS stimator. 18

22 Q1 R1 L R14 R Q R L M M O M R41 R4 L Q4 1 4 Ω= (18) whr Q j σ j 0 L 0 0 σ j 0 L = M M O σ j (19) and R ij δij 0 L 0 0 δ 0 L M M O δij ' ij = Rji = (0) Following Brusch and Pagan (1980), w can construct th following Lagrang multiplir tst to formally tst for non-zro corrlations btwn th disturbanc trms in th four quations: i i 1 λ = T ρ () m= 1 n= 1 mn whr ρ mn is th stimatd corrlation btwn th OLS rsiduals of th i=4 quations and T is th numbr of obsrvations in ach quation. Th tsts statistic is distributd asymptotically as a chi-squar random variabl with i(i-1)/ dgrs of frdom undr th null hypothsis of zro corrlation btwn th disturbanc trms W assum th numbr of obsrvations on ach quation is th sam, which accounts for th slight diffrnc in th tim subscripts for th data associatd with horizons h=1,, 3 (t=,..., T) and horizon h=4 (t=1,..., T-1). 19 Th assumptions undrlying th spcification of Ω ar broadly consistnt with th data. Our dcision not to incorporat additional own- and cross-covarianc procsss was basd on furthr inspction of th OLS rsiduals as wll as dgrs of frdom considrations. 19

23 4. Empirical Rsults A. Masurs of Expctd Inflation, Disagrmnt and Uncrtainty Thr is on additional fatur of th SPF inflation data that mrits spcial attntion. Spcifically, thr hav bn priodic changs in th numbr of intrvals and thir widths in th SPF s survy instrumnt. As shown in Tabl 1, th survy initially providd 15 intrvals. From 1981:Q3-1991:Q4, howvr, th numbr of intrvals was rducd to 6. Sinc 199:Q1, thr hav bn 10 intrvals. Th intrval widths also varid from 1 prcntag point bfor 1981:Q3 and aftr 1991:Q4 to prcntag points in th intrvning priod. Th prsnc of varying intrval widths poss a particular concrn bcaus it will impact on som of our summary masurs and thir movmnts across sub-priods. Thrfor, w rdfin th intrvals to impos a common prcntag point width throughout th whol sampl priod. 0 To undrstand th importanc of this considration, th uppr and lowr panls of Figur 1 dpict, rspctivly, th ntropy of th aggrgat dnsity forcast distribution and th ntropy of th aggrgat point forcast distribution using both th raw and adjustd data. 1 Th profils for th ntropy diffr markdly bfor 1981:Q3 and aftr 1991:Q4, spcially in th cas of th aggrgat dnsity forcast distribution. Thus, th us of th raw data would rsult in an artificial incras in th ntropy during th sub-priods associatd with th narrowr intrval widths. 0 Du to th odd numbr of intrvals usd ovr th sub-priod 1968:Q4-1981:Q, w us a unit intrval lngth for th middl intrval. As an altrnativ to imposing a common % width, on might think about rdfining th intrvals from 1981:Q3-1991:Q4 to hav a unit intrval width. W found this adjustmnt procdur to b much lss satisfactory du to th difficulty of dtrmining how to allocat th probabilitis across th subdividd intrvals. 0

24 As shown in Figur, th changs in intrval widths also affct th profil for th stimatd varianc of th aggrgat dnsity forcast distribution usd for th Wallis dcomposition. Spcifically, whil th stimats of th varianc ar gnrally highr during th 1980s, th us of th adjustd data partly rducs th diffrntial during th pr-1981:q3 and post-1991:q4 sub-priods. Consquntly, th us of th raw data would lad to lowr stimats of inflation uncrtainty during th pr-1981:q3 and post-1991:q4 sub-priods. Figurs 3-5 prsnt th tim profils for th masurs of disagrmnt, uncrtainty and xpctd inflation usd in th mpirical analysis. As shown in Figur 3, th bhavior of th two disagrmnt masurs is qualitativly similar and indicats a gratr divrsity of opinion about xpctd inflation during th arlir part of th sampl priod. Th ntropy-basd masur ( H ) s displays slightly mor variability, although th cross-sctional varianc of th point forcast ( s ) f is charactrizd by occasional spiks in disagrmnt. Th sawtooth pattrn vidnt in both masurs spaks to th gratr unanimity across point forcasts as th forcast horizon shrinks. In contrast to th masurs of disagrmnt, thr is a markd diffrnc in th faturs of th avrag uncrtainty masurs across th two data approachs. In particular, th Wallis-basd masurs ar gnrally highr and mor variabl than th ntropy-basd masur. 3 Nvrthlss, both masurs dpict a dclin in inflation 1 Th missing obsrvations in Figur 1 (as wll as in subsqunt figurs) rflct th xcludd survy dats discussd in Sction.C. W only display on sris during th middl priod du to th coincidnc of th raw and adjustd data. Th impact of th changs in intrval widths on th stimatd man of th aggrgat dnsity forcast distribution turns out to b ngligibl. Th masur of disagrmnt using th point forcast data is not affctd by changs in intrval width. 3 Th Wallis-basd masur of uncrtainty is about twic as high on avrag with a standard dviation that is mor that twic that for th ntropy-basd masur. 1

25 uncrtainty starting around As xpctd, thy also tnd to rflct a gratr disprsion of intraprsonal probabilistic blifs as th forcast horizon incrass, although thr is a surprising slight dclin at th h=4 quartr horizon. Comparing th avrag lvls of disagrmnt and uncrtainty across th sam data approach, disagrmnt undrstats uncrtainty to a considrabl xtnt. Spcifically, th uncrtainty masur is largr by a factor of fiv using th Wallis approach and is narly twic as larg using th ntropy approach. Whil uncrtainty displays gratr variability than disagrmnt for th Wallis-basd masurs, th opposit is tru for th ntropybasd masurs. Whn w xamin th masurs of xpctd inflation in Figur 5, howvr, w obsrv that th sris display a high dgr of conformity and ar practically indistinguishabl from ach othr. Th two inflation xpctations sris display th sam pronouncd ris and subsqunt dclin as actual inflation during th cours of th sampl priod. B. Estimatd Rlationships Th sampl covrs th survys conductd from 1968:Q4 through 003:Q3, so that th valus on th ralizd annual rat of inflation covr th priods through W bgin by xamining corrlations and goodnss-of-fit masurs from OLS stimation of quations (1) (14) rportd, rspctivly, in Tabls Bcaus w will subsquntly addrss th issu of stimation fficincy, w dfr for th momnt from any discussion of statistical significanc and instad focus our initial attntion on th conomic significanc of th rlationships.

26 As shown, th variabls display a positiv association across all of th rlationships. With rgard to any systmatic pattrn to th corrlations, thy do not bhav in a monotonic fashion as th forcast horizon incrass. Rathr, th corrlations tnd to b highst at th h = and 3 quartr horizons. Thr ar, howvr, svral othr notabl findings that that mrg from th analysis. On is immdiatly struck by th xtrmly low xplanatory contnt of disagrmnt for movmnts in avrag uncrtainty whn disagrmnt is masurd by th varianc of th point forcasts. With th xcption of th rgrssion associatd with th h=3 quartr horizon in th uppr panl of Tabl, disagrmnt accounts for lss than 10 prcnt of th variation in uncrtainty. 5 Th rsults ar qualitativly similar whn w turn to th linkag btwn xpctd inflation and th Wallis-basd masur of uncrtainty in th uppr panl of Tabl 3. Th lack of any maningful co-movmnt btwn th variabls suggsts th issu of th statistical significanc of ths rlationships is largly irrlvant for th rmaining analysis. Equally striking, howvr, is th markd incras in th strngth of ths sam rlationships whn th ntropy-basd masurs of disagrmnt and uncrtainty ar usd in th rgrssions. For xampl, th corrlations xcd 0.6 at th h= and 3 quartr horizons in th lowr panl in Tabl and Tabl 3, with th othr corrlations of modrat siz. Last, an xamination of Tabl 4 indicats th rlationship btwn disagrmnt and xpctd inflation is much mor robust to th construct of th masur of forcast disprsion. Whil th corrlations indicat a somwhat wak rlationship at 4 W rcogniz thr is littl diffrnc in th information convyd by th rportd corrlations and th R s, as th lattr simply involvs squaring th formr and adjusting for dgrs of frdom. Nvrthlss, w rport both statistics to allow for a basis of comparison to th rsults of othr studis. 5 Rcall that Södrlind and Giordani focus thir analysis on th data for th h=3 quartr horizon. 3

27 th h=1 quartr horizon, th othr horizons display rasonably strong corrlations that ar comparabl to thos associatd with th ntropy basd masurs in Tabls and 3. To addrss th issu of statistical significanc in th rlationships, w initially applid th Brusch-Pagan (1980) tsting procdur to th OLS rsiduals within ach systm of four quations. As shown by th valus of th tst statistic rportd in th last column of Tabls -4, thr is significant corrlation btwn th quations disturbanc trms associatd with th sam inflation targt. Th on xcption is th rlationship btwn disagrmnt and xpctd inflation using th varianc of th point forcasts to masur disprsion. Consquntly, w rtain th mthod of OLS for stimation in this cas. In all othr cass, w will stimat th rlationships using th mthod of SUR. Tabls 5-7 rport th stimats of th paramtrs and th corrsponding standard rrors. 6 Bcaus th dfinition of an R statistic is not obvious in th cas of SUR stimation, w do not attmpt to rport any typ of goodnss-of-fit statistic. Morovr, bcaus most rsarchrs typically posit a positiv rlationship btwn th variabls, w conduct a on-taild tst for statistical significanc. Th conclusions, howvr, gnrally will not dpnd on th choic of a on- vrsus two-taild tst for statistical significanc. As shown, th findings typically documnt a statistically significant positiv association btwn th variabls in th rlationships. Not surprisingly, th qualitativ faturs of th rsults paralll thos from th prvious analysis in trms of forcast horizon and data construct. That is, th statistical significanc of th rlationships btwn disagrmnt and uncrtainty as wll as btwn xpctd inflation and uncrtainty is lss robust using th Wallis-basd masurs than th ntropy-basd 6 Thr ar an unqual numbr of obsrvations across th quations. Howvr, w ignord this diffrnc and calculatd th own- and cross-covariancs using all availabl obsrvations. 4

28 masurs. Morovr, th rlationship btwn disagrmnt and xpctd inflation rmains highly statistically significant across both data constructs. Takn togthr, thr ar svral conclusions that can b drawn from th rportd rsults in Tabls -7. Thr is mixd support for th propositions that gratr disagrmnt is indicativ of hightnd uncrtainty and that highr xpctd inflation is accompanid by incrasd uncrtainty. Th lack of robustnss of ths rsults likly stms from th gratr variability of th Wallis-basd masur of uncrtainty rlativ to th ntropy-basd masur of uncrtainty (as wll as th corrsponding masurs of disagrmnt). In light of th conflicting vidnc, it is natural to ask if on st of rsults might b viwd as mor prsuasiv. Th answr to this qustion would b guidd by slcting th approach that provids th bttr approximation to th masurs of intrst. Bcaus w s advantags and disadvantags to ach approach that ar roughly qual on balanc, w ar unabl to offr any rsolution to this mattr at prsnt. Whil w cannot rsolv th disparity in th rsults, w can still commnt on th rsults associatd with a particular data approach. In this rgard, th stimatd rlationship btwn disagrmnt and inflation uncrtainty using th Wallis-basd masurs has particular rlvanc. This is bcaus almost all mpirical studis using disagrmnt as a proxy for uncrtainty hav masurd disagrmnt by th varianc (or standard dviation) of point forcasts. W find, howvr, that among all of th stimatd rlationships, th association btwn this masur of disagrmnt and inflation uncrtainty is th wakst in trms of conomic and statistical significanc. 7 7 As discussd in Sction.B, th rsults for th disagrmnt-inflation uncrtainty and disagrmntxpctd inflation rlationships using th Wallis-basd masurs ar slightly wakr if w adopt th uniform assumption for th location of th probability mass within intrvals. Thus, ths rsults ar bing prsntd in a mor favorabl light. 5

29 Thus, thr appars to b littl justification for this convntional masur of disagrmnt to srv as a proxy for uncrtainty. Th stimatd rlationship btwn disagrmnt and uncrtainty using th Wallis-basd masurs also allows for a rasonabl basis of comparison to th rsults of Zarnowitz and Lambros (1987) and Giordani and Södrlind (003). Our findings ar closr to thos rportd by Zarnowitz and Lambros and contrast sharply with th conclusions of Giordani and Södrlind. With rgard to th lattr study, thir rsults likly diffr bcaus of a smoothr masur of uncrtainty du to th us of normal approximation mthods as wll as a masur of disagrmnt that prtains to th dnsity forcast data. As notd in th txt and Appndics, w hav discussd various concrns about th logic and statistical basis of thir mthodology. On th othr hand, th vidnc is much lss ambiguous and quit favorabl about a positiv co-movmnt btwn disagrmnt and xpctd inflation. Whn w rstrict our attntion to th Wallis-basd masurs to allow for a basis of comparison to Zarnowitz and Lambros, our findings ar much strongr in trms of conomic and statistical significanc. This may b a consqunc of th longr sampl priod usd in our analysis. It is also intrsting to not that, of th thr rlationships xamind in th papr, th linkag btwn disagrmnt and xpctd inflation has rcivd th last attntion on th part of rsarchrs. V. Conclusion Our study uss matchd point and dnsity forcasts of inflation from th Survy of Profssional Forcastrs to rvisit qustions concrning th co-movmnt btwn aggrgat xpctd inflation, th dgr of disagrmnt among individual inflation 6

30 forcasts, and th lvl of avrag inflation uncrtainty. W attmpt to improv upon prvious studis in trms of th construction of th masurs usd for th mpirical analysis as wll as th statistical mthods usd to assss th natur of th rlationships. As such, w driv masurs of disagrmnt and uncrtainty by using a statistical framwork rcntly proposd by Wallis (004, 005) as wll as by drawing upon th concpt of ntropy from th mor stablishd litratur on information thory. W also adopt a smingly unrlatd rgrssion framwork to xploit fficincy gains that ar affordd by th rcurrnt dclining forcast horizon of th SPF inflation data. Th variabls gnrally display a statistically significant association, although this fatur varis somwhat across th particular rlationships. Spcifically, th incidnc and lvl of statistical significanc is highst for th linkag btwn disagrmnt and xpctd inflation, and somwhat lowr for th othr rlationships. In trms of conomic significanc, howvr, w obtain markdly diffrnt rsults across th rlationships and data constructs. W documnt that movmnts in disagrmnt and xpctd inflation display rasonably strong positiv corrlations, and would contnd that an adquat modl of xpctations formation must b abl to account for this co-movmnt. On th othr hand, th vidnc of a maningful rlationship btwn disagrmnt and uncrtainty as wll as btwn xpctd inflation and uncrtainty ssntially disappars whn w switch from th ntropy-basd masurs of uncrtainty and disagrmnt to th Wallis-basd masurs. Th lack of robustnss of ths rsults lads us to conclud that th rlvanc of on of th positd channls of ffct of xpctd inflation on ral activity rmains an opn qustion. Th sam holds tru concrning th validity of using disagrmnt as a proxy for uncrtainty. With rspct to 7

31 th last point, w ar spcially cautious about th conclusions of mpirical studis that hav usd convntional forcast disprsion masurs to proxy inflation uncrtainty. 8

32 Rfrncs Brusch, Trvor S., and Adrian R. Pagan. Th Lagrang Multiplir Tst and Its Application to Modl spcification in Economtrics. Rviw of Economic Studis 47 (January 1980): Englbrg, Josph, Charls F. Manski, and Jard Williams. Comparing th Point Prdictions and Subjctiv Probability Distributions of Profssional Forcastrs. Working papr, Northwstrn Univrsity: 006. Giordani, Paul, and Paul Södrlind. Inflation Forcast Uncrtainty. Europan Economic Rviw 47 (Dcmbr 003), Lahiri, Kajal, and Christi Tigland. On th Normality of Probability Distributions of Inflation and GNP Forcasts. Intrnational Journal of Forcasting 3 (1987): Mankiw, N. Grgory, Ricardo Ris, and Justin Wolfrs. Disagrmnt about Inflation Expctations. in NBER Macroconomics Annual 003, vol. 18: Mankiw, N. Grgory, and Ricardo Ris. Sticky Information Vrsus Sticky Prics: A Proposal to Rplac th Nw Kynsian Phillips Curv. Quartrly Journal of Economics 117 (Novmbr 00), Rich, Robrt, and Josph Tracy. Sticky Information and Htrognity in Forcast Bhavior: Furthr Implications and Evidnc from th Survy of Profssional Forcastrs. Working papr, Fdral Rsrv Bank of Nw York: 004. Wallis, Knnth F. Forcast Uncrtainty, Its Rprsntation and Evaluation. Tutorial Lcturs, IMS Singapor, May Wallis, Knnth F. Combining Dnsity and Intrval Forcasts: A Modst Proposal. Oxford Bulltin of Economics and Statistics 67 (005 Supplmnt): Zarnowitz, Victor and Louis A. Lambros. Consnsus and Uncrtainty in Economic Prdiction. Journal of Political Economy 95 (Jun 1987):