Structural Shocks and the Comovements Between Output and Interest Rates

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Structural Shocks and the Comovements Between Output and Interest Rates Elmar Mertens 2010-21 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Structural Shocks and the Comovements between Output and Interest Rates Elmar Mertens∗ February 18, 2010 Abstract StylizedfactsonU.S.outputandinterestrateshavesofarprovedhardtomatchwithDSGE models. Butmodelpredictionshingeonthejointspecificationofeconomicstructureandaset of driving processes. In a model, different shocks often induce different comovements, such thattheoverallpatterndependsasmuchonthespecifiedtransmissionmechanismsfromshocks tooutcomes, aswellasonthecompositionofthesedrivingprocesses. Iestimatecovariances between output, nominal and real interest rate conditional on several shocks, since such evidencehaslargelybeenlackinginpreviousdiscussionsoftheoutput-interestratepuzzle. Conditional on shocks to neutral technology and monetary policy, the results square with simple models, like the standard RBC model or a textbook version of the New Keynesian model. Inaddition,newsaboutfutureproductivityhelptoexplaintheoverallcounter-cyclical behavioroftherealrate. A sub-sample analysis documents also interesting changes in these pattern. During the GreatInflation(1959–1979),permanentshockstoinflationaccountedforthecounter-cyclical behavior of the real rate and its inverted leading indicator property. Over the Great Moderation(1982–2006),neutraltechnologyshocksweremoredominantinexplainingcomovements betweenoutputandinterestrates,andtherealratehasbeenpro-cyclical. JELClassification: C32,E32andE43. Keywords: InterestRates,BusinessCycles,BandpassFilter,StructuralVAR,NewsShocks ∗The views in this paper do not necessarily represent the views of the Federal Reserve Board, or any other person in the Federal Reserve System or the Federal Open Market Committee. Any errors or omissions should be regardedassolelythoseoftheauthor. BoardofGovernorsoftheFederalReserveSystem,WashingtonD.C.20551. elmar.mertens@frb.gov 1

1 Introduction Understanding the relationship between output and interest rates is important to macroeconomists and policymakers alike. But basic stylized facts on their comovements in U.S. data have proved difficulttomatchwithinavarietyofsimpleDSGEmodels. Forinstance, KingandWatson(1996) study three models: a real business cycle model, a sticky price model, and a portfolio adjustment cost model. They report that this battery of modern dynamic models fails to match the business cyclecomovementsofrealandnominalinterestrateswithoutput: While the models have diverse successes and failures, none can account for the fact that real and nominal interest rates are “inverted leading indicators” of real economic activity.1 Calling interest rates inverted leading indicators refers to their negative correlation with future output. These correlations are typically measured once the series have been passed through a business cycle filter.2 Amongst the diverse failures mentioned by King and Watson, RBC models generatemostlyapro-cyclicalrealrate. But in the data, the real rate is clearly counter-cyclical, it is negatively correlated with current output. Asmentionedalready,itisalsoanegativeleadingindicator. Thiscommonlyfoundpattern ofcorrelationbetweenbc-filteredoutputandshort-terminterestratesisdepictedinFigure 1.3 What is the correct conclusion from a mismatch between implications from a dynamic model and stylized facts? Modern dynamic models always involve a joint specification of fundamental economicstructure anddrivingprocesses. Modeloutcomes, such asthe output-interest ratecorre- 1KingandWatson(1996,p.35). Theinvertedleadingindicatorpropertyhasbeenthesubjectofvariousempirical studies, for example Sims (1992) and Bernanke and Blinder (1992). The expression “negative leading indicator” is synonymous. 2Whenitcanbeappliedwithoutconfusion,Iusethephrase“businesscyclefilter”,orshort“bc-filter”,todescribe the bandpass filters developed and applied in Baxter and King (1999) and Stock and Watson (1999) or the filter of HodrickandPrescott(1997,“HP”)sinceeacheliminatesnonstationaryandotherlowfrequencycomponentsfroma time series. These filters differ mainly in that the typical bandpass filters eliminates not only cycles longer than 32 quartersbutalsothoseshorterthan6quarters,whilethislatterhigh-frequencycomponentisretainedintheHPfilter. 3Thisevidenceisbroadlyinlinewithpreviousstudies, seeforinstancethestylizedfactscollectedbyStockand Watson(1999,Table2)forbandpass-filteredU.S.data.Thefactsarealsosignificantascanbeseenfromtheconfidence intervalsplottedinPanelb)ofFigure3. 2

Figure1: Lead-lagCorrelationsforOutputandInterestRates 1.0 nominal rate real rate 0.5 0.0 −0.5 −1.0 −8 −4 0 4 8 Note: cor(y˜,x˜ )wherey˜ isbandpass-filteredper-capitaoutputandx˜ isbandpass-filterednominal, respectively t t−k t t realrate. Thisex-anterealrateisconstructedfromtheVARdescribedinSection3asr = i −E π . Quarterly t t t t+1 lagsonthex-axis. U.S.data1959–2006. lation,involvethecompoundeffectofthesetwofeatures. Yet,when“puzzling”findingsaretaken as evidence against a particular structural feature – such as sticky prices or portfolio adjustment costs–itistypicallynotacknowledgedthattheeconomymightalternativelybedrivenbydifferent typesofshocksthatyielddifferenteffectswithinthegivenstructure. Yet,morecarefully,itissimply unclear whether dynamic models fail (or succeed) because of their transmission mechanisms orbecauseofthenatureoftheirdrivingforces. To shed more light on this important issue, I provide empirical evidence about output-interest rate comovement conditional on various types of shocks: Neutral technology shocks, monetary shocks, investment specific shocks and news about future productivity and permanent inflation shocks. ThefirsttwoofthesealsodrivethemodelsofKingandWatson(1996). Thedecomposition is applied both to a continuous sample of postwar data ranging from 1959–2006 as well as as 3

to two sub-samples, commonly associated with the Great Inflation (1959–1979) and the Great Moderation(1982–2006)4,extendingtheoriginaldatasetofKingandWatson(1996)bymorethan ten years.5 There are striking results of my decomposition, which are reported in Section 4 using plotsanalogoustoFigure 1: • After conditioning on neutral technology shocks, the real rate is pro-cyclical and a positive leading indicator – just the opposite of its unconditional behavior. In response to such permanent growth shocks, this is a common outcome for variants of the neoclassical growth model, be they of the RBC or the New Keynesian variety (King and Watson, 1996; Gal`ı, 2003;Walsh,2003;Woodford,2003). • Conditional on monetary shocks, the real rate is counter-cyclical and a negative leading indicator,whichsquareswithsimpleNew-Keynesianmodels,too.6 • A strong counter-cyclical, and negatively leading behavior of the real rate is attributed to newsaboutfutureproductivity;andtoalesserextentalsotoinvestmentspecificshocks. • As in the full sample, the real rate has been counter-cyclical during the Great Inflation, mostly due to permanent inflation shocks. However, during the Great Moderation the real ratehasbeenpro-cyclical,withtechnologyshocksplayingamoreprominentroleinaccountingforitscomovementswithoutput. Thus,forthefullsamplethe“output-interestratepuzzle”isalreadydefusedbyconditioningon two widely-studied shocks: Technology and monetary shocks, which counteract each other. Such opposing effects of shocks to “supply” and “demand” are a general theme in Keynesian models (Be´nassy, 1995). This result also carries over to the sub-sample analysis, which is motivated by growing evidence, documenting substantial changes in the behavior of macroeconomic data since 4Neithersampleincludesmoredatafor2007andlater,sincethedisruptionsinfinancialmarketsintroducenumerousissueswhicharebeyondthescopeofthispaper. 5KingandWatson(1996)usedatafrom1959–1992. 6Money is neutral in RBC models, so they have not much to say here. Conditional on monetary shocks, output remainsinsteadystateandcorrelationsarezero. 4

Figure2: CyclicalbehaviorofOutputandInterestRates 5 y r 0 −5 1965 1970 1975 1980 1985 1990 1995 2000 5 y i 0 −5 1965 1970 1975 1980 1985 1990 1995 2000 Note: Time series of bandpass-filtered output, nominal and real interest rates. The real rate is constructed from the VARdescribedinSection3asr =i −E π . t t t t+1 the late 1970s and early 1980s.7 In particular, since the beginning of the 1980s—and at least until 2006—macroeconomic data has been characterized by a decrease in overall volatility, called the GreatModeration(Bernanke,2004),whereasthe1970shavebeencharacterizedbyhighandrising inflationrates,alsoknownastheGreatInflation(Sargent,1999;StockandWatson,2007). To foreshadow some of the sub-sample results, consider the time series of bandpass-filtered output and interest rates shown in Figure 2. As can be seen from the figure’s upper panel, the counter-cyclicality of the real rate is most prevalent over the Great Inflation, while it displays mostly positive comovements with output over the later part of the sample. The nominal rate behaves consistently pro-cyclical throughout the entire postwar period. What remains to be seen 7Amongst others, these changes have been attributed to differences in the conduct of monetary policy (Clarida etal.,2000;LubikandSchorfheide,2004)orvariationsinthenatureofshockshittingtheeconomy(SimsandZha, 2006). 5

from the sub-sample decompositions presented in Section 4 is how these changes can mostly be attributed to changes in the relative importance of shocks, rather than substantial changes in the conditional comovements. This result is consistent with the study of Sims and Zha (2006), who attribute changes in macroeconomic dynamics over the postwar period mostly to changes in the amplitudesofshocksratherthanchangesintheirtransmissionmechanism. ThebackboneofmycalculationsisaVARforthejointprocessof(unfiltered)output,nominal and real interest rate. The VAR serves both as a platform for identifying the structural shocks and to model the bc-filtered covariances and correlations. The identified shocks are shocks to the unfiltered data. For instance, both technology shocks have permanent effects on output but they might also have important effects on economic fluctuations. The point of bc-filtered statistics is to judge models solely on those cyclical properties, not on their implications for growth (Prescott, 1986). Inthisvein,theVARisusedtotraceouttheeffectsofshockstothebc-filteredcomponents of output and interest rates. This is done analytically using a frequency domain representation for the VAR and the bc-filters. Rotemberg (1996), Gal`ı (1999) and den Haan (2000) stress the importance of looking at conditional comovements in the context of the comovements of output with either prices or hours. In applying this general idea to output and interest rates, my specific approach is motivated by the fact that the “puzzle” in this area is typically expressed in terms of bc-filtereddata. The remainder of this paper is structured as follows: Related literature is briefly discussed in Section 2. Section 3 lays out my VAR framework for the identification of shocks as well as for decomposing the filtered covariances. Results are presented in Sections 4. Concluding comments aregiveninSection 5. 2 Related Literature To overcome the output-interest rate puzzle, Beaudry and Guay (1996) and Boldrin et al. (2001) propose models with habit preferences and frictions to capital accumulation respectively sectoral 6

factor immobility. This matches the real rate evidence by tweaking the transmission mechanism for a single kind of shock, namely technology. But the evidence presented in this study, suggests thatthestandardRBCmechanismfortechnologyworksfine.8 Itisrathertheinteractionofseveral shocksleadingtothe“puzzling”evidence.9 In this spirit, Rotemberg and Woodford (1997) report success with decision lags in a sticky price model10. The only structural shock they identify are disturbances to monetary policy. But theirsolutiontotheoutput-interestratepuzzleisbasedontheinteractionwithothershocks,which are left unidentified. This is revealed by their impulse response functions (Rotemberg and Woodford, 1997, Figure 1). Following a monetary shock, their model’s output responses are negative (respectively zero) at all lags whilst they are positive for the nominal rate. Since conditional leadlag covariances are just convoluted impulse responses, they are negative (respectively zero) at all leadsandlags. Thiscontrastswiththechangingsignsintheunconditionalcovariancesdepictedin myFigure1respectivelytheirFigure2. Likewise, Fuhrer and Moore (1995) model the inverted leading indicator property of interest rates with multiple, non-structural shocks and couch their analysis just in terms of unconditional statistics. Thispaperisanempiricalattempttodisentangletheunderlyinginteractionofthevarious structuralshocks. 8BeaudryandGuay(1996)recognizetheimportanceofconditioningontechnology,too. Theyusecointegrating propertiesbetweenoutput,consumptionandinvestmentderivedbyKingetal.(1991),whicharesimilarinspirittomy specificationdescribedinSection3.1.Whenconditioningonthesepermanentshocks,theyreportnegativecorrelations betweenoutputgrowthandtheunfilteredrealrate. Sincegrowthratesamplifyhigh-frequencyfluctuationsinsteadof focusing on business cycle characteristics, these results are not directly comparable to my approach and the puzzle framedbyKingandWatson(1996). 9Another line of attack in this area has been opened by Dotsey et al. (2003) by pointing out that the real rate evidenceissensitivetothechoiceofpricedeflatorusedforconstructingtherealrate. ThewidelyreportedcountercyclicalityoftherealrateisparticularlystrongwhendeflatingwiththeCPIwhichisusedinthispaper. Itisa-cyclical orweaklypro-cyclicalusingthedeflator forpersonalconsumptionexpenditures(PCE). Icanreplicatethiswithmy VAR,too. However,thebasicresultsforconditionalcomovementsbetweenoutputandrealrateremainvalid. These alternativeresultsforthePCEdeflatorareavailablefromtheauthoruponrequest. 10RotembergandWoodford(1997)lookonlyatoutputandthenominalrate. Theyuselineardetrendinginsteadof thestochasticproceduresconsideredhere. Stilltheyfindsimilarpatternsofcovariationandjuxtaposetheirresultsto thepuzzleposedbyKingandWatson(1996). 7

3 Empirical Methodology The variables of interest for this paper are the logs of per-capita output11, the nominal as well as (cid:183) (cid:184) 0 ˜ the real interest rate: Y = y i r . Let us call their bc-filtered component Y . The goal is to t t t t t ˜ modelandestimatehowstructuralshocksinducecomovementsbetweentheelementsof Y . t The backbone of the calculations in this paper is a VAR. Since the real interest rate, Y is not t fully observable, the VAR cannot be run directly over Y but rather over a vector of observables t X . Tosufficientlycaptureawide-rangingsetofstructuralshocks,thelogsofthefollowingeleven t variables are included in X : the change in the relative price of investments, the growth rate in t labor productivity, inflation, the ratios of consumption and investment to output, hours worked, wagemarkups,capacityutilization,thenominalinterestrate,thevelocityofmoneyandthespread betweennominallong-andshort-terminterestrates: (cid:183) (cid:184) X = ∆pI ∆a π c −y x −y l µ u i v iL −i (1) t t t t t t t t t t t t t t t where a = y − l . Apart from adding the long-term bond spread to the list of observables, this t t t specification corresponds to the VAR used by Altig et al. (2005). Details of the data construction aredescribedinSection 4. Thedynamicsof X arecapturedbya p-thorderVAR:12 t A(L)X = e = Qε (2) t t t (cid:80) where A(L) = p A Lk, A = I and E ε = 0, E ε ε0 = I. The coefficients A and k=0 k 0 t−1 t t−1 t t k forecast errors e can be estimated using OLS. Identification of the structural shocks ε will be t t concerned with pinning down Q. Since fewer shocks are identified than the VAR has equations, thereremainsanunidentifiedcomponentwithoutstructuralinterpretation. TherealrateiscomputedfromtheFisherequationr = i −E π whereinflationexpectations t i t t+1 11Allquantityvariablesshallbeper-capitawithoutfurthermention. 12For convenience, I dropped the constants such that X is mean zero. This is without loss of generality since t estimatingaVARfromdemeaneddataisequivalenttorunningaVARwithconstants. 8

aregivenbytheVAR.So Y canbeconstructedfrom X byapplyingalinearfilter: t t     y (1−L)−1h +h  t  a l      Y t =  i t   = H(L)X t where H(L) =   h i       (cid:161)(cid:80) (cid:162) r h −h p A Lk−1 t i π k=1 k andwhereh ,h andh areselectionvectorssuchthat ∆a = h X andsoon. a i π t a t The remainder of this section describes the following: First, how the structural shocks are identified (Section 3.1). This gives us Q and the conditional dynamics of the unfiltered variables can be computed from Y = H(L)A(L)−1Qε . Second, how to apply a bc-filter to the structural t t componentsof Y toobtainthedecompositionoftheirauto-covariances(Section 3.2). t 3.1 Identification of Structural Shocks In order to identify a variety of structural shocks, which are commonly investigated by theoretical and empirical studies,13 this paper applies long-, medium- and short-run restrictions to the estimated VAR model. For the benchmark VAR, whose eleven variables are specified in (1), four structuralshocks are identified: Investment-specificand neutral technologyshocks, monetary policy shocks and news shocks about future productivity. When studying the sub-sample of the data associated with the Great Inflation (1959–1979), the VAR is respecified to allow for a common trend in inflation and nominal interest rates. For this model, innovations to this nominal trend are estimated as a fifth structural shock. Since all VAR models estimated here have more observables than identified shocks, there remains also an unidentified component, which is orthogonal to the identifiedshocks. Theseidentificationschemesaredescribedbelow. 3.1.1 Investment-SpecificandNeutralTechnologyShocks Two types of technology shocks are identified from long-run restrictions as it has been done before by Fisher (2006), Altig et al. (2005), Gali and Gambetti (2009). Investment-specific shocks 13SeeforexampleRotembergandWoodford(1997),Altigetal.(2005)orSmetsandWouters(2007). 9

are identified as the sole source of variations in the permanent component of the relative price of investments, pI. Neutral technology shocks are identified as driving the permanent component of t labor-productivity (output per hour) beyond what is explained by the investment-specific technology shocks.14 Since hours are assumed to be stationary, the identification scheme used here is actually equivalent to identifying (neutral) technology shocks from the permanent component of outputasin ShapiroandWatson (1988).15 TheidentifyingrestrictionforbothtechnologyshocksimposeszerorestrictionsonA(1)−1Q:16   a 0 ... ... 0  11    a a 0 ... 0  21 22    A(1)−1Q =   · · · · ·   (3)    . . . .  . .   · · · · · The shocks are signed by imposing a < 0 and a > 0; a is unrestricted.17 Together with 11 22 21 the orthogonality of the structural shocks, this identifies the first two column of Q, which is then computedasinBlanchardandQuah (1989).18 Thestandardizedshockstoinvestmentspecificand neutraltechnologywillbedenoted εI andεa,respectively. t t 14This builds on the identification strategy of Gal`ı (1999) who identified technology shocks as the sole driver of variationsinthepermanentcomponentoflaborproductivity. 15Thisiseasytoseefromy = (y −l )+l wherel areloghours(percapita). Astochastictrendinoutputwill t t t t t be identical to the one of labor productivity if l ∼ I(0). While both the measurement of hours and the treatment t oftheirstationarityhavebeenfoundtobecontentiousissues,seeforexampleFrancisandRamey(2005),Christiano et al. (2003) and Gal`ı and Rabanal (2004), my results for bc-filtered comovements are robust to whether hours are includedinlevelsordifferences. Thecovariate-augmentedDickey-FullertestofHansen(1995)alsorejectstheunitroothypothesisforhoursinmysampleatthe1%level,whenusingtheVARdataascovariates. 16Dotsrepresentotherwiseunrestrictednumbers. 17Thenegativesignofa associatestheinvestmentspecificshockwithapermanentdecreaseintherelativeprice 11 ofinvestments. 18Analternativemethod,yieldingthesameresults,wouldbetheinstrumentalvariablesregressionsofShapiroand Watson(1988),whicharealsousedbyAltigetal.(2005). 10

3.1.2 MonetaryPolicyShocks Shocks to monetary policy are defined as unexpected deviations from endogenous policy. As in Christianoetal.(1999)orRotembergandWoodford(1997),theshort-terminterestrateisassumed tofollowalinearpolicyrule i = γ Z +γ εa +γ εI +θ(L)X +σ εm t Z t a t I t t−1 m t whereZ isavectorofcontainingallobservablevariablesX ,whichappearbeforei in(1),19 The t t t standardized policy shock εm is assumed to be uncorrelated with the other variables in the policy t rule. εm corresponds to the projection of the VAR’s forecast error in the interest rate equation off t theinnovationsofZ andoffthetwotechnologyshocks. Thevelocityofmoney,v ,andtheinterest t t ratespread,iL−i ,havebeenincludedintheVARspecification(1)toensurethatthisnullspacehas t t (atleast)arankofone.20 Apreviousworkingpaperversionofthispaper(Mertens,2007)usedthe shock series constructed by Romer and Romer (2004) for each FOMC meeting from 1966–1996 andfoundresultssimilartothosereportedhere. 3.1.3 NewsShocksaboutFutureProductivity News shocks have lately attracted considerable interest in theoretical and empirical work, with at least parts of the literature suggesting that they might play an important role in business cycle fluctuations (Beaudry and Portier, 2006; Schmitt-Grohe and Uribe, 2008; Jaimovich and Rebelo, 2009). Barsky and Sims (2009) and Sims (2009) identify news shock as “the shock orthogonal to technology innovations that best explains future variation in technology”. While they use direct data on total factor productivity, their definition is applied here to labor productivity, as measured by the second element in the vector of VAR observables defined in (1). In addition to requiring that news have no contemporaneous impact on labor productivity, news shocks are assumed to be (cid:163) (cid:164) 19ThevectorX definedin(1)canthusbewrittenasX = Z0 i v iL−i ’andZ contains∆pI,∆a ,π , t t t t t t T t t t t c −y ,x −y ,l ,µ andu . t t t t t t t (cid:163) (cid:164) 20AbsentanexactcollinearityintheVARresiduals,thenullspaceof Z0 εI εa isexactlyofrankone. t t t 11

orthogonal to the other shocks identified in this study. Computational details of the identification strategyaredescribedinAppendix B 3.1.4 NominalTrendShocks When estimating the VAR over the “Great Inflation” sample (1959–1979), nominal trend shocks areidentifiedaswell. AsdescribedinSection4.3below,forthissub-sampletheVARisadaptedto allow for a common trend in inflation and nominal rates by replacing π in (1) with ∆π and i by t t t i −π . The nominal trend shocks are then identified as the third element in the Blanchard-Quah t t factorization of the spectral density at frequency zero shown in (3).21 Such a trend could capture slow-moving fluctuations in policymakers’ preferences for inflation—be these variations in their actualinflationtargetasinIreland(2007)orself-fullfilingmarketperceptionsasinAlbanesietal. (2003)orSargent(1999). 3.2 Decomposition of BC-Filtered Covariances Summarizing the previous discussion, the impulse responses of the unfiltered variables in Y are t given by H(L)A(L)−1Q. These do not only trace out the business cycle responses of Y to the t structural shocks ε , but also how the shocks induce growth as well as high-frequency variations. t Themotivationforbc-filteringistofocusonlyonthebusinesscycleeffects.22 Formally,itremains 21TheidentificationofBlanchardandQuah(1989)reliesonaCholeskifactorizationofthespectraldensityofX t at frequency zero. In (3), the two technology shocks are identified as the first two elements of this lower triangular factorization.ThisfactorizationallowsbothtechnologyshockstohavepermanenteffectsonthethirdVARelement— inflationinthiscase—aswell. However,thelong-runeffectsofbothtechnologyeffectsoninflationareinsignificant. Thishasbeenverifiedbytwotypesofstatisticaltests,bothrelyingoninstrumentalvariablesregressionsasinShapiro andWatson(1988). First, aWaldtestfailstorejectthatbothtechnologycoefficientsarejointlyzerowithap-value of67%,whenestimatingtheunrestrictedinflationequationasinBlanchardandQuah(1989). Secondly,theinflation equationcouldalsobeestimatedwhileimposingtheover-identifyingofzerolong-runeffectsfromtechnologyshocks on inflation. The J-statistic of Hansen (1982) fails to reject the overidentifying restrictions with a p-value of 42%. To ensure that the covariance decompositions add up exactly, I employ the Blanchard-Quah factorization which is equivalenttotheunrestrictedinflationequation,fordetailsonthisequivalenceseeforexampleFrancisetal.(2003). 22Business cycle filters have also been criticized for creating spurious cycles, originally by Harvey and Jaeger (1993) and followed by Cogley and Nason (1995) as well as in the discussion between Canova 1998a; 1998b and Burnside(1998).WhilstmostofthesepapersfocusedontheHPfilter,theiranalysisalsoappliestothebandpassfilter. Butthebc-filteredstatisticsemployedherecanbeperfectlyjustifiedfromtheperspectiveofmodelevaluationinthe frequencydomain: Thegoalisnottomatchdataandmodeloverallspectralfrequencies,butonlyoverasubsetwhich isassociatedwith“businesscycles”. FortheUnitedStatesthisistypicallytakentobe6to32quartersfollowingthe 12

to apply a bc-filter and to decompose the filtered lead-lag covariancesinto the contributionsof the structuralshocks. Thecomputationsarestraightforwardtoperforminthefrequencydomainanda briefoverviewisgivenin A. ˜ Since the bc-filtered variables in Y are covariance-stationary, their lead-lag covariances exist t and so does their spectrum. They can be computed from the VAR parameters and the filters H(L) andB(L). Toeasenotation,theimpulseresponsesof Y afterapplyingthebc-filterarewrittenas C ˜ (L) ≡ B(L)H(L)A(L)−1Q so that the bc-filtered spectrum can be expressed as S (ω) = C ˜ (e−iω)C ˜ (e−iω)0. For each fre- Y˜ quencyω,thisissimplyaproductofcomplex-numberedmatrices. Thelead-lagcovariancematri- ˜ cesofY canberecoveredfromthespectruminwhatisknownasaninverseFouriertransformation t (cid:90) 1 π Γk ≡ EY ˜ Y ˜ = S (ω)eiωkdω (4) Y˜ t t−k 2π Y˜ −π Since the structural shocks are orthogonal to each other, the decomposition of the covariances Γk is straightforward. First, the spectrum is computed conditional on each shock. Then, the Y˜ conditional lead-lag covariances follow from an inverse Fourier transformation, analogously to equation (4). To fix notation, the shocks are indexed by s and J is a square matrix, full of zes ros except for a unit entry in its s’th diagonal element. The spectrum conditioned on shock s is (cid:80) S (ω) = C ˜ (e−iω) J C ˜ (e−iω)0. Since J = I the conditional spectra add up to S (ω). This Y˜|s s s s Y˜ carries over to the coefficients Γk ≡ E(Y ˜ Y ˜ |s) from the inverse Fourier transformation of Y˜|s t t−k (cid:80) S (ω)suchthat Γk = Γk. Y˜|s s Y˜|s Y˜ NBERdefinitionsofBurnsandMitchell(BaxterandKing,1999;StockandWatson,1999).Formalconceptsofmodel evaluationinthisveinhavebeenadvancedbyWatson(1993),Dieboldetal.(1998),aswellasChristianoandVigfusson (2003). Using the concept of the pseudo-spectrum this extends also to nonstationary variables, notwithstanding the analysisofHarveyandJaeger(1993). 13

4 Decomposition Results This section presents the results for the VAR described in the previous section. The VAR is estimated using quarterly data for the United States from 1959 to 2006. The construction of the data is described in C. The lag-length of each VAR used in this paper has been selected based on a comparison of information criteria, bootstrapped Portmanteau tests and the ability of the VAR to replicatebc-correlationcomputedfromapplyingtheBandpassfilterofBaxterandKing(1999)directlytothedata. Detailscanbefoundintheweb-appendixforthispaper,whichhasbeenattached attheendofthisdocument. To assess the statistical significance of the results, bootstrapped confidence intervals are computed for each shock. As discussed by (Sims and Zha, 1999) these are best interpreted as the posterior distributions from a Bayesian estimation with flat prior. The small sample adjustment of Kilian (1998) is used to handle the strong persistence of the VAR23. In a first round, the small sample bias of the VAR coefficients is estimated from 1,000 Monte Carlo draws. In the second round, the posterior distribution is constructed from 2,000 draws using the VAR adjusted for the smallsamplebias. The web-appendix documents the robustness of some of the key results reported below, when applyingthedecompositiontoamuchsmaller,trivariateVAR,usingonlydataonoutput,inflation andthenominalinterestrate: • Conditional on (neutral) technology, the real rate is pro-cyclical and a positive leading indicator. • Technology shocks play a more substantial role over the Great Moderation, accounting for anunconditionallypro-cyclicalrealrate. • The counter-cyclical, negatively leading behavior of the real rate during the Great Inflation isexplainedbypermanentshockstoinflation. 23Estimating the VAR over the full sample of postwar data, the largest root of the VAR amounts to 0.9905, see. Whenperformingthebootstraps,arejectancesamplingisappliedconsideringonlystableVARssuchthatthelongrun restrictionscanbeapplied. 14

4.1 Full Sample Aresultwhichconsistentlyholdsoverthevarioussamplesconsideredhereisthatneutraltechnologyshocksinduceastronglypro-cyclicalrealratewhichisalsoapositiveleadingindicatorforup tooneyear. Forthefullsampleofdata(1959–2006),thisisdepictedintheupperpanelofFigure3 which decomposes the filtered covariances between output and the real rate. Covariances add linearly, so they are a natural measure for the decomposition. The total covariances in Figure 3 are justarescalingofthecorrelationsreportedinFigure1above. Figure3showsfurtherthatmonetary shocks and news about future productivity induce negative covariances at leads between zero and uptotwoyears. (Toamuchlesser,andlargelyinsignificantextent,thesameistrueforinvestment specifictechnologyshocks.) Turningtothenominalrate,allidentifiedshockscontributetoitsprocyclicalandpositivelyleadingcomovementwithoutput. Forbrevity,theseresultsaredocumented inaweb-appendix,attachedattheendofthisdocument. When adding up the conditional covariances induced by the four structural shocks, there remains an unidentified component, inducing a counter-cyclical real rate, comoving negatively with output at leads and lags of up to a year. Whilst monetary policy and news shocks help to account for the counter-cyclical, negatively leading real rate, their strength does at best offset the procyclical and positively leading behavior induced by technology shocks. Still, the results clearly demonstrate that conditional comovements are far from being as puzzling as suggested by King andWatson(1996). The comovements induced by monetary policy and news shocks are also significant as can be seen from the lower panel of Figure 3. There is much larger uncertainty and small sample bias associated with the technology shocks, reflecting that they are identified from the more uncertain, long run behavior of the data (Christiano et al., 2006). The small bias becomes apparent when comparing the point estimates of the VAR (dashed line) with the mean of the bootstrapped small sample distribution.24 When considering shape and location of the bootstrapped distributions the 24Unfortunately,thesmallsampleadjustmentusedfromKilian(1998)isonlypartiallysuccessfulhere. Thebootstrapsaregeneratingbytreatingthepointestimatesastruevalues. Asacounterfactualexercise,thebootstrapshave alsobeenrepeated,generatinglongersamples,withtheresultthatthemeanofthebootstrappeddistributionsbecomes 15

Table1: VarianceDecomposition Shocks Variables Mon. Policy N-Tech I-Tech News Nom. Trend Rest FullSample(1959–2006) Output 6.36 15.09 7.29 26.70 44.56 Nom. rate 11.24 11.04 3.72 43.91 30.09 Realrate 9.92 31.43 3.27 17.71 37.67 Inflation 4.30 2.60 13.24 29.06 50.80 Hours 5.11 16.39 6.93 31.16 40.40 GreatModeration(1982–2006) Output 0.88 56.50 5.39 15.23 22.00 Nom. rate 3.31 14.24 2.18 55.83 24.45 Realrate 3.54 16.85 6.84 43.56 29.21 Inflation 0.52 18.22 1.43 14.81 65.02 Hours 3.54 28.49 4.39 37.06 26.52 GreatInflation(1959–1979) Output 3.74 5.57 12.41 18.16 30.59 29.53 Nom. rate 7.49 5.63 6.28 14.67 39.32 26.60 Realrate 11.76 4.25 22.75 6.84 28.06 26.33 Inflation 3.71 4.83 15.82 14.36 36.20 25.08 Hours 4.74 2.73 11.58 22.67 30.83 27.44 Note: Variancedecompositionforbandpass-filteredvariablescomputedfromtheVARsdescribedinSections3(Full Sample and Great Moderation) and 4.3 (Great Inflation). “Mon. Policy” is the monetary policy shock, “N-Tech” and “I-Tech” refer to the neutral, respectively investment specific technology shocks and “News” refers to the news shocksaboutfutureproductivitywhoseidentificationisdescribedinSection3.1.AsdescribedinSection4.3,theVAR estimatedfortheGreatInflationallowsforacommontrendininflationandrealrates. Shockstothistrendarelabeled “Nom. Trend”above. conditional correlations for neutral technology shocks, it is however reassuring that they are tilted towardspro-cyclicalpointestimates. Since covariances can be negative as well as positive, it is ambiguous to put a number like “percentage explained” on their decomposition. Table 1 reports variance shares for the bandpassfiltered fluctuations induced by each shocks for output, inflation, interest rates and hours. For the full sample of postwar data, the identified shocks account for about half the business cycle fluctuations in output, inflation and hours and up to two-thirds of the variations in nominal and real interest rates — mostly due to neutral technology shocks and news about future productivity. identicaltothepointestimates,justassuggestedbylargesampletheory. 16

In the sub-sample analysis discussed below, the unexplained share of variations will mostly drop belowathirdofthetotal. Sofarthecyclicalbehaviorofoutputandinterestrateshasbeendescribedintermsofbandpassfiltered covariances. Figure 4 documents the unfiltered impulse responses estimated for the full sample,andtheresultsresonateverywellwiththefilteredcomovementsdiscussedabove. Because of the non-trivial effects of bc-filtering25 it is however not a foregone conclusion, that the picture emergingfromtheimpulseresponsesshouldmirrortheresultsforthebc-filteredcomovements. A monetary policy shocks increases the nominal rate for about a year, causing a prolonged contractioninoutputforuptofouryearsaswellasareductionininflation,26. Insum,therealrate riseswhiletheeconomycontractsafteramonetaryshock. Byconstruction,theneutraltechnology shock raises output permanently. As would be predicted by standard neoclassical growth models, e.g. such as the ones considered in King and Watson (1996), the technology induced growth in outputisaccompaniedbyanincreaseintherealrateforuptotwoyears. Corroboratingtheresults of Sims (2009), output and hours drop in response to a news shock while the consumption output ratio increases.27 After an initial increase, the real rate drops as output recovers, in line with their counter-cyclicalcomovementsdocumentedforthebusinesscyclefrequenciesinFigure3. Thefull sampleresponsesofrealrateandoutputtoinvestmentspecificshocksarelargelyinsignificant.28 25ForacriticaldiscussionseeforinstanceCanova(1998a)orKingandRebelo(1993). 26The reduction in inflation follows after an initial upwards blip in the first period after the shocks, mirroring previousresultsofapricepuzzle,whichisforexampleevidentintheestimatesofAltigetal.(2005)aswell. 27By construction, the response of output per hours (labor productivity) to a news shocks is zero. In addition, the sign of the news shock is identified such that “positive” news lift the consumption output ratio. The sign of the (identical)impactresponsesofoutputandhoursishoweverunrestricted. 28After an initial increase in output, the estimated output responses to an investment specific shock displays even aprolonged, buttemporarydeclineinrealactivity. Thelong-runeffectofaninvestmentspecificshockonoutputis positive,butonlyathorizonsbeyondthetenyearsshowninFigure4. Re-estimatingseparateimpulseresponsesfor theGreatModerationandtheGreatInflation, theoutputresponsestoinvestmentspecificshocksaregenerallymore positive,mirroringalsotheresultsreportedbyFisher(2006)whousesasimilarsub-samplesplit. 17

Figure3: OutputandRealRates: ConditionalComovements (a)CovarianceDecomposition (b)ConditionalCorrelations Note: Bandpass-filtered moments, Cov(y˜,r˜ ), for U.S. data (1959–2006) computed from the VAR described in t t−k Section3. Quarterlylagsonthex-axis. Lowerpanel: Bootstrappedstandard-errorsbandswithKilian(1998)’ssmall sampleadjustment(1000drawsinfirstround, 2000drawsinsecond). Percentilesareshaded: 95%(light)and68% (dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR. 18

Figure4: ImpulseResponses Note: Estimates for U.S. data (1959–2006) from VAR, equation (1) described in Section 3. Responses of unfiltered variables to a one-standard deviation shock (H(L)A(L)−1Q).Bootstrappedstandard-errorsbandswithKilian(1998)’ssmallsampleadjustment(1000drawsinfirstround,2000drawsinsecond). Percentiles areshaded:95%(light)and68%(dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR.Quarterlylagsonthex-axis. 19

4.2 Great Moderation Reestimating the VAR described in Section 3 with data from the Great Moderation (1982–2006) yieldsconsiderablydifferentresultsthanwhathasbeenfoundaboveforthefullsampleofpostwar data. However, just as in the full sample analysis, neutral technology shocks induce a pro-cyclical andpositivelyleadingrealrate. Strikingly,theunconditionalbehavioroftherealratehasbeenprocyclical and a positive leading indicator of output during the Great Moderation—just the opposite ofwhatKingandWatson(1996)havebeenpuzzledaboutusingdatafrom1959–1992. Toagoodextenttheshiftintheoverallcyclicalityofrealratescanbeattributedtoanincreasein therelativeimportanceofneutraltechnologyshocksasasourceofbusinesscyclefluctuations(see the middle panel of Table 1). Similarly, shocks to investment specific technology and monetary policyinduceapro-cyclicalrealrate,howeverwithoutbeingquantitativelyrelevant. Thesecondfactorbehindthechangeintheunconditionalcomovementsisadifferentbehavior in the responses to news shocks, which are estimated to induce a pro-cyclical real rate over the Great Moderation. Similar to the full sample results, the unfiltered impulse responses to news shocks still display a drop in aggregate activity while the real rate increases on impact before it drops shortly thereafter. However, in subsequent periods the movements in the real rate follow more closely the subsequent recovery in aggregate activity, compared to the more prolonged drop of the real rate after a news shock in the full sample estimates shown in Figure 4. (The impulse responsesforthesub-samplesarereportedinaweb-appendixattachedattheendofthisdocument.) Thesesubtlechangesintheunfilteredresponsescauseasignificantshiftinthecyclicalcomponents ofrealrateandoutputaswitnessedbythedecompositionresultsshowninFigure 5. 20

Figure5: OutputandRealRates: ConditionalComovements(GreatModeration) (a)CovarianceDecomposition (b)ConditionalCorrelations Note: Bandpass-filteredmoments,Cov(y˜,r˜ ),forU.S.data(1982–2006)computedfromVARdescribedinSect t−k tion 3. Quarterly lags on the x-axis. Lower panel: Bootstrapped standard-errors bands with Kilian (1998)’s small sampleadjustment(1000drawsinfirstround, 2000drawsinsecond). Percentilesareshaded: 95%(light)and68% (dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR. 21

4.3 Great Inflation A notable feature of the Great Inflation (1959–1979) is the highly persistent rise in inflation, consistent with a common stochastic trend in inflation and nominal interest rates.29 Accordingly, the VAR specification described in Section 3 has been adapted to allow for such a common trend, by replacing the inflation rate with the change in inflation, and by replacing the nominal interest rate with the spread between the nominal interest rate and the contemporaneous inflation rate. This allowstoidentifypermanentshockstothecommontrendininflationandnominalinterestratesas describedinSection 3.1. In addition, the velocity of money, capacity utilization and wage markups have been dropped from the VAR, reflecting their very persistent behavior over this limited sample.30 The vector of VARobservablesusedfortheGreatInflationsampleisthus (cid:183) (cid:184) X = ∆pI ∆a ∆π c −y x −y l i −π iL −i t t t t t t t t t t t t t In general, the real rate decomposition for the Great Inflation resembles what has been found above for the full sample: The real rate has been pro-cyclical and a negative leading indicator, conditional on neutral technology shocks it has been pro-cyclical and positively leading, and vice versa when conditioning on monetary policy shocks. However, both shocks play quantitatively a smaller role in accounting for the comovements between output and the real rate over the Great Inflation. As before, part of the real rate’s counter-cyclical behavior is accounted for by news shocks. 29SeeforexampleCampbellandShiller(1987),StockandWatson(2007)orCogleyetal.(2010). 30Sincetherearelessobservables,andanadditionallong-runshock,onecannotimposeasmanyzerorestrictions on the impact responses to monetary policy as in the larger VAR specified in equation (1). As a compromise, the impactresponsesofthepriceofinvestments,aswellastheratiosofconsumptionandinvestmenttooutputhavebeen leftunrestricted(asbefore,theresponseofthetermspreadremainedunrestrictedaswell). FortheGreatInflation,the identifiedshocksmaythusalsoreflectsomeofthesystematicresponsesofmonetarypolicytoaggregateactivity. But theimpulseresponsesreportedintheweb-appendix,stilldisplaythetypicallycontractionarybehaviorofactivityand inflationafterapolicyshocks. 22

Figure6: OutputandRealRates: ConditionalComovements(GreatInflation) (a)CovarianceDecomposition (b)ConditionalCorrelations Note: Bandpass-filtered moments, Cov(y˜,r˜ ), for U.S. data (1959–1979) computed from VAR allowing for a t t−k commontrendininflationandnominalratesasdescribedinSection4.3. Quarterlylagsonthex-axis. Lowerpanel: Bootstrapped standard-errors bands with Kilian (1998)’s small sample adjustment (1000 draws in first round, 2000 draws in second). Percentiles are shaded: 95% (light) and 68% (dark). The sold line is the mean of the bootstraps. ThedashedlineisthepointestimateoftheVAR. 23

What is new, is that a further part of the hitherto unexplained, counter-cyclical remainder of the real rate, can now be attributed to permanent inflation shocks. In response to a permanent increaseininflation,outputrisestemporarilywhilstnominalratesdonotsufficientlyincreasewith inflation, causing a drop in real rates. These result are consistent with the (unfiltered) impulse responses found by Ireland (2007) when estimating a New-Keynesian DSGE model with random walk shocks to an exogenous inflation target.31 The web-appendix reports similar decomposition resultswhenallowingforanominaltrendovertheentiresampleofpostwardata. 5 Conclusions Aneconomicmodelspecifiesrestrictionsonhowtheeconomyrespondstoexogenousforces. Data maynotconformtothesepredictions,eitherbecausethespecifiedresponsesarewrong,orbecause the set of forces considered in the model does not sufficiently capture those impinging on the real world (or both).32 King and Watson (1996) report an output-interest rate puzzle, because of discrepanciesintheunconditionalcorrelationsofoutputandinterestratesinU.S.dataandavariety of calibrated models. But it appears in a different light, once the bc-statistics are conditioned on structural shocks. At the root of the “puzzle” are not so much the transmission mechanisms of their models, but rather the interaction of several shocks and their relative importance in different sub-samplesofthedata. Threepointsstandout: Conditional on technology shocks, the comovements between output and real rate lines up fairly well with standard models, be it the standard RBC model or the technology channel of textbook New-Keynesian models as studied by Gal`ı (2003), Walsh (2003) or Woodford (2003). For all specifications considered, the contemporaneous correlation between (bc-filtered) real rate andoutputispositive. Likewise,therealrateisapositiveleadingindicatorofoutputforalmostone 31Ireland(2007)usesdatafrom1959–2004. 32A case in point is how Christiano and Eichenbaum (1992) add government spending shocks to RBC theory to resolvetheDunlop(1938)/Tarshis(1939)/Keynes(1939)debateontheoverallcyclicalityofrealwages, theissueis also summarized by Sargent (1987, p. 487). In a similar vein, Baxter and King (1991) enrich the RBC model with demandshocks. 24

year. Unconditionally, the real rate is widely reported to be just the opposite – namely counter- or a-cyclical and a negative leading indicator. Attempts to match this with technology shocks appear tobegoinginthewrongdirection.33 The overall behavior of the real rate must be the outcome of an interaction of several shocks. Indeed: When conditioning on monetary shocks, the real rate is counter-cyclical and a negative leadingindicatoraspredictedbythesimpleNew-Keynesianmodels. Inaddition,newsaboutfuture productivity contribute to the counter-cyclical behavior of the real rate over the postwar period. Such opposing responses to “supply” and “demand” shocks are a general theme in Keynesian models(Be´nassy,1995). The comovements between output and the real rate differ when comparing the Great Inflation (1959–1979)withtheGreatModeration(1982–2006). Asinthefullsample,therealratehasbeen counter-cyclical during the Great Inflation, mostly due to permanent inflation shocks. However, during the Great Moderation the real rate has been pro-cyclical, with technology shocks playing a moreprominentroleinaccountingforitscomovementswithoutput. Acknowledgments This paper is based on the first chapter of my dissertation at the University of Lausanne. I would liketothankthemembersofmythesiscommitteeJean-PierreDanthine(chair),PhilippeBacchetta, Robert G. King and Peter Kugler, the editor, two anonymous referees as well as V.V. Chari, Jordi Gali, Jean Imbs, Jinill Kim, Andrew Levin, David Lopez-Salido and seminar participants at the BoardofGovernors,theFederalReserveBankofMinneapolis,theUniversityofLausanneandthe SwissNationalBankfortheirmanyhelpfulcommentsandsuggestions. Allremainingerrorsareof coursemine. ThemainpartofthispaperhasbeenwrittenwhileworkingatStudyCenterGerzensee 33SeeforinstancetheRBCmodificationsofBeaudryandGuay(1996)andBoldrinetal.(2001)withhabitpreferencesandfrictionslikecapitalaccumulationandsectoralfactorimmobility. BeaudryandGuay(1996)recognizethe importance of conditioning on technology. But since they use a quite different detrending method their results of a counter-cyclicalrealrateevenafterconditioningontechnologyarehardtocomparewiththeresultsinthisstudy. See alsoFootnote8. 25

andIwouldliketothanktheCenterforitsgeneroussupportduringmydissertation. Thisresearch has been carried out within the National Center of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK). NCCR FINRISK is a research program supported by theSwissNationalScienceFoundation. Earlierdraftsofthispaperhavealsobeencirculatedunder the title “Puzzling Comovements between Output and Interest Rates? Multiple Shocks are the Answer.” Appendix A Applying BC-Filters to VAR Model Conceptually, the analysis in this paper is applicable to a wide class of bc-filters, including the HP-Filter,theapproximatebandpassfilterofBaxterandKing(1999)aswellastheexactbandpass filter. A classic reference for the necessary tools is Priestley (1981), and similar techniques are employed by Altig et al. (2005) and Chari et al. (2007). For the computations it is key that the bc-filter can be written as a linear, two-sided, infinite horizon moving average whose coefficients (cid:80) sumtozero:34 Y ˜ ≡ B(L)Y whereB(L) = ∞ B Lk andB(1) = 0. t t k=−∞ k The bandpass-filter is a such a symmetric moving average. It is explicitly defined in the frequencydomainandmostofmycalculationsarecarriedoutinthefrequencydomain. Forfrequencies ω ∈ [−π,π], evaluate the filter at the complex number e−iω instead of the lag operator L. This is also known as the Fourier transform of the filter which represents it as a series of complex numbers (one for each frequency ω). Requiring B(1) = 0 sets the zero-frequency component of the filtered time series to zero. For instance, the bandpass filter passes only cycles between two 34Ofcourse,somecoefficientsB canbezero. SoB(L)couldalsobethefirst-differencefilter. Butmeaningfulbck filtersshouldalsobesymmetric,suchthattheyhaveazerophaseshift. Otherwise,comovementsoveronefrequency band,saybusinesscycles,couldbeattributedbythefiltertootherfrequencies,likegrowth. 26

andahalfandeightyears. Formonthlydata,itisspecifiedasfollows:   (cid:163) (cid:164)  1 ∀|ω| ∈ 2π , 2π B(e−iω) = 8·12 2.5·12   0 otherwise This VAR framework is also capable of handling unit roots in Y . By construction, H(L) and t thusY hasaunitrootsuchthatH(1)isinfinite. Whencomputingthebc-filteredspectrumS (ω), t Y˜ B(1) = 0 takes precedence over this unit root. It is straightforward to check that H(eiω) is well defined everywhere, except at frequency zero. So we can think of the nonstationary vector Y as t havingapseudo-spectrumS (ω) = C(e−iω) C(e−iω)0 whereC(L) = H(L)A(L)−1Qandwhich Y existsforeveryfrequencyontheunitcircleexceptzero. B Identification of News Shocks This section describes the identification of news shocks about future productivity. The identification strategy follows Barsky and Sims (2009) and Sims (2009) who identify news shock as “theshockorthogonaltotechnologyinnovationsthatbestexplainsfuturevariationintechnology”. While they use direct data on total factor productivity, their definition is applied here to labor productivity, as measured by the second element in the VAR observables defined in (1). In addition to requiring that news have no contemporaneous impact on labor productivity, they are assumed to be orthogonal to the other shocks identified in this study (neutral and investment-specific technology shocks as well as monetary policy shocks.35 This orthogonality constraint can be easily accommodatedbyextendingthemethodof Uhlig(2004)asexplainedbelow. Forthismethoditisconvenienttoexpresstheidentificationintermsofanorthonormalmatrix ˜ Q and not in terms of the matrix of impact coefficients Q defined in equation (2) above. These two are related via the Cholesky decomposition of the VAR’s forecast error variance, Σ = Ee e0: t t 35When applying the VAR decomposition to the Great Inflation as discussed in Section 4.3, the news shocks are alsorequiredtobeorthogonaltothenominaltrendshocks. 27

Ψ ≡ chol(Σ) and Q ˜ ≡ Ψ−1Q. By construction we have Q ˜ Q ˜0 = I.We seek the column of ˜ Q, associated with the news shock. Denoting this column as q˜, it solves the following variance maximizationproblem (cid:195) (cid:33) (cid:195) (cid:33) (cid:88)50 (cid:88)50 max h0 C Ψq˜q˜0Ψ0C0 h = q˜0 C0Ψ0h0 h ΨC q˜ (5) a k k a k a a k q˜ k=0 (cid:124) k=0 (cid:123)(cid:122) (cid:125) ≡S subjectto q˜0q˜= 1 and (Ψ−1Q ¯ )0 q˜= 0 (6) whereC arethecoefficientsoftheVAR’svectormovingaveragerepresentation,C(L) = A(L)−1, k ¯ h selectsthegrowthrateoflaborproductivityfromtheVARdefinedin(1). Qisamatrixwithfour a columns,containingthepreviouslyidentifiedthreecolumnsofQaswellasafourthcolumnensuring that news shocks have no contemporaneous effect on labor productivity. This fourth column contains the slopes from regression the VARs forecast errors on the projection of the innovation in labor productivity off the space of forecast errors spanned by the previously identified three shocks.36 Uhlig (2004) solves the above problem without the orthogonality constraint (6). In this case the problem reduces to finding the largest eigenvector of the positive definite matrix S defined in (5)withq˜beingitsnormalizedeigenvector. Apreviousworkingpaperversionofthispaper(Mertens,2007)hasextendedUhlig’scomputationstohandletheorthogonalityconstraint(6)asfollows:37 LetB beanorthonormalbasisforthe nullspaceofΨ−1Q ¯ . Thesetofpermissiblevectorsq˜isthen{q˜: q˜= Bz ∀ z ∈ Rn}wherenisthe dimensionofthenullspace. Reparametrizedintermsofz,theproblemreducestosetz equaltothe normalizedeigenvectorofofB0SB associatedwithitslargesteigenvalue,denotedz∗. Thesignof z∗ (and thus the sign of the news shock) is determined by making it raise the consumption-output ratio on impact. It should be noted that this sign restriction only affects the sign of first moments 36ThefourthcolumnofQ¯ thuscontainstheslopesfromregressingtheforecasterrorse ontotheresiduale˜a inthe t t regressionea =β εI +β εa+β εm+e˜a. t 1 t 2 t 3 t t 37BarskyandSims(2009)proceedsimilarly. 28

(e.g. impulseresponses),butnotthesignofsecondmoments(e.g. conditionalcovariances),which areattheheartofthispaper. C Data The dataset used for this study has been collected from FRED38 and the Federal Reserve Board’s G17 releases, which are publicly available. All data is quarterly and expressed in logs.39 Quantity variableshavebeenconvertedtoper-capitaunitsusingtheBureauLaborStatistics(BLS)measure ofthecivilianpopulationover16,interestratesandinflationrateshavebeenannualized. The data is largely identical to the time series used by Altig et al. (2005) for the sample 1959 to 2006. The only major difference is the use of NIPA data to construct the relative price of investments as in Gali and Gambetti (2009) and Justiniano and Primiceri (2008), who construct the (log) relative price of investments as a weighted average of the (log) deflators of nondurables and services consumption minus the weighted average of the (log) deflators for investment and durable consumption, with the weights given by the relative (nominal) shares of each spending category,40 whereasAltigetal. (2005)usedataconstructedby Fisher(2006). In detail, output (y ) is measured by real GDP per-capita and inflation is measured by the t log-difference in the CPI index. As in Altig et al. (2005), the (log-) ratios of consumption and investmenttooutput,c −y andx −y ,arecomputedfromnominalconsumptionandinvestments t t t t as well as nominal GDP.41 Hours per capita, l are calculated from the BLS measure of nonfarm t business hours. Wage markups, µ = y$ −h −w$, are computed from the difference of nominal t t t t GDP less hours worked, less the BLS measure of nominal compensation per hour in the nonfarm business sector. Capacity utilization, u , is measured for the manufacturing industry as reported t 38FederalReserveEconomicData,maintainedbytheFederalReserveBankofSt. Louis. 39The nominal interest rate is computed from the log of the annualized gross interest rate. i = log(1+I /100) t t whereI istheannualizedpercentageratequotedbyFRED. t 40The deflators are constructed as the ratios of nominal to real expenditure in each category, using the followingformulas: PCDG/PCDGCC96(durables),PCND/PCNDGC96(nondurables),PCESV/PCESVC96(services)and FPI/FPIC1(investment). 41Consumption is the the sum of nondurables, services and government consumption. Investment is the sum of householdconsumptionofdurablesandgrossprivatedomesticinvestment. 29

by the Federal Reserve Board’s G17 release.42 The quarterly average of nominal yields on three month Treasury Bills is used for the nominal short-term interest rate, i . The term spread of nomt inal interest rates is computed from the short-term interest rate and the 10-year Treasury constant maturity rate. The velocity of money, v = y$ −m , is computed from the log-difference between t t t nominal GDP and the money-at-zero-maturity measure computed by the Federal Reserve Bank of St. Louis. Web-Appendix This is the web-appendix with supplementary details of the VAR lag-length selection (Section I), additional results for the large VARs, which are also used in the main paper (Section II) as well as results for smaller, trivariate VARs, which use only data on output, inflation and the nominal short-terminterestrate(Section III). I Lag-Length Selection Specification of the VAR’s lag-length is based on a comparison of information criteria, bootstrapped Portmanteau tests and the ability of the VAR to replicate bc-correlation computed from applyingtheBandpassfilterofBaxterandKing(1999)directlytothedata.43 Lag-Lengthselection tablesandplotsoftheautocovariancefunctionsfortheVARsusedinthemainpaperarepresented below. 42Anindexoftotalcapacityutilizationisonlyavailablesince1967. 43Since the large sample distribution of the Portmanteau tests tend to over-reject for U.S. macro data, the critical valuesarebootstrappedasinAltigetal.(2005). 30

TableA.2: Lag-LengthSelectionCriteriafor(LargeVAR,FullSample) VAR InformationCriteria PortmanteauTestsforlags... lag T-K llf/T maxroot AIC HQIC SIC 4 8 10 12 16 18 20 1 178 −11.05 0.9881 23.49 24.41 25.75† 688.21∗∗ 1297.69∗∗ 1580.05∗∗ 1867.45∗∗ 2387.65∗∗ 2659.14∗∗ 2916.48∗∗ 2 166 −10.25 0.9899 23.19 24.94 27.53 458.73∗∗ 1024.55∗∗ 1303.42∗∗ 1590.83∗∗ 2092.52∗∗ 2357.49∗∗ 2607.01∗∗ 3 154 −9.48 0.9905 22.93 25.54 29.37 281.81 830.16∗∗ 1092.80∗∗ 1374.52∗∗ 1863.32∗∗ 2133.33∗∗ 2374.64∗∗ 4 142 −8.87 0.9918 23.03 26.50 31.59 207.07 721.60 988.32∗ 1254.53∗ 1745.60∗ 2009.33∗ 2261.62∗ 5 130 −8.24 0.9977 23.11 27.44 33.79 165.41 651.57 928.10 1200.37 1715.08∗ 1977.33∗ 2238.14∗ 6 118 −7.55 1.0055 23.06 28.26 35.89 177.79 600.30 874.35 1157.61 1680.78 1948.61 2209.05 7 106 −6.90 1.0022 23.12 29.20 38.12 194.70 571.37 856.21 1119.12 1620.36 1874.85 2124.78 8 94 −6.16 0.9940 23.02 29.98 40.19 241.23 610.63 914.81 1179.07 1676.77 1915.52 2146.17 9 82 −5.21 1.0031 22.50 30.35 41.86 299.36 635.32 870.95 1140.96 1649.86 1896.51 2136.42 10 70 −3.97 0.9993 21.43 30.18 43.01 394.48 828.13 1032.45 1316.03 1802.79 2064.12 2307.86 12 46 −1.08 1.0030 18.51† 29.07 44.56 591.20 1122.76 1375.56 1631.89 2125.80 2390.27 2624.10 Note: Model chosen with lag-length 3. † denotes minimum IC. Q-statistics for Portmanteau test. ∗ denotes significance at the 10%, ∗∗ at the 5%, ∗∗∗ at the 1% level of bootstrapped distribution (2000 draws). The column labeled T −K reports the degrees of freedom in each estimation, measured by the number of observationsafterdroppinginitialvalues(T)lessthenumberofVARcoefficientstobeestimated(K). llf/T reportsthelog-likelihoodofeachVARscaledbythe numberofobservations. AIC,HQICandSICdenotetheAkaikeInformationCriteria,Hannan-QuinnandSchwarzInformationCriteria,respectively. 31

FigureA.7: BC-Moments: FilteredVARversusData(LargeVAR,FullSample) cov(y, y ) cor(y, i ) cor(y, r ) cor(y, π ) cor(y, h ) t t−k t t−k t t−k t t−k t t−k 5 1 1 1 1 0 0 0 0 0 −5 −1 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, i ) cov(i, i ) cor(i, r ) cor(i, π ) cor(i, h ) t t−k t t−k t t−k t t−k t t−k 2 5 1 1 1 0 0 0 0 0 −2 −5 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, r ) cov(i, r ) cov(r, r ) cor(r, π ) cor(r, h ) t t−k t t−k t t−k t t−k t t−k 1 1 2 1 1 0 0 0 0 0 −1 −1 −2 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, π ) cov(i, π ) cov(r, π ) cov(π, π ) cor(π, h ) t t−k t t−k t t−k t t−k t t−k 2 2 1 5 1 0 0 0 0 0 −2 −2 −1 −5 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, h ) cov(i, h ) cov(r, h ) cov(π, h ) cov(h, h ) t t−k t t−k t t−k t t−k t t−k 5 2 1 2 5 0 0 0 0 0 −5 −2 −1 −2 −5 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 Note: Bandpass-filteredmoments, EY˜Y˜0 . ThethicklinesplotunconditionalcorrelationscomputedfromfilteringtheVAR-Spectrum. Thethinlinesaretheir t t−k analogues computed from filtering the data first and then taking sample correlations. Data for the ex-ante real rate is constructed from fitted values of the VAR for expected inflation. The exact bandpass was used for the VAR-based correlations, and the Baxter-King approximation for the data (Baxter and King (1999) recommend a lag truncation of 12 in quarterly data, with monthly data, 12 × 4 = 48 is used here). (The thin lines plot mostly underneath the thick ones). Correlationsonupperdiagonal. 32

TableA.3: Lag-LengthSelectionCriteriafor(LargeVAR,GreatModeration) VAR InformationCriteria PortmanteauTestsforlags... lag T-K llf/T maxroot AIC HQIC SIC 4 8 10 12 16 18 20 1 84 −7.58 0.9854 17.92 19.34† 21.44† 552.92∗∗ 1074.39∗ 1306.45 1573.62 2029.30 2260.28 2471.24 2 72 −5.68 0.9787 16.68 19.43 23.48 427.30 933.69 1174.00 1464.41 1915.26 2126.01 2360.92 3 60 −4.07 1.0218 16.10 20.18 26.22 361.85 892.27 1140.30 1432.11 1945.63 2200.74 2468.93 4 48 −2.74 0.9788 16.12 21.56 29.60 348.57 851.17 1140.38 1431.08 1927.28 2176.18 2455.66 5 36 −0.17 0.9989 13.72 20.54 30.61 428.87 960.38 1281.58 1593.88 2148.05 2422.73 2723.53 6 24 2.43 1.0126 11.33† 19.53 31.66 634.86 1142.21 1417.21 1698.30 2253.84 2553.62 2845.22 Note: Model chosen with lag-length 2. † denotes minimum IC. Q-statistics for Portmanteau test. ∗ denotes significance at the 10%, ∗∗ at the 5%, ∗∗∗ at the 1% level of bootstrapped distribution (2000 draws). The column labeled T −K reports the degrees of freedom in each estimation, measured by the number of observationsafterdroppinginitialvalues(T)lessthenumberofVARcoefficientstobeestimated(K). llf/T reportsthelog-likelihoodofeachVARscaledbythe numberofobservations. AIC,HQICandSICdenotetheAkaikeInformationCriteria,Hannan-QuinnandSchwarzInformationCriteria,respectively. 33

FigureA.8: BC-Moments: FilteredVARversusData(LargeVAR,GreatModeration) cov(y, y ) cor(y, i ) cor(y, r ) cor(y, π ) cor(y, h ) t t−k t t−k t t−k t t−k t t−k 1 1 1 1 1 0 0 0 0 0 −1 −1 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, i ) cov(i, i ) cor(i, r ) cor(i, π ) cor(i, h ) t t−k t t−k t t−k t t−k t t−k 1 2 1 1 1 0 0 0 0 0 −1 −2 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, r ) cov(i, r ) cov(r, r ) cor(r, π ) cor(r, h ) t t−k t t−k t t−k t t−k t t−k 1 2 1 1 1 0 0 0 0 0 −1 −2 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, π ) cov(i, π ) cov(r, π ) cov(π, π ) cor(π, h ) t t−k t t−k t t−k t t−k t t−k 0.5 1 1 1 1 0 0 0 0 0 −0.5 −1 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, h ) cov(i, h ) cov(r, h ) cov(π, h ) cov(h, h ) t t−k t t−k t t−k t t−k t t−k 2 2 2 1 5 0 0 0 0 0 −2 −2 −2 −1 −5 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 Note: Bandpass-filteredmoments, EY˜Y˜0 . ThethicklinesplotunconditionalcorrelationscomputedfromfilteringtheVAR-Spectrum. Thethinlinesaretheir t t−k analogues computed from filtering the data first and then taking sample correlations. Data for the ex-ante real rate is constructed from fitted values of the VAR for expected inflation. The exact bandpass was used for the VAR-based correlations, and the Baxter-King approximation for the data (Baxter and King (1999) recommend a lag truncation of 12 in quarterly data, with monthly data, 12 × 4 = 48 is used here). (The thin lines plot mostly underneath the thick ones). Correlationsonupperdiagonal. 34

TableA.4: Lag-LengthSelectionCriteriafor(LargeVAR,GreatInflation) VAR InformationCriteria PortmanteauTestsforlags... lag T-K llf/T maxroot AIC HQIC SIC 4 8 10 12 16 18 20 1 71 −6.85 0.9847 15.51 16.37† 17.65† 271.51∗ 538.25∗ 654.48 796.43 1112.04∗ 1254.15∗ 1385.88∗ 2 62 −6.08 0.9918 15.60 17.23 19.68 197.50 477.95 603.71 733.81 1012.81 1155.47 1285.33 3 53 −5.13 0.9869 15.39 17.80 21.43 148.08 428.01 551.13 677.43 944.63 1075.99 1204.03 4 44 −4.27 0.9767 15.40 18.61 23.44 136.64 413.19 538.61 669.48 941.23 1076.62 1205.73 5 35 −2.99 0.9981 14.60 18.62 24.66 146.57 423.39 555.64 691.19 970.83 1127.91 1288.69 6 26 −1.41 1.0138 13.28† 18.12 25.39 250.10 533.02 649.68 804.84 1097.23 1248.38 1414.31 Note: Model chosen with lag-length 2. † denotes minimum IC. Q-statistics for Portmanteau test. ∗ denotes significance at the 10%, ∗∗ at the 5%, ∗∗∗ at the 1% level of bootstrapped distribution (2000 draws). The column labeled T −K reports the degrees of freedom in each estimation, measured by the number of observationsafterdroppinginitialvalues(T)lessthenumberofVARcoefficientstobeestimated(K). llf/T reportsthelog-likelihoodofeachVARscaledbythe numberofobservations. AIC,HQICandSICdenotetheAkaikeInformationCriteria,Hannan-QuinnandSchwarzInformationCriteria,respectively. 35

FigureA.9: BC-Moments: FilteredVARversusData(LargeVAR,GreatInflation) cov(y, y ) cor(y, i ) cor(y, r ) cor(y, π ) cor(y, h ) t t−k t t−k t t−k t t−k t t−k 5 1 1 1 1 0 0 0 0 0 −5 −1 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, i ) cov(i, i ) cor(i, r ) cor(i, π ) cor(i, h ) t t−k t t−k t t−k t t−k t t−k 5 5 1 1 1 0 0 0 0 0 −5 −5 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, r ) cov(i, r ) cov(r, r ) cor(r, π ) cor(r, h ) t t−k t t−k t t−k t t−k t t−k 1 0.5 0.5 1 1 0 0 0 0 0 −1 −0.5 −0.5 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, π ) cov(i, π ) cov(r, π ) cov(π, π ) cor(π, h ) t t−k t t−k t t−k t t−k t t−k 5 5 0.5 5 1 0 0 0 0 0 −5 −5 −0.5 −5 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 cov(y, h ) cov(i, h ) cov(r, h ) cov(π, h ) cov(h, h ) t t−k t t−k t t−k t t−k t t−k 5 5 1 5 10 0 0 0 0 0 −5 −5 −1 −5 −10 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 Note: Bandpass-filteredmoments, EY˜Y˜0 . ThethicklinesplotunconditionalcorrelationscomputedfromfilteringtheVAR-Spectrum. Thethinlinesaretheir t t−k analogues computed from filtering the data first and then taking sample correlations. Data for the ex-ante real rate is constructed from fitted values of the VAR for expected inflation. The exact bandpass was used for the VAR-based correlations, and the Baxter-King approximation for the data (Baxter and King (1999) recommend a lag truncation of 12 in quarterly data, with monthly data, 12 × 4 = 48 is used here). (The thin lines plot mostly underneath the thick ones). Correlationsonupperdiagonal. 36

II Additional Results for Large VARs Thissectionreportsthefollowingresults: • ImpulseResponsesestimatedfortheGreatModerationandtheGreatInflation • Decompositionsforcomovementsbetweenoutputandthe nominal rate • ResultsobtainedfromestimatingaVARallowingforacommonstochastictrendininflation and nominal rates over the full sample of postwar data. In addition to what is described in Section 4.3 of the main paper, the VAR includes also the velocity of money as well as the ratiooftotalreservestothepricelevel. 37

FigureA.10: ImpulseResponses(LargeVAR,GreatModeration) Note: EstimatesforU.S.data(1982–2006)fromVAR,equation(1)inSection3ofthemainpaper. Responsesofunfilteredvariablestoaone-standarddeviation shock (H(L)A(L)−1Q). Bootstrapped standard-errors bands with Kilian (1998)’s small sample adjustment (1000 draws in first round, 2000 draws in second). Percentilesareshaded: 95%(light)and68%(dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR.Quarterlylagson thex-axis. 38

FigureA.11: ImpulseResponses(LargeVAR,GreatInflation) Note: Estimates for U.S. data (1959–1979) from VAR with nominal trend, described in Section 4.3 of the main paper. Responses of unfiltered variables to a one-standarddeviationshock(H(L)A(L)−1Q).Bootstrappedstandard-errorsbandswithKilian(1998)’ssmallsampleadjustment(1000drawsinfirstround,2000 drawsinsecond). Percentilesareshaded:95%(light)and68%(dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR. Quarterlylagsonthex-axis. 39

FigureA.12: OutputandNominalRates(LargeVAR,FullSample) cov(y, i ): t t−k 2 1.5 Total MP N−Tech 1 I−Tech News Rest 0.5 0 −0.5 −1 −1.5 −8 −4 0 4 8 (a)CovarianceDecomposition MP N−Tech I−Tech 1 1 1 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 News Total 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 (b)ConditionalCorrelations Note: Bandpass-filteredmoments,Cov(y˜,r˜ ),forU.S.data(1959–2006)computedfromVARdescribedinSect t−k tion3ofthemainpaper. Quarterlylagsonthex-axis. Lowerpanel: Bootstrappedstandard-errorsbandswithKilian (1998)’s small sample adjustment (1000 draws in first round, 2000 draws in second). Percentiles are shaded: 95% (light)and68%(dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR. 40

FigureA.13: OutputandNominalRates(LargeVAR,GreatModeration) cov(y, i ): t t−k 0.8 0.6 Total MP N−Tech 0.4 I−Tech News Rest 0.2 0 −0.2 −0.4 −0.6 −8 −4 0 4 8 (a)CovarianceDecomposition MP N−Tech I−Tech 1 1 1 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 News Total 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 (b)ConditionalCorrelations Note: Bandpass-filteredmoments,Cov(y˜,r˜ ),forU.S.data(1982–2006)computedfromVARdescribedinSect t−k tion3ofthemainpaper. Quarterlylagsonthex-axis. Lowerpanel: Bootstrappedstandard-errorsbandswithKilian (1998)’s small sample adjustment (1000 draws in first round, 2000 draws in second). Percentiles are shaded: 95% (light)and68%(dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR. 41

FigureA.14: OutputandNominalRates(LargeVAR,GreatInflation) cov(y, i ): t t−k 3 2 Total MP N−Tech I−Tech 1 News Nom. Trend Rest 0 −1 −2 −3 −8 −4 0 4 8 (a)CovarianceDecomposition MP N−Tech I−Tech 1 1 1 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 News Nom. Trend Total 1 1 1 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 (b)ConditionalCorrelations Note: Bandpass-filtered moments, Cov(y˜,r˜ ), for U.S. data (1959–1979) computed from VAR allowing for a t t−k commontrendininflationandnominalratesasdescribedinSection4.3ofthemainpaper.Lowerpanel:Bootstrapped standard-errorsbandswithKilian(1998)’ssmallsampleadjustment(1000drawsinfirstround,2000drawsinsecond). Percentilesareshaded:95%(light)and68%(dark). Thesoldlineisthemeanofthebootstraps. Thedashedlineisthe pointestimateoftheVAR.Quarterlylagsonthex-axis. 42

FigureA.15: ImpulseResponses(FullSamplew/NominalTrend) y MP N−Tech I−Tech News Nom. Trend 0.4 1 0.5 0.4 0.4 0.2 0.2 0.2 0 0.5 0 0 0 −0.2 −0.2 0 −0.5 −0.2 −0.4 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 i 0.4 0.2 0.4 0.4 0.2 0 0 0.2 0 0.2 −0.2 0 −0.5 −0.2 −0.4 −0.2 0 −1 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 r 0.5 0.5 0.4 0.2 0.2 0.2 0 0 0 0 0 −0.5 −0.2 −0.2 −0.2 −0.4 −0.4 −0.4 −0.5 −1 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 π 0.6 0.5 0 0 0.5 0.4 0 0.2 0 −0.5 −0.5 −0.5 0 −1 −1 −1 −0.2 −0.5 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 h 0.6 0.5 0.5 0.4 0.2 0.4 0.2 0 0 −0.2 0. 0 2 0 0 −0.2 −0.4 −0.4 −0.2 −0.5 −0.5 −0.6 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Note: EstimatesforU.S.data(1959–1979)fromVARwithnominaltrend. Responsesofunfilteredvariablestoaone-standarddeviationshock(H(L)A(L)−1Q). Bootstrappedstandard-errorsbandswithKilian(1998)’ssmallsampleadjustment(1000drawsinfirstround,2000drawsinsecond). Percentilesareshaded: 95% (light)and68%(dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR.Quarterlylagsonthex-axis. 43

FigureA.16: OutputandRealRates(FullSamplew/NominalTrend) cov(y, r ): t t−k 0.6 0.4 0.2 0 −0.2 Total −0.4 MP N−Tech I−Tech −0.6 News Nom. Trend Rest −0.8 −8 −4 0 4 8 (a)CovarianceDecomposition MP N−Tech I−Tech 1 1 1 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 News Nom. Trend Total 1 1 1 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 (b)ConditionalCorrelations Note: Bandpass-filtered moments, Cov(y˜,r˜ ), for U.S. data (1982–2006) computed from VAR with nominal t t−k trend. Quarterly lags on the x-axis. Lower panel: Bootstrapped standard-errors bands with Kilian (1998)’s small sampleadjustment(1000drawsinfirstround, 2000drawsinsecond). Percentilesareshaded: 95%(light)and68% (dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR. 44

FigureA.17: OutputandNominalRates(FullSamplew/NominalTrend) cov(y, i ): t t−k 2 1.5 Total MP N−Tech 1 I−Tech News Nom. Trend 0.5 Rest 0 −0.5 −1 −1.5 −8 −4 0 4 8 (a)CovarianceDecomposition MP N−Tech I−Tech 1 1 1 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 News Nom. Trend Total 1 1 1 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 −1 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 (b)ConditionalCorrelations Note: Bandpass-filtered moments, Cov(y˜,r˜ ), for U.S. data (1982–2006) computed from VAR with nominal t t−k trend. Quarterly lags on the x-axis. Lower panel: Bootstrapped standard-errors bands with Kilian (1998)’s small sampleadjustment(1000drawsinfirstround, 2000drawsinsecond). Percentilesareshaded: 95%(light)and68% (dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR. 45

III Results for Small VARs Asarobustnesscheck,thisappendixreportsdecompositionresultsobtainedfromsmaller,trivariate VARs,usingthebareminimumofvariables,whichareneededtomodeloutputandtherealinterest rate: output growth, inflation and the nominal interest rate. As in the main paper, a common trend in inflation and nominal rate is allowed for when estimating the VAR for the Great Inflation. The trivariate VAR cannot be used to identify as many shocks as with the larger VAR used in the main paper. Attentionislimitedheretotheidentificationofneutraltechnologyshocks,asbeingthesole driver of the permanent component in output, as well as nominal trend shocks (when estimating the VAR for the Great Inflation). Confirming results from the main paper, the key findings from this exercise are that the real rate is pro-cyclical when conditioned on technology shocks, and counter-cyclicalwhenconditionedonnominaltrendshocksduringtheGreatInflation. In the main paper, the permanent component of labor productivity is driven by neutral and investment-specific technology shocks. Since hours are assumed to be stationary, the permanent component of labor productivity is identical to the permanent component of output and the technologyshocksidentifiedherecorrespondtoamixtureofbothtypesoftechnologyshocksidentified in the main paper. In all three samples considered (Full, Great Moderation and Great Inflation), therealrateispro-cyclicalwhenconditionedontechnologyshocks. FortheVARspecificationwithstationaryinflation,resultsarealsoreportedfor“MonetaryPolicy”whicharesimplyidentifiedbyprojectingtheinnovationofthenominalrateoffthetechnology shocksandtheinnovationininflation,whichisunlikelytoproperlyaccountforendogenouspolicy responsestovariationsinaggregateactivityandinflation. Still,theresultsmirrortheconditionally counter-cyclicalbehavioroftherealratereportedinthemainpaperforthelargerVARs. References StefaniaAlbanesi,V.V.Chari,andLawrenceJ.Christiano. Expectationtrapsandmonetarypolicy. TheReviewofEconomicStudies,70(4):715–741,October2003. David Altig, Lawrence Christiano, Martin Eichenbaum, and Jesper Linde. Firm-specific capital, 46

FigureA.18: OutputandRealRatedecomposedwithSmallVAR(FullSample) cov(y, r ): t t−k 2 1 0 −1 Total MP −2 Tech −3 Rest −4 −8 −4 0 4 8 (a)CovarianceDecomposition MP Tech 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 Rest Total 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 (b)ConditionalCorrelations Note: Bandpass-filteredmoments, Cov(y˜,r˜ ), forU.S.data(1959–2006)computedfromatrivariateVARusing t t−k outputgrowth, inflationandthenominalinterestrate. Lowerpanel: Bootstrappedstandard-errorsbandswithKilian (1998)’s small sample adjustment (1000 draws in first round, 2000 draws in second). Percentiles are shaded: 95% (light)and68%(dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR. Quarterlylagsonthex-axis. 47

FigureA.19: OutputandRealRatedecomposedwithSmallVAR(GreatModeration) cov(y, r ): t t−k 2 Total 1.5 MP 1 Tech Rest 0.5 0 −0.5 −1 −8 −4 0 4 8 (a)CovarianceDecomposition MP Tech 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 Rest Total 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 (b)ConditionalCorrelations Note: Bandpass-filteredmoments, Cov(y˜,r˜ ), forU.S.data(1982–2006)computedfromatrivariateVARusing t t−k outputgrowth, inflationandthenominalinterestrate. Lowerpanel: Bootstrappedstandard-errorsbandswithKilian (1998)’s small sample adjustment (1000 draws in first round, 2000 draws in second). Percentiles are shaded: 95% (light)and68%(dark). Thesoldlineisthemeanofthebootstraps. ThedashedlineisthepointestimateoftheVAR. Quarterlylagsonthex-axis. 48

FigureA.20: OutputandRealRatedecomposedwithSmallVAR(GreatInflation) cov(y, r ): t t−k 0.6 0.4 0.2 0 −0.2 −0.4 Total Pi −0.6 Tech Rest −0.8 −8 −4 0 4 8 (a)CovarianceDecomposition Pi Tech 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 Rest Total 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −8 −4 0 4 8 −8 −4 0 4 8 (b)ConditionalCorrelations Note: Bandpass-filtered moments, Cov(y˜,r˜ ), for U.S. data (1959–1979) computed from trivariate VAR using t t−k output growth, the change inflation and the difference between the nominal interest rate and inflation, which allows for a common trend in inflation and the nominal rate. Quarterly lags on the x-axis. Lower panel: Bootstrapped standard-errorsbandswithKilian(1998)’ssmallsampleadjustment(1000drawsinfirstround,2000drawsinsecond). Percentilesareshaded:95%(light)and68%(dark). Thesoldlineisthemeanofthebootstraps. Thedashedlineisthe pointestimateoftheVAR. 49

nominal rigidities and the business cycle. NBER Working Papers 11034, National Bureau of EconomicResearch,Inc,January2005. RobertB.BarskyandEricR.Sims. Newsshocks. NBERWorkingPapers15312,NationalBureau ofEconomicResearch,Inc,September2009. Marianne Baxter and Robert G. King. Productive externalities and business cycles. Discussion Paper53,FederalReserveBankofMinneapolis,InstituteforEmpiricalMacroeconomics,1991. Marianne Baxter and Robert G. King. Measuring business cycles: Approximate band-pass filters foreconomictimeseries. TheReviewofEconomicsandStatistics,81(4):575–593,1999. PaulBeaudryandAlainGuay. Whatdointerestratesrevealaboutthefunctioningofrealbusiness cyclemodels? JournalofEconomicDynamicsandControl,20(9-10):1661–1682,1996. Paul Beaudry and Franck Portier. Stock prices, news, and economic fluctuations. The American EconomicReview,96(4):1293–1307,September2006. Jean-Pascal Be´nassy. Money and wage contracts in an optimizing model of the business cycle. JournalofMonetaryEconomics,35(2):303–315,April1995. Ben S. Bernanke. The Great Moderation. Remarks by Governor Ben S. Bernanke at the meetings oftheEasternEconomicAssociation,WashingtonD.C.,February2004. Ben S Bernanke and Alan S Blinder. The federal funds rate and the channels of monetary transmission. TheAmericanEconomicReview,82(4):901–21,1992. Olivier Jean Blanchard and Danny Quah. The dynamic effects of aggregate demand and supply disturbances. TheAmericanEconomicReview,79(4):655–673,September1989. Michele Boldrin, Lawrence J. Christiano, and D.M. Fisher. Habit persistence, asset returns, and thebusinesscycle. TheAmericanEconomicReview,91(1):149–166,March2001. Craig Burnside. Detrending and business cycle facts: A comment. Journal of Monetary Economics,41(3):513–532,1998. John Y. Campbell and Robert J. Shiller. Cointegration and tests of present value models. The JournalofPoliticalEconomy,95(5):1062–1088,October1987. Fabio Canova. Detrending and business cycle facts. Journal of Monetary Economics, 41(3):475– 512,1998a. Fabio Canova. Detrending and business cycle facts: A user’s guide. Journal of Monetary Economics,41(3):533–540,1998b. V.V.Chari, PatrickJ. Kehoe, and Ellen R. McGrattan. Business cycleaccounting. Econometrica, 75(3):781–836,May2007. LawrenceJ.ChristianoandMartinEichenbaum.Currentreal-businesscycletheoriesandaggregate labor-marketfluctuations. TheAmericanEconomicReview,82(3):430–450,June1992. 50

Lawrence J. Christiano and Robert J. Vigfusson. Maximum likelihood in the frequency domain: theimportanceoftime-to-plan. JournalofMonetaryEconomics,50(4):789–815,2003. LawrenceJ.Christiano,MartinEichenbaum,andCharlesL.Evans. Monetarypolicyshocks: What havewelearnedandtowhatend? In TaylorandWoodford(1999),chapter2. Volume1A. Lawrence J. Christiano, Martin Eichenbaum, and Robert Vigfusson. What happens after a technologyshock? WorkingPaper9819,NBER,July2003. Lawrence J. Christiano, Martin Eichenbaum, and Robert Vigfusson. Assessing structural vars. In Daron Acemoglu, Kenneth Rogoff, and Michael Woodford, editors, NBER Macroeconomics Annual2006,chapter1.MITPress,Cambridge,MA,2006. RichardClarida,JordiGali,andMarkGertler. Monetarypolicyrulesandmacroeconomicstability: Evidence and some theory. The Quarterly Journal of Economics, 115(1):147–180, February 2000. TimothyCogleyandJamesM.Nason. Effectsofthehodrick-prescottfilterontrendanddifference stationary time series implications for business cycle research. Journal of Economic Dynamics andControl,19(1-2):253–278,1995. Timothy Cogley, Giorgio E. Primiceri, and Thomas J. Sargent. Inflation-gap persistence in the U.S. AmericanEconomicJournal: Macroeconomics,2(1):43–69,January2010. WouterJ.denHaan. Thecomovementbetweenoutputandprices. JournalofMonetaryEconomics, 46(1):3–30,2000. Francis X Diebold, Lee E Ohanian, and Jeremy Berkowitz. Dynamic equilibrium economies: A frameworkforcomparingmodelsanddata. ReviewofEconomicStudies,65(3):433–51,1998. MichaelDotsey,CarlLantz,andBrianScholl. Thebehavioroftherealrateofinterest. Journalof MoneyCreditandBanking,35(2):91–110,February2003. John T. Dunlop. The movement of real and money wage rates. The Economic Journal, 48(191): 413–434,September1938. JonasD.M.Fisher. Thedynamiceffectsofneutralandinvestmentspecifictechnologyshocks. The JournalofPoliticalEconomy,114(3):413–451,October2006. NevilleFrancis andValerieA. Ramey. Measuresof percapita hours andtheir implications forthe technology-hoursdebate. WorkingPaper11694,NBER,October2005. NevilleR.Francis,MichealT.Owyang,andAthenaT.Theodorou. Theuseoflong-runrestrictions for the identification of technology shocks. The Federal Reserve Bank of St. Louis Review, 85 (6):53–66,November/December2003. Jeffrey C. Fuhrer and George R. Moore. Monetary policy trade-offs and the correlation between nominal interest rates and real output. The American Economic Review, 85(1):219–239, March 1995. 51

Jordi Gal`ı. Technology, employment, and the business cycle: Do technology shocks explain aggregatefluctuations? TheAmericanEconomicReview,89(1):249–271,1999. Jordi Gal`ı. New perspectives on monetary policy, inflation, and the business cycle. In Mathias Dewatripont,LarsPeterHansen,andStephenJ.Turnovsky,editors,AdvancesinEconomicsand Econometrics: Theory and Applications, Eighth World Congress of the Econometric Society, chapter6,pages151–197.CambridgeUniversityPress,Cambridge,UK,2003. Jordi Gali and Luca Gambetti. On the sources of the Great Moderation. American Economic Journal: Macroeconomics,1(1):26–57,January2009. Jordi Gal`ı and Pau Rabanal. Technology shocks and aggregate fluctuations: How well does the RBCmodelfitpostwarU.S.data? NBERWorkingPaper10636,July2004. Bruce E. Hansen. Rethinking the univariate approach to unit root testing: Using covariates to increasepower. EconometricTheory,11(5):1148–71,1995. Lars P. Hansen. Large sample properties of generalized method of moments estimators. Econometrica,50(4):1029–1054,July1982. A.C. Harvey and A. Jaeger. Detrending, stylized facts and the business cycle. Journal of Applied Econometrics,8(3):231–47,1993. Robert J. Hodrick and Edward C. Prescott. Postwar U.S. business cycles: An empirical investigation. JournalofMoneyCreditandBanking,29(1):1–16,February1997. Peter N. Ireland. Changes in the federal reserve’s inflation target: Causes and consequences. JournalofMoney,CreditandBanking,39(8):1851–1882,December2007. Nir Jaimovich and Sergio Rebelo. Can news about the future drive the business cycle? The AmericanEconomicReview,99(4):1097–1118,September2009. Alejandro Justiniano and Giorgio E. Primiceri. The time-varying volatility of macroeconomic fluctuations. TheAmericanEconomicReview,98(3):604–41,June2008. J. M. Keynes. Relative movements of real wages and output. The Economic Journal, 49(193): 34–51,March1939. Lutz Kilian. Small-sample confidence intervals for impulse response functions. The Review of EconomicsandStatistics,80(2):218–230,May1998. Robert G. King and Sergio T. Rebelo. Low frequency filtering and real business cycles. Journal ofEconomicDynamicsandControl,17(1-2):207–231,1993. Robert G. King and Mark W. Watson. Money, prices, interest rates, and the business cycle. The ReviewofEconomicsandStatistics,78(1):35–53,February1996. Robert G. King, Charles I. Plosser, James H. Stock, and Mark W. Watson. Stochastic trends and economicfluctuations. TheAmericanEconomicReview,81(4):819–40,1991. 52

Thomas A. Lubik and Frank Schorfheide. Testing for indeterminacy: An application to U.S. monetarypolicy. TheAmericanEconomicReview,94(1):190–217,March2004. ElmarMertens. Puzzlingcomovementsbetweenoutputandinterestrates? Multipleshocksarethe answer. WorkingPaper05.05,StudyCenterGerzensee,2007. EdwardC.Prescott.Theoryaheadofbusinesscyclemeasurement.Carnegie-RochesterConference SeriesonPublicPolicy,25:11–44,1986. M.B. Priestley. Spectral Analysis and Time Series. Probability and Mathematical Statistics. ElsevierAcademicPress,Amsterdam,1981. Christina D. Romer and David H. Romer. A new measure of monetary policy: Derivation and implications. TheAmericanEconomicReview,94(4):1055–1084,September2004. JulioJ.Rotemberg. Prices,output,andhours: Anempiricalanalysisbasedonastickypricemodel. JournalofMonetaryEconomics,37(3):505–533,1996. Julio J. Rotemberg and Michael Woodford. An optimization-based econometric framework for the evaluation of monetary policy. In Ben S. Bernanke and Julio J. Rotemberg, editors, NBER MacroeconomicsAnnual1997,pages297–346.TheMITPress,Cambridge(MA),1997. Thomas J. Sargent. The Conquest of American Inflation. Princeton University Press, Princeton (NJ),1999. Thomas J. Sargent. Macroeconomic Theory. Economic Theory, Econometrics, and Mathematical Economics.AcademicPress,NewYork,2ndedition,1987. Stephanie Schmitt-Grohe and Martin Uribe. What’s news in business cycles. NBER Working Papers14215,NationalBureauofEconomicResearch,Inc,August2008. Matthew D. Shapiro and Mark W. Watson. Sources of business cycle fluctuations. In Stanley Fischer, editor, NBER Macroeconomics Annual 1988, pages 111–148. The MIT Press, Cambridge (MA),1988. Christopher A. Sims. Interpreting the macroeconomic time series facts: The effects of monetary policy. EuropeanEconomicReview,36(5):975–1000,1992. Christopher A. Sims and Tao Zha. Error bands for impulse responses. Econometrica, 67(5): 1113–1156,1999. Christopher A. Sims and Tao Zha. Were there regime switches in U.S. monetary policy? The AmericanEconomicReview,96(1):54–81,March2006. Eric R. Sims. Expectations driven business cycles: An empirical evaluation. University of Notre Dame,mimeo,May2009. FrankSmetsandRafaelWouters. ShocksandfrictionsinU.S.businesscycles: AbayesianDSGE approach. TheAmericanEconomicReview,97(3):586–606,June2007. 53

JamesH.StockandMarkW.Watson. WhyhasU.S.inflationbecomehardertoforecast? Journal ofMoney,CreditandBanking,39(S1):3–33,February2007. James H. Stock and Mark W. Watson. Business cycle fluctuations in U.S. macroeconomic time series. InTaylorandWoodford(1999),chapter1. Volume1A. LorieTarshis. Changesinrealandmoneywages. TheEconomicJournal,49(193):150–154,March 1939. John B. Taylor and Michael Woodford, editors. Handbook of Macroeconomics. Number 15 in HandbooksinEconomics.Elsevier,Amsterdam,1999. Harald Uhlig. What moves GNP? Econometric Society 2004 North American Winter Meetings, EconometricSociety,August2004. CarlE.Walsh. MonetaryTheoryandPolicy. TheMITPress,Cambridge,MA,2ndeditionedition, 2003. Mark W Watson. Measures of fit for calibrated models. The Journal of Political Economy, 101 (6):1011–41,1993. MichaelWoodford. InterestandPrices. PrincetonUniversityPress,Princeton,NJ,2003. 54