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2 Measuremen Errors and Moneary Policy: Then and Now Pooyan Amir-Ahmadi Goehe Universiy Frankfur Chrisian Mahes Federal Reserve Bank of Richmond Mu-Chun Wang Universiy of Hamburg Working Paper No November 5, 215 Absrac Should policymakers and applied macroeconomiss worry abou he difference beween real-ime and final daa? We ackle his quesion by using a VAR wih ime-varying parameers and sochasic volailiy o show ha he disincion beween real-ime daa and final daa maers for he impac of moneary policy shocks: The impac on final daa is subsanially and sysemaically differen (in paricular, larger in magniude for differen measures of real aciviy) from he impac on real-ime daa. These differences have persised over he las 4 years and should be aken ino accoun when conducing or sudying moneary policy. Keywords: real-ime daa, ime-varying parameers, sochasic volailiy, impulse responses We would like o hank Fabio Canova, Tim Cogley, and Dean Croushore as well as paricipans a he 215 IAAE conference and he Cirano workshop on real-ime daa in Monreal for heir commens. Miki Doan, Marisa Reed, and Daniel Trach provided excellen research assisance. The views expressed in his paper are hose of he auhors and do no necessarily reflec hose of he Federal Reserve Bank of Richmond or he Federal Reserve Sysem. 1

3 1 Inroducion When moneary policymakers evaluae he effecs of heir mos recen policy decisions (say, o prepare for he nex round of moneary policy decisions) hey only have access o preliminary real-ime esimaes of macroeconomic daa ha have been colleced afer he policy decision hey wan o evaluae. Is he difference beween realime and final macroeconomic daa imporan enough o be considered when analyzing and conducing moneary policy? We revisi his quesion asked by Croushore & Evans (26) in ligh of recen evidence (Aruoba (28)) ha he measuremen errors in macroeconomic daa are far from saisfying he properies of classical measuremen errors and evidence ha here is subsanial ime variaion in he dynamics of U.S. macroeconomic ime series, as emphasized by Cogley & Sargen (25) and Primiceri (25). We use a vecor auoregression (VAR) wih ime-varying parameers and sochasic volailiy esimaed on daa ha includes real-ime and final releases of macroeconomic daa o uncover subsanial ime variaion in he dynamics of measuremen errors: We find ha he measuremen errors are significanly correlaed for some variables, feaure subsanial changes in volailiy and can be differen from zero for long periods of ime wih magniudes ha are economically meaningful. We use a model wih ime-varying parameers and sochasic volailiy because ime variaion in he dynamics and volailiy of final daa has been idenified as imporan for (final) U.S. daa by Cogley & Sargen (25), Primiceri (25), and Canova & Gambei (29), among ohers. Our paper shows ha hese feaures carry over o real-ime daa as well. By using sign resricions o idenify a moneary policy shock, we esablish ha policymakers should indeed care abou measuremen errors. Differences beween he impulse responses of real-ime and final daa on measures of real aciviy are significan and persis over ime. As hese differences are persisen over ime, policymakers should ake hem ino accoun. Our work is relaed o he lieraure on ime variaion in macroeconomic dynamics such as Cogley & Sargen (25), Primiceri (25), and Gali & Gambei (29). The 2

4 model we use o analyze ime variaion in our daa is borrowed from hose papers. 1 As pioneered by Canova & Nicolo (22), Faus (1998) and Uhlig (25), we use sign resricions o idenify moneary policy shocks. Canova & Gambei (29) use sign resricions o idenify moneary policy shocks in a VAR wih ime-varying parameers and sochasic volailiy, bu hey do no consider real-ime daa. Croushore & Evans (26) ackle issues similar o ours, hough in he conex of a fixed coefficien VAR using eiher recursive or long-run resricions. Their model of measuremen error is less general han ours. For example, heir models no only use fixed coefficien models, bu also do no allow for biases (non-zero inerceps) in he relaionship beween differen vinages of daa. We find ha hese feaures maer, reinforcing he resuls by Aruoba (28), and beyond ha esablish ha measuremen errors feaure sochasic volailiy and are correlaed across variables. In conras o our work and Croushore & Evans (26), he large majoriy of papers on real-ime daa focuses on saisical models of measuremen error ha do no idenify effecs of srucural shocks. Jacobs & van Norden (211) are moivaed by he evidence in Aruoba (28) and build a flexible model for a univariae measuremen error series. In conras o us, hey model inermediae daa releases, bu do no consider he relaionship of measuremen errors across variables, ime variaion in he parameers, or sochasic volailiy. Jus as he model used in Jacobs & van Norden (211), our model is general enough o allow for measuremen errors ha are correlaed wih eiher only final daa ( news ) or correlaed only wih he real-ime daa ( noise ) as well as inermediae cases, as we show in he model secion. Jacobs, Sarferaz, van Norden & Surm (213) build a mulivariae version of Jacobs & van Norden (211), bu sill absrac from sochasic volailiy and ime-varying parameers. Boh Jacobs & van Norden (211) and Jacobs e al. (213) do no sudy he response of he economy o srucural shocks, which is our main focus. D Agosino, Gambei & Giannone (213) use a VAR wih ime-varying parameers and sochasic volailiy on real-ime daa o sudy he forecasing abiliy of models in his class. 1 An overview of his lieraure is given in Koop & Korobilis (21). 3

5 Fixed-coefficien VARs using various vinages of real-ime daa have previously been used o improve forecasing abiliy by Kishor & Koenig (29) and Carriero, Clemens & Galvao (215), for example. While we focus on pos-wwii daa, he issue of mismeasured daa is of umos imporance when sudying long-run hisorical daa. While scholars using hisorical daa usually do no have access o revised daa for he enire sample, hey someimes explore overlapping daa sources - Cogley & Sargen (214) do his in a model for US inflaion ha feaures sochasic volailiy for rue daa, bu in conras o our approach heir model does no feaure sochasic volailiy for he measuremen error. Croushore & Sill (214) esimae a dynamic sochasic general equilibrium (DSGE) model on final daa and hen use he approach of Schorfheide, Sill & Kryshko (21) o link real ime daa o he sae variables of he esimaed DSGE model. Similar o our findings, heir findings show boh ha here are subsanial differences beween real-ime and final daa responses and ha final daa responses end o be larger in absolue value. In he nex secion we describe our model. We hen show how i can be moivaed by a DSGE model wih asymmeric informaion before urning o resuls for our benchmark specificaion. Finally, we show ha our findings are robus o alernaive specificaions: (i) using an alernaive measure of real aciviy, employmen growh, (ii) using an alernaive idenificaion scheme o idenify moneary policy shocks, and (iii) using an alernaive definiion of final daa. 2 The Model We joinly model he dynamics of he firs release of any daa poin published - we call his real-ime daa - and he laes vinage available a he ime of he wriing of his paper - which we use as a proxy for final daa. Throughou his paper, we sudy 4

6 he dynamics of vecors of he following form: y = π real π final x real x final (1) i where π denoes inflaion, i he nominal ineres rae, and x a measure of real aciviy. In our benchmark, x will be GDP growh, bu we also sudy employmen growh. 2 A superscrip real denoes real ime daa, whereas he superscrip final denoes final daa. Throughou he paper real-ime daa refers o he firs available release of a daa poin. We wan o recover he join dynamics of real-ime and final daa and ask wha hose dynamics ell us abou he effecs of moneary policy shocks on boh real-ime and final daa. The dynamics of y are given by y = µ + L A j, y j + e (2) j=1 where he inerceps µ, he coefficiens on lagged observables A j,, and he covariance marix Ω of e are allowed o vary over ime. Following mos of he lieraure ha has used hese models on quarerly daa such as Del Negro & Primiceri (215) and Amir-Ahmadi, Mahes & Wang (214), we se he number of lags L = 2. By wriing down a model for real-ime and final versions of he same daa series, we have also implicily defined a model of he measuremen errors η π and η x : ηπ η x = πreal x real πfinal x final = Sy (3) 2 Joinly modeling he dynamics of real-ime and final inflaion, GDP growh, employmen growh, and he nominal ineres rae leads o issues of numerical insabiliies in he Gibbs sampler we use o esimae he model. We hus sudy differen varians of he model including one indicaor of real aciviy a a ime. We could have reduced he lag lengh, bu ha would have made our resuls less comparable o ohers in he lieraure. Similar issues are documened in Benai (214), for example. For he same reason we also refrain from including inermediae daa revisions as observables. 5

7 where S is a selecion marix. 3 We hus use a flexible ime series model for he measuremen errors ha does no impose srong resricions on he measuremen errors - hey can be correlaed, have non-zero means, and feaure subsanial ime variaion in condiional momens. 4 This is imporan since Aruoba (28) has found ha daa revisions are no necessarily well behaved. To see ha our model can capure measuremen errors ha feaure boh news and noise componens (i.e. he measuremen errors can be correlaed wih boh final and real-ime daa), we can use a oy version of our model for a generic scalar variable c wihou ime variaion, sochasic volailiy, or any dynamics: cfinal c real = e = e 1, e 2, (4) where e N(, Ω c ). The measuremen error η c = c real e 2, e 1,. Consider as an example he following model for e : c final is hen defined as e 1, = w and e 2, = w + v where w and v are independen Gaussian random variables. Then we have η c = v, which is independen of he final daa c final = w. Reversing he roles of c final and c real shows ha measuremen error can be independen of real-ime daa in our framework. To see an inermediae case where he measuremen error is correlaed wih boh real-ime and final daa, assume ha e 1, = w, as before, bu now e 2, = 2w + v so ha η c = w + v, which is correlaed wih boh real-ime and final daa. To concisely describe he model we use o sudy ime variaion in he parameers 3 S = ( 1 1 ) 4 The measuremen errors inheri hese feaures from he variables in he VAR. 6

8 of he model, we define X I (1, y..., y L ) and rewrie (2):5 : y = X θ + e (5) θ = θ + u (6) Following Primiceri (25), i is convenien o break he covariance marix of he reduced-form residuals ino wo pars as implied by he following equaion: e = Λ Σ ε (7) where ε is a vecor of independenly and idenically disribued (iid) Gaussian innovaions wih mean and covariance marix I. Λ is a lower riangular marix wih ones on he main diagonal and represenaive non-fixed elemen λ i. Σ is a diagonal marix wih represenaive non-fixed elemen σ j. Those elemens vary over ime according o: λ i = λ i + ζ i (8) log σ j = log σ j + ν j (9) All innovaions are normally disribued wih covariance marix V, which, following Primiceri (25), we resric as follows: V = V ar ε u ζ ν = I Q T W (1) T is furher resriced o be block diagonal, which simplifies inference. ζ and ν are vecors ha collec he corresponding scalar innovaions described above. We esimae his model using he Gibbs sampling algorihm described in Del Negro & 5 I denoes he ideniy marix. 7

9 Primiceri (215) 6. We follow Primiceri s choice of priors, adjused for he size of our raining sample. The Gibbs sampler we use is oulined in deail in Del Negro & Primiceri (215). In conras o Cogley & Sargen (25), we do no impose any resricions on he eigenvalues of he companion form marix of he VAR. We do so boh on empirical grounds (in Amir-Ahmadi e al. (214) we show ha here is a subsanial probabiliy of emporarily explosive dynamics in US daa) and heoreical grounds (Cogley, Mahes & Sbordone (215) show ha emporarily explosive dynamics can emerge naurally in micro-founded dynamic equilibrium models when agens are learning). In order o ascerain wheher or no moneary policy shocks affec real-ime and final daa differenly and if hose effecs have changed over ime, we idenify moneary policy shocks using our VAR models. As our benchmark, we use sign resricions. An idenificaion scheme of his sor has been used in ime-varying parameer VARs wih sochasic volailiy by Benai & Lubik (214), Canova & Gambei (29), and Amir-Ahmadi e al. (214), among ohers. Srucural models used by macroeconomiss give us a good sense of he signs of he effecs of moneary policy shocks on final daa. The corresponding effecs on realime daa are less clear, and depend on he specifics of any paricular DSGE model wih boh real-ime and final daa (we presen one such model in he nex secion). This consideraion leads us o only use sign resricions on final daa, no on real-ime daa. We are hus no imposing any resricions on he impulse response funcions of real-ime daa. We resric he nominal ineres rae o no decrease afer a posiive moneary policy shock and boh final inflaion and final GDP growh o no increase afer a posiive moneary policy shock. We impose hose resricions on impac and for he firs wo periods afer impac - his is he same number of periods as chosen by Benai (21), for example. For he model wih employmen growh, we impose ha, in addiion o he resricions on inflaion and he nominal ineres rae, final employmen growh can no increase afer a posiive moneary policy shock. 6 We use 25, poserior draws, ou of 2, are used as burn-in. We hen keep every 1h draw of he remaining 5,, resuling in 5 sored draws. We have assessed and ensured convergence of he Markov Chain using he sandard diagnosics. 8

10 The equaion for he nominal ineres rae ha is recovered using our idenificaion scheme gives, by consrucion, he nominal ineres rae as a funcion of lagged realime and final daa. The lagged final daa is no direcly observable by he cenral bank when i makes is decisions every period. As such, we do no direcly inerpre he nominal ineres rae equaion as a moneary policy rule (in conras o Canova & Gambei (29)), bu insead inerpre i as he cenral bank responding o observables such as survey and forecas daa, which in urn depend on boh real-ime and final daa. This assumpion can be jusified by referring o micro-founded srucural models where he privae secor has an informaional advanage and hus knows he final daa before he cenral bank does. The nex secion describes one DSGE model wih hese feaures. Tha model shares feaures wih work by Aoki (23), Nimark (28), Lubik & Mahes (214), and Svensson & Woodford (24), for example. 7 Given ha, for compuaional reasons, we can no include addiional observables such as inermediae daa releases, he only viable alernaive would have been o resric he cenral bank o only reac o lagged real-ime (i.e. firs release) observables. This approach would have subsanially underesimaed he informaion available o he cenral bank. We hink our approach beer approximaes he acual (large) informaion ses considered by cenral banks when making heir decisions. 8 3 The Choice of Variables for Our VAR - A DSGEbased Moivaion In his secion, we describe one (admiedly, along many dimensions, very simple) DSGE model ha delivers, as is reduced form, a VAR in he same sae variables 7 To keep heir models racable, hose papers eiher assume relaively simple sochasic processes for he measuremen error ha can no mach our findings on he properies of measuremen errors, or hey assume ha he rue realizaion of he daa is observed by he cenral bank, bu only wih a lag. 8 In fuure work, we plan o relax his assumpion and insead of using final daa use he laes available vinage of daa each quarer as new daa becomes available. We view his as a separae projec since in such a VAR we could no longer sudy he join dynamics of real ime and final daa, which is our main goal in he curren paper. 9

11 ha we use in our empirical analysis. We do his o moivae our choice of variables, bu also o highligh ha we can indeed idenify moneary policy shocks using he observables described above. 9 The DSGE model does no feaure ime-varying dynamics or sochasic volailiy - hose feaures could be added by inroducing learning along he lines of Cogley e al. (215), for example. For simpliciy, he DSGE model presened here has a VAR of order 1 as is reduced form, whereas we work wih a VAR of order 2 in he empirical analysis. Addiional lags could easily be inroduced in he DSGE model, bu would no add any insigh o he exposiion. We direcly presen he linearized version of he model. The firs wo equaions give he dynamics of our real variable x and inflaion π, condiional on iid shocks z and g as well as he nominal ineres rae i : x = a x E x +1 + b x (i E π +1 ) + g + c x x (11) π = a π E π +1 + b π x + z + c π π (12) Following he lieraure cied in he previous secion, he privae secor in his model has access o final daa and hus he IS and Phillips curves are sraigh ou of sandard New Keynesian models. The variable x in micro-founded DSGE models is usually he oupu gap. Real-ime measures of he oupu gap are unforunaely chronically unreliable (Orphanides & van Norden (22)), so we choose o use alernaive measures of real aciviy in our empirical analysis insead. This lack of reliabiliy does no come from he real-ime naure of oupu daa, bu raher from he large uncerainy surrounding rend esimaes of GDP in real ime. Thus, i seems likely ha real ime measures of he oupu gap would no be used in moneary policy decisions. 9 This is no immediaely obvious since our sign resricions imply ha he nominal ineres rae is he moneary policy insrumen. The nominal ineres rae in our VAR reacs o final daa lagged once and wice, which is no in any cenral bank s informaion se. The DSGE model in his secion shows how our approach can be jusified. 1

12 Nex, we implicily define he measuremen errors: x real = x + η y (13) π real = π + η π (14) The wo non-sandard feaures of his economy are he moneary policy rule and he dynamics of he measuremen error, which are subsanially more general han wha is usually assumed in he lieraure: i = a i x real + b i π real + c i i + d i E x + e i E π + ε i (15) η = [η x η π ] = M 1 [y π i ] + M 2 [x π i ] + ρη + M 3 ε real (16) Noe ha he nominal ineres rae only reacs o real-ime daa as well as survey measures of expecaions gahered in he previous period. The moneary policy shock is denoed ε i. While his is cerainly resricive (cenral banks have access o revised daa for pas periods), his assumpion respecs he informaion consrains of acual cenral banks in ha he cenral bank can no reac o final daa for recen periods. Daa is revised for muliple years (as menioned before, Aruoba (28) uses a window of hree years o define final daa for he Unied Saes, for example), whereas here are muliple policy meeings per year for all major cenral banks. The definiion of he measuremen error dynamics (we denoe he vecor of measuremen errors by η ) allows for dependence on lagged measuremen errors as well as lagged and conemporaneous final daa. We do no sae a heory ha delivers hese dynamics, bu insead wan o show ha even wih general dynamics like hese, he dynamics are fully capured by he variables we use in our VAR. To solve he model, we can firs reduce he sysem and use he implici definiion of he measuremen errors o eliminae η. We are hen lef wih a sysem ha does include E π and E x as sae variables. Solving his model using sandard mehods for linear raional expecaions models such as Gensys (Sims (22)), we ge he 11

13 following reduced form: x π x real π real i E x +1 = A x π x real π real i E x + B g z ε real ε i (17) E π +1 E π where A and B are marices ha are reurned by he soluion algorihm for linear raional expecaions models. If we ake ime 1 condiional expecaions of his sysem, he resuling firs wo equaions of ha sysem give E x and E π as a linear funcion of x, π, x real, π real, i, and E x and E π. Those wo equaions can hus be solved for E x and E π as a funcion of x, π, x real, π real and i. Doing his and replacing he expecaion erms in he sysem above gives he reduced form dynamics in erms of he variables ha we use in our VAR: y π y real π real i = A 2 y π y real π real i + B 2 g z ε real ε i (18) where A 2 and B2 are he marices corresponding o A and B in he original soluion afer he lagged expecaions erms have been subsiued ou. In his model, he forecas error in he nominal ineres rae equaion is he moneary policy shock (all oher righ-hand side variables in he moneary policy rule are predeermined), even hough some of he righ hand-side variables in he reduced form nominal ineres rae equaion are no in he cenral bank s informaion se. If we ook his model lierally, we could be emped o use his insigh o direcly esimae he moneary policy error. Insead, we see his DSGE model as one possible 12

14 model ha delivers a reduced form in line wih our empirical specificaion. Thus, we choose o use sign resricions o idenify he moneary policy shock insead. 4 Daa As our benchmark, we use he Philadelphia Fed s real-ime daabase (Croushore & Sark (21)) o consruc a sample of annualized quarerly real-ime and final inflaion (based on he GNP/GDP deflaor) and annualized quarerly real-ime and final real GNP/GDP growh 1. As a robusness check, we also use annualized quarerly real-ime and final employmen growh (based on nonfarm payroll employmen). 11 Real-ime growh raes are calculaed using all available daa when an esimae of he laes level of he corresponding series is firs available - he growh rae of GDP over he las year a any poin in ime is defined as he raio of he laes real GDP release o he mos curren available vinage of real GDP one quarer earlier, for example. As a proxy for final daa, we use he mos recen vinage available o us. Oher approaches are cerainly possible - Aruoba (28) defines he final daa as he vinages available afer a fixed lag (for mos variables 3 years). In our secion on robusness checks we presen addiional resuls ha use his alernaive definiion of final daa. The real-ime daa is available saring in he fourh quarer of The las vinage we use is from he second quarer of 214 (incorporaing daa up o and including he firs quarer of 214). We use 4 observaions o iniialize he prior for our ime-varying-var model. For he nominal ineres rae (which is measured wihou error), we use he average effecive Federal Funds rae over each quarer. Figure 1 plos he real-ime daa and he measuremen error as defined in he previ- 1 From now on we will refer o his variable as GDP growh. 11 The employmen series is originally monhly. We define employmen wihin a quarer as he average employmen over he hree monhs belonging o ha quarer. The real-ime (or firs available) esimae is defined as he esimae available in he middle monh of he following quarer (in line wih he definiion of he oher quarerly variables). We also esimaed versions of our model using he realime and final unemploymen rae, bu here are subsanially fewer revisions in he unemploymen rae by he very naure in which he daa for he unemploymen rae is colleced - i is a survey-based measure. More deails on he difference beween real-ime and final daa for a broader se of variables can be found in Aruoba (28). 13

15 ous secion. To ge he final daa, we have o subrac he measuremen error from he real-ime daa - posiive measuremen error implies ha he real-ime measuremen is higher han he final daa. To convince yourself ha he difference beween realime and final daa can be meaningful, i suffices o look a he mid-197s: Real-ime GDP growh and employmen growh acually were lower han in he mos recen recession, bu here were subsanial revisions o ha daa laer on - he measuremen errors associaed wih hose errors is negaive, meaning ha daa was revised upward subsanially. 12 In he following secion we will analyze he ime series properies of he measuremen errors and check how heir behavior has changed over ime. We will see ha i is indeed imporan o allow for ime variaion in he dynamics of hese series. 1 Inflaion 2 Measuremen Error Inflaion Employmen Growh Measuremen Error Employmen Growh GDP Growh 5 5 Measuremen Error GDP Growh FFR Figure 1: Real-ime daa and oal revision 12 Lubik & Mahes (214) use a learning model o model he choices of a cenral bank ha only has access o real-ime daa as i makes is decisions. Jus as Orphanides (22), hey highligh ha mis-measured daa had a big impac on U.S. moneary policy in he 197s. In he curren paper, we insead focus on he impac of moneary policy shocks on boh real-ime and final daa. 14

16 5 Resuls 5.1 The Time-Varying Properies of Measuremen Errors Firs, we wan o describe how he properies of he measuremen errors in inflaion and GDP growh have changed over ime. Cogley & Sargen (25) have pioneered he use of local o ime momens o sudy changes in he dynamics of VARs wih ime-varying parameers and sochasic volailiy. In shor, hey calculae (uncondiional) momens of he daa governed by equaion 2 a each poin in ime assuming ha he coefficiens will remain fixed over ime. Tha way hey recover a sequence of momens over ime. This is feasible in heir seup because hey impose resricions on he eigenvalues of he companion form marix of he VAR. We, on he oher hand, do no impose any such resricions for he reasons menioned before. Insead, we sudy forecass from our model based on smoohed or full-sample parameer esimaes (assuming, similar o Cogley & Sargen (25), ha he coefficiens will no change in he fuure) and calculae he momens of forecass of he measuremens errors. I is imporan o emphasize ha we use hese momens of forecass as lowdimensional summary saisics ha capure he dynamics of our model. We can hink of hese momens as finie horizon versions of he summary saisics used by Cogley & Sargen (25). We focus here on one-year ahead forecass. Increasing he forecas horizon subsanially would increase he uncerainy surrounding he esimaed forecas momens exacly because we do no impose any resricions on he dynamics of he VAR. Figure 2 plos he median and 68 % poserior bands for he one-year ahead forecass of he measuremen error based on he model wih GDP growh. 13 Our model predics subsanial measuremen errors one year in advance. We can hink of hese one-year ahead forecass as a proxy for rends or more generally he persisen pars of he measuremen errors. 14 Our resuls confirm hose in Aruoba (28), who finds ha measuremen errors in many variables do no have a mean of zero. Throughou 13 The dae on he x-axis represens he dae of he condiioning informaion. 14 Inerpreing forecass as rends has a long radiion in empirical macroeconomics going back o Beveridge & Nelson (1981). 15

17 5 Forecas of Measuremen Error in Inflaion Forecas of Measuremen Error in GDP Growh Figure 2: One-year ahead forecass of measuremen errors mos he 198s he one-year ahead forecas in he measuremen error of inflaion is posiive, of an economically meaningful size, and borderline saisically significan - inflaion was iniially overesimaed during ha period. During he 199s and up o he financial crisis, inflaion insead ended o be iniially underesimaed. During he financial crisis, inflaion was subsanially overesimaed iniially. The measuremen error in real GDP growh is negaive during he 198s (meaning ha GDP growh was iniially esimaed o be lower han he final daa suggess), before urning saisically insignifican during he 199s. From 2 o he financial crisis we see an iniial overesimaion of GDP growh (wih a subsanial overesimaion during he financial crisis). We now urn o higher momens of he forecass. Figure 3 plos he volailiies of he one-year ahead forecass of measuremen errors and he associaed correlaion beween he forecased measuremen errors. Boh volailiies share a similar paern I is common in models of he ype we use here, used in conjuncion wih he ype of daa ha we analyze, ha he volailiies of he variables in he VAR share a similar paern - see for example Del Negro & Primiceri (215). 16

18 (a) Volailiy of Measuremen Error in Inflaion (b) Volailiy of Measuremen Error in GDP Growh (c) TVP Correlaion of Measuremen Error Figure 3: Volailiy and correlaion of forecased measuremen errors - high volailiy in he 197s and early 198s, a decline aferward and a noiceable upick in volailiy during he recen financial crisis. Ineresingly, he correlaion beween he measuremen errors is significanly negaive hroughou our sample, bu has an upward rend for mos of our sample ha is only broken during he early 2s. A negaive correlaion implies ha an increase in he measuremen error of GDP growh (real-ime GDP growh becomes larger relaive o final GDP growh) is associaed wih a decrease in he measuremen error in inflaion (final inflaion becomes larger relaive o he real-ime measuremen), so ha an iniial overesimaion of GDP growh ends o be associaed wih an underesimaion of inflaion. Since he magniude of he correlaion decreased subsanially over ime, his paern has become weaker over ime. To summarize, a simple model of measuremen errors ha models hem as being independen across variables and having consan innovaion variance can miss imporan feaures of observed measuremen errors. 17

19 5.2 The Effecs of Moneary Policy Shocks Over Time We firs show impulse responses for differen periods. We follow he sandard approach in he lieraure o consruc hese impulse responses: For each ime period, we draw parameers from he poserior disribuion for ha period and hen keep hese coefficiens fixed as we race ou he effecs of a moneary policy shock. We focus on impulse responses a shor horizons because ha is where we find he larges difference beween real-ime and final daa. Since we are ineresed in he differences beween he effecs on real-ime and final daa (raher han changes in he impulse responses funcions over ime per se), we use one sandard deviaion shocks, where he sandard deviaion changes over ime. This will give us a sense of how he impac of a usual shock has changed over ime. Figure 4 plos he evoluion of he nominal ineres rae o such a shock. The black line gives he poinwise median response and he gray bands cover he area from he 15h o he 85h percenile of he response wih each of he 5 shades of gray covering he same probabiliy. We can see ha here are differences on impac over ime (in paricular, he sandard deviaion of moneary policy shocks decreases), bu he overall median paern remains sable over ime. In conras, here is subsanial ime variaion in he uncerainy surrounding he median response. A some poins in ime here are some draws ha imply explosive behavior of he nominal ineres rae. Figure 5 plos he responses of real-ime and final inflaion o he same moneary policy shock. The black line and gray areas correspond o he median and he 15h o 85h perceniles of real-ime daa responses, whereas he red lines represen he responses of final daa. The bold red line is he median and he ouer dashed red bands correspond o he same perceniles as he ouermos error bands for real-ime daa (he 15h and 85h perceniles). For he mos par he responses of real-ime and final inflaion are very similar, especially afer 4 o 5 periods. The sign resricions are mosly saisfied by responses of real-ime inflaion even hough we do no impose hose resricions. Noneheless, we do find significan differences. For example, in 1979 he median impac response of real-ime inflaion is wice as large as ha of final daa. Broadly speaking, we see a larger difference (on impac) for he firs par 18

20 IRF in 1976:Q IRF in 1979:Q IRF in 1984:Q IRF in 1989:Q IRF in 1994:Q IRF in 1999:Q IRF in 24:Q IRF in 29:Q IRF in 213:Q Figure 4: Impulse response funcions for he nominal ineres rae o a one sandard deviaion moneary policy shock. of our sample (hrough he 198s). Subsanial differences in he responses beween real-ime and final inflaion are presen in he lae 197s o he lae 198s. The impulse responses for GDP growh in figure 6 show a differen paern wih more pronounced differences. On impac and for he firs few periods afer he shock his, final GDP growh is lower han real-ime GDP growh. This paern is mos pronounced in 1984 and 1989, bu persiss hroughou our sample. The magniude of hose differences is economically significan - i maers if he response o a conracionary moneary policy shock on impac is a reducion of.25 percenage poins in annualized GDP growh or.75 percenage poins (hese are roughly he magniudes in 1984:Q4). We can also see ha he sign resricions we impose on final daa are also me for mos draws of he real-ime daa response. 19

21 IRF in 1976:Q IRF in 1989:Q IRF in 1979:Q IRF in 1994:Q IRF in 1984:Q IRF in 1999:Q IRF in 24:Q4.1.1 IRF in 29:Q4.1 IRF in 213:Q Figure 5: Impulse response funcions for real-ime (gray/black) and final (red) inflaion o a one sandard deviaion moneary policy shock. So far we have sudied he marginal disribuions of he impulse responses o real-ime and final daa and compared hem o each oher. We are also ineresed in he evoluion of he join disribuion of impulse responses across real-ime and final daa. Our esimaion algorihm allows us o sudy he join poserior of impulse responses for a given horizon a each poin in ime. For each of hose ime/horizon pairs, we calculae an esimae of he join poserior of real-ime and final impulse responses (for each horizon and dae his can be hough of as a scaerplo). We call r real,i (j) he impulse response a horizon j of real-ime variable i (i {π, GDP, emp}) calculaed using a draw of VAR coefficiens a ime and r final,i (j) he response of he corresponding final variable (boh calculaed using he same parameer draw). We firs plo he median and he 15h and 85h percenile bands for he difference 2

22 IRF in 1976:Q IRF in 1989:Q IRF in 24:Q IRF in 1979:Q IRF in 1994:Q IRF in 29:Q IRF in 1984:Q IRF in 1999:Q IRF in 213:Q Figure 6: Impulse response funcions for real-ime (gray/black) and final (red) real GDP growh o a one sandard deviaion moneary policy shock. beween final daa and real-ime impulse responses of GDP growh 16 on impac (i.e. a horizon ): r final,gdp () r real,gdp (). A negaive number means ha he final daa response is smaller han he corresponding real-ime response. Figure 7 reveals ha he median difference has been negaive hroughou our sample wih a maximum of -.1 and a minimum of -.5 percenage poins, 17 meaning ha he response of final daa is larger in magniude han he response of real-ime daa as he final response is resriced o be negaive on impac and he real-ime response is negaive for mos draws. This again emphasizes ha he differences are economically meaningful - cenral banks would care abou hese magniudes. The 85h percenile of he differ- 16 The median difference for inflaion is cenered a for mos of he sample, so we omi i here. This finding is also eviden from figure Remember ha we use annualized values hroughou his paper. 21

23 ence hovers around. Thus, here is a posiive probabiliy ha he difference is a any poin in our sample as can be seen by he poin-wise error bands 18. However, he fac ha he median difference is negaive hroughou and of a economically significan magniude leads us o believe ha i is indeed imporan o ake he difference beween real-ime and final daa seriously. Furhermore, policymakers regularly worry abou wors case oucomes. We can see ha he difference beween he impac response for final and real-ime daa could be subsanially larger in magniude han wha is suggesed by he median numbers. IRF Difference (Final GDP Growh,Real GDP Growh) Figure 7: Differences beween GDP growh impulse responses on impac: The disribuion of r final,gdp () r real,gdp () over ime. For each period in our sample, we hen regress he real-ime responses a ha 18 Noe ha he error bands are calculaed based on he marginal disribuion of he differences each period. They do no direcly ake ino accoun informaion abou he difference in he proceeding and following periods (i.e. he join disribuion of he difference across periods). The bands based on he marginal disribuion only ake ino accoun informaion abou oher periods in an indirec fashion since hey are based on smoohed (full sample) parameer esimaes. For fixed coefficien VARs wih sign resricions, issues wih poinwise error bands have been highlighed by Inoue & Kilian (213), for example. I is no clear how o exend heir mehods o VARs wih ime-varying coefficiens and sochasic volailiy in general and o our quesion a hand in paricular. 22

24 poin in ime on he final responses a he same poin in ime and a consan: r real,i () = α i + βr i final,i () + u (19) Thus, each hypoheical scaerplo is summarized by wo numbers, he consan α i and he slope β. i We focus on he conemporaneous response since he differences are larges for small horizons. Since he sample size for each regression is given by he number of draws we use o calculae he impulse responses, we do no repor sandard errors for he coefficiens - hese sandard errors would be iny. If responses based on real-ime daa are jus a noisy version of he responses based on final daa, we would expec he inercep α i o be zero and he coefficien on he responses for final daa β i o be 1. Oherwise here is a bias in he real-ime daa responses relaive o he responses based on final daa ha economiss are acually ineresed in. Figure 8 shows how he inercep and he coefficien on he final-daa response vary over ime for he case of he conemporaneous response o a moneary policy shock. The gray line represens he slope of he regression β i (righ axis) and he red line represens he inercep α i (lef axis). Boh pahs show a similar paern: Unil 198 here is a clear bias. Afer 198 he coefficiens quickly move oward values ha imply no bias. Figure 9 shows he resuls for he same regressions in he case of he conemporaneous response of real-ime and final GDP growh. We see a broadly similar paern for he inercep ha moves oward zero afer 198. There is no subsanial shif in he behavior of he slope, hough. The slope is never as small as he minimum slope for inflaion, bu i also does no subsanially move oward 1 afer 198. Real-ime GDP growh responds differenly han final GDP growh o a moneary policy shock on impac in sysemaic fashion hroughou our sample. We hink of hese resuls as a cauionary ale abou he informaion conen of real-ime daa releases of GDP growh. I is imporan o remember here ha we ry o recover he rue response of real-ime daa o a moneary policy shock, no he response o a moneary policy shock ha can be recovered in real-ime. 23

25 Consan Slope Time Figure 8: Relaionship beween inflaion real-ime and final daa based impac impulse responses over ime. Inercep α π in red and slope β π in gray. 6 Robusness Checks In his secion, we presen addiional resuls: (i) a model wih employmen growh insead of GDP growh, (ii) impulse responses idenified using a recursive idenificaion scheme, and (iii) a model wih an alernaive definiion of final daa, following Aruoba (28). 6.1 Resuls for he Model Wih Employmen Growh Turning o a model wih employmen growh raher han GDP growh, we find ha he forecased measuremen error in inflaion is very similar across he wo specificaions, as shown in figure 1. The error bands for he forecas error in employmen growh do no conain for more periods han in he GDP growh case. The case for non-zero measuremen errors is hus a leas as srong for he employmen growh case as for he GDP growh case. Turning o second momens in figure 11, we see 24

26 Consan Slope Time Figure 9: Relaionship beween GDP growh real-ime and final daa based impac impulse responses over ime. Inercep α GDP in red and slope β GDP in gray. he same paern for he volailiy of he inflaion measuremen error as in he case of he model wih GDP growh. The evoluion of volailiy of he measuremen error in employmen growh is broadly similar o ha of GDP growh. The correlaion srucure, on he oher hand, is quie differen from he case of he GDP growh VAR. There is no subsanial rend in he correlaion; he correlaion is smaller in magniude and is very close o zero for subsanial periods of ime. While he assumpion of uncorrelaed measuremen errors is less of a problem when using employmen and inflaion daa, our resuls for his case sill show ha he dynamics of measuremen errors call for a more complex model han wha is ofen assumed. We find ha sochasic volailiy and a bias in he measuremen errors are presen. We nex urn o sudying he effecs of moneary policy shocks on employmen growh daa. The responses of he nominal ineres rae as well as real-ime and final inflaion are very similar o he benchmark model, so we omi hem here. For he impulse response of employmen growh o a moneary policy shock, a 25

27 Forecas of Measuremen Error in Inflaion Forecas of Measuremen Error in Employmen Figure 1: Forecased measuremen errors for employmen growh model (a) Volailiy of Measuremen Error in Inflaion (b) Volailiy of Measuremen Error in Employmen (c) TVP Correlaion of Measuremen Error Figure 11: Volailiy and correlaion of forecased measuremen errors for employmen growh model qualiaively similar picure o real GDP growh emerges in figure 12. The larges differences appear on impac. Those differences are economically significan and he final daa responses are larger in absolue value han hose of he real-ime daa. 26

28 Ploing he differences beween conemporaneous responses of real-ime and final IRF in 1976:Q IRF in 1979:Q IRF in 1984:Q IRF in 1989:Q4 IRF in 1994:Q4 IRF in 1999:Q IRF in 24:Q4.1 IRF in 29:Q4.1 IRF in 213:Q Figure 12: Impulse response funcions for real ime (gray/black) and final (red) employmen growh o a one sandard deviaion moneary policy shock. employmen growh over ime, a similar picure o real GDP growh emerges. Figure 13 shows ha he median difference is negaive hroughou and he 85h percenile band is hovering around, which implies ha here is a subsanial probabiliy a any poin in ime ha he difference in responses is negaive and ha he values of ha difference are economically meaningful. Finally, we reurn o using regressions on draws for he real-ime and final daa conemporaneous responses. Since he responses o inflaion urned ou o be similar o hose obained using he VAR wih GDP growh (wih even smaller differences beween real-ime and final daa responses), we focus on he responses of employmen growh. 27

29 .2 IRF Difference (Final Empl,Real Time Empl) Figure 13: Differences beween employmen growh impulse responses on impac Consan Slope Time Figure 14: Relaionship beween real-ime and final daa employmen growh based impac impulse responses over ime. Inercep α emp in red and slope β emp in gray. 28

30 Figure 14 shows ha he movemens in he slope and inercep coefficiens are smaller han for GDP growh. The slope coefficien is no oo far from 1, bu he inercep is always larger han, so here is sill a bias in he relaionship of realime and final daa responses on impac. There is a break in he 198s oward less bias in he relaionship beween he responses, bu he break happens laer han in he case of he GDP growh VAR (around 1985). Mos imporanly, he responses of final employmen growh are larger in magniude han hose of he real-ime daa, jus as we found in he model wih GDP growh. 6.2 A Recursive Idenificaion Scheme The resuls so far all use sign resricions o idenify moneary policy shocks. To check wheher or no our resuls are robus o oher idenificaion schemes (especially idenificaion schemes ha impose he same resricion on real-ime and final daa), he same exercise is carried ou using a recursive idenificaion scheme wih he nominal ineres rae ordered las. Condiional on reduced-form parameer esimaes, he exac ordering of he oher variables does no maer for he impulse response o a moneary policy shock (Chrisiano, Eichenbaum & Evans (1999)). 19. Thus, his recursive idenificaion scheme imposes he same resricions on he responses of real-ime and final daa - hey are ordered before he moneary policy variable. For he sake of breviy, we focus on he response o real GDP growh. I should be noed, hough, ha in he case of he recursive idenificaion scheme he response of inflaion displays a price puzzle, which is one reason why we prefer he sign resricion approach. Figure 15 displays he responses of real GDP growh. We sill see subsanial differences beween real-ime and final daa responses (wih final daa responses showing some erraic behavior in 1985 and 2). While we do no pu a lo of faih in a lieral inerpreaion of he resuls of his recursive idenificaion scheme, i is noneheless imporan o ou ha jus as in he case wih sign resricions, he differences beween real-ime and final daa responses are sill 19 The ordering of variables can in heory maer for he esimaion of he reduced-form parameers in he class of models we use - see Primiceri (25) for a discussion. 29

31 larges on impac (because of he recursive idenificaion scheme, a moneary policy shock only impacs he oher variables a horizon 1) and he response of final daa is smaller on impac (excep for he firs wo ime periods shown, where he impac responses are similar). Thus, he difference on impac in he sign resricion case hus does no seem o be an arifac of imposing resricions on only final daa responses. IRF in 1976:Q IRF in 1989:Q IRF in 24:Q IRF in 1979:Q IRF in 1994:Q IRF in 29:Q IRF in 1984:Q IRF in 1999:Q IRF in 213:Q Figure 15: Impulse response funcions for real-ime (gray/black) and final (red) real GDP growh o a one sandard deviaion moneary policy shock, using he recursive idenificaion scheme. 6.3 An Alernaive Definiion of Final Daa Aruoba (28) argued for a definiion of final daa ha uses daa published afer a fixed lag (roughly 3 years for mos of his variables). We now follow his procedure (wih a lag of 3 years) and replicae our benchmark analysis. For he sake of breviy, 3

32 we focus on he response o moneary policy shocks (he reduced form evidence on he dynamics of measuremen errors is in line wih he benchmark case). The responses of boh he nominal ineres rae as well as boh he real-ime and final inflaion raes are also similar o he benchmark case and are hus omied here. IRF in 1976:Q IRF in 1989:Q IRF in 24:Q4 IRF in 1979:Q IRF in 1994:Q IRF in 29:Q4.5.5 IRF in 1984:Q IRF in 1999:Q Figure 16: Impulse response funcions for he real GDP growh rae o a one sandard deviaion moneary policy shock, using he alernaive definiion of final daa. Imporanly, he response of real GDP growh (figure 16) shows he same bias as in he benchmark case: On impac, he response of final daa is subsanially larger in absolue magniude. The differences in his case are more shor-lived, bu also more pronounced on impac relaive o he benchmark. 31

33 7 Conclusion Measuremen errors are pervasive in real-ime macroeconomic daa. We exend he insighs of Aruoba (28) o incorporae ime varying dynamics and documen ha hese measuremen errors feaure subsanial ime-varying volailiy, can be correlaed wih a ime-varying correlaion, and are no cenered around zero. Thus, modeling real-ime daa as he sum of he final daa and a simple independen noise process can miss imporan feaures of he daa. We show ha hese facs are no a curiosiy, bu have policy implicaions: (i) These differences beween real-ime daa and final daa manifes hemselves in he subsanially differen ways ha real-ime and final daa respond o moneary policy shocks, and (ii) he real-ime responses can be subsanially biased. As such, he responses of various measures of real aciviy are larger in magniude for final daa. 32

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