OBSERVED EXPECTATIONS, NEWS SHOCKS, AND THE BUSINESS CYCLE

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1 OBSERVED EXPECTATIONS, NEWS SHOCKS, AND THE BUSINESS CYCLE FABIO MILANI & ASHISH RAJBHANDARI Universiy of California, Irvine Absrac. This paper explois informaion from he erm srucure of survey expecaions o idenify news shocks in a a DSGE model wih raional expecaions. We esimae a srucural business-cycle model wih price and wage sickiness. We allow for boh unanicipaed and anicipaed componens ( news ) in each srucural disurbance: neural and invesmen-specific echnology shocks, governmen spending shocks, risk premium, price and wage markup shocks, and moneary policy shocks. We show ha he esimaion of a sandard DSGE model wih realized daa obfuscaes he idenificaion of news shocks and yields weakly or non-idenified parameers peraining o such shocks. The idenificaion of news shocks grealy improves when we re-esimae he model using daa on observed expecaions regarding fuure oupu, consumpion, invesmen, governmen spending, inflaion, and ineres raes - a horizons ranging from one-period o five-periods ahead. The news series hus obained largely differ from heir counerpars ha are esimaed using only daa on realized variables. Moreover, he resuls sugges ha he idenified news shocks explain a sizable porion of aggregae flucuaions. News abou invesmen-specific echnology and risk premium shocks play he larges role, followed by news abou labor supply (wage markup) and moneary policy. Keywords: News Shocks, Esimaion of DSGE Model wih Survey Expecaions, News in Business Cycles, Idenificaion in DSGE Models, Raional Expecaions. JEL classificaion: E32, E5. Dae: Ocober, 212. Address for correspondence: Fabio Milani: Deparmen of Economics, 3151 Social Science Plaza, Universiy of California, Irvine, CA Phone: Fax: fmilani@uci.edu. URL: hp:// fmilani. Ashish Rajbhandari: Deparmen of Economics, 3151 Social Science Plaza, Universiy of California, Irvine, CA Phone: Fax: arajbhan@uci.edu.

2 OBSERVED EXPECTATIONS, NEWS SHOCKS, AND THE BUSINESS CYCLE 1 1. Inroducion The key role of expecaions in driving or amplifying aggregae economic flucuaions was recognized a long ime ago. Pigou (1927) poined o excesses of opimism and pessimism by businessmen as causes of flucuaions in economic aciviy. Keynes (1936) aribued a large porion of flucuaions o he acion of invesors animal spiris. A renowned survey of business cycle heories wrien in he 193s by Haberler (1937) also assigned a cenral role o expecaions, including discussions of how expecaions may represen sources of shocks o he economy. Wih he raional expecaions revoluion in he 197s, however, he funcion of expecaions in macroeconomic models has changed. Expecaions sill remain key in he propagaion of macroeconomic shocks. Bu under he assumpion of raional expecaions, expecaions generally no longer consiue auonomous sources of flucuaions. 1 Expecaional errors can be expressed as unique funcions of srucural innovaions. The majoriy of macroeconomic models wih raional expecaions, herefore, absracs from expecaion shocks ha canno be explicily reconduced o fundamenals. The mos popular conemporaneous heories of he business cycle imply ha flucuaions are driven by unanicipaed fundamenal shocks, mos ofen o echnology (Hicks-neural or invesmen-specific) or o demand condiions (such as preference shocks ha affec consumers s uiliy, exogenous shifs in governmen spending, and so forh). Theories of expecaions-driven business cycles, however, have araced much renewed aenion recenly. On he heoreical side, Beaudry and Porier (26) and Jaimovich and Rebelo (29) presen models in which news abou fuure echnology shocks is a primary source of business cycle flucuaions, leading o comovemen in oupu, consumpion, invesmen, and labor hours. These heories imply ha news abou he fuure is able o generae realisic boom-bus cycles even if no change in echnology maerializes ex-pos. 2 Recenly, he ineres has urned oward evaluaing empirically heories based on news and quanifying he conribuion of news o aggregae flucuaions. Beaudry and Porier (26) are he firs o provide favorable empirical evidence in he conex of srucural VARs. They show ha a shock ha doesn affec echnology in he shor-run, bu ha is correlaed wih echnology in he long-run, accouns for a large share of flucuaions. Given is properies, he shock can be inerpreed as reflecing news abou fuure echnology. Beaudry and Lucke (21) find similar evidence using more comprehensive VAR and VECM specificaions, including a variey of idenified shocks. 1 An excepion is he lieraure on sunspos, equilibrium indeerminacy, and animal spiris, in raional expecaions models (e.g., Benhabib and Farmer, 1999). In such cases, expecaional errors depend no only on fundamenal innovaions, bu also on sunspos shocks, which are unrelaed o fundamenals. Sunspo shocks can induce flucuaions and increase volailiy in such models. 2 Lorenzoni (211) presens a review of he mechanisms a work in microfounded business cycle models wih news.

3 2 FABIO MILANI & ASHISH RAJBHANDARI Anoher sraegy o invesigae he imporance of news consiss of uilizing fully-fledged srucural models as opposed o aheoreical VARs. Schmi-Grohé and Uribe (212) esimae a DSGE model wih flexible prices, which incorporaes news abou fuure neural and invesmen-specific echnology, preference, governmen spending, and wage mark-up shocks, and conclude ha news accouns for roughly half of oupu movemens. Oher papers, however, follow similar sraegies o esimae DSGE models ha are exended o include sicky prices, sicky wages, and a larger menu of srucural disurbances (e.g., Fujiwara e al., 211, Khan and Tsoukalas, 212), bu find only a modes role for news. The wide range of resuls is no necessarily surprising. The idenificaion of wha should be defined as news from macroeconomic daa is complicaed. The srucural shocks ha ener business cycle models are already unobserved o he economerician. When news is added, boh he unanicipaed and he anicipaed (he news) componens in he srucural shocks are reaed as unobserved and need o be inferred from a ypically limied se of macroeconomic ime series. The separaion of he wo componens ress on he propery ha news affecs fuure expecaions of he srucural shocks, which in urn affec consumpion, invesmen, price seing, and oher opimizing decisions, while unanicipaed componens do no influence fuure forecass. Empirical papers on news, however, ypically do no have available or do no employ informaion on privae secor s anicipaions. VAR sudies use sock prices as a proxy forward-looking variable ha is mean o capure news abou fuure echnology. Oher forward-looking variables have also been used (e.g., consumer confidence, slope of he erm srucure) wih mixed conclusions. DSGE models, insead, have lagged behind in he use of similar forward-looking variables (wih sock prices being a parial excepion, since hey are occasionally used in robusness check exercises as an addiional observable). Paper s Conribuion. This paper aims o advance he empirical lieraure on he imporance of news in business cycles by exploiing he exensive, bu underused, informaion conained on he available observed expecaions daa. We exploi he erm srucure of expecaions, obained from he Survey of Professional Forecasers, in he esimaion of a DSGE model, while reaining he convenional assumpion of raional expecaions. Observed expecaions provide addiional key informaion ha can consrain he compuaion of raional expecaions hrough addiional measuremen equaions ha are appended o he model, and ha can help he economerician disenangle unanicipaed shocks and news over he business cycle. We esimae a popular DSGE model wih sicky prices and wages, based on Smes and Wouers (27), using full-informaion Bayesian mehods. We exploi expecaions a he one, wo, hree, four, and five-quarer-ahead horizons on oupu, consumpion, invesmen, governmen spending,

4 OBSERVED EXPECTATIONS, NEWS SHOCKS, AND THE BUSINESS CYCLE 3 inflaion, and ineres raes, o inform he exracion of news shocks. Given our focus on he idenificaion of news over he sample, we find i worhwhile using real-ime daa for our macroeconomic series of ineres in he esimaion. We show, however, ha he conclusions are robus o he use of revised, curren-vinage, daa series. In erms of mehodological choices, we believe ha an advanage of our approach is ha i can fully reain he assumpion of raional expecaions, ye i forces expecaions o be consisen wih he available observed expecaion series. Even under he assumpion of raional expecaions, expecaions-driven business cycles may arise here because of he exisence of news. News abou fuure shocks, and subsequen revisions in hose news, can consiue a source of aggregae flucuaions and creae addiional volailiy in he economic sysem. In addiion, he use of a srucural heory-based model, raher han a VAR, is moivaed, among oher hings, by he well-known inveribiliy problem ha affecs VARs when anicipaions are presen (e.g., Leeper and Walker, 28). Leeper and Walker discuss how he differen informaion ses available o he agens in he economy and o he economerician esimaing he VAR, which exis when anicipaions are an imporan componen of he daa, preven economericians from correcly idenifying he srucural shocks, and consequenly lead o misleading impulse responses and variance decomposiion shares. In our empirical analysis, we compare he news shocks and heir imporance for business cycles wih hose esimaed wihou using any informaion from expecaions. We also re-esimae he model wihou news and wih revised, raher han real-ime, daa o check he conribuion of each modeling and esimaion elemen o he final resuls. When he model is esimaed omiing daa on expecaions, i is unclear wheher news shocks acually play a major role in he economy. Firs, he poserior means of he sandard deviaions of news shocks move closer o zero if compared wih he corresponding prior means. The vas majoriy of he 95% credible ses for he news parameers conain he value of zero, which would indicae ha he specific news is empirically unimporan. The main finding, however, is ha, when expecaions daa are no used in he esimaion, several parameers relaed o news shocks are very weakly idenified or non-idenified. In many cases, he priors are no really updaed, as he poserior disribuions for he news sandard deviaions overlap wih he priors, or, if no overlapping, he wo disribuions closely resemble each oher. When he model is re-esimaed exploiing daa on observed expecaions, he idenificaion of news subsanially improves. The poserior disribuions for he news coefficiens now ypically fall furher from he priors, and become narrower around heir means. Moreover, he daa ofen

5 4 FABIO MILANI & ASHISH RAJBHANDARI sugges values for he sandard deviaions of news ha are significanly higher han prior means; in mos cases, he credible ses are in sricly posiive range. In he baseline esimaion, he empirical resuls indicae(unanicipaed) invesmen-specific echnology shocks as he main drivers of business cycles, a finding ha is in line wih recen evidence by Jusiniano, Primiceri, and Tambaloi (211), among ohers. Such shocks explain beween 3 and 4% of real GDP growh (forecas error) variance. Bu news shocks are also imporan: he fracion of aggregae economic flucuaions ha can be aribued o news also falls beween 3 and 4%. News abou he invesmen-specific echnology shock a shor-erm horizons accouns for he larges share; shor-erm news abou moneary policy and longer-horizon news abou he risk-premium and wage markup shocks also have nonrivial roles. The inclusion of expecaions and news in he esimaion also leads o changes in he poserior esimaes for coefficiens ha are unrelaed o news. The degree of real fricions, such as habi formaion in consumpion and invesmen adjusmen coss, subsanially falls. The degree of nominal fricions, such as rigidiy in wages and prices, and indexaion o pas inflaion, are also reduced. Therefore, he evidence suggess ha news and subjecive expecaions work o creae persisence in he sysem, so ha he role of some popular fricions is diminished. Relaed Lieraure. The paper mainly aims o add o he emerging lieraure focused on esing he empirical imporance of news over he business cycle. While he previously-discussed resuls by Beaudry and Porier (26) and Beaudry and Lucke (29) sugges a major role for news in VAR models, ohers (e.g., Forni e al., 211, using a facor-augmened VAR) disagree. Theoreical work and he early empirical papers have mosly focused on news abou echnology. Schmi-Grohé and Uribe (212) esimae a RBC-ype model and allow for news in a wider range of disurbances. Fujiwara e al. (211), Khan and Tsoukalas (212), esimae DSGE models wih New Keynesian feaures similar o he one we use here. Again, here is conrasing evidence. Schmi- Grohé and Uribe (212) uncover a significan role of news over he business cycle. Fujiwara e al. (211) and Khan and Tsoukalas (212), on he oher hand, find only limied conribuions. Milani and Treadwell (212) consider news regarding fuure moneary policy choices, possibly indicaing cenral bank announcemens or simply privae secor s aemps a anicipaions, and show ha anicipaed moneary policy innovaions play a larger role over he business cycle han moneary policy surprises. Rajbhandari (212) esimaes an open economy model wih news and finds a limied ransmission of news across counries, while news plays an imporan role domesically. Wihin he lieraure on esimaed DSGE models wih news, his paper has also poins of conac wih Adrjiev (211), who suggess using sock prices in he esimaion of DSGE models wih news o beer capure forward-looking informaion. Our paper differs, because we use an exensive se of

6 OBSERVED EXPECTATIONS, NEWS SHOCKS, AND THE BUSINESS CYCLE 5 expecaions, direcly regarding mos variables ha ener he model. Adrjiev sudies he effecs of adding sock prices in differen esimaed specificaions of a flexible price model, while we consider a possibly more convenional sicky-price sicky-wage model of he U.S. economy. Moreover, our use of expecaions abou a large se of macroeconomic variables, raher han a sock price index as a single forward-looking variable, has he advanage of shielding us from he well known difficuly of general equilibrium models o simulaneously explain he real and financial sides of he economy. The resuls in he wo papers, however, can usefully complemen each oher. 3 In erms of mehodology, he paper shows how he inclusion of expecaions daa can be useful o preven raional expecaions from falling oo far from he available observaions on macroeconomic expecaions. The approach used here, herefore, is no resriced o applicaions focused on news, bu i can be generally exploied in he esimaion of any DSGE model, wih or wihou raional expecaions. 4 There is a long hisory of ineres in he use of survey daa on expecaions (as exemplified, for example, by he survey by Pesaran and Weale, 26). Bu heir use in he esimaion of DSGE models has sared only more recenly. Del Negro and Eusepi (211) quesion wheher ypical models wih raional expecaions can mach he dynamics of observed inflaion expecaions. Ormeno (211) uses inflaion expecaions daa in he esimaion of a model wih learning. Milani (211) uses daa on observed oupu, inflaion, and ineres rae expecaions in a model wih learning, showing ha idenified expecaion shocks accoun for roughly half of U.S. business cycle flucuaions; Milani (212b) exends he analysis o a medium-scale model. This paper, insead, uses a much larger se of expecaions daa han hose precursors, and is novely lies in exploiing hem o insruc he exracion of news. 2. Model Framework 2.1. A Sicky-Price Sicky-Wage DSGE Model. We use a popular medium-scale DSGE model, based on Smes and Wouers (27) and Chrisiano e al (25), o characerize he dynamics of he U.S. economy a business cycle frequencies. The model includes a number of real and nominal rigidiies, which have been shown o be useful in fiing macroeconomic ime series. Prices and nominal wages are sicky à la Calvo. Capial 3 A conemporaneous paper, which has evolved enirely independenly from our work, by Hirose e al (212) has similar scopes o ours. They esimae a small-scale hree-equaion New Keynesian model using forecass daa from he Survey of Professional Forecasers. We focus on a more convenional, larger-scale, business cycle model of he U.S. economy and we exploi a much larger se of expecaions series, which allow us o beer exrac and disenangle news abou echnology, risk-premia, markup-shocks, and so forh. There are also differences in some of he economeric choices, which will be discussed laer in he paper. Again, he resuls in he wo papers can usefully complemen and reinforce each oher. 4 The use of survey expecaions o inform and consrain he esimaion of raional expecaions models has been advocaed, for example, in Milani (212a) and Milani and Rajbhandari (212).

7 6 FABIO MILANI & ASHISH RAJBHANDARI adjusmen decisions are subjec o adjusmen coss and he capaciy uilizaion rae can be varied depending on he renal rae of capial. Consumers are assumed o maximize a uiliy funcion ha is non-separable in consumpion (subjec o exernal habi formaion) and labor. The model is consisen wih a balanced seady-sae growh pah driven by a deerminisic rae of progress in echnology. The log-linearized model equaions are as follows 5 y = c y c +i y i +u y u +g y g (2.1) c = c 1 c 1 +(1 c 1 )E c +1 +c 2 (l E l +1 ) c 3 (r E π +1 +b ) (2.2) i = i 1 i 1 +(1 i 1 )E i +1 +i 2 q +φ (2.3) q = q 1 E q +1 +(1 q 1 )E r+1 k (r E π +1 +b ) (2.4) y = Φ p (αk s +(1 α)l +a ) (2.5) k s = k 1 +u (2.6) u = u 1 r k (2.7) k = k 1 k 1 +(1 k 1 )i +k 2 φ i (2.8) µ p = α(k s l )+a w (2.9) π = π 1 π 1 +π 2 E π +1 π 3 µ p +νp (2.1) r k = (k l )+w (2.11) ( ( µ w 1 = w σ l l + c h )) 1 h/γ γ c 1 (2.12) w = w 1 w 1 +(1 w 1 )E (w +1 +π +1 ) w 2 π +w 3 π 1 w 4 µ w +ν w (2.13) R = ρ R R 1 +(1 ρ R )[r π π +r y (y y )+r y( y y )]+mp. (2.14) The composie coefficiens in he previous equaions are given by: c y = 1 i y g y ; i y = δk y ; u y = rk k y; c 1 = h/(1+h); c 2 = (σ c 1)(W h L /C )/σ c (1+h); c 3 = (1 h)/[σ c (1+h)]; i 1 = 1/(1+β); i 2 = 1/[(1+β)ϕ]; q 1 = β(1 δ); u 1 = ψ/(1+ψ); k 1 = 1 δ; k 2 = δ(1+β)ϕ; π 1 = ι p /(1+βι p ); π 2 = β/(1+βι p ); π 3 = [1/(1+βι p )][(1 βξ p )(1 ξ p )/(ξ p (φ p 1)ε p +1)]; w 1 = 1/(1+β); w 2 = (1+βι w )/(1+β); w 3 = ι w /(1+β); w 4 = [1/(1+β)][(1 βξ w )(1 ξ w )/(ξ w (φ w 1)ε w +1)]. Equaion (2.1) is he economy s aggregae resource consrain. Oupu, denoed by y, equals he sum of consumpion c, invesmen i, governmen spending g, and he resource cos of varying 5 For he ineresed reader, a deailed derivaion of he model equaions is available in a echnical appendix as supplemen o Smes and Wouers (27), and hence no repeaed here.

8 OBSERVED EXPECTATIONS, NEWS SHOCKS, AND THE BUSINESS CYCLE 7 capial uilizaion, where u is he capial uilizaion rae. The parameers c y, i y, g y, denoe he shares of consumpion, invesmen (which, in urn, is a funcion of he capial depreciaion rae δ and of he capial-o-oupu raio k y ), and governmen spending, o oupu, in seady-sae, and u y denoes he seady-sae rae of capial uilizaion, which depends on he seady-sae renal rae of capial rk and capial-o-oupu raio. Equaion (2.2) is he log-linearized consumpion Euler equaion. Curren consumpion depends on boh pas consumpion, hrough he assumpion of exernal habi formaion in he uiliy funcion, and expecaions abou fuure consumpion, on curren and expeced labor supply, where l denoes hours of work, and on he ex-ane real ineres rae (i E π +1 ). Consumpion is also affeced by a disurbance b, which is a risk-premium shock. The composie coefficiens c 1, c 2, and c 3 are funcions of he srucural parameers h 1, which denoes he degree of habi formaion, σ c >, he inverse of he elasiciy of ineremporal subsiuion, σ l >, he inverse of he Frisch elasiciy of labor supply, and γ, he economy s seady-sae growh rae, as well as of he seady-sae levels of wages, hours, and consumpion. Equaion (2.3) describes he dynamics of invesmen. Curren invesmen i depends on lagged invesmen, which eners hrough he assumpion of adjusmen coss relaed o changes in he rae of invesmen, expeced fuure invesmen, and on he real value of he capial sock q. The sensiiviy of invesmen o q is an inverse funcion of he elasiciy of he cos of capial adjusmen ϕ in he capial adjusmen funcion; he coefficien β denoes, insead, he household s discoun facor. The erm φ denoes an invesmen-specific echnology shock. The value of he capial sock evolves as indicaed by equaion (2.4). The curren value is affeced by fuure expecaions for he value of capial, fuure expecaions for he renal rae r, k and by he ex-ane real ineres rae. The risk-premium shock also eners he equaion for q, helping he model accoun for he comovemen beween consumpion and invesmen. Equaion (2.5) denoes he aggregae producion funcion. Capial services k s and labor l are used o produce oupu. The erm a is he oal facor produciviy shock. The parameer Φ p is equal o one plus he fracion of fixed coss in producion, while α denoes he share of capial. Equaion (2.6) expresses capial services as a funcion of he pas capial sock k 1 and he uilizaion rae. The capial uilizaion rae is a funcion of he renal rae of capial, as indicaed by Equaion (2.7). Theparameer u 1 is an inverse funcion of he elasiciy of he capial uilizaion adjusmen cos funcion, which is governed by parameer ψ. Equaion (2.8) describes he capial accumulaion process. Capial is expressed as a funcion of he pas capial sock, curren invesmen, and is affeced by he invesmen-specific echnology shock.

9 8 FABIO MILANI & ASHISH RAJBHANDARI Equaion(2.9) expresses he price markup as he difference beween he marginal produc of labor and he real wage. The price markup is a driver of inflaion. Inflaion dynamics is characerized by equaion (2.1), which is a New Keynesian Phillips curve. Curren inflaion depends on lagged inflaion, hrough he assumpion of indexaion o pas inflaion in price seing, expeced inflaion, on he price markup, and on a price markup shock ν p. The degree of backward-lookingness in inflaion is a posiive funcion of ι p 1, he degree of auomaic price indexaion, while he slope of he curve is an inverse funcion of he degree of price sickiness, which depends on he Calvo parameer ξ p ; oher coefficiens ha affec he slope are φ p and ɛ p, he seady-sae price markup and he curvaure of he Kimball aggregaor funcion. Equaion (2.11) expresses he renal rae of capial as a posiive funcion of he real wage and a negaive funcion of he capial-labor raio. In equaion (2.12), he wage markup is expressed as equal o he difference beween he real wage and he marginal rae of subsiuion beween consumpion and leisure. Equaion(2.13) describes he evoluion of he real wage, which depends on lagged wages, a erm ha arises from he assumpion ha nominal wages, when no re-opimized, are indexed o he pas aggregae inflaion rae, on he expeced fuure real wage, on he wage markup, on pas, curren, and fuure inflaion, and on he wage markup shock ν w. The reducedform parameers are a funcion of he degree of wage sickiness ξ w, wage indexaion o pas inflaion ι w, he discoun facor, he seady-sae wage markup φ w, and he Kimball curvaure ɛ w. Finally, i is common o assume ha moneary policy can be approximaed by a Taylor rule as in equaion (2.14). The policy insrumen is parially adjused depending on movemens in he levels of inflaion and he oupu gap, and in he growh rae of he oupugap. The oupugap is defined as he deviaion of oupu from is poenial level y, where poenial oupu is denoed as he level of oupu ha would prevail in he same economy, bu under flexible prices and wages (and no markup shocks). Deviaions from sysemaic moneary policy are capured by he moneary policy shock mp. To highligh he imporance of exploiing expecaions in he esimaion of models wih news, we have chosen a DSGE model ha serves as benchmark for much of he empirical macroeconomic lieraure. We have no alered preferences o limi he wealh effec on labor supply as in Jaimovich and Rebelo (29) or Schmi-Grohé and Uribe (212). The change in preferences helps he model in capuring he comovemen of macro variables, bu, as discussed in Lorenzoni (211), a he cos of generaing posiive income effecs on labor supply (which are inconsisen wih he micro evidence) or large increases in ineres raes and plummeing asse prices afer posiive news shocks (which seem counerinuiive). We adop a preference srucure ha is, insead, more sandard for esimaed DSGE models. I seems more naural, we believe, o inroduce nominal rigidiies, for

10 OBSERVED EXPECTATIONS, NEWS SHOCKS, AND THE BUSINESS CYCLE 9 which he micro evidence is more clearly favorable (e.g., Nakamura and Seinsson, 28, for prices, and Baraieri e al, 21, for wage rigidiy), o obain he necessary comovemen over he business cycle condiional on a news shock Srucural and News Shocks. We assume ha expecaions, denoed by he mahemaical expecaion operaore, arefullyraional. Weexend, however, hemodeloallow foranicipaions regarding fuure shocks, i.e. news or news shocks. We assume ha news can affec each disurbance in he model: i becomes an empirical maer o disinguish among hose for which news maers, and hose for which he inroducion of news is superfluous. There are seven exogenous disurbances: governmen spending g, risk-premium b, invesmenspecific φ, echnology a, price markup ν p, wage markup νp, and moneary policy mp. The shocks evolve as AR(1) processes (we assume also AR(1) markup shocks, which represen a minor difference from he ARMA(1,1) processes used in Smes and Wouers, 27); as in Smes and Wouers (27), we deal wih he poenial endogeneiy of governmen spending by allowing i o respond o conemporaneous echnology innovaions. Wih he inclusion of news, now he disurbances laws of moion become: g = ρ g g 1 +ε g +g a(ε a + H b = ρ b b 1 +ε b + H h=1 φ = ρ φ φ 1 +ε φ + H a = ρ a a 1 +ε a + ν π h=1 H h=1 = ρ π ν π 1 +επ + H ν w = ρ w ν w 1 +εw + h=1 H h=1 mp = ρ mp mp 1 +ε mp + h=1 η a,h h )+ H h=1 η g,h h (2.15) η b,h h (2.16) η φ,h h (2.17) η a,h h (2.18) η π,h h (2.19) η w,h h (2.2) H h=1 η mp,h h, (2.21) where he erms η j,h h denoe news ha becomes known in h abou shocks ha maerialize only h periods ahead, and where H is he maximum news horizon, which will be assumed equal o 5 in he baseline esimaion, o mach forecass daa availabiliy. Our horizon srucure is, herefore, denser han he one in Schmi-Grohé and Uribe (212), who impose news a horizons 4 and 8

11 1 FABIO MILANI & ASHISH RAJBHANDARI only, bu wih he drawback ha we do no include longer-erm news. The erms ε j, j = g,...,mp, insead, denoe he unanicipaed shocks of ype j ha are ypically included in DSGE models. The lieraure has more ofen emphasized news abou fuure echnology, as in(2.18), and recenly also in he form of news abou fuure invesmen-specific, raher han neural, echnology shocks, as in (2.17). Bu news regarding oher disurbances, even relaed o demand, is equally plausible (Schmi-Grohé and Uribe, 212, and Khan and Tsoukalas, 212, allow for news abou a variey of supply and demand shocks). For example, news abou governmen spending can allow researchers o capure he consequences of governmen announcemens abou spending or ax policy changes ha will be implemened only a a fuure dae. News abou governmen spending has also been sudied in Schmi-Grohé and Uribe (212), Khan and Tsoukalas (212), and, in a SVAR conex, in Merens and Ravn (211). In a similar way, allowing for news relaed o fuure moneary policy decisions yields a way o model he increasingly common cenral banks announcemens abou he fuure direcion of policy or, a leas, o capure anicipaions by he privae secor abou fuure cenral bank s decisions, wheher correc or no (as analyzed in Milani and Treadwell, 212). Here, we consider news abou all seven disurbances. The scope is o le he daa choose he naure of news ha is more empirically relevan, raher han imposing dogmaic a priori resricions. If some of he news shocks ha are included aren empirically relevan, hey will end up being eiher unidenified or heir sandard deviaions would sele around zero in he esimaion. As cusomary in he lieraure, all unanicipaed and anicipaed innovaions are assumed o be independen. All innovaions are i.i.d. and follow Normal disribuions. For each disurbance, assuming horizons h = 1,2,3,4,5, news shocks are insered in he sae space model as follows (we presen here he illusraive case for he moneary policy shock): mp η mp,5 mp 1 ρ mp η mp,5 η mp,5 1 1 η mp,5 η mp, η mp,5 η mp, η mp,5 1 η mp, η mp,5 η mp,4 5 η mp,4 η mp, η mp,4 η mp,4 2 = 2 1 η mp,4 + η mp, η mp,3 η mp,4 4 η mp,3 1 η mp,3 1 1 η mp,3 1 η mp,3 2 2 η mp,2 η mp,3 3 η mp,2 1 η mp,2 1 1 η mp,2 η mp,1 2 η mp,1 1

12 + OBSERVED EXPECTATIONS, NEWS SHOCKS, AND THE BUSINESS CYCLE 11 σ mp σ ηmp,5 σ ηmp,4 σ ηmp,3 σ ηmp,2 σ ηmp,1 η mp,2 η mp,1 where he σ coefficiens denoe sandard deviaions of he news shocks, and he η erms simply ε mp η mp,5 η mp,4 η mp,3 (2.22). redefine he original news denoed by η. Therefore, for each disurbance, he addiion of news wih horizons 1 o 5, leads o an expansion of he sae space dimension from 1 o 16. In DSGE models, he idenificaion of news shocks works hrough heir effecs on expecaions. News affecs expecaions abou fuure srucural disurbances, which, in urn, are relevan for expecaions abou fuure aggregae macroeconomic variables ha are needed o solve households and firms maximizing decisions. By consrucion, he srucural innovaions ha are ypically included in macroeconomic models are, insead, unpredicable and, hence, hey do no affec fuure expecaions. Expecaions abou fuure moneary policy shocks, for example, in he model wih news equal E mp +1 mp +2 mp +3 mp +4 mp +5 mp +6 = ρ mp mp +η mp,5 4 +ηmp,4 3 +ηmp,3 2 +ηmp,2 1 +ηmp,1 ρ mp E mp +1 +η mp,5 3 +ηmp,4 2 +ηmp,3 1 +ηmp,2 ρ mp E mp +2 +η mp,5 2 +ηmp,4 1 +ηmp,3 ρ mp E mp +3 +η mp,5 1 +ηmp,4 ρ mp E mp +4 +η mp,5 ρ mp E mp +5. (2.23) Therefore, expecaions abou one-period-ahead moneary policy deviaions from he Taylor rule incorporae conemporaneous news, news obained in he previous period, up o news obained four periods in advance; wo-quarer-ahead expecaions add o he + 1 moneary policy expecaions news ha was obained saring from hree quarers in advance and relaed o he +2 quarer, and so forh, up o five-quarer-ahead expecaions, which revise he four-quarer-ahead expecaions only o incorporae conemporaneous news; expecaions abou six or more quarers ahead are unaffeced by news (excep hrough shorer-erm news ha implicily eners E mp +5 ) in our framework. In he empirical secion, he idenificaion of news regarding fuure moneary policy shocks will be informed, firs and more direcly, by he use of survey expecaions abou fuure shor-erm ineres raes, bu also by he use of all oher observed expecaions series. The idenificaion will work in similar ways for news abou oher disurbances. Expecaions will be used as observable variables ha he esimaion will ry o mach; he ineracion beween expecaions and realized variables should provide addiional resricions ha can be exploied o idenify news. Schmi-Grohé and Uribe (212) presen Mone Carlo evidence showing ha unanicipaed and news shocks are idenified in a sylized example. Idenificaion, alhough heoreically possible,

13 12 FABIO MILANI & ASHISH RAJBHANDARI may no be sraighforward in a complicaed model as he one used here. We believe ha using a variey of expecaions series a differen horizons will help he inference regarding news shocks. The news srucure used in he paper can also accoun for revisions of previous privae secor s anicipaions (which may have failed o fully maerialize). For example, economic agens receive in 5 he news η mp,5 5 regarding a shock anicipaed for ime. In 4, he news shock η mp,4 4 can be inerpreed as a revision of he original news η mp,5 5 (since hey boh reflec informaion abou shocks expeced o maerialize a ime-): he anicipaion abou fuure policy shocks is now ( ) given by η mp,4 4 +ηmp,5 5. In 3, he erm η mp,3 3, can be inerpreed as a revision of he previous ( ) anicipaion η mp,4 4 +ηmp,5 5, and so forh. 3. Srucural Esimaion wih Observed Expecaions 3.1. Observed Expecaions and Real-Time Daa. We esimae he DSGE model using fullinformaion Bayesian mehods. In he baseline esimaion of he paper, we use real-ime daa, for a sample spanning he period beween 1981:III and 211:II (he saring dae corresponds o he firs quarer of availabiliy of mos of our expecaions series). 6 The daa frequency is quarerly. We use eigh realized variables as observables: real oupu growh, real consumpion growh, real invesmen growh, real governmen spending growh, real wage growh, log hours, and a shor-erm nominal ineres rae. Daa Appendix A provides more deails on he series and daa ransformaions ha we have used. The real-ime series are obained from he Real Time Daa Se for Macroeconomiss, made available by he Federal Reserve Bank of Philadelphia (wih he excepion of ineres raes, since heir series is no revised). The se of observables corresponds o he seven variables in Smes and Wouers (27), bu wih he addiion of he growh rae of governmen spending as an addiional observed realized variable here. Oher differences wih he daa se used in Smes and Wouers (27) are our use of he hree-monh Treasury Bill yield as our ineres rae in place of he Federal Funds rae (a choice ha is moivaed by he availabiliy of forecass daa for he hree-monh rae, bu no for he Federal Funds rae) and our real wage series. Here, we choose o use he only wage series ha is available in real-ime, which corresponds o oal wage and salary disbursemens (privae indusries), raher han Smes and Wouers definiion. To mainain consisency in he esimaion, herefore, we favor using all real-ime series, raher han a mixofreal-imeandrevisedseriesha wouldbenecessaryokeep hesamewageseries asdefinedin Smes and Wouers (27). We compue real wages as oal wage and salary disbursemens divided by oal aggregae hours and he GDP deflaor. The observable is hen he log firs difference of he 6 Specifically, for each variable, we use he firs daa release as our relevan ime series. We do no aemp o model, insead, he muliple revision process from he dae of firs release o he final revised vinage. Incorporaing revisions in our esimaion is complicaed by he already large dimensionaliy of our sae space sysem.

14 OBSERVED EXPECTATIONS, NEWS SHOCKS, AND THE BUSINESS CYCLE 13 derived real wage series. Hours are also compued using he real-ime aggregae weekly hours index divided by civilian noninsiuional populaion. Observables for oupu, consumpion, invesmen, governmen spending, are obained as he log firs difference of he corresponding variable in real erms. Inflaion is calculaed as he log firs difference of he implici GDP price deflaor. The ineres rae is used in levels, bu ransformed ino quarerly, raher han yearly, raes, o mach he definiion in he model. In addiion o he eigh realized variables, a leas in he main esimaion of ineres in he paper, we exploi all available relevan daa on expecaions: we include expecaions abou fuure oupu growh, fuure consumpion growh, fuure invesmen growh, fuure governmen spending growh, fuure inflaion, and fuure ineres raes, for horizons ranging from one quarer ahead o five quarers ahead. Therefore, we exploi a oal of hiry expecaion series. All expecaions series are obained from he Survey of Professional Forecasers, published by he Federal Reserve Bank of Philadelphia. Our series correspond o he mean across forecasers (again, Appendix A provides deails on he series). 7 Daa on forecass for hours and wages are no available and, hence, no included in he esimaion. In principle, forecass regarding employmen growh would be available, bu heir availabiliy sars only from 23, and, herefore, we omi hem. Figure 1 shows he realized and expecaion series ha are used in he esimaion. To avoid cluer in he figure, we only plo expecaions a he one-quarer and four-quarer-ahead horizons for each variable, raher han he full se of expecaions. A sylized fac abou expecaions is ha hey are generally smooher han he forecased series; as expeced, longer-horizon expecaions are considerably smooher han one-quarer-ahead expecaions Sae-Space Sysem. The sae space expands considerably wih he inclusion of news. For each news shock up o horizon H, he sae space expands is size by H h=1h: in our case, news a horizons 1 o 5 abou each srucural disurbanceadds 15 rows o he sae-space sysem, for a oal of 15 new rows (wih coefficien marices composed by ones and zeros only). The log-linearized equaions, along wih he laws of moion for he disurbances and news can be wrien as Γ ξ = Γ 1 ξ 1 +Ψω +Πζ, (3.1) where ξ collecs he foureen endogenous variables in he model, a subse of he corresponding variables in he associaed flexible price economy (necessary o compue he poenial oupu erm ha eners he Taylor rule), he seven srucural disurbances, he expecaion erms, and all he news componens, ω collecs he i.i.d. innovaions, and ζ is a vecor of expecaional errors, 7 Mansky (21) emphasized how he use of he mean across forecasers may creae a composiion bias, caused by he enry and exi of differen forecasers over he sample. While we are sympaheic o he use of individual forecasers daa, raher han summary saisics, we absrac from his issue here. We believe ha he level of aggregaion we impose here is consisen wih ypical pracice in empirical macroeconomics.

15 14 FABIO MILANI & ASHISH RAJBHANDARI η = ξ E 1 ξ, such ha E 1 η =. The model can be solved, under he assumpion of raional expecaions and following he approach laid ou in Sims (2), o obain which gives he sysem ransiion equaion. ξ = Fξ 1 +Gω, (3.2) The link beween observable variables and he heoreical variables in he model is capured by he following se of observaion equaions: Y obs γ C obs I obs γ G obs γ L obs γ W obs l π obs γ π R obs r + π E Y+1 obs γ E Y+5 obs γ E C+1 obs γ E C+5 obs = +H γ E I+1 obs γ E I+5 obs γ E G obs +1 γ E G obs +5 γ E π+1 obs π E π+5 obs π E R+1 obs r + π E R+5 obs r + π which we can wrie more compacly as y y 1 c c 1 i i 1 g g 1 l w w 1 π R E [y +1 y ]... E [y +5 y +4 ] E [c +1 c ]... E [c +5 c +4 ] E [i +1 i ]... E [i +5 i +4 ] E [g +1 g ]... E [g +5 g +4 ] E π E π +5 E R E R +5 [ ξ ] +Ω o y o E y +1 o E y +2 o E y +3 o E y +4 o E y +5, (3.3) OBS = H +Hξ +Ωo. (3.4) The marices H and Ω are selecion marices, composed of ones and zeros: H selecs he observable ] variables wihin he sae vecor ξ (where [ ξ conains hose sae variables in he vecor ξ for which no observables are available), while Ω selecs he measuremen errors o ener he observaion equaions for realized real GDP growh and for he five real GDP growh expecaions series. We allow for measuremen error erms in he oupu equaions o accoun for possible differences in he definiion of oupu growh in he model and in he daa (also necessary since expors and impors are no explicily modeled) and o break he igh link implied by he resource consrain

16 OBSERVED EXPECTATIONS, NEWS SHOCKS, AND THE BUSINESS CYCLE 15 equaion(2.1). Given ha we use observables for oupu, consumpion, invesmen, and governmen spending, a failure o allow for measuremen errors would spuriously assign o he cos of varying capial uilizaion, u, any difference beween hese heoreical variables and heir relaionship in he daa; his would also cause bias in he oher esimaed relaions in he model. There are no measuremen errors for all he oher relaions. 8 Thevecor H conains, insead, seady-sae values: γ will be esimaed, while we will fix l, π, and r, o heir sample averages. The reamen of rends follows Smes and Wouers (27): we impose a common rend γ in oupu, consumpion, invesmen, governmen spending, and he real wage. 9 For now, we assume ha agens, when forming expecaions, recognize he correc values of he economy s balanced growh rae, γ, he seady sae values of inflaion, labor hours, real ineres raes, and so forh. Such assumpion seems consisen wih he overall assumpion of raional expecaions. The main novely in his paper is he use of exensive informaion from he erm srucure of survey expecaions. We exploi informaion on one-period-ahead o five-period-ahead oupu, consumpion, invesmen, and governmen spending growh, inflaion, and ineres rae. As made clear by (3.3), observed expecaions are, herefore, assumed equal o he raional expecaion for he corresponding variable from he model plus a measuremen error erm in he case of real GDP growh expecaions. To summarize, in he esimaion scenario wihou expecaions daa, we shall consider seven srucural shocks ha mirror hose in Smes and Wouers (27), one measuremen error o accoun for differences beween our daa on real GDP growh and he model definiion, and news shocks a horizons one o five for each of he seven srucural shocks. In he main esimaion of ineres, wih expecaions reaed as observables, we add measuremen errors for real GDP growh forecass, for he reason oulined above. The increase in observables is no associaed o an equivalen increase in he number of shocks: he exisence of several news shocks guaranees ha he model is no affeced by sochasic singulariy (even wihou he use of measuremen errors). News shocks are now idenified from resricions imposed by changes in expecaions a differen horizons. In paricular, he idenificaion of news is made possible hrough differences beween one, wo, and more, period ahead expecaions, all formed a he same ime, bu also from revisions in forecass formed in +1 abou he variable in +2 compared wih he previous period forecas abou he variable in +2, and so forh. 8 Our sraegy here differs from he one used in Hirose e al. (212): we do no add measuremen errors o each observed expecaions series. he addiion of news shocks already guaranees ha he sysem does no suffer from sochasic singulariy. 9 Regarding governmen spending, we have performed he esimaion, eiher imposing he same rend, or allowing is rend o differ, in order o accoun for he declining share of governmen expendiures in GDP. The resuls are no affeced.

17 16 FABIO MILANI & ASHISH RAJBHANDARI We remark ha while he sae space dimension expands considerably, we exploi a large number of new observable variables wihou adding a large number of parameers (only he sandard deviaions for he GDP growh measuremen errors). Therefore, even if he esimaion is compuaionally more burdensome, he idenificaion is faciliaed by he use of addiional hiry-eigh observable series Priors. The prior disribuions for he srucural parameers mirror in mos cases hose in Smes and Wouers (27), wih some modes differences, and are shown in Table 1. 1 We revise downward he prior for he habi formaion coefficien, which has a mean of.5, raher han.7 as in Smes and Wouers (27). We selec priors for he Calvo price and wage sickiness coefficiens o mach he micro evidence on price rigidiy: he prior means equal.66, implying prices and wages on average fixed for 9 monhs (a duraion ha is consisen wih he findings in Nakamura and Seinsson, 21), wih a sandard deviaion of.6. The prior in Smes and Wouers (27) implied less rigid prices (mean.5). We choose a Gamma prior wih mean equal o 1.5 and sandard deviaion.375 for σ c, he inverse of he ineremporal elasiciy of subsiuion coefficien. Schmi-Grohé and Uribe (212) fix, insead, he coefficien o equal 1 in heir esimaion. Given he imporance of he coefficien for business cycle analysis, we esimae his coefficien as well. The elasiciy of labor supply is capured by he parameer σ l : we assume a Gamma prior wih mean equal o 2 and sandard deviaion.4. The prior selecions ha are mos relevan here, however, concern he sandard deviaions of unanicipaed and news shocks. We follow Schmi-Grohé and Uribe (212) in assuming ha he sandard deviaions follow Gamma priors. Gamma priors wih equal values for he mean and sandard deviaions ensure ha values close o are assigned higher probabiliy han posiive and larger values and ha is also a value wih posiive probabiliy mass. Given ha he model has more shocks han observables, his choice ensures ha he daa pick he mos influenial shocks, raher han spuriously forcing each shock o have a posiive sandard deviaion. 11 As in Schmi- Grohé and Uribe (212), he unanicipaed shocks are assumed o accoun for 75% of he a priori variance, while he five news shocks for each disurbance accoun for he remaining 25%; herefore, 1 Some of he coefficiens are fixed o he same values chosen by Smes and Wouers (27): hese are he quarerly depreciaion rae δ =.25, he seady-sae price and wage markup parameers φ p = φ w = 1.5, and he Kimball curvaure parameers ɛ p = ɛ w = 1; we fix he share of governmen spending in GDP g y o equal.21, our sample mean, raher han.18. We also fix he discoun facor β =.99 and he share of capial in producion α =.3; l, π, and r are equal o he variables sample means. 11 Wehavealsoexperimenedwihpossiblyless informaiveuniformdisribuionsandinversegammadisribuions of he ype IG(ɛ, ɛ), wih ɛ a small posiive number, for sandard deviaions in he esimaion. Gelman (26) discusses how uniform disribuions may be unexpecedly informaive, wih miscalibraion oward posiive values, when he sandard deviaions are close o zero, and how resuls under he previous inverse gamma prior are usually very sensiive o he choice of ɛ. Therefore, we choose here a prior ha seeks o impar parsimony by assigning higher probabiliy o sandard deviaion values near zero (hus poenially shuing down some shocks).

18 OBSERVED EXPECTATIONS, NEWS SHOCKS, AND THE BUSINESS CYCLE 17 for each disurbance j, we selec he prior mean so ha make sure ha news shocks are no unduly favored. σ 2 j σ 2 j + 5 h=1 σ2 η j,h =.75. The priors, herefore, Finally, measuremen error erms, when presen, are assumed o be i.i.d. We assume ha he sandard deviaion coefficiens follow Inverse Gamma prior disribuions wih mean equal o.25 and sandard deviaions equal o 1. In our empirical analysis, he measuremen error sandard deviaions appear very well idenified and prior choices do no affec in any way he corresponding poserior esimaes. In he robusness secion, we also repea some of he esimaions by fixing he measuremen errors o levels ha force hem o explain less han 1% of realized or forecased oupu growh variances Esimaion Sraegy. Before urning o he analysis of he main model wih expecaions daa and news, we would like o undersand he conribuion of each of our auxiliary choices o he final resuls. To his scope, we perform a sequence of inermediae esimaions before focusing on our baseline model. Firs, we re-esimae he model in Smes and Wouers using heir original daa se, bu wih 1981:III as he saring dae, o be consisen wih our subsequen esimaions. Their sample ends in 24:IV. We hen exend he Smes and Wouers model o include news and re-esimae he model on heir original, alhough pos-1981, daa se. In our main esimaion, however, we will use real-ime, raher han revised, daa. Therefore, o single ou he effec of real-ime daa, we also re-esimae he Smes and Wouers model wih and wihou news, on our real-ime daa se, wih he updaed 1981:III-211:II sample. Besides he real-ime naure of he daa used in his esimaion, he mos imporan difference is he addiion of governmen spending o he se of observables, which limis he flexibiliy of he governmen spending shock o shif around o mask misspecificaion in he model and o fi oher real variables. Finally, we urn o our main esimaion of ineres: he esimaion of a DSGE model wih news shocks and using daa on a large se of observed expecaions. This case is similar o he previous esimaion wih real-ime daa, bu wih he addiion of survey expecaions as observables ha he esimaion under raional expecaions will be forced o mach. In he laer case, he model is expressed as in (3.2) and (3.4); he previous four inermediae cases are simplified versions of he same sae-space sysem. For mos esimaions, we generae one million draws using he Meropolis-Hasings algorihm. For he baseline case, given he expanded size and complexiy, we use a longer chain o make sure ha we have no seled on a local mode. We repor poserior esimaes based on he las 5, draws. We have repeaed he esimaion saring from differen iniial condiions and compared he similariy of poserior esimaes. To check convergence, we use race plos and we check recursive means of he draws. Mos poserior

19 18 FABIO MILANI & ASHISH RAJBHANDARI disribuions have a unique mode, bu in few cases, he parameers poserior disribuions appear bimodal: we will highligh bimodaliy issues when hey exis in our discussion of resuls in he following secion. 4. Empirical Resuls 4.1. Poserior Esimaes. Table 1 shows he esimaion resuls. Columns (1) and (2) display he poserior esimaes for he Smes and Wouers daa se, resriced o he pos-1981 period, wihou news and wih news shocks. Column (3) refers o he case in which we repea he esimaion wih he use of real-ime, firs-vinage, daa series, and wih he inclusion of governmen spending as an observable variable. Column (4) shows he esimaion resul on he same real-ime daa se as case (3), bu now allowing for news shocks. Finally, column (5) refers o he baseline esimaion in he paper. The same specificaion esimaed in column (4) is now required o mach a large se of survey expecaions ha are added o he lis of observables. The informaion conained in expecaions is exploied o improve he exracion of he news componen over he business cycle. The main comparison of ineres in he paper is beween cases (2), (4) and (5): he firscorresponds o he mos common pracice of exracing news from revised daa and wih no informaion from available expecaions, he second adds a beer approximaion of real-ime privae-secor knowledge o he esimaion, while he hird improves over he firs wo cases, by exrapolaing news from expecaions daa, exploiing how expecaions vary across horizons in he same quarer, how expecaions a he same horizon are revised from one quarer o he nex, and how hey inerac wih realized macroeconomic observaions. For he esimaion on he revised Smes and Wouers sample shown under column (1), mos of he resuls are consisen wih Smes and Wouers (27) esimaes. One issue o poin ou in his esimaion is ha here is a clear bimodaliy in he coefficiens reflecing he serial correlaion of he risk premium shock and he degree of habi formaion in consumpion: one mode is characerized by high serial correlaion in he exogenous risk premium shock and relaively low habi formaion, while he oher is characerized by low serial correlaion and high habi formaion. The mode wih high habis - low serial correlaion, however, achieves a subsanially higher poserior probabiliy (which, however, does no preven he Markov chain from ofen visiing he second mode as well). Smes and Wouers choice of prior mean equal o.7 for he habi coefficien would work o reduce he imporance of he second mode, while our prior les he daa more freedom o explore and pick any of he wo modes We have also esimaed he model using he same prior mean as Smes and Wouers: in ha case, he poserior mean esimae for habi formaion is higher, while he esimae for he risk-premium AR coefficien falls around.2.