A Structural Measure of the Shadow Federal Funds Rate

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) A Structural Measure of the Shadow Federal Funds Rate Callum Jones, Mariano Kulish, James Morley 2021-064 Please cite this paper as: Jones, Callum, Mariano Kulish, and James Morley (2021). “A Structural Measure of the Shadow Federal Funds Rate,” Finance and Economics Discussion Series 2021-064. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2021.064. 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.

A Structural Measure of the Shadow Federal Funds Rate* Callum Jones† Mariano Kulish‡ James Morley§ July 2021 Abstract We propose a shadow policy interest rate based on an estimated structural model that accountsforthezerolowerbound. Thelowerboundconstraint,ifexpectedtobind,iscontractionary and increases the shadow rate compared to an unconstrained systematic policy response. By contrast, forward guidance andother unconventionalpolicies that extend the expected duration of zero-interest-rate policy are expansionary and decrease the shadow rate. By quantifying these distinct effects, our structural shadow federal funds rate better capturesthestanceofmonetarypolicygiveneconomicconditionsthanashadowratebased onlyonthetermstructureofinterestrates. Keywords: zerolowerbound;forwardguidance;shadowrate;monetarypolicy JELclassifications: E52;E58 *This research was supported by Australian Research Council grant DP190100537. We thank Heather Anderson, Anthony Brassil, Isabel Cairó, Etienne Gagnon, Michaela Haderer, Matteo Iacoviello, Adrian Pagan, and seminar participants at the Federal Reserve Board, Monash University, and the Reserve Bank of Australia for helpful comments. The latest structural shadow rate series is available for download at https://github.com/callumjones/shadow-rate. The views expressed are those of the authors and not necessarilythoseoftheFederalReserveBoardortheFederalReserveSystem. †FederalReserveBoard. Email: callum.j.jones@frb.gov. ‡SchoolofEconomics,UniversityofSydney. Email: mariano.kulish@sydney.edu.au. §SchoolofEconomics,UniversityofSydney. Email: james.morley@sydney.edu.au. 1

1 Introduction In response to the Great Recession and the COVID-19 crisis, the Federal Reserve, like many other central banks, cut its policy interest rate close to zero. When this happens, the lower bound constraint on nominal interest rates makes it difficult to determine the stance of monetary policy given prevailing economic conditions from the observed policy rate alone. In an influential paper, Wu and Xia (2016) use a term structure model to construct a ‘shadow’ policy rate intended to quantify the interest-rate-equivalent stance of policy at the zero lower bound (ZLB). The basic idea is that the shadow rate reflects the effects of unconventional policies in termsofahypotheticalunconstrainedshort-terminterestrate. Having a shadow rate that captures monetary policy at the ZLB is useful for two reasons. First, policymakers can gauge the scale of unconventional policy actions with a comparable measure to policy conducted during conventional times. Second, researchers can extend empiricalanalysisusinglinearmodelsintoperiodsinwhichtheobservedpolicyrateisattheZLB, as done with the Wu-Xia shadow rate in studies such as Avdjiev et al. (2020) and Anderson et al.(2017). Weargue,however,thatshadowratesderivedfromtermstructuremodels,includingthose developed in Ichiue and Ueno (2013) and Krippner (2013), fail to accurately reflect the stance of monetary policy because they do not disentangle movements in the term structure due to unconventionalpoliciesfromthoseduetoothershocks.1 Termstructuremodelscanbeusedto extract market expectations about the duration of the ZLB, as in Ichiue and Ueno (2015). Yet, to capture the policy stance given economic conditions, it is critical to uncover the underlying determinants of the duration. Specifically, is the policy rate expected to be zero because deteriorating economic conditions suggest the ZLB constraint is likely to bind for a long time? Or does the expected duration reflect unconventional ‘lower-for-longer’ zero-interest-rate policy beyondwhattheZLBconstraintwouldimplyonitsown? Shadowratetermstructuremodels(SRTSMs)donotaccountforthisdistinctionintermsof why the policy rate is expected to be zero. To do so, it is necessary to have a more structural model, which is what we propose in this paper. In the SRTSM approach, the actual policy rate 1BauerandRudebusch(2016)cautionagainstusingshadowratesfromtermstructuremodelsforthealternative reason that they are sensitive to model specification. Meanwhile, Johannsen and Mertens (2021) argue that it is importanttoaugmenttermstructuremodelswithmacroeconomicvariables. However,wenotethat,likewiththe more basic term structure models, their shadow rate is restricted to be non-positive and they do not separately identifytheeffectsofunconventionalpoliciesandothershocks. 2

follows i = max(i ∗ ,0), where i ∗ is the shadow policy rate. Therefore, it is impossible to have t t t the shadow rate greater than zero when i = 0. In our approach, a binding constraint is equivt alent to a contractionary policy shock in an unconstrained system that increases the shadow raterelativetothelevelimpliedbythesystematicpolicyresponsetoeconomicconditions. Our structuralmeasureoftheshadowratethusallowsthepossibilitythati ∗ > i = 0,whichwould t t occur, for example, if the unconstrained systematic policy response is to set the policy rate exactly at zero, but the ZLB constraint is expected to bind in the future. Meanwhile, forward guidanceandotherunconventionalpoliciesthatextendtheexpecteddurationofzero-interestratepolicyacttooffsettheeffectsoftheconstraint,decreasingtheshadowrate. The overall level of our shadow rate at any point in time reflects the net effects of the ZLB constraint and unconventional policies. Depending on the implied systematic policy response and how much unconventional policies offset a binding ZLB constraint, our shadow rate can bepositiveornegativewhentheactualpolicyrateiszero. IntheSRTSMapproach,theshadow rate is estimated from yield-curve data and a 3-month yield is typically used as the short-term rate. Consequently,itistechnicallyfeasibleforanSRTSMmeasuretobegreaterthanthefederal fundsrate,asinFigure4ofWuandXia(2016)atthestartoftheZLBin2009whenthe3-month yield is still positive. But, unlike with our approach, the reason is not directly related to the extenttowhichtheZLBconstraintbindsforthepolicyrate. Section2describeshowourstructuralshadowrateisconstructed,illustratingitsdirectlink to the stance of monetary policy with a simple analytical example. Section 3 presents the estimated structural shadow federal funds rate for the Smets and Wouters (2007) model allowing for the ZLB. Section 4 shows how our shadow rate closely aligns with notable unconventional policy actions in the aftermath of the Great Recession, compares it with an SRTSM measure in a VAR with data from the ZLB, and extends the sample to look at monetary policy during the recentpandemic. Section5concludes. 2 The Structural Shadow Rate 2.1 Decomposition of ZLB Durations The structural shadow rate can be constructed given a structural model that accounts for the ZLB. In the next section, we consider the Smets and Wouters (2007) model allowing for the lower bound on nominal interest rates, but here we first describe how the structural shadow rateisconstructedingeneral. Becausethestructuralshadowrateisbasedondecomposingthe 3

expecteddurationofhowlongtheZLBwillholdintoitsunderlyingsources,webeginwiththe detailsofthisdecomposition,asproposedinJonesetal.(2021). Assumethepolicyrate,i ,eitherfollowsanunconstrainedTaylor-typepolicyruleorisfixed t whenattheZLB.Forexpositionalsimplicity,wesetthefixedleveltozero,althoughitcouldbe any feasible value, including an alternative effective lower bound. Thus, the policy rate can be describedby (cid:40) policyrule, I = 0 i = t (1) t 0, I = 1, t where the indicator I keeps track of the policy-setting regime. When I = 0, the policy rate t t followstheunconstrainedpolicyruleandlinearizedstructuralequationsaregivenby Ax = C+Bx +D E x +Fε , (2) t t−1 t t+1 t where x is an n×1 vector of model variables, ε is an l ×1 vector of structural shocks, and A, t t C,B,D,andFareconformablematrices. Thereduced-formsolutiontothe I = 0regimeis t x = J+Qx +Gε , (3) t t−1 t whereJ,Q,andGarethestandardlinearrationalexpectationssolutionmatrices. When I = 1, t thestructuralequationsaredescribedby A¯ x = C¯ +B¯x +D¯E x +F¯ε , (4) t t−1 t t+1 t wheretheonlyequationthatchangesfrom(2)istheonecorrespondingtothepolicyrate,which now is fixed at zero.2 As discussed in Jones (2017) and Kulish et al. (2017), the solution when I = 1is,followingKulishandPagan(2017),atime-varyingVARoftheform t x = J +Q x +G ε , (5) t t t t−1 t t where J , Q , and G are time-varying matrices that depend on the expected duration of the t t t fixed-interest-rate regime, denoted d . To keep track of the reduced-form that prevails at each t point time, we allow T¯ to be an arbitrarily large upper-bound duration and find the sequences {J }T¯ ,{Q }T¯ ,and{G }T¯ suchthatJ ,Q ,andG arereduced-formmatricescorrespondd d=1 d d=1 d d=1 d d d ing to a particular expected duration d. The prevailing reduced-form in a given period can be 2It is possible to allow other structural equations to change in the fixed-interest-rate regime, such as would be the case if the propagation of certain shocks changes under the ZLB. However, we focus on the case where therelevantstructuralchangeisintermsofsettingthepolicyratetoalignwithourapplicationtotheSmetsand Wouters(2007)model. 4

relabeled as J = J , Q = Q , and G = G and, noting that d = 0 in periods where I = 0 t dt t dt t dt t t and J = J, Q = Q, and G = G, the reduced-form over the full sample is also given by 0 0 0 (5), providing the basis of estimation. At the estimated parameter values with the estimated structuralshocks,(5)returnsbackthedata. Following Jones et al. (2021), the duration of a fixed-interest-rate regime d corresponds to t the actual duration expected by agents, which is not necessarily the same duration prescribed by the policy rule given the ZLB constraint. With the occasionally-binding-constraint solution of Jones (2017), we can find the duration in each period t prescribed by the policy rule – i.e. the expected duration implied by the estimated state x , the estimated non-policy structural t−1 shocks, and a given lower bound.3 This duration corresponds to monetary policy following exactly max(policyrule,0) in projecting when the policy rate lifts off from the ZLB. We denote this expected duration by dlb and refer to it as the lower-bound duration. In the absence of t future shocks, this duration is expected to fall by one period at a time as the effects of current shocksunwind. Again following Jones et al. (2021), for each period of the fixed-interest-rate regime, we fg definetheforward-guidanceduration,d ,asthedifferencebetweentheactualdurationd and t t thelower-bounddurationdlb,sothattheactualdurationisdecomposedas t d = dlb+d fg . (6) t t t The forward-guidance duration captures announcements and other factors that change the actual duration beyond the lower-bound duration.4 This decomposition is important because changesind thatoriginatefromdlb havedifferentimplicationsfortheobservedvariablesthan t t fg those from d . For example, output and inflation might fall after a negative shock that makes t the ZLB constraint bind for longer. But output and inflation might increase after a ‘lower-forlonger’announcementextendingthefixed-interest-rateregimebeyondwhattheZLBconstraint implies. 3Also see Guerrieri and Iacoviello (2015) on solving rational expectations models with occasionally-binding constraints. 4Inadditiontoexplicitcentralbankcommunicationsaboutdurations,theforward-guidancedurationcouldbe influencedbyotherunconventionalpolicies,suchaslargescaleassetpurchases,orbypublicexpectationsthatthe centralbankmightdeviatefromitspolicyrule,suchasbyraisingthepolicyratesoonerthanimpliedbytheZLB constraint. For example, markets reassessed the likely timing of liftoff as the Fed was tapering the rate of bond purchasesin2013andthiscanbethoughtofashavingdecreasedtheforward-guidanceduration. 5

2.2 Constructing the Structural Shadow Rate To construct the structural shadow rate, we find shocks under the unconstrained-policy-rule regime I = 0 that would replicate outcomes in the data under the fixed-interest-rate regime t I = 1. Specifically,weaugmentthevectorofestimatedstructuralshocksε withashadowrate t t shock, ε ∗ , to a hypothetical policy rate for the periods when I = 1. The shadow rate shocks m,t t arechosensothatoutcomesinashadoweconomybasedonthestructurein(3) x ∗ = J+Qx ∗ +Gε ∗ , (7) t t−1 t approximate outcomes for at least some variables in x observed in the actual economy (5). t When I = 0, the structural shocks ε ∗ are the same as ε . The structural shadow rate, i ∗ , is t t t t definedasthepolicyinterestratethatprevailsintheshadoweconomy(7). Formally,letιdenoteavectorthatselectswhichvariablestotargetinmatchingtheoutcomes across systems (5) and (7), with ∆ = ι(x −x ∗) denoting the difference between the observed t t t targetedvariablesandthesamevariablesastheyevolveintheshadoweconomy.5 Theshadow ∗ rateshock ε ischosenforeachperiodtosolve m,t ∆(cid:48) ∆ min W , (8) t t ε∗ m,t where W is a diagonal weighting matrix that reflects the volatility of the targeted variables, with the diagonal elements being the inverse of the variance of each corresponding variable. Thisweightingschemeeffectivelystandardizesthedataand,asaresult,impliesequalweights ∗ in matching each of the targeted variables. The shadow rate i is then constructed using the t unconstrained policy rule with the shadow rate shock that solves the minimization problem (8). Thenextsubsectionpresentsasimplemodeltoshowanalyticallyhowchangesinthelength oftheZLBconstraintandforwardguidancemapintocontractionaryandexpansionaryshadow rateshocksandhowtheconstructedshadowratereflectsthestanceofmonetarypolicy. 5The choice of which variables to target potentially matters because, in principle, shocks to the hypothetical policyrateinalinearsystemthatreplicatethedynamicsofonevariableneednotbethesameasthosethatreplicate thedynamicsofanother. However,inourapplicationtotheSmetsandWouters(2007)model,wechoosetomatch all of the observed variables other than interest rates and find we can closely match all of the targeted variables simultaneously. 6

2.3 Analytical Results for a Simple Model We consider a simple three-equation New Keynesian model that provides analytical solutions fortheshadowrateshockandtheshadowrate. Themodelis yˆ = − (cid:0) i −i¯−E πˆ (cid:1) +E yˆ +ε (9) t t t t+1 t t+1 y,t πˆ = β E πˆ +κyˆ (10) t t t+1 t i = max(0,i¯+φπˆ +ε ), (11) t t m,t where yˆ is output in deviation from steady state, i is the nominal interest rate in levels with t t steady-state value i¯ > 0 and a ZLB constraint i ≥ 0, πˆ is inflation in deviation from steady t t state,andε andε aremean-zeroserially-uncorrelatedshocks,withpositivediscountfactor, y,t m,t β > 0,positiveslopeofthePhillipscurve,κ > 0,andamorethanone-for-onesystematicpolicy responsetoinflation,φ > 1. Conditionaloni > 0,mean-zeroshocksimply E πˆ = E yˆ = t t t+1 t t+1 0andthesolutionis 1 yˆ = (ε −ε ) (12) t 1+κφ y,t m,t κ πˆ = (ε −ε ) (13) t 1+κφ y,t m,t κ 1 i = i¯+φ ε + ε . (14) t 1+κφ y,t 1+κφ m,t Lower Bound. Suppose ε = 0 and the shock ε is such that the ZLB constraint binds, m,t y,t whichfrom(14)occurswhen ε ≤ −i¯(1+κφ) . Inthiscase, i = 0andthesolutionis y,t κφ t yˆlb = i¯+ε (15) t y,t πˆlb = κ(i¯+ε ). (16) t y,t A binding ZLB introduces a kink in the slope of the policy functions for output and inflation, asacomparisonof(12)with(15)and(13)with(16)shows. ∗ Our procedure for generating the shadow rate is to find a hypothetical policy shock ε in m,t theno-ZLBsolution(12)to(14)thatreplicatestheoutcomesyˆlb andπˆlb. Wecandosofollowing t t ∗ ∆(cid:48) ∆ ourprocedurebychoosing ε tominimize W from(8),whichinthiscaseis m,t t t (cid:16) (cid:17)2 (cid:16) (cid:17)2 w yˆ ∗ −yˆlb +w πˆ ∗ −πˆlb , y t t π t t where w and w are implicit weights. Because πˆ = κyˆ and πˆlb = κyˆlb, minimizing ∆(cid:48) W ∆ y π t t t t t t is equivalent to minimizing either (cid:0) yˆ ∗ −yˆlb (cid:1)2 or (cid:0) πˆ∗ −πˆlb (cid:1)2 . This can be done exactly with t t t t 7

∆ = 0bysetting yˆ ∗ = yˆlb,or,equivalently, t t t 1 (ε −ε ∗ ) = i¯+ε , 1+κφ y,t m,t y,t whichimplies ε ∗ = −i¯(1+κφ)−κφε ≥ 0. (17) m,t y,t Theequalityin(17)showsthecontractionaryshadowrateshockrequiredtogeneratethesame equilibrium outcomes as those obtained under the binding ZLB. This shadow rate shock and (14) imply the shadow rate is i ∗ = 0, which is higher than the negative level implied by the t systematic monetary policy response to economic conditions if the inequality ε < −i¯(1+κφ) y,t κφ is strict, i.e. i = i¯(1+κφ)+κφε < 0, where i ≡ i¯+φπˆ . Meanwhile, as t,systematic y,t t,systematic t illustratednext,persistencecangeneratepositiveshadowratesattheZLB. Expected Negative Shock at the Lower Bound. Assume now that E ε = −i¯(1+κφ) , so t y,t+1 κφ that agents expect a negative shock tomorrow will continue to cause the ZLB to bind beyond the negative shock ε = −i¯(1+κφ) that hits today. E yˆ and E πˆ are determined from (15) y,t κφ t t+1 t t+1 and(16)and,given i = 0, t (2+κ) yˆlb2 = −i¯ t κφ (2+κ+β) πˆlb2 = −i¯ . t φ Unlike the previous case, πˆlb2 (cid:54)= κyˆlb2, so there is not necessarily a shadow rate shock that t t wouldleadtoanexactmatchwhentargetingbothoutputandinflation. Numerically,wecould minimize (cid:16) (cid:17)2 (cid:16) (cid:17)2 w yˆ ∗ −yˆlb2 +w πˆ ∗ −πˆlb2 . (18) y t t π t t ∗ However, for analytical tractability, suppose we choose ε to match inflation only (i.e. let m,t w = 0).6 This match can be done exactly by setting πˆ∗ = πˆlb2. In this case, the shadow rate y t t shockis i¯ ε ∗ = (1+κφ)(1+κ+β) > 0, m,t κφ and,from(14),theshadowrateis i¯ i ∗ = (1+κ+β) > 0. t κφ 6If, instead, weweretomatchoutputonly, wewouldgetsimilar, butnotidentical, analyticalexpressionsfor theshadowrateshockandshadowrate,withthesameimplicationsintermsoftheirsigns. However,theshadow ratewouldnotbeexactlyequaltothesumoftheimpliedsystematicpolicyresponseandtheshadowrateshock. 8

Thiscaseillustrateshowpersistentnegativeshocksgeneratecontractionaryexpectationstoday that can push the shadow rate into positive territory, even though the actual interest rate is at zero, while the level implied by the systematic monetary policy response is negative, i.e. i = −i¯(1+κ + β) < 0. Thus, unlike the SRTSM approach in which the actual short t,systematic rate is equal to i = max(i ∗ ,0), the shadow rate in our approach can be above zero when t t i = 0becausetheZLBconstraintbeingexpectedtobindinthefutureactslikeacontractionary t monetary policy shock in the shadow economy given lower expected future inflation raising thecurrentrealinterestrate. Forward Guidance. Suppose, instead, that the ZLB constraint binds in period t because of a negative shock ε = −i¯(1+κφ) , but, in addition to setting i = 0, the central bank credibly y,t κφ t announces it will continue to hold the interest rate at zero in the next period, E i = 0. Then, t t+1 E yˆ = i¯and E πˆ = κi¯. Inthiscase, t t+1 t t+1 yˆ fg = i¯(2+κ)+ε t y,t πˆ fg = κi¯(2+κ+β)+κε . t y,t The ‘lower-for-longer’ announcement boosts output and inflation today relative to the lowerboundsolution(15)and(16).7 Inconstructingtheshadowrate,similartothecaseofanexpected negative shock, it is analytically convenient to match inflation only, which again can be done exactly with a perfect match ∆ = 0 by setting πˆ∗ = πˆ fg . Rearranging to solve for the shadow t t t rateshockgives ε ∗ = −i¯(1+κφ)(1+κ+β) < 0. m,t Thus, forward guidance maps into an expansionary shadow rate shock and, from (14), the shadow rate would be less than the positive level implied by the systematic monetary policy responsetoeconomicconditions,i.e. i = κφi¯(1+κ+β) > 0,asitisstrictlynegative: t,systematic i ∗ = −i¯(1+κ+β) < 0. t Inthisway,theshadowrateisabletoreflectthepolicystancewhenthereisforwardguidance.8 7Wenotethedifferencesinthestimulatoryeffectofforwardguidancefromthestimulatoryeffectofaconventional expansionary monetary policy shock, which reflects the fact that forward guidance operates by affecting expectations. 8This contrasts with Hills and Nakata (2018), who define a shadow rate in a New Keynesian model as correspondingtothepolicyratethatwouldbesetaccordingtothepolicyruleintheabsenceoftheZLBconstraintor monetarypolicyshocks. Specifically, theirshadowrateislinkedtotheobservedratevia i = max(i∗,0). Inthis t t case, a negative shadow rate i∗ < 0 simply reflects how much and for how long the constraint binds given the t policyruleandotherstructuralshocks,ratherthanprovidingaguidetothestanceofmonetarypolicy. 9

3 The Shadow Rate from an Estimated Structural Model 3.1 A Medium-Scale New Keynesian Model Toestimatethestructuralshadowfederalfundsrate,weconsidertheSmetsandWouters(2007) model allowing for the ZLB, as in Kulish et al. (2017). The model is estimated using U.S. data over1984Q1to2019Q4andwemakethefollowingchangestoSmetsandWouters(2007): First, similar to Kulish et al. (2017), we expand the set of observables to include the 1-year and 5year Treasury yields. Second, to capture the trend decline in interest rates over this period, we allow for a decline in trend growth. In particular, motivated by the evidence of structural breakintrendgrowthinthe2000sdocumentedinanumberofstudiesincludingFernald(2012), Antolin-Diaz et al. (2017), and Eo and Morley (2020), we allow for a one-time change in trend growth at a magnitude and date to be estimated. This captures the possibility that a decline in trend growth lowers the equilibrium level of the policy rate, which could cause the ZLB to be visited more frequently. Finally, we calibrate the inflation target to a 2% annualized rate to reflecttheFed’sinflationobjective. WefollowSmetsandWouters(2007)inallotherrespects,includingtheremainingobserved variables and their construction, the set of estimated parameters, and priors. Motivated by the resultsinKulishetal.(2017),weusemodalreportedvaluesofdurationsfromBlueChipFinancialForecastsandtheNewYorkFedSurveyofPrimaryDealerstomeasureexpecteddurations ofzero-interest-ratepolicyduringtheZLB.Fulldetailsofthemodelareintheappendix. 3.2 Estimated Shadow Rate TheshadowratefortheSmetsandWouters(2007)modelisconstructedasdescribedinSection 2.2. Wetargetalloftheobservedvariablesexcepttheinterestrates.9 Figure 1 plots the estimated structural shadow federal funds rate and the shadow rate shocks. The posterior mean of the shadow rate reported in the top panel deviates from the observed federal funds rate when it hit the ZLB in 2009Q1, taking on a value of 1.6% with precise 90% posterior bands. This initial positive value for the shadow rate illustrates the contractionaryeffectsoftheZLBconstraintbeingexpectedtobindpersistentlygiventhelargenegative 9The appendix contains plots of the paths of targeted variables both in the data and in the shadow economy. Theplotsshowthatthevariablesintheshadoweconomyareveryclosetotheactualdata.Theshadowrateshocks arethusabletocapture,withgreataccuracy,thedynamicsofallofthetargetedvariables. Wefindthatresultsare similarifwealsotargetlonger-terminterestrates. 10

Figure1: TheStructuralShadowFederalFundsRate (a)ShadowRate,i∗,andSystematicPolicyResponse,%Annualized t (b)ShadowRateShocks,ε∗ m,t Notes: Panel(a)plots,inannualizedpercentageterms,theestimatedshadowrate,alongwiththeimpliedsystematicpolicyresponsetoprevailingeconomicconditionsbasedonthepolicyrulewithoutmonetarypolicyorshadowrateshocksbutallowingforinterestratesmoothing, i.e. it,systematic = i¯+(1−αi )αpπˆt +(1−αi )αyy˜t +α∆y ∆y˜t +αi (it−1,systematic −i¯),wheretheα’sarethemonetarypolicyresponsecoefficients andy˜tistheoutputgapfromtheflexiblepriceequilibriumfortheSmetsandWouters(2007)modeldetailedintheappendix.Panel(b)plots, inannualizedpercentagepointterms,theshadowrateshocks. Thelinescorrespondtoposteriormeans,whilethebandsshow90percent equal-tailedposteriorintervalsfortheshadowrateandshadowrateshocksduringtheZLBperiod. 11

shocksthattriggeredtheGreatRecession. Thecontractionarystanceofmonetarypolicydueto the ZLB constraint is especially clear in comparison to the quick decline in the posterior mean of the systematic policy response implied by the policy rule and prevailing economic conditions to values below -3%.10 However, despite the ZLB constraint, the estimated shadow rate implies relatively expansionary monetary policy compared to the systematic policy response from the beginning of 2011 and declines to about -2.4% by 2012Q4 given implementation of various unconventional policies including forward guidance that allowed the Fed to achieve the equivalent of an unconstrained negative rate in the shadow economy. After bottoming out in2012,theestimatedshadowrateincreasedbacktowardsthelevelimpliedbysystematicpolicy at around -0.5% in 2013Q2 and reaches zero in 2015Q1, just before liftoff. These shifts in the policy stance are reflected in the shadow rate shocks reported in the bottom panel. After the initial contractionary effects of the ZLB constraint being expected to bind for a number of quarters,theshadowrateshocksaretypicallyestimatedtobenegativethroughouttheremainingZLBperiodandquantifytheinterest-rate-equivalenteffectsofforwardguidanceandother unconventionalpolicies. There are important differences in the estimates of our shadow rate and other measures. In the appendix, we compare our shadow rate to the one constructed by Wu and Xia (2016). After being slightly more positive in 2009Q1, our shadow rate falls below the Wu-Xia measure in 2010 and has more pronounced fluctuations that appear closely related to forward guidance announcements, as discussed in more detail in Section 4. At its trough in 2012Q4, just before the taper tantrum, our shadow rate is more than a percentage point below the Wu-Xia measure. A stark gap then opens up between our shadow rate and the Wu-Xia shadow rate over 2013through2015. Relativetoafairlyflatimpliedsystematicpolicyresponsetoeconomicconditions at the time, our measure suggests that monetary policy was becoming relatively less accommodativeintheleaduptoliftofffromtheZLB,whiletheWu-Xiameasurefellbyalmost 2percentagepointstoalowof-2.9%by2014Q2,suggestingpolicywasbecomingmoreexpansionary.11 In Section 4, we compare the performance of both measures in a VAR that controls for economic conditions in terms of inflation and output growth when identifying monetary 10Theimpliedsystematicpolicyresponseiscalculatedusingtheprevailingvaluesofthetargetvariablesinthe policyrule,butsettingthemonetarypolicyshocksandshadowrateshockstozeroandallowingforinterestrate smoothingaccordingtotheestimatedpolicyrule. ThepolicyruleintheSmetsandWouters(2007)modelassumes systematicresponsestoinflation,theoutputgap,andchangesintheoutputgap. 11Sims and Wu (2020) show that the Wu-Xia measure is correlated with the Fed’s balance sheet during this period, but acknowledge that the “shadow rate series is based on empirical term structure models that do not haveanexplicitmappingbackintostructuraleconomicmodelsorparticularunconventionaltools”. 12

policy shocks and find that our measure performs better due to the different signals it gives aboutthestanceofpolicyduringtheZLB. 3.3 No Forward Guidance Counterfactual To understand the effects of unconventional policies, we explore the counterfactual scenario of whatwouldhavehappenedintheaftermathoftheGreatRecessionwithoutforwardguidance. We construct this counterfactual using the solution (5) implied by the expected durations d to t obtain an estimate of the structural shocks ε and then feeding the estimated structural shocks t throughtheoccasionally-binding-constraintsolutionofJones(2017)tosolveforthepathofthe economy if monetary policy simply followed the prescription of the policy rule constrained by the ZLB. In this case, the durations of the ZLB expected by agents are dlb in the notation of t Section2. Figure 2 plots the counterfactual paths of output, inflation, and the shadow rate removing forward guidance. These paths imply that unconventional policies extending expected durations raised the level of output by as much as 4 percent in 2012Q4, while inflation was less affected due to a strong degree of nominal rigidities according to the parameter estimates.12 Using these variables together with the other non-interest-rate observables as targets in ∆ = t ι(x −x ∗), the counterfactual shadow interest rate is positive and not just in the early stages of t t the ZLB. Instead, it remains non-negative throughout the ZLB and rises somewhat above zero around2012whenoutputwouldhavebeenmostdepressedwithoutunconventionalpolicies. 4 Applications of the Structural Shadow Rate Weexplorethreeapplicationsofourstructuralshadowrate. Thefirsttworeflectthetypicaluse of a policy rate in linear setups, while the third considers monetary policy during the recent pandemic. 4.1 Shock Decomposition Thestructuralshadowratecanbeusedtoperformashockdecompositioninordertoassessthe contributionofpolicyshockstoobservedvariables. Specifically,weobtainsmoothedestimates 12Thelargereffectsonrealactivitythaninflationareinlinewithcounterfactualsfortheunemploymentrateand inflationinEberlyetal.(2020)basedonastructuralVARwithexternalinstruments. Wenotetoothat, asshown intheappendix,ourshadowrateestimatesarerobusttoanalternativecalibrationofCalvoparametersmotivated byfindingsinFitzgeraldetal.(2020). 13

Figure2: CounterfactualRemovingForwardGuidance Notes: Forthedatainsolidblueandacounterfactualremovingtheeffectsofforwardguidanceinblack,panel(a)plotsanindexofoutput normalizedto1inthebaseyearof2008andpanel(b)plots,inannualizedpercentageterms,year-on-yearinflation.Panel(c)plots,inannualizedpercentageterms,theshadowratecomputedforthiscounterfactualpath.Theblacklinesforthecounterfactualscorrespondtoposterior means,whilethebandsshow90percentequal-tailedposteriorintervals. 14

of historical structural shocks, including the shadow rate shock. We then feed each shock oneby-oneintothemodeltocalculatetheeffecteachshockhasontheobservedvariables. Figure 3 plots the path of the change in the annualized 5-year yield under the shadow rate shock alone, noting some key events related to unconventional policies. After initial contractionaryeffectsfromtheZLBconstraintatthebeginningof2009,theshadowrateshockslargely act to reduce the long rate during the ZLB. Of particular note, the black dashed vertical lines correspond to quarters in which calendar-based forward guidance announcements were made in FOMC statements. The contribution of the shadow rate shocks to lowering the long rate closelyalignswiththequartersoftheseannouncements. Forexample,calendar-basedforward guidancewasinitiatedin2011Q3whentheFOMCannouncedthatthefederalfundsratewould beheldatzerountil“atleastthroughmid-2013”. In2012Q1,thisdatewasextendedto“atleast through late-2014”. In 2012Q4, the FOMC introduced threshold-based forward guidance, announcingthatthefederalfundsratewouldnotberaiseduntilcertainvaluesforunemployment andinflationwereachieved. Thesethreequarters–2011Q3,2012Q1,and2012Q4–sawthreeof thefourlargestcontributionsoftheshadowrateshocktoloweringthe5-yearyield. Meanwhile, the red dashed vertical lines correspond to notable events when shadow rate shocks increased the 5-year yield. 2013Q2 covers the taper tantrum, when markets interpreted remarks by the Fed as a signal that it would slow asset purchases. 2015Q1 covers the removal of references by the FOMC in its statement to maintaining the federal funds rate at the lower bound for a “considerabletime”followingtheendofitsassetpurchaseprogram. 4.2 VAR Analysis Another use of a shadow rate is in VAR analysis when the sample covers the ZLB. We considerourshadowrateinathree-variableVARthatalsoincludesquarterlyinflationandoutput growth. TheVARisestimatedover2009Q1to2015Q3andweincludeonelagbasedondiagnosticssuggestingtheforecasterrorsareseriallyuncorrelated. Forcomparison,wealsoconsidera versionoftheVARwiththeWu-Xiashadowrateinsteadofourshadowrate. Weemployastandardidentificationofmonetarypolicyshocksbyorderingtheshadowrate last and using a Cholesky factorization of the forecast-error variance-covariance matrix in order to calculate impulse responses to a one-standard-deviation monetary policy shock under theassumptionthattheshadowratecanrespondtocontemporaneousinformationaboutinflation and output growth, but only affects them with a lag. Figure 4 plots the impulse response 15

Figure3: ContributionofShadowRateShockstoChangesinthe5-YearYield Notes: The bars are annualized percentage point contributions of shadow rate shocks to changes in the 5-year yield based on smoothed estimates. Dashedverticallinesrepresentthefollowingdates: 2009Q2: QE1; 2010Q4: QE2; 2011Q3: calendar-basedforwardguidance“at leastthroughmid-2013”;2012Q1: calendar-basedforwardguidance“atleastthroughlate2014”;2012Q3: calendar-basedforwardguidance “atleastthroughmid-2015”;2012Q4:threshold-basedforwardguidance;2013Q2:tapertantrum;2014Q1:removalofthreshold-basedforward guidance;2015Q1:removalofreferencestocalendar-basedforwardguidanceandtothemaintenanceofinterestratesatthelowerboundfora “considerabletime”followingtheendoftheassetpurchasingprogram. functions for a monetary policy shock in the two cases of using our shadow rate and the Wu- Xia shadow rate. Given a contractionary shock, we find significant declines in both inflation and output growth within a one-year horizon when using our shadow rate to identify policy shocks. By contrast, there is a ‘price puzzle’ in the case of the Wu-Xia shadow rate in the form of initial positive (but insignificant) responses of inflation to a contractionary shock, although theydoturnnegativeatlongerhorizons. One interpretation of the price puzzle is that a VAR with Cholesky factorization does not cleanly identify monetary policy shocks, but mixes them with endogenous responses of monetary policy to other shocks with inflationary effects. The Wu-Xia shadow rate tends to fall whenever there is a decline in long-term interest rates, regardless of the reason for the decline. If those declines reflect a deterioration in inflation expectations rather than more expansionary unconventional policies, then it will overstate how accommodative the policy stance has actually become. The identified policy shock in the VAR system will be negative when inflation expectationsfall,thusleadingtoapositivecorrelationbetweentheidentifiedpolicyshockand inflation, i.e. the price puzzle. By contrast, our shadow rate should provide a more accurate reading of the policy stance relative to economic conditions and, therefore, can better avoid mixingpolicyshockswithendogenousresponsestoothershockswithinflationaryeffects. Our approach identifies whether a decline in long-term interest rates is due to a deterioration in in- 16

Figure4: ImpulseResponsesinaThree-VariableVAR (a)EstimatedresponsesusingtheStructuralShadowRate (b)EstimatedresponsesusingtheWu-XiaShadowRate Notes:Thisfiguredisplaysresponsesofinflationandoutputgrowthinquarterlypercentagetermsandtheshadowrateinannualizedpercentagetermstoaone-standarddeviationmonetarypolicyshockinathree-variableVARwithquarterlyinflation,outputgrowth,andtheshadow rateestimatedovertheZLBperiod,2009Q1to2015Q3. Theshadowrateisorderedlastforidentificationofmonetarypolicyshocksusinga Cholskyfactorizationoftheforecast-errorvariance-covariancematrix. Panel(a)reportsresultsforthestructuralshadowrateandpanel(b) reportsresultsfortheshadowratemeasureconstructedbyWuandXia(2016).Theimpulseresponsesarecomputedusingabootstrapproceduredrawingresidualswithreplacement.Theblacklinescorrespondtomeanresponsesandthebandsshow90percentequal-tailedbootstrap confidenceintervals. flation expectations increasing the expected duration of the ZLB because of the constraint or a changeinforwardguidanceincreasingtheexpectedduration,withacorrespondingincreaseor decrease in the shadow rate, respectively. Unlike with the Wu-Xia shadow rate, the identified policyshockintheVARusingourshadowratewouldthenbepositivewheninflationexpectations fall, thus leading to a negative correlation between identified policy shock and inflation, i.e. avoidingthepricepuzzle. These VAR results support the idea that the structural shadow rate can better reflect the stance of policy than SRTSM measures, suggesting it can be employed in empirical analysis using linear models when the sample period covers the ZLB. While the structural model can be directly used to consider the effects of monetary policy, as in Kulish et al. (2017), it may be useful to have a simple summary of the policy stance during the ZLB when conducting other 17

Figure5: TheStructuralShadowFederalFundsRateincludingtheCOVID-19Crisis Notes:Thisfigureplots,inannualizedpercentageterms,theestimatedshadowrateforasamplethatendsin2021Q1,alongwiththeimplied systematicpolicyresponse. ThereddashedlineshowstheestimatedshadowrateoverthesecondZLBepisode,startingfrom2020Q2. The impliedsystematicpolicyresponsetoeconomicconditionsisbasedonthepolicyrulewithoutmonetarypolicyorshadowrateshocksbut allowingforinterestratesmoothing,i.e. it,systematic = i¯+(1−αi )αpπˆt +(1−αi )αyy˜t +α∆y ∆y˜t +αi (it−1,systematic −i¯),wheretheα’sarethe monetarypolicyresponsecoefficientsandy˜t istheoutputgapfromtheflexiblepriceequilibriumfortheSmetsandWouters(2007)model detailedintheappendix.Thelinescorrespondtoposteriormeans. empiricalanalysis. Byprovidingamoreaccuratemeasureofmonetarypolicy,ourshadowrate canserveasabettercontrolorindicatorinsuchanalysis. 4.3 Monetary Policy during the Pandemic Ourlastapplicationextendsthesampletocoverthelargeeconomicfluctuationsassociatedwith therecentpandemic. Givenoutliersinsomevariables,wedonotreestimatethemodeloverthe extended sample. Instead, we use parameter estimates for data from 1984Q1 to 2019Q4, but extend the sample to 2021Q1 to find structural shocks. During the second ZLB episode that starts in 2020Q2, we again use expected durations based on the modal reported values from the New York Fed Survey of Primary Dealers. The survey implies durations of 8 quarters in 2020Q2,14quartersin2020Q3,16quartersin2020Q4,and12quartersin2021Q1. Figure 5 plots the estimated structural shadow federal funds rate given the updated data. The red dashed line highlights the shadow rate since 2020Q2. The estimated shadow rate 18

jumped to 7.7% in 2020Q2, reflecting the highly contractionary effects of a persistent expected ZLB constraint when the systematic policy response implied by the policy rule fell to below -10%. However, the estimated shadow rate quickly reversed to almost -5% in 2020Q3, close to the systematic policy response implied by the unconstrained policy rule. This reversal is line withanimmediate,albeitpartial,recoveryineconomicconditions,alongwiththeFed’simplementation of a number of extraordinary unconventional policies, including forward guidance related to a shift in the monetary policy framework to consider average inflation in August 2020, with a corresponding doubling of expected durations from 2 years in 2020Q2 to 4 years in2020Q4. 5 Conclusion IdentifyingthestanceofmonetarypolicyattheZLBrequiresastructuralmodel. Termstructure modelscanproduceestimatesoftheexpecteddurationofzero-interest-ratepolicy. Butastructural model is needed to uncover the reasons behind this expectation. Deteriorating economic conditions that make the ZLB constraint bind for longer are equivalent to tighter monetary policy, while unconventional policies that extend the duration correspond to more expansionary policy. In the SRTSM approach, the short rate follows i = max(i ∗ ,0), which constrains t t the behavior of the shadow rate to be non-positive when the short rate is at the ZLB. Given persistent large negative shocks that extend the ZLB constraint enough to make our structural shadow rate positive, the SRTSM measure would imply the stance of monetary policy is more expansionary than it actually is. By contrast, our structural shadow rate accurately reflects the stance of policy, as is evident in its strong coherence with announcements by the Fed related to forward guidance, as well as its performance in VAR analysis when the sample covers the ZLB.Thestructuralshadowfederalfundsratealsosuggestsalargeandquickreversalfroman initially high positive level of interest-rate-equivalent policy reflecting a severe binding of the ZLB constraint with the onset of the COVID-19 crisis to an even more negative level than ever occurredduringtheGreatRecessionoritsaftermath. The structural shadow rate is, by its nature, dependent on the model used in its estimation. This is no different than the SRTSM approach, which is sensitive to model specification, as highlighted in Bauer and Rudebusch (2016). However, to the extent that expected durations arepinneddownbysurveydataandthecomponentofadurationrelatedtotheZLBconstraint is identified by reasonable estimates of structural shocks and the monetary policy rule, results 19

should be fairly robust to a range of related models. Importantly, our approach can be applied to any structural model that accounts for the ZLB. Thus, an interesting extension for future research would be to consider a model in which quantitative easing plays a distinct role, such asinGertlerandKaradi(2011). 20

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Appendix For Online Publication A Description of the Smets and Wouters Model We list here the linearized equations of the Smets and Wouters (2007) model. We use similar notationforvariablesandparametersasintheirpaper,butwesubstitute i forr whenreferring to the nominal interest rate and ik for i when referring to investment. The model variables are presented in terms of deviations from steady state, but with hats suppressed for simplicity. A full description of the model is available in Smets and Wouters (2007) and its accompanying onlineappendix. A.1 Sticky Price Economy Factorprices: mc = αrk +(1−α)w −ε t t t a,t rk = w +l −ks t t t t z = 1−ψ rk t ψ t Investment: (cid:16) (cid:17) ik = 1 ik +βγ E ik + 1 q +ε t 1+βγ t−1 t t+1 γ2φ t i,t q = 1−δ E q + R¯k E rk −i +E π + σc (1+λ/γ) ε t 1−δ+R¯k t t+1 1−δ+R¯k t t+1 t t t+1 1−λ/γ b,t Consumption: c = λ/γ c + 1 E c + (σc −1)W∗L∗/C∗ (l −E l )− 1−λ/γ (i −E π )+ε t 1+λ/γ t−1 1+λ/γ t t+1 σc (1+λ/γ) t t t+1 σc (1+λ/γ) t t t+1 b,t Resourceconstraint: y = c c +ikik +z z +ε t t y t y t y g,t Productionfunction: y = φ (αks +(1−α) l +ε ) t p t t a,t ks = z +k t t t−1 23

Monetarypolicyrule: (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) f f f i t = (1−α i )α p π t +(1−α i )α y y t −y t +α∆y y t −y t − y t−1 −y t−1 +α i i t−1 +ε m,t Longerterminterestrates: i = ε +η 4,t η,t 4,t i = ε +η 20,t η,t 20,t Evolutionofcapital: k = (1−δ) k + (γ−1+δ) ik + (γ−1+δ) φγ2ε t γ t−1 γ t γ i,t PriceandwagePhilipscurves: (cid:18) (cid:19) π = 1 βγ E π +ι π + (1−ξp )(1−βγξp ) 1 mc +ε t 1+βγιp t t+1 p t−1 ξp 1+(φp −1)εp t p,t w = w w +(1−w )E (w +π )−w π +w π − t 1 t−1 1 t t+1 t+1 2 t 3 t−1 (cid:18) (cid:19) 1 λ/γ w σl + c − c −w +ε , 4 l t 1−λ/γ t 1−λ/γ t−1 t w,t where w = 1 , w = w (1+βγι ), w = w ι ,and w = w (1−ξw )(1−βγξw ) . 1 1+βγ 2 1 w 3 1 w 4 1ξw (1+(φw −1)εw ) A.2 Flexible Price Economy Thecorrespondingequationsdefiningtheflexiblepriceeconomyare αr k,f +(1−α)w f = ε t t a,t r k,f = w f +l f −k f t t t t z f = 1−ψ r k,f t ψ t f f f k = z +kp t t t−1 (cid:16) (cid:17) i k,f = 1 i k,f +βγ E i k,f + 1 q f +ε t 1+βγ t−1 t t+1 γ2φ t i,t q f = 1−δ E q f + R¯k E r k,f −i f + σc (1+λ/γ) ε t 1−δ+R¯k t t+1 1−δ+R¯k t t+1 t 1−λ/γ b,t c f = λ/γ c f + 1 E c f + (σc −1)W∗L∗/C∗ (cid:16) l f −E l f (cid:17) − 1−λ/γ i f +ε t 1+λ/γ t−1 1+λ/γ t t+1 σc (1+λ/γ) t t t+1 σc (1+λ/γ) t b,t y f = c f c +i k,f i k,f +z f z +ε t t y t y t y g,t (cid:16) (cid:17) f f f y = φ αk +(1−α) l +ε t p t t a,t k p,f = (1−δ) k p,f + (γ−1+δ) i k,f + (γ−1+δ) φγ2ε t γ t−1 γ t γ i,t w f = σl f + 1 c f − λ/γ c f . t l t 1−λ/γ t 1−λ/γ t−1 24

A.3 Exogenous Processes Letting ζ denoteani.i.d. standardnormalinnovation,theexogenousprocessesare ε = ρ ε +σ ζ a,t a a,t−1 a a,t ε = ρ ε +σ ζ b,t b b,t−1 b b,t ε = ρ ε +σ ζ +ρ σ ζ g,t g g,t−1 g g,t ga a a,t ε = ρ ε +σ ζ i,t i i,t−1 i i,t ε = ρ ε +σ ζ m,t m m,t−1 i m,t ε = ρ ε +η −µ η p,t p p,t−1 p,ma,t p p,ma,t−1 η = σ ζ p,ma,t p p,t ε = ρ ε +η −µ η w,t w w,t−1 w,ma,t w w,ma,t−1 η = σ ζ w,ma,t w w,t ε = ρ ε +σ ζ η,t η η,t−1 η η,t η = σ ζ 4,t i,4 4,t η = σ ζ . 20,t i,20 20,t A.4 Measurement Equations Finally,themeasurementequationsare dy = γ¯ +y −y t t t−1 dc = γ¯ +c −c t t t−1 dik = γ¯ +ik −ik t t t−1 dw = γ¯ +w −w t t t−1 πobs = π¯ +π t t iobs = i¯+i t t iobs = i¯+i¯ +i 4,t 4 4,t iobs = i¯+i¯ +i 20,t 20 20,t lobs = l¯+l . t t 25

B Data, Parameter Estimates, and Additional Results B.1 Data Sources and Mapping to Model Weusethefollowingdataseriesandsources(withFREDmnemonicsinparentheses): • RealGrossDomesticProduct(GDPC1) • FixedPrivateInvestment(FPI) • PersonalConsumptionExpenditures(PCEC) • Inflation: GrossDomesticProduct,ImplicitPriceDeflator(GDPDEF) • NonfarmBusinessSector: AverageWeeklyHours(PRS85006023) • NonfarmBusinessSector: CompensationPerHour(COMPNFB) • FederalFundsRate(FEDFUNDS) • 1-YearTreasuryConstantMaturityRate(GS1) • 5-YearTreasuryConstantMaturityRate(GS5) • PopulationLevel(CNP16OV) • EmploymentLevel(CE16OV) • ZLB Durations: following Kulish et al. (2017), we use the ZLB durations extracted from the New York Fed Survey of Primary Dealers, conducted eight times a year from 2011Q1 onwards and the Blue Chip Financial Forecasts survey before 2011.13 For our measure of anexpectedduration,wetakethemodeofthedistributionimpliedbythesesurveys. Wemaptheseseriestoourobservedvariablesinthefollowingway: CNP16OV CNP16OV_idx = CNP16OV 1992Q3 13See the website https://www.newyorkfed.org/markets/primarydealer_survey_questions.html for more information on the Primary Dealers survey. For example, in the survey conducted on January 18 2011, one of the questionsaskedwas: “Ofthepossibleoutcomesbelow,pleaseindicatethepercentchanceyouattachtothetiming of the first federal funds target rate increase” (Question 2b). Responses were given in terms of a probability distributionacrossfuturequarters. 26

CE16OV CE16OV_idx = CE16OV 1992Q3 PCEC 1 dc = 100×∆ × t GDPDEF CNP16OV_idx GDPC1 dy = 100×∆ t CNP16OV_idx FPI 1 dik = 100×∆ × t GDPDEF CNP16OV_idx COMPNFB dw = 100×∆ t GDPDEF (cid:18) (cid:19) CE16OV_idx lobs = 100×log PRS85006023× . t CNP16OV_idx Wedemean lobs over1984Q1to2019Q4. t B.2 Parameter Estimates The prior and posterior distributions of the estimated parameters are given in Table B.1. The Calvo price parameter is centered around a value of 0.93, indicating the aggregate data prefers strong nominal rigidities and a relatively flat Phillips curve, which helps to rationalize a relatively stable inflation rate with a large output and employment gap in the post-2009 sample. The Calvo wage parameter is centered around a value of 0.36, in line with the estimates from Fitzgerald et al. (2020) that use relative US state-level data. In Appendix C below, we explore the robustness of our results to calibrating both the Calvo wage and price parameters to their estimatesinFitzgeraldetal.(2020),namelyaCalvowageparameterof0.4andaCalvopriceparameterof0.6. Theresultsintermsoftheshadowratearehighlyrobust. Theposteriorestimates of the response of the policy interest rate to inflation and output fluctuations are slightly lower than those reported in Smets and Wouters (2007), noting that their sample ends in 2004. The posteriorestimatefortrendgrowthiscenteredaround0.57%perquarter. Weobtainaprecisely estimated one-time decline in trend growth of -0.18% in 2003Q4, consistent with other studies noted above that find a decline in trend growth around that time. This decline translates into a difference in the annual rate of trend growth of 2.3% before 2003Q4 to 1.5% thereafter. Given ourposteriorestimateofthediscountfactor,theestimateddeclineintrendgrowthimpliesthat the annualized steady-state nominal interest rate falls from 5.2% to 4.4%. In simulations, this decline in the steady-state nominal interest rate has the effect of raising the fraction of time spentattheZLBfromabout5%toabout15%. 27

TableB.1: ParameterEstimates Prior Posterior Parameter Type Mean 5% 95% Mode Median 5% 95% φ N 4.0 1.5 6.5 6.98 6.79 5.05 8.74 σ N 1.5 0.9 2.1 1.08 1.09 0.90 1.30 c λ B 0.7 0.5 0.9 0.27 0.26 0.19 0.35 ξ B 0.5 0.3 0.7 0.33 0.36 0.25 0.50 w σ N 2.0 0.8 3.2 0.89 0.95 0.56 1.49 l ξ B 0.5 0.3 0.7 0.93 0.93 0.91 0.95 p ι B 0.5 0.3 0.7 0.37 0.40 0.18 0.65 w ι B 0.5 0.3 0.7 0.17 0.19 0.09 0.34 p ψ B 0.5 0.3 0.7 0.74 0.74 0.58 0.87 φ N 1.2 1.0 1.5 1.50 1.50 1.37 1.64 p α N 1.5 1.3 1.7 1.64 1.65 1.50 1.80 p α B 0.8 0.6 0.9 0.79 0.76 0.57 0.90 i α N 0.1 0.0 0.2 0.06 0.06 0.05 0.08 y α∆y N 0.1 0.0 0.2 0.17 0.17 0.13 0.20 100 (cid:0) β−1−1 (cid:1) G 0.2 0.1 0.4 0.16 0.17 0.08 0.30 γ N 0.4 0.2 0.6 0.58 0.57 0.53 0.62 α N 0.3 0.2 0.4 0.15 0.15 0.13 0.18 i¯ N 0.1 0.0 0.3 0.04 0.04 −0.00 0.08 4 i¯ N 0.5 0.1 0.9 0.19 0.19 0.12 0.26 20 l¯ N 0.0 −2.5 2.5 2.31 2.25 0.14 3.91 PersistenceandVariancesofExogenousProcesses ρ B 0.5 0.2 0.8 0.93 0.93 0.91 0.95 a ρ B 0.5 0.2 0.8 0.98 0.98 0.97 0.99 b ρ B 0.5 0.2 0.8 0.99 0.99 0.98 1.00 g ρ B 0.5 0.2 0.8 0.82 0.81 0.73 0.88 i ρ B 0.5 0.2 0.8 0.80 0.79 0.69 0.87 p ρ B 0.5 0.2 0.8 0.99 0.99 0.99 1.00 w µ B 0.5 0.2 0.8 0.72 0.69 0.52 0.81 p µ B 0.5 0.2 0.8 0.74 0.73 0.56 0.86 w ρ N 0.5 0.2 0.8 0.48 0.47 0.35 0.59 ga ρ B 0.5 0.2 0.8 0.78 0.78 0.69 0.87 η σ IG 0.1 0.0 0.3 0.42 0.42 0.38 0.47 a σ IG 0.1 0.0 0.3 0.04 0.04 0.03 0.05 b σ IG 0.1 0.0 0.3 0.37 0.36 0.33 0.40 g σ IG 0.1 0.0 0.3 0.22 0.22 0.19 0.27 i σ IG 0.1 0.0 0.3 0.13 0.13 0.11 0.14 m σ IG 0.1 0.0 0.3 0.11 0.10 0.09 0.12 p σ IG 0.1 0.0 0.3 0.49 0.50 0.42 0.62 w σ IG 0.1 0.0 0.3 0.06 0.06 0.05 0.07 η σ IG 0.1 0.0 0.3 0.01 0.01 0.01 0.02 i,4 σ IG 0.1 0.0 0.3 0.09 0.09 0.08 0.10 i,20 ChangeinTrendGrowth ∆γ¯ N 0.0 −0.4 0.4 −0.18 −0.18 −0.23 −0.13 Dateof∆γ¯ U 2000Q1 1994Q3 2006Q3 2003Q4 2003Q4 2001Q1 2005Q2 28

Figure B.1 plots the convergence of the two chains along the chain using the Gelman R2 diagnostic. The R2 diagnostic lies below the value of 1.1 for all parameters by the end of the chain,indicatingconvergence,acrosschains,oftheposteriordistributions. FigureB.1: ConvergenceofMCMCChains B.3 Additional Results FigureB.2plotsthepathofthevariablestargetedintheconstructionoftheshadowinterestrate in the first six panels, in the data (in blue), and in the shadow economy (in red) given by the system (7). The plot shows that the paths of these variables under the shadow rate shocks are veryclosetotheobservedpaths. Thenumericalmatchingprocedureweusetofindtheshadow rate shocks is thus able to replicate the data. For completeness, the path of the policy rate and theshadowrateisalsogiveninthelastpanel. Figure B.3 plots the ZLB durations decomposed into the lower-bound component, dlb, and t fg the forward-guidance component, d . The sum of these two components gives the actual dut ration expected by agents in the economy, denoted by d in the text. The figure shows how the t 29

FigureB.2: PathsofTargetsintheShadowEconomy 30

FigureB.3: ZLBDurationsatthePosteriorMode forward-guidance durations were initially quite short, but increased over 2011 and 2012. The forward-guidance duration was even briefly negative in 2009Q1, reflecting an estimated belief by economic agents that the Fed would deviate from the policy rule and begin raising rates one quarter before the ZLB constraint was expected to stop binding for the policy rule. From 2013 to 2015, as the federal funds rate moved closer to liftoff, the actual and forward-guidance durationsfellbacktowardszero. Figure B.4 plots the paths of all the variables in the data and in the counterfactual scenario wheretheforward-guidancedurationsaresettozero. C Robustness Figure C.1 plots the shadow rate estimates when the Calvo price and wage parameters are calibrated instead of estimated. The estimates are very similar to those reported in the main text. FigureC.2plotstheshadowrateconstructedwhen,inadditiontothemacroeconomicaggregates,wealsotargetthe1-yearand5-yearyields. Thefiguredisplaystheshadowrateestimates including the pandemic and shows that there are almost no differences when compared to the baselineshadowrateweconstructusingmacroeconomicaggregatesalone. FiguresC.3andC.4 plot the paths of variables in the shadow economy given the alternative targets including the 31

FigureB.4: CounterfactualPathsRemovingForwardGuidance 32

FigureC.1: ShadowFederalFundsRategivenCalibratedCalvoParameters (a)ShadowRate,i∗ t (b)ShadowRateShocks,ε∗ m,t Notes: Panel(a)plots,inannualizedpercentageterms,theestimatedshadowrateandpanel(b)plots,inannualizedpercentagepointterms, theshadowrateshocksforestimationoftheSmetsandWouters(2007)modelwheretheCalvopriceandCalvowageparametersarecalibrated toξp =0.6andξw =0.4,respectively. Thedashedblacklinescorrespondtoposteriormeans,whilethebandsshow90percentequal-tailed posteriorintervals. 33

FigureC.2: ShadowFederalFundsRateAdditionallyTargetingOtherInterestRates 1-yearand5-yearyieldsandshowthatthefitofthemacroeconomicaggregatesisvirtuallyunchanged to the baseline, while the interest rates, especially the 1-year yield, behave differently in the unconstrained shadow economy. This robustness exercise makes clear how our identificationoftheshadowrateisdrivenbythemacroeconomicaggregates,notthetermstructureof interestrates,whichisakeypointofcontrasttotheexistingliteratureonshadowpolicyrates. D Comparison with the Wu-Xia Shadow Rate WecompareourstructuralshadowratewiththeshadowrateofWuandXia(2016). FigureD.1 plots the mean of our shadow rate series across posterior draws on the same axes as the Wu- Xia measure. Both shadow rates initially start above zero. As explained in the main text, our measure can be above zero when the federal funds rate is at zero because of shocks interacting withtheZLBgenerateadditionalnonlinearcontractionaryeffectsthatmapintocontractionary shadow rate shocks. In contrast, the Wu-Xia shadow rate can only be above zero when the federal funds rate is at zero if the measure of the short rate used in estimation – the 3-month 34

FigureC.3: PathsofTargetsintheShadowEconomyAdditionallyTargeting1Y,5YYields 35

FigureC.4: PathsofTargetsintheShadowEconomyAdditionallyTargeting5YYield 36

FigureD.1: ComparisonofDifferentMeasuresoftheShadowFederalFundsRate forward rate – is positive. Thus, both measures have similar paths before 2010, but for completelydifferentreasons. The two measures both decline through to 2013, although our measure is about half a percentage point more negative at the mean across posterior draws. Our measure also displays more volatility. Larger differences become apparent after 2013, when our measure increases fromitstroughof-2.4%overthecourseof2013toabout-0.5%,whereitstaysbeforeincreasing to zero by the time of liftoff in 2015. The Wu-Xia measure shows a notable contrast, falling to almost-3%bytheendof2014,despitethetapertantrumin2013andtheremovalofthresholdbasedforwardguidancein2014,withamuchmoresuddenrisebacktozeroin2015. OneillustrativewaytocontrastourapproachwiththatofWu-Xiaistosimulatedatafroma structural model and use the two methods to construct the respective implied shadow interest rates. To do so, we take the Smets and Wouters (2007) model estimated in this paper on the 1984Q1 to 2019Q4 sample and additionally add yield curve variables up to 10-years. This allows us to simulate a yield curve with the following interest rates – 1Q, 2Q, 1Y, 2Y, 5Y, 7Y, 10Y – that we can use to estimate a term structure model. We simulate a long sample of the model 37

FigureD.2: SimulationtoCompareDifferentApproachestoMeasuringtheShadowRate and choose a period of the simulation where the ZLB is a significant constraint. For the most contrastbetweenmethods,weabstractfromforwardguidance. The top two panels of Figure D.2 plot the simulated yield curve and the simulated output series. Also plotted in the output panel is the output series if the ZLB were removed as a constraint on monetary policy. Comparing this with output under the ZLB illustrates the contractionary forces that the ZLB can induce on the economy. The simulated federal funds rate is showninthebottomtwopanelsinblue–theZLBbindsonandoffoverthesample. The shadow rate constructed using our approach is shown in the dashed red series in the bottom left panel. In order to replicate the contractionary effects of the ZLB, our procedure finds contractionary shadow rate shocks, which work to push the shadow interest rate well 38

abovezerointheperiodsthattheZLBbinds. TheconstructedWu-Xiashadowrate,presentedinthefinalpanel,providesastarkcontrast, with the shadow rate measure falling well below zero and almost reaching -8%. Clearly, the inferences that would be made from this measure of the shadow rate would not line up with the conceptual basis of the shadow rate, which is that it reflects the policy stance of the central bank. In this simulation, the central bank does not react at all to shocks that occur at the ZLB. The Wu-Xia shadow rate instead simply reflects the behavior of the yield curve, but the yield curvecandeclineeitherbecauseofnegativeshocks(whichisthecasehere)orbecauseofpolicy actions (which is not the case here). The Wu-Xia measure is not able to discriminate between thetwosourcesofdecline. 39