Working Paper The role of expectations in U.S. inflation dynamics. Working paper series // Federal Reserve Bank of Boston, No.

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1 econsor Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Fuhrer, Jeffrey C. Working Paper The role of expecaions in U.S. inflaion dynamics Working paper series // Federal Reserve Bank of Boson, No. - Provided in Cooperaion wih: Federal Reserve Bank of Boson Suggesed Ciaion: Fuhrer, Jeffrey C. (2) : The role of expecaions in U.S. inflaion dynamics, Working paper series // Federal Reserve Bank of Boson, No. - This Version is available a: hp://hdl.handle.ne/49/55653 Nuzungsbedingungen: Die ZBW räum Ihnen als Nuzerin/Nuzer das unengelliche, räumlich unbeschränke und zeilich auf die Dauer des Schuzrechs beschränke einfache Rech ein, das ausgewähle Werk im Rahmen der uner hp:// nachzulesenden vollsändigen Nuzungsbedingungen zu vervielfäligen, mi denen die Nuzerin/der Nuzer sich durch die erse Nuzung einversanden erklär. Terms of use: The ZBW grans you, he user, he non-exclusive righ o use he seleced work free of charge, erriorially unresriced and wihin he ime limi of he erm of he propery righs according o he erms specified a hp:// By he firs use of he seleced work he user agrees and declares o comply wih hese erms of use. zbw Leibniz-Informaionszenrum Wirschaf Leibniz Informaion Cenre for Economics

2 No. - The Role of Expecaions in U.S. Inflaion Dynamics Jeffrey C. Fuhrer Absrac: A growing body of lieraure examines alernaives o he raional expecaions hypohesis in applied macroeconomics. This paper coninues his srand of research by examining he role survey expecaions play in he inflaion process and repors hree principal findings. One, shor-run inflaion expecaions appear o play a significan role in explaining U.S. inflaion over he pas 2 25 years. Two, long-run expecaions generally do no appear o have a direc influence on U.S. inflaion over he same period, alhough hese longer expecaions ener indirecly as a key deerminan of he shor-run expecaions. The resricions implied by rend inflaion models of inflaion are generally rejeced in he daa. Three, by employing a survey operaor, his paper develops a firs pass a a srucural model ha incorporaes he feaures discussed above and assesses is performance in explaining inflaion in he poswar period. JEL Classificaions: E3, E52 Jeffrey C. Fuhrer is an execuive vice presiden and senior policy advisor a he Federal Reserve Bank of Boson. His address is jeff.fuhrer@bos.frb.org. The views expressed in his paper are hose of he auhor and do no necessarily represen hose of he Federal Reserve Bank of Boson or he Federal Reserve Sysem. This paper, which may be revised, is available on he web sie of he Federal Reserve Bank of Boson a hp:// /economic/wp/index.hm. This version: Ocober 26, 2

3 Inroducion Once one assumes ha prices are sicky and assume is he appropriae word, for he deeper micro foundaions ha would moivae price-seers o hold nominal prices fixed for exended periods remain elusive in mos applied macro models inflaion expecaions ake on a prominen role in price modeling. The inuiion is sraighforward: if prices are expeced o remain fixed for some period hen he economic condiions ha are expeced o prevail during ha period mus be aken ino accoun if prices are o be se opimally. To dae, he bulk of research n inflaion has used he raional expecaions assumpion o generae he required expecaions (here are imporan and noable excepions: see Robers (997); Nunes (2)\; and Adam and Padula (2)). The raional expecaions assumpion imposes significan resricions on price-seing models wih sicky prices, as has been well-documened. In paricular, he canonical formulaions of priceseing under he Calvo (983), Roemberg (982), or Taylor (98) assumpions can imply a very flexible rae of inflaion in he sense ha inflaion can jump in response o shocks o marginal cos. Equivalenly, any persisence in observed inflaion mus be aribued o he persisence of he driving process. The daa for mos (and perhaps all) of he poswar period are difficul o reconcile wih such a model, as i appears ha inflaion behaves as if i boh inheris he persisence of he driving process and adds some of is ownso-called inrinsic inflaion persisence (Fuhrer 26, 2). John Robers work in he 99s suggesed ha i migh be how expecaions are characerized, raher han he mechanics of price-seing, ha leads o such sicky inflaion behavior. By subsiuing survey expecaions (semiannual observaions from he Philadelphia Fed s Livingson Survey), Robers found considerable success in explaining he behavior of U.S. inflaion hrough he early 99s. However, subsequen work on DSGE models largely coninued wih he raional expecaions assumpion, adoping a variey of augmenaions o he Calvo-influenced models ha allowed hem o replicae he key dynamic feaures of inflaion, such as indexaion (Chrisiano, Eichenbaum, and Evans 25), rule-of-humb pricing (Galí and Gerler 999), or serial correlaion in he shocks o inflaion (Roemberg 997; Smes and Wouers 23; Chrisiano, Eichenbaum, and Evans 25). Some more recen work revisis he role of survey expecaions as a measure of he expecaions employed by price-seers (see Nunes (2); Del Negro and Eusepi (2); and Adam and Padula (2)). These auhors arrive a somewha conradicory conclusions abou such expecaions measures, wih Nunes (2) finding lile empirical role for survey expecaions, while

4 Adam and Padula find survey expecaions o be an imporan deerminan of inflaion for U.K. daa. This paper coninues his srand of research by examining he role of survey expecaions in he inflaion process. The paper develops four principal findings. One, shor-run inflaion expecaions appear o play a significan role in explaining U.S. inflaion over he pas 2 25 years his resul is suggesed bu no fully explored in Fuhrer, Olivei and Tooell (forhcoming). Two, long-run expecaions generally do no appear o have a direc influence on U.S. inflaion over he same period, alhough hese ener indirecly as a key deerminan of he shor-run expecaions. The resricions implied by rend inflaion models of inflaion are generally rejeced in he daa. Three, when nesed in a model ha allows expecaions o be deermined by a linear combinaion of raional expecaions and survey expecaions, he daa srongly favor survey expecaions, and in general assign a small and saisically insignifican role o raional expecaions. This finding is robus across a variey of esimaion mehods. Four, as a parial explanaion of hese findings, several exercises sugges ha he raional expecaions of inflaion generaed by dynamic sochasic general equilibrium (DSGE) models wih a New Keynesian Phillips curve differ significanly from hose measured by he survey. In an imporan sense, he survey expecaions reflec an expecaions process ha is no wellcharacerized by he raional expecaions derived from simple DSGE models. Finally, he paper develops a firs pass a a srucural model ha incorporaes he feaures discussed above, and assesses is performance in explaining U.S. inflaion in he poswar period. Survey and Raional Expecaions in Reduced-Form and Srucural Models of Inflaion This paper employs quarerly daa for inflaion, oupu, unemploymen, real marginal cos (he convenional labor share measure), and survey expecaions from he Survey of Professional Forecasers (SPF). The paper uses boh shor-erm (four-quarer) and long-erm (he -year average inflaion rae) expecaions from he SPF survey. The -year expecaions are aken o be a proxy for long-erm or rend inflaion expecaions. In his paper we use he Board of Governor s esimae of he -year expecaions as implemened in heir FRB/US model. 2 a. Reduced-form models Mos of he empirical work in his area uses he SPF four-quarer inflaion expecaion as a proxy for he one-quarerahead inflaion expecaion used in sandard Phillips curves. The one-quarer-ahead SPF expecaion, which is available a he Philadelphia Fed s websie, would seem more naural. The esimaes presened below have been duplicaed using his expecaional proxy, and he resuls are essenially he same. 2 The model variable is PTR. See he documenaion a hp://fweb.rsma.frb.gov/mq/frbus/. 2

5 Ordinary leas squares (OLS) regressions can serve o moivae many of he key resuls in he paper. The regression akes he core inflaion rae (here proxied by he consumer price index excluding food and energy) as he dependen variable, and regresses i on lagged inflaion (wo quarerly lags are included), he four-quarer SPF inflaion expecaion colleced a ime, he longrun SPF inflaion expecaion daed ime, and he unemploymen gap, which uses he CBO s esimae of he NAIRU. 3 The sum of he coefficiens on lagged inflaion and he wo expecaions variables is consrained o one. The esimaion period is 982:Q o 2:Q3, and he resuls are summarized below in able. Table Reduced-Form Phillips Curve Esimaes, OLS esimaion SPF,4q year π = aπ bπ cπ du Variable Coefficien Sandard error Lagged inflaion (sum).63. SPF four-quarer expecaion year expecaion (PTR).32.2 Unemploymen gap R 2 =.72. These unconsrained resuls sugges a srong and significan correlaion beween core inflaion and he four-quarer expecaion, lile dependence on lagged inflaion, and a modes correlaion wih he -year expecaion. The unemploymen gap eners quie significanly, wih a coefficien ha is sizable by recen sandards. Bu underlying his full-sample esimae lies an array of resuls for subsamples wihin he 3- year span. As figures and 2 sugges, lagged inflaion is srongly correlaed wih core inflaion in he early par of he sample, and he four-quarer and -year expecaions measures rade off in size and significance over he sample. The p-value for he unemploymen gap lags, no shown, is ypically below.5, alhough his is no he case early in he sample. These resuls sugges some combinaion of mulicollineariy and subsample insabiliy, a leas in his reduced-form regression. 4 More progress can be made in making sense of his inflaion daa by imposing a modes amoun of srucure on i. To ha end, we esimae Phillips curve regressions ha impose wo resricions: () he sum of he coefficiens on lagged inflaion and he four-quarer survey 3 We employ he four-quarer SPF expecaion because ohers in he lieraure commonly use his as a proxy for nex period s inflaion. Resuls ha use nex quarer s expecaion yield very similar resuls. 4 The ime-varying simple correlaion beween he wo survey expecaions measures is fairly seady a.9 across mos of he sample, falling o.4 for he mos recen 5 years. 3

6 expecaion is consrained o one, as hese measures normally serve as proxies for expeced inflaion in he nex period, and (2) he long-run survey expecaion is no included in ha resricion, as we consider i more likely serves as a proxy for rend inflaion in he sense of Cogley and Sbordone (28). If ha assumpion is reasonable, hen he esimaed coefficien(s) on his long-run rend inflaion proxy should be zero. To see his, consider a rend inflaion model ha wries he Phillips curve in erms of he inflaion gap, or in deviaions of all inflaion erms from rend inflaion. Denoing he rend inflaion proxy by π, he Phillips curve as modified from able above akes he form SPF,4q ˆ π ˆ ˆ = aπ + ( a) π + du SPF,4 q SPF,4q ˆ π ; ˆ ˆ π π π π π ; π π π (). We can esimae his equaion in unconsrained form, allowing us o es he resricions implied by he rend inflaion model: SPF,4q π = aπ + ( a) π + bπ + b2π + du. (2) If he consrains are saisfied, he one should no be able o rejec he consrain b + b 2 =. 5 Figures 3 and 4 display he resuls from esimaing equaion (2) on a rolling sample of 6- quarer windows from 982 o he presen. Here we obain more consisen resuls across he sample. The four-quarer expecaion is is esimaed very precisely, wih an economically significan coefficien ha varies beween.6 and across he enire sample; lagged inflaion is esimaed wih a modes coefficien ha is marginally significan for par of he sample. Moreover, he -year expecaion varies in sign across he sample, and in some cases he hypohesis ha he sum of coefficiens is zero is no rejeced, a resul ha is consisen wih he rend inflaion model. However, one can almos never rejec he hypohesis ha boh lags are insignifican, and conribue nohing o his simple Phillips regression. Ineresingly, he p-value for he unemploymen gap rarely falls below.5, which could imply ha he four-quarer expecaion capures he dependence of inflaion on a gap variable. We will reurn o he implicaions of his resul below. The paern will play ou in ess of more srucural versions of he inflaion equaion laer in he paper. These same resuls are replicaed for a differen measure of rend inflaion (he Cogley- Sbordone filered VAR esimae), for headline CPI, and for he GDP deflaor. In addiion, he iming of he four-quarer survey variable is shifed back one quarer, in par because he appropriae 5 Rearranging he inflaion gap equaion (), one can see ha he coefficiens on π and π are, respecively, a and a. 4

7 expecaions dae ( or - ) is no uniform in he lieraure, and in par because one migh worry abou simulaneiy bias in his simple regression. 6 The resuls are qualiaively similar in mos respecs: in all cases, he four-quarer SPF expecaion eners significanly hroughou he sample; he lags ener wih a bes modes significance and size; and he long-run inflaion proxies are rarely if ever significan conribuors o he regression. One excepion is he GDP deflaor, which shows a significan effec of he -year SPF expecaion in he las 5 years or so of he sample. However, he coefficien esimaes over his par of he sample cener on, which is difficul o reconcile wih any reasonable inflaion model. b. Models wih explici raional expecaions In Fuhrer, Olivei, and Tooell (forhcoming), we es he nesed model ha allows survey expecaions, raional expecaions, and lagged inflaion o ener a New Keynesian Phillips curve, and examine he relaive conribuions each makes o inflaion dynamics for recen decades of U.S. inflaion daa. Here we replicae he resuls of ha paper, and augmen he esimaion wih wo GMM esimaors o es he robusness of he resul o mehodological differences. The es equaion is SPF,4q π π = a( π π ) + b( Eπ+ π+ ) + ( a b)( π π+ ) + cs, (3) where he inflaion variables are as defined above, s is a proxy for real marginal cos (he labor share), which is more commonly used in raional expecaions models of inflaion, he uni sum consrain is again imposed across he variables ha proxy for one-period-ahead expecaions, and E denoes he model-consisen expecaion. Following he example of he preceding secion, we also examine less-consrained versions of he equaion ha allow he rend inflaion variables o ener in an unconsrained fashion (as in he previous secion), so as o allow esing of he resricions imposed by he rend inflaion model. 7 6 In fac, he SPF surveys are colleced mid-quarer, a which poin forecasers would have a mos one monh of CPI daa (depending on he precise dae). In any even, he maximum likelihood esimaes presened laer on, which ake accoun of any simulaneiy in he daa, sugges ha his should be of lile concern in hese simple OLS esimaes. 7 A se of resuls ha use he unemploymen gap are presened in able A. in he appendix. These are qualiaively similar o he resuls ha employ real marginal cos. In all bu he simple GMM esimaor, he four-quarer expecaion plays a dominan role as he inflaion proxy. In he simple GMM case, he Sock-Yogo es for insrumen relevance suggess ha he insrumens do a poor job spanning he relevan endogenous variables. The resricions imposed by imposing rend inflaion, measured eiher by he -year expecaion or he Cogley-Sbordone measure, are eiher rejeced ourigh, or he rend variable can be excluded from he specificaion enirely wihou any damage o he specificaion. Files Phil_es_ml_U.m and Phil_es_gmm_U.m produce he parallel ses of resuls. 5

8 For he maximum likelihood esimaes of equaion 3, we include vecor auoregressive equaions ha allow us o form raional expecaions of inflaion (and rend inflaion) wihou imposing furher srucure on he model. 8 The rend inflaion variable π is assumed o follow a random walk. For he GMM esimaes, we use an insrumen se ha includes wo lags for each of he survey expecaions variables, real marginal cos, core inflaion, and he oupu gap as defined by he CBO s esimae of poenial oupu. Noe ha we use wo varians of GMM, a convenional ieraed esimaor wih a sandard weigh marix, and an opimal insrumens GMM esimaor (Fuhrer and Olivei 24) ha imposes more srucure on he insrumens ha are chosen o form he unobserved expecaions in he model. The laer is shown o have superior finie-sample properies, a characerisic shared wih he maximum likelihood esimaor and demonsraed for an invenory model applicaion in Fuhrer, Moore, and Schuh (995). Table 2, shown on he following page, summarizes he esimaion and es resuls for he hree esimaors, and includes he p-value for he es of he resricions imposed by he rend inflaion model. 9 The esimaes ell a nearly consisen sory he excepion is he convenional GMM esimae, which implies a much higher weigh on raional expecaions han he ohers, as has been found in previous papers. For his case he Sock and Yogo (25) es for weak insrumens reurns a value of.5, well below he 5 percen criical values for he es. Thus he mos likely explanaion for he discrepancy beween he simple GMM esimaes and he ohers is ha he insrumens are weak, and canno idenify boh lagged and expeced inflaion effecs on curren inflaion. A similar se of resuls are repored in Fuhrer and Olivei (24). We repor he simple GMM esimae for compleeness, bu he bulk of he evidence poins oward a prominen role for he survey expecaions, and very lile for he model-consisen or raional expecaion. 8 The influence of he long-erm expecaion proxy on four-quarer inflaion is exremely srong, whereas i eners insignificanly in he oher VAR equaions. As a consequence, we excluded he long-erm expecaion from all he reduced-form equaions excep for he four-quarer expecaion. In addiion, he four-quarer expecaion eners insignificanly in he oupu gap and marginal cos equaions, so i is excluded from hese equaions. 9 The maximum likelihood esimaes ake he esimaed coefficiens in he VAR equaions as fixed a heir OLS esimaes. This resricion is aken for convenience (o minimize he number of esimaed parameers) and does no qualiaively affec he oher esimaed parameers in he model. This finding may reconcile our resuls OLS, maximum likelihood, and opimal insrumens esimaor wih hose of Nunes e al., who find a less significan role for survey expecaions using a convenional GMM esimaor. 6

9 Table 2 Esimaion Resuls for Equaion 3, Various Esimaors, 99:Q 2:Q3 Maximum likelihood (ML) Trend inflaion = SPF -year expecaion Coefficien Esimae Sandard error a -saisic Lagged inflaion Raional expecaion. - - Survey expecaion Tes of rend inflaion resricions Tes of rend inflaion resricions:.44 Tes for exclusion of -year expecaion and RE from model:.44 Maximum likelihood Trend inflaion = Cogley-Sbordone rend inflaion measure Coefficien Esimae Sandard error a -saisic Lagged inflaion Raional expecaion. - - Survey expecaion Tes of rend inflaion resricions Tes of rend inflaion resricions:.6 Tes for exclusion of Cogley-Sbordone rend inflaion measure and RE from model:.99 Robusness check: ML esimae including 98s Trend inflaion = SPF -year expecaions Coefficien Esimae Sandard error a -saisic Lagged inflaion Raional expecaion Survey expecaion Opimal insrumens GMM Trend inflaion imposed Coefficien Esimae Sandard error a -saisic Lagged inflaion Raional expecaion Survey expecaion Convenional GMM, Trend inflaion imposed Coefficien Esimae Sandard error a -saisic Lagged inflaion Raional expecaion Survey expecaion a BHHH sandard errors Overall, he esimaes from hese mehods are broadly consisen wih he simple OLS esimaes presened above and sugges ha: The one-year expecaion plays a prominen role in deermining U.S. inflaion; The long-run expecaions do no affec inflaion direcly, eiher as consrained along he lines of a rend inflaion model, or unconsrained; 7

10 The resricions imposed by he rend inflaion model are rejeced, eiher because he coefficien resricions are direcly rejeced, or because one canno rejec he hypohesis ha eiher rend inflaion proxy should be dropped from he model; The raional expecaions erm develops a coefficien ha is insignificanly differen from zero in almos all cases. Noe ha he GMM resuls echo hose in Nunes (2), bu hese appear o be an arifac of he esimaor. An alernaive mehod of momens esimaor produces resuls ha conform more closely o he maximum likelihood and OLS esimaes. How Differen are he Survey Expecaions from Raional Expecaions? The evidence in able 2 above suggess ha survey expecaions dominae raional expecaions in explaining consumer price inflaion. This raises he quesion of wha are he differences beween he survey expecaions and model-consisen expecaions of inflaion for comparable horizons. Are he survey expecaions raional? How much do hey deviae from modelconsisen expecaions of inflaion? This secion addresses hese quesions. By now i is well-documened ha survey expecaions ofen fail ess of efficiency and unbiasedness (see Thomas (999); Mehra (22); Bachelor (986); Bryan and Gavin (986); and he references herein). A brief confirmaory exercise is summarized in able A.2 in he appendix. As he able indicaes, survey expecaions are clearly no raional in his simple sense. Ye hey ener as significan predicors in inflaion equaions. So despie heir irraionaliy, i behooves us o deermine how bes o characerize he informaion conained in survey expecaions. In a recen paper, Del Negro and Eusepi (2) examine he abiliy of a DSGE model o produce inflaion expecaions ha mach survey expecaions. Using varians of fairly sandard DSGE models and esimaing key parameers ha are well wihin he range of esimaes produced in he exan lieraure, hey find i quie difficul o mimic he behavior of survey expecaions. This secion expands on heir work by conducing wo similar exercises:. Esimaing he parameers of a simple DSGE model wih a minimum disance esimaor ha uses he disance beween he survey expecaions and he raional expecaions implied by he model as he esimaion crierion. To enable his comparison, we augmen he DSGE model wih equaions ha define he model s implied four-quarer and 4- quarer average inflaion expecaions; and 8

11 2. Using an unconsrained VAR o model he evoluion of key macro variables. The VAR is augmened jus as in he DSGE model, and we examine he disance beween he VAR s implied four-quarer and 4-quarer inflaion expecaions and he corresponding survey measures. We begin wih he more ighly consrained DSGE model ha incorporaes Calvo pricing wih price indexaion, an oupu equaion derived from he consumpion Euler equaion wih habi formaion, and a Taylor rule for he policy rae. The inflaion equaion allows for he presence of ime-varying rend inflaion as in Cogley and Sbordone (28). Real marginal cos is linked o oupu and oher model variables by a reduced-form equaion. Thus he key model equaions are π π = µπ ( π ) + ( β µ ) E ( π π ) + γs + ε y y Ey f E π + + y = µ y + ( β µ y) + σ( π+ ρ) + ε f = ω + ( ω)[ π ( π π) + y + π + ρ] + ε f f a ay π y s s = BZ + ε ; Z = f s Z = AZ + e π π + π + π π 4, e.25 E( π+ 4 i ye, D E π+ i + D i= + D π = + π δπ ε The firs exercise akes he esimaed parameers from Del Negro and Eusepi (2) ha correspond o he key parameers in he DSGE model, and simulaes he model o obain modelimplied inflaion expecaions. These are compared wih he survey expecaions for he four-quarer and -year average measures from he SPF. The resuls are displayed in figure 5. As he figure suggess, heir esimaed parameers achieve only modes success in replicaing he shor- and longerm inflaion expecaions measured by he surveys. The baseline parameers are summarized in Table 3. Table 3 Parameer Seings for Figures 5 and 6 Parameer Baseline Alernaive µ.22.7 γ.3. ) (4) 9

12 a π D 4 The nex exercise alers some of he key parameers in he DSGE model (he Alernaive column in able 3). Figure 6 displays he model s implied values for shor- and long-erm inflaion expecaions for a variey of parameer seings. Raising he degree of indexaion in he model improves he coherence beween he model-implied one-year expecaions and he survey measure in he early par of he sample, bu no in he laer half. There is no parameer combinaion considered ha leads o an implied -year expecaion ha closely maches he survey daa. Recall ha a ime-varying rend inflaion measure, which declines significanly in he firs half of he sample, is incorporaed ino he model. Ineresingly, shorening he duraion of he -year modelimplied expeced inflaion measure o only en quarers, which is equivalen o fron-weighing nearer-erm expeced inflaion in his measure, achieves he bes (alhough sill only parial) success in maching he decline in -year survey expecaions in he 98s, bu i is only parial. Because he exercise above suggess ha some parameer variaions can improve he model s abiliy o mach he survey daa, we exend his exercise by esimaing he model s parameers using a disance crierion ha minimizes he difference beween model-implied inflaion expecaions and survey expecaions daa: 4 4, S 2 y ys, 2 ( + i + i) ( ) ( + i + i ). (5) C = ω Eπ π + ω Eπ π We begin by seing he weighs [ ω, ( ω) ] on he four-quarer and -year expecaions o onehalf, respecively. Preliminary esimaion aemps sugges ha his esimaion crierion does no conain enough informaion (in his sample) o esimae all eigh of he key parameers. As a consequence, we consrain γ o.7 and σ o.9. Alernaive values for hese parameers can significanly worsen he esimaion crierion, bu here appears lile value in opimizing furher wih respec o hese wo parameers. The resuls of his exercise are summarized in able 4 and figure 7. The Hessian for his esimaion problem has rank six, and a singular value decomposiion of he Hessian suggess a srong dependence among γ, a and σ. π,

13 Table 4 Esimaes of DSGE Model Parameers Using he Minimum-Disance Crierion Parameer Opimized value Sandard error Opimized value, 99-2 Sandard error Memo: Del Negro and Eusepi parameers, baseline parameers μ γ ω a π a y μ y σ D The good news in hese esimaes is ha he fi of he model-implied inflaion expecaions, shown in figure 7, is significanly improved over he esimaes from Del Negro and Eusepi The bad news is ha he parameers required o achieve his fi, summarized in able 4 above, while fairly precisely esimaed, are in some cases difficul o reconcile wih he received wisdom regarding he srucure of he macroeconomy. For example, he smoohing parameer for he policy rule, ω, is esimaed a.3, as compared o common esimaes in he range of.7 and above. The weigh on he habi parameer in he oupu equaion, μ y, is implausibly large a.99 (he upper limi of he feasible parameer se). The duraion of he long-run inflaion expecaion is under four years, which is quie shor for a model variable ha is designed o mach up wih he en-year average inflaion expecaion derived from he SPF survey. As a firs check on hese resuls, we re-esimae he parameers from 99 o he presen. As he second se of columns in able 4 suggess, his sample change suggess some imporan changes in some parameers, and hese cener srongly on he moneary policy rule. The degree of ineresrae smoohing and he relaive emphasis on inflaion versus oupu shif significanly in his more recen sample. This re-esimaion suggess ha aking accoun of shifs in he sysemaic componen of moneary policy will be imporan in undersanding he behavior of he survey. As a second check on his exercise, we ake he esimaed parameers in able 4 and use hese o simulae acual inflaion over he sample period. Figure 8 compares simulaed inflaion from his

14 exercise o he simulaed values for inflaion a he baseline parameers. As he figure indicaes, he parameers required o mach he model s expeced inflaion series wih he survey expecaions imply a vasly differen rajecory of inflaion han he baseline parameers, which rack inflaion reasonably well. VAR-based expecaions of inflaion and survey expecaions The nex exercise is a less consrained version of he previous one we ake an unconsrained esimaed VAR, append equaions ha define four-quarer and -year (duraion) inflaion expecaions, and compue he VAR-implied inflaion expecaions. The VAR includes inflaion, expressed as he deviaion of he core inflaion measure used above from Cogley and Sbordone s rend inflaion measure, he oupu gap, he federal funds rae, real marginal cos, and he log change in he relaive price of oil. The duraion of he long-run inflaion expecaion is se o years. Figure 9 displays he resuls for he sample As he figure indicaes, he abiliy of he unconsrained model o replicae he shorer-erm inflaion expecaions is no grea, bu beer han he DSGE model a is baseline parameer values, and nearly as good as he DSGE model wih parameers opimized o fi he expecaions daa. Unlike he shor-run expecaions, he fi of he long-run expecaionsis poor. Alering he duraion of he (implied) long-run expecaion o he value of 5 found in he DSGE opimizaion above improves he fi somewha bu no dramaically (no shown). Taking a lesson on he imporance of ime-variaion in he policy rule in he DSGE exercise, we re-esimae he VAR over he pos-989 period, hopefully conrolling for he possibiliy of a shif in underlying moneary policy behavior. This exercise, no shown, resuls in a bes a very modes improvemen in he fi o he long-run survey expecaion. We draw some enaive conclusions from hese exercises in maching consrained and unconsrained model-consisen inflaion expecaions o he survey measures:. One can compue a model-based inflaion expecaion ha is fairly close o he fourquarer survey expecaion. However, o do so, one needs o dramaically aler he parameers of a sandard srucural model in ways ha cause i o behave quie poorly wih respec o he oher variables in he model; 2. One can achieve reasonably close fi using an unconsrained VAR, a leas for he shorrun expecaion. This suggess ha, wih regard o he shor-run expecaion, he 2

15 difficuly in replicaing he survey daa lies in he resricions imposed by he srucural model. 3. Wih regard o long-run expecaions, i is difficul o closely mach he long-run survey expecaions wih eiher consrained or unconsrained models. This is rue even when allowing for a ime-varying inflaion rend ha is aken as exogenous o he model. This difficuly represens a significan unresolved challenge. Overall, he resuls sugges ha we have some way o go before we can model he kind of expecaions ha appear empirically relevan for he deerminaion of inflaion. The paper moves parly oward his direcion in he nex and final secion. Endogenizing Shor-Term Expecaions We now rever o he somewha reduced-form modeling echniques used in he earlier secions of his paper o invesigae he key deerminans of he shor-erm expecaions ha we have found o be of value in explaining inflaion. As a saring poin, consider he reduced-form regression linking he four-quarer expecaions, he -year expecaions, lagged inflaion, and he unemploymen gap. For noaional convenience, he erm π represens he sum of he four quarerly lags included in he regression, he erm represens he sum of he wo quarerly lags 4,e of he unemploymen gap included in he regression, and π represens he curren and single lag of he -year (4-quarer) SPF inflaion expecaion. We include he conemporaneous and lagged values of he -year SPF expecaion in order o compare hese wih he nex specificaion, which imposes ha he four-quarer expecaion is properly hough of as a deviaion from his proxy for he rend in inflaion. The consrained version of he equaion may be wrien as 4, e 4, e π = aπ + ( a) π cu (6) 4, e 4, e 4, e π = π + a( π π ) cu. In all cases, he sum of he esimaed coefficiens is significanly differen from zero; obviously he null hypohesis es ha all he coefficiens are joinly zero is rejeced wih overwhelming confidence. In able 5 he second panel considers he same regression wih he consrain ha he -year ener as a rend variable; ha is, ha he coefficiens on lagged and - year expeced inflaion sum o one. While his consrain booss he conribuion of he unemploymen gap erm, he uni sum consrain is rejeced overwhelmingly. U 3

16 Table 5 Esimaes of reduced-form equaion for four-quarer SPF expecaions 4, e 4, e π = aπ + bπ cu. Unconsrained Coefficien p-value Lagged inflaion (4 quarerly lags).4. -year SPF expecaion (PTR).4. Unemploymen gap Consrained (deviaions form, a+b=) Lagged inflaion (4 quarerly lags).42. -year SPF expecaion (PTR).58. Unemploymen gap -.4. p-value for uni sum consrain:. Like he reduced-form Phillips curves esimaed above, here is evidence of significan ime variaion in hese simple relaionships. Figures 3 summarize he evidence for he unconsrained and consrained versions of he regressions, using a rolling esimaion window of 6 quarers. These figures sugges a less idy picure for he four-quarer expecaions han was obained for he Phillips curves. While all hree variables generally conribue o he fi of he shor-run expecaion, lagged inflaion loses is significance oward he middle of he sample, and he uni sum consrain is ofen rejeced. The unconsrained regressions show why he es ends o rejec: he sum of he lags and he -year expecaion fall significanly shor of one in he firs par of he sample, and significanly exceed one in he laer hird of he sample. Ineresingly, he unemploymen gap is significan hroughou, in conras o he Phillips curve regressions above. This suggess ha he four-quarer expecaion influences inflaion in par because i capures he dependence of inflaion on he unemploymen gap, an influence ha was clear in he unconsrained Phillips curves presened above. A more srucural model of inflaion wih survey expecaions Using he evidence gahered so far, we consruc a quasi-srucural model of inflaion wih he following elemens. We begin wih he survey-based New Keynesian Phillips curve, allowing for he presence of indexaion, SPF,4 π = aπ + ( a) π bu (7) π = aπ + ( a) S π bu, + where we have re-wrien he equaion in he second line o inroduce a survey expecaions operaor. Solving equaion (7) for he unobserved expecaion in period + is sraighforward under 4

17 raional expecaions. Endogenizing he survey expecaions requires addiional assumpions abou how he survey expecaions are formed. We will assume ha, while clearly differen from he raional expecaions operaor, he S operaor mainains he following properies:. I can be ieraed forward consisenly; 2. I exhibis he simple propery ha he sum of he -period expecaion of and he previous periods variables equals he realizaion of hese variables. 3. In addiion, we will assume ha as he expecaion s ime horizon increases, he expecaions formed by he S operaor converge o hose formed by he E operaor. In he weak form, his assumes ha survey expecaions properly esimae he long-run equilibrium value of he variable. In he somewha sronger form, his assumes ha survey expecaions capure he laer sages of he reversion o equilibrium in he same way as raional expecaions. As a pracical maer, his assumpion is also moivaed by he limiaions of he survey daase: forecass are available for he curren and nex four quarers, and for he curren and nex year. 2 Taking hese properies on board, one can hen wrie he equaion for he survey-based expecaion of nex period s inflaion as Sπ+ = asπ b ωisu + i i=. (8) This expression is convenien, as we observe he SPF survey of fuure unemploymen raes for four quarers beyond he survey dae, and for he curren year and he nex year. If in addiion we assume ha S π = π, we can hen approximae (8) by adding he raional expecaion of unemploymen gaps beyond he observable survey horizon: 8 Sπ = aπ b ωsu b δ EU + i + i i + i i= i= 9 Equaion (9) assumes ha he raional expecaion of he unemploymen gap for he horizon beyond he nex several quarers is a reasonable proxy for he survey expecaion. In many cases, his assumpion will be reasonable, as he unemploymen gap will be expeced o reurn o is equilibrium (zero) a his horizon. For many iniial condiions, his approximaion simply assumes ha he survey expecaions ge he mean of he forecased variable righ. For cases in which he. (9) 2 In he las several years of he SPF daase, he wo-year-ahead forecas is published, bu due o is very shor span of availabiliy, we canno use i for his paper. 5

18 unemploymen gap has been more dramaically displaced from is long-run value, we assume ha he survey and raional expecaions converge a hese longer horizons. This approximaion is implemened by adding he auxiliary equaion ha defines he model s expecaions of unemploymen pas he survey horizon as U + = δu 9 ( δ) EU , () and equaion (9) becomes 8 + π+ = π ωi + i i= S a b S U bu. () These assumpions do no compleely close he model, as i has no ye been specified how o deermine unemploymen, a ask ha is required o form he model-consisen forecass of unemploymen ha ener U +. For his paper, we ake an agnosic view on he deerminaion of unemploymen, modeling i wih a reduced-form VAR equaion in hree observables (lags of he funds rae, inflaion, and unemploymen). The SPF unemploymen gap forecass are assumed o rever oward zero a a rae ha is deermined by a second-order auoregression. The empirical implemenaion of equaion () uses quarerly values of he curren-year and nex-year forecas of he unemploymen rae, from which he CBO s esimae of he NAIRU for he corresponding year is subraced o form an oupu gap. In addiion, in recogniion of he imporan role played by rend inflaion in he preliminary regressions repored in figures -3, we express he inflaion variables as deviaions from rend inflaion as Y + π+ = π+ ( ) π ωi + i i= S a a b S U cu. Preliminary esimaes sugges ha i may be difficul o idenify he ω i independenly, so we condense he equaion o Y Y + Sπ+ = aπ + ( a) π b( SU + SU + ) + cu. (3) Finally, allowing for he possibiliy ha survey expecaions are formed in a way consisen wih a more explici hybrid model of inflaion, we inroduce wo lags of he unemploymen gap ino he expecaion equaion, viz Y Y + Sπ+ = aπ + ( a) π bl( U + U 2) b( SU + SU + ) + cu. (3 ) (2) 6

19 For reference, his equaion s OLS esimae for he sample 982 2, excluding he modelconsisen expecaions of he unemploymen gap, yields (HAC sandard errors in parenheses) 3 S π =.22π +.78π.33U +.7U Y Y + + (.49) (.49) (.) (.) The model comprises equaions (7) and (3 ), he definiion of. U + from equaion (), and VAR equaions o forecas he unemploymen gap. 4 The sample begins in 982:Q, he firs quarer for which he expecaions variables (accouning for lags) are available. We employ a 6-quarer esimaion window, alhough he resuls are no paricularly sensiive o he window size. Figure 4 presens rolling-sample ML esimaes of he key equaions (7) and (3), while figure 5 displays he p-values for he Wald ess of he null hypoheses ha a model parameer (or wo parameers joinly) are zero. 5 The es uses he BHHH esimae of he inverse of he covariance marix of he parameers, which we denoe Ω. Denoing he esimaed and consrained parameer vecors by ˆβ and C β, he es W is ˆ C ( ) ( ˆ C W = β β Ω β β ). The ML esimaes for he survey expecaions Phillips curve are reasonably sable across ime. As in all he resuls presened above, he esimaed coefficien on he one-year SPF expecaion varies beween.5 and, wih an average value of abou.7. This implies a modes coefficien of abou one-hird for lagged inflaion in he Phillips curve. As indicaed in figure 5, he p-value for he Wald es ha he lagged inflaion coefficien is zero is rejeced overwhelmingly hroughou he sample. The unemploymen gap, he red line in figures 4 and 5, generally eners wih he correc sign and quie significanly, excep for he early par of he sample. The average coefficien across all rolling samples is -.2. Turning o he equaion ha explains he shor-erm inflaion expecaion, figure 4 shows ha curren inflaion and he rend inflaion proxy ener wih roughly equal weighs, alhough he coefficiens flucuae somewha over he sample (recall ha hese are consrained o sum o one). The coefficien is esimaed precisely, as indicaed by he middle panel in figure 5. The influence of unemploymen, he hird panel in figure 4, is generally significan and negaive, consisen wih he 3 The uni sum consrain is rejeced over he full sample, as i is in he preliminary regressions above. 4 These equaions include wo quarerly lags of he unemploymen gap, he core inflaion rae, and he federal funds rae. 5 The parameers of he VAR equaion and he second-order auoregressions are also esimaed for each sample in he rolling window, joinly wih he oher srucural parameers in he Phillips curve and he inflaion expecaions equaion. 7

20 basic logic of he Phillips Curve. Summing he SPF forecass and he model-consisen forecass, he average unemploymen effec across all esimaes is However, wo caveas are worh noing : Firs, he overall effec of unemploymen (he sum of he SPF unemploymen gap forecass and he U + erm) urns posiive in several of he samples oward he end of he esimaion period, which is difficul o inerpre. Second, he influence of he SPF unemploymen forecass (no shown) varies in sign and significance over he esimaion samples. This could reflec he presence of speed limi (change in expeced unemploymen) effecs on inflaion expecaions, bu i also suggess ha some cauion should be exercised in inerpreing he resuls. Conclusions As suggesed in he inroducion, hese resuls suppor a renewed emphasis on using nonraional expecaions measures in inflaion modeling. The reduced-form and moderaelyconsrained esimaes sugges a srong role for survey expecaions, paricularly he one-year (alhough hese resuls are replicaed for one-quarer-ahead expecaions as well). The esimaed conribuion of a srucural model s raional expecaion is almos always dominaed by he conribuion from survey expecaions. The resricions implied by he rend inflaion model are almos uniformly rejeced. The aemp o ariculae a model ha endogenizes he survey expecaions provides some ineresing resuls. While long-run expecaions may no ener he Phillips curve direcly, hese sill serve as an anchor for he shor-run survey expecaions in all he esimaes presened. A more careful modeling of he survey expecaions ha imposes more srucural expecaional assumpions is parly successful, as i finds some role for expeced unemploymen gaps, and in a manner consisen wih convenional heories. However, in his regard he empirical resuls are no uniform across all samples. The sign of he unemploymen gap effec someimes flips o posiive, and he survey s unemploymen forecass do no always ener significanly. While he resuls presened in his paper are neiher conclusive nor free of problems, hey do sugges ha he use of survey expecaions may presen an imporan direcion for inflaion and DSGE modelers. While he simpliciy of working wih raional expecaions is sacrificed o some exen, i is possible o use survey daa on inflaion expecaions while mainaining a reasonable blend of heoreical and empirical rigor. 8

21 References. Adam, Klaus, and Mario Padula. 2. Inflaion Dynamics and Subjecive Expecaions in he Unied Saes. Economic Inquiry 49(): Bachelor, Roy A Quaniaive v. Qualiaive Measures of Inflaion Expecaions. Oxford Bullein of Economics and Saisics 48(2):99 2. Bryan, Michael F., and William T. Gavin Models of Inflaion Expecaions Formaion: A Comparison of Household and Economis Forecass. Journal of Money, Credi and Banking 8(4): Calvo Guillermo Saggered Prices in a Uiliy-Maximizing Framework. Journal of Moneary Economics 2(3): Chrisiano Lawrence, Marin Eichenbaum, and Charles Evans. 25. Nominal Rigidiies and he Dynamic Effecs of a Shock o Moneary Policy. Journal of Poliical Economy 3(): 45. Cogley, Timohy, and Argia M. Sbordone. 28. Trend Inflaion, Indexaion, and Inflaion Persisence in he New Keynesian Phillips Curve. American Economic Review 98(5): Del Negro, Marco, and Sefano Eusepi. 2. Fiing Observed Inflaion Expecaions. Federal Reserve Bank of New York Saff Repor No New York: Federal Reserve Bank of New York. Fuhrer, Jeffrey. 26. Inrinsic and Inheried Inflaion Persisence. Inernaional Journal of Cenral Banking 2(3): Fuhrer, Jeffrey. 2. Inflaion Persisence. In Handbook of Moneary Economics, Vol. 3A, ed. Benjamin M. Friedman and Michael Woodford, Amserdam: Norh-Holland. Fuhrer, Jeffrey C., George R. Moore, and Sco D. Schuh Esimaing he Linear-Quadraic Invenory Model: Maximum Likelihood versus Generalized Mehod of Momens. Journal of Moneary Economics 35(): Fuhrer, J. and G. Olivei. 24. Esimaing Forward-Looking Euler Equaions wih GMM and Maximum Likelihood Esimaors: An Opimal Insrumens Approach. In Models and Moneary Policy: Research in he Tradiion of Dale Henderson, Richard Porer, and Peer Tinsley, ed, Jon Faus, Ahanasios Orphanides, and David Reifschneider. Washingon, DC: Board of Governors of he Federal Reserve Sysem. Available a hp:// Olivei.pdf. Fuhrer, Jeffrey, Giovanni Oliviei, and Geoffrey Tooell. Forhcoming. Inflaion Dynamics When Inflaion is Near Zero. Journal of Money, Credi and Banking. Galí, Jordi, and Mark Gerler Inflaion Dynamics: A Srucural Economeric Analysis. Journal of Moneary Economics 44(2):

22 Mehra, Yash P. 22. Survey Measures of Expeced Inflaion: Revisiing he Issues of Predicive Conen and Raionaliy. Federal Reserve Bank of Richmond Economic Quarerly 88(3): Nunes, R. 2. Inflaion Dynamics: The Role of Expecaions, Journal of Money, Credi and Banking.42(6): Robers, John M Is Inflaion Sicky? Journal of Moneary Economics 39(2): Roemberg, Julio J Aggregae Consequences of Fixed Coss of Price Adjusmen. American Economic Review 73(3): ( Roemberg, Julio J Towards a Compac, Empirically-Verified Raional Expecaions Model for Moneary Policy Analysis: A Commen. Carnegie-Rocheser Conference Series on Public Policy 47: Smes, Frank, and Raf Wouers. 23. An Esimaed Dynamic Sochasic General Equilibrium Model of he Euro Area. Journal of he European Economic Associaion(5): Sock, James H., and Moohiro Yogo 25. Tesing for Weak Insrumens in Linear IV Regression. In Idenificaion and Inference for Economeric Models, ed. Donald W. K. Andrews, 8 8. New York: Cambridge Universiy Press. Taylor, John B. 98. Aggregae Dynamics and Saggered Conracs. Journal of Poliical Economy 88(): 23. Thomas, Lloyd B. Jr Survey Measures of Expeced U.S. Inflaion. Journal of Economic Perspecives 3(4):

23 Figure.4 Coefficiens.2 Lags(sum) 4-qr. -year (sum) :Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period Sources: Auhor s calculaions Figure 2 p-values Lags(sum) 4-qr. -year (sum) Unemp. gap :Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period Sources: Auhor s calculaions 2

24 Figure 3.2 Coefficiens Lags SPF 4-qr. -year :Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period Sources: Auhor s calculaions Figure 4 p-values Lags SPF 4-qr. -year Unemp. gap :Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period Sources: Auhor s calculaions 22

25 Figure quarer Survey Baseline year Year Sources: Survey of Professional Forecasers (4-quarer expecaions), Board of Governors of he Federal Reserve Sysem, FRB/US model (-year expecaions), auhor s calculaions 23

26 Figure quarer year Survey Baseline µ=.7 D= Year Sources: Survey of Professional Forecasers (4-quarer expecaions), Board of Governors of he Federal Reserve Sysem, FRB/US model (-year expecaions) auhor s calculaions 24

27 Figure 7 Fied values of survey variables from minimum-disance esimaion 8 6 SPF 4-qr. Fied SPF -year Fied Year Sources: Survey of Professional Forecasers (4-quarer expecaions), Board of Governors of he Federal Reserve Sysem, FRB/US model (PTR)auhor s calculaions 25

28 Figure 8 Simulaed inflaion values a baseline and opimized parameers 2 Acual inflaion Opimized parameers Baseline parameers Year Sources: Bureau of Labor Saisics (acual inflaion), auhor s calculaions 26

29 Figure 9 VAR-implied inflaion expecaions 7 6 SPF 4q VAR-implied PTR VAR-implied Year Sources: Survey of Professional Forecasers (4-quarer expecaions), Board of Governors of he Federal Reserve Sysem, FRB/US model (PTR), auhor s calculaions 27

30 Figure.4.2 Lagged inflaion -year ugap Sum of lagged π, -year Unconsrained regressions Coefficiens :Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period Sources: Auhor s calculaions Figure.25.2 p-values Lagged inflaion -year ugap :Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period Sources: Auhor s calculaions 28

31 Figure 2.2 Consrained regressions Coefficiens Lagged inflaion -year ugap sum of lags and -year :Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period Sources: Auhor s calculaions Figure p-values Lagged inflaion -year ugap p-value of uni sum consrain Sources: Auhor s calculaions 29

32 Figure 4 Rolling ML esimaes, Window = 6 Phillips Curve.5 SPF 4-qr. Unemp. gap :Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period S π + equaion π π -bar.5 82:Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period S π + equaion Unemploymen gap effec (SPF + U + ) :Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period Sources: Auhor s calculaions 3

33 Figure 5 p-values for Wald ess of parameer resricions Tes for null ha indicaed parameer(s) = p-values, Phillips Curve parameers SPF 4-qr. Unemp. gap 82:Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period p-values, S π + equaion.8 π :Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period p-values, S π + equaion Join U effecs 82:Q 84:Q3 87:Q 89:Q3 92:Q 94:Q3 97:Q Saring dae of esimaion period Sources: Auhor s calculaions Appendix Maximum likelihood esimaes for Table 2, using he unemploymen rae Table A. 3