Learning and Misperception: Implications for Price-Level Targeting

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Learning and Misperception: Implications for Price-Level Targeting Martin Bodenstein, James Hebden, and Fabian Winkler 2019-078 Please cite this paper as: Bodenstein, Martin, James Hebden, and Fabian Winkler (2019). “Learning and Misperception: Implications for Price-Level Targeting,” Finance and Economics Discussion Series 2019-078. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2019.078. 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.

Learning and Misperception: Implications for Price-Level Targeting Martin Bodenstein James Hebden Fabian Winkler∗ October 25, 2019 Abstract Monetarypolicystrategiesthattargetthepricelevelhavebeenadvocatedasamoreeffectivewaytoprovideeconomicstimulusinadeeprecessionwhenconventionalmonetary policyislimitedbythezerolowerboundonnominalinterestrates. Yet,theeffectivenessof thesestrategiesdependsonacentralbank’sabilitytosteeragents’expectationsaboutthe future path of the policy rate. We develop a flexible method of learning about the central bank’spolicyrulefromobservedinterestratesthattakesintoaccountthelimitedinformationalcontentatthezerolowerbound. Whenagentslearn,switchingfromaninflationtargetingtoaprice-leveltargetingstrategyattheonsetofarecessiondoesnotyieldthedesired stabilizationbenefits.Thesebenefitsonlymaterializeafterthepolicyrulehasbeeninplace forasufficientlylongtime. Temporaryprice-leveltargetingstrategiesarelikelytobemuch lesseffectivethantheirpermanentcounterparts. ∗FederalReserveBoard,20thStandConstitutionAveNW,WashingtonDC20551.Ouremailaddresses aremartin.r.bodenstein@frb.gov,james.s.hebden@frb.govandfabian.winkler@frb.gov.WethankEtienne Gagnon,ChristopherGust,DavidLopez-Salido,DirkNiepelt,BobTetlow,andseminarparticipantsatthe EuropeanCentralBankforhelpfulcomments. TheviewsexpressedinthispaperaresolelytheresponsibilityoftheauthorsandshouldnotbeinterpretedasreflectingtheviewsoftheBoardofGovernorsofthe FederalReserveSystemoranyotherpersonassociatedwiththeFederalReserveSystem. 1

1 Introduction A number of ongoing structural developments have tested the ability of central banks around the globe to achieve their goals. The neutral real interest rate has likely fallen, implyinglessleewaytolowerinterestratesintheeventofarecessionbecauseofthezero lowerbound(ZLB).Thisdevelopmentposesachallengeforthepredominantmonetary policy approach of flexible inflation targeting to manage future recessions and has led some central banks, including the Federal Reserve, to review their existing monetary policy frameworks.1 Academics and policymakers have explored “makeup strategies” thataimtooffset,atleastinpart,pastmissesofinflationfromitslong-runtarget,incontrasttoflexibleinflationtargetingwherethehistoryofpastdeviationsofinflationfrom thistargetisirrelevant. Makeupstrategiesarethoughttohavelargestabilizationbenefitsthatstemfromtheireffectonexpectations: Acommitmenttomakeupforinflation shortfalls through lower interest rates in the future raises near-term inflation expectations and lowers real interest rates which then stimulate the economy even when the short-termnominalinterestrateisconstrainedbytheZLB.Aparticulartypeofmakeup strategiesthathasreceivedmostoftheattentionisprice-leveltargeting.2 Whenassessingtheeffectsofprice-leveltargeting,itiscommonlyassumedthatprivate sector agents know the strategy pursued by the central bank and fully believe policymakers’ commitment to this strategy. While these assumptions may be reasonable when the monetary policy strategy has been stable over time, they are questionable whenthecentralbankchangestoastrategywithouthistoricalprecedentandthepublic has no experience with the new strategy. A switch to price-level targeting or a similarmakeupstrategywouldconstitutearadicaldeparturefromcurrentpracticeandthe public would require time to learn and to trust this new approach to monetary policy. 1Compare the Minutes of the Federal Open Market Committee, September 17-18, 2019 available at www.federalreserve.gov/monetarypolicy/files/fomcminutes20190918.pdf. 2Eggertsson and Woodford (2004) show that price-level targeting is the optimal commitment policy in the textbook New Keynesian model subject to the ZLB. Reifschneider and Williams (2000), Hebden andLo´pez-Salido(2018),Bernankeetal.(2019),andMertensandWilliams(2019)discussthestabilizationpropertiesofvariousstrategiesthatseektostabilizethepricelevelinquantitativemodels. Seealso Svensson(2019)forarecentreview. Foradiscussionofotherbenefitsofprice-leveltargeting,seeSvensson(1999),Vestin(2006);Ambler(2009)andHatcherandMinford(2016)offerextensivesurveys. 2

Afterall, eveninthecomparativelybenigncaseoftheshiftinmonetarypolicystrategy by Federal Reserve Chairmen Volcker and Greenspan towards a low inflation target, it tookconsiderabletimeforthepublictounderstandthenewstrategy,asarguedinErceg and Levin (2003). Since the effectiveness of price-level targeting and other makeup strategies hinges crucially on expectations, we think that a complete evaluation of the merits of these strategies must consider deviations from the full information, rational expectations paradigm, taking into account the public’s uncertainty about the central bank’sstrategy. There exists an active emerging literature on the effect of agents’ cognitive limitationsontheeffectivenessofmonetarypolicy,includingGabaix(2016),FarhiandWerning(2017),Woodford(2018),andAngeletosandLian(2018). Thesestudieshavefocused onmitigatingtheso-called“forwardguidancepuzzle”bywhichtheeffectofannouncementsoffuturemonetarypolicyoninflationandoutputtodayincreasewiththehorizon at which the changes are expected to occur.3 However, these cognitive frictions impair theeffectivenessofprice-leveltargetingstrategiesaswell,sincethesefrictionsgenerally limit the effect of future monetary policy actions on expectations. Eusepi and Preston (2018) study the effectiveness of price-level targeting strategies under adaptive learning and find that it retains its stabilization properties.4 All of these studies, however, consider cases in which a policy strategy is in place indefinitely, and misperceptions causedbyagents’cognitivelimitationsalsopersistindefinitely. Theliteraturetherefore fallsshortonthequestionofhowexpectationscanaffectthetransition betweenpolicy strategies. Inthispaper,wedevelopamethodoflearningaboutthecentralbank’spolicystrategy that can be applied to models with a ZLB constraint. Agents hold subjective beliefsabouttheparametersinthecentralbank’srulefortheshort-termnominalinterest rate—thepolicyrate—andtheyupdatethesebeliefssolelyfromobservationsofthepolicyrateandtheinputstothepolicyrulesuchasinflationandtheoutputgap. Contrary 3Del Negro et al. (2012) coined the term “forward guidance puzzle” in work showing that standard medium-scaleDSGEmodelsgrosslyoverestimatetheimpactofforwardguidanceonthemacroeconomy. 4Meleetal.(2018)arguethatprice-leveltargetingneednotbeoptimalwhenarationalcentralbank interactsstrategicallywithalearningprivatesector. 3

tothelearningenvironmentsfeaturedinEvansandHonkapohja(2001)andEusepiand Preston (2018), the entire structure of the economy is common knowledge and agents makerationalforecastsconditionalontheirperceivedpolicyrule. Ifthetruepolicyrule stays in place over a sufficiently long time horizon, then the learning equilibrium converges to the full information, rational expectations equilibrium. However, when the central bank switches to a new strategy, it takes time for agents to learn this new strategy. Duringthistransition,misperceptionofthepolicystrategybytheprivatesectorcan have unintended consequences for economic outcomes. The convergence of beliefs can be hampered in particular when the economy is at the ZLB, because the observed policyratecarrieslittleinformationaboutthetrueruleparametersinthiscase. To assess the implications of our learning framework for monetary policy, we consider the textbook New Keynesian model. The central bank sets the policy rate i act cording to a simple rule that can react to an inflation gap and/or a price-level gap. We focusonaswitchfroma(flexible)inflationtargetingtoa(flexible)price-leveltargeting strategy. Our analysis suggests that under learning the switch to price-level targeting falls short of delivering the stabilization benefits that are found under full information inademand-drivenrecessionwithabindingZLB. When the central bank switches to price-level targeting at the onset of a demanddriven recession, the switch mitigates the loss in output and the shortfall in inflation under rational expectations and full information. By contrast, when agents are learning,outputandinflationinitiallyfalljustasmuchasundertheinflationtargetingstrategy despite the switch. Because agents do not immediately understand the switch in the policy rule, they initially attribute the differences in the policy rate resulting under the price-level targeting strategy to a series of discretionary policy shocks rather than theswitchinstrategy. Agentsthereforefailtoanticipatethemoreaccommodativepolicy associated with the price-level targeting strategy. The learning problem is further complicated by the fact that the policy rate quickly reaches the ZLB. At the ZLB the interest rate prescribed by the policy rule is censored as the actual policy rate cannot fall below zero and the private sector agents receive little information about the true rule parameters. As a result, under learning, the central bank is left with a much larger 4

negativeprice-levelgapthanunderfullinformation,andthushastoallowforsubstantial overshooting of inflation after the recession to deliver on its promise of price level stabilization. The costs of this overshooting are incurred without having accrued any stabilizationbenefitsinthemidstoftherecession. In order for the stabilization benefits of price-level targeting to materialize, pricelevel targeting should be introduced in relatively calm times—that is, when inflation is not persistently undershooting its long-run target and the federal funds rate is not constrained by the ZLB—to give agents the opportunity to learn the new policy strategy. Whenputinplaceforasufficientlylongtime,systematicprice-leveltargetingthen becomessuperiortoinflationtargeting,justasunderfullinformation. Wealsoshowthataprice-leveltargetingstrategythatispermanentlyinoperationis preferable to a temporary price-level targeting strategy of the type suggested by Evans (2012)andBernankeetal.(2019). Undertemporaryprice-leveltargeting,monetarypolicy falls into two regimes: The central bank targets the price level when the ZLB binds butswitchesbacktotargetinginflationoncetheprice-levelgapaccumulatedduringthe ZLB episode has been closed. Consequently, the price-level targeting regime is active precisely when it is difficult for agents to infer changes in monetary policy. Agents are unlikelytoanticipatethatmonetarypolicywillbemoreaccommodativeinthisregime, a failure that renders the strategy ineffective. This result echoes the concern voiced by Svensson(2019)thattheeffectivenessofmakeupstrategies“probablyrequiresthateconomicagentsneedtoseethepolicypracticedanditsprinciplesobeyedforsometime, inordertobelievethatitwillbemaintainedandbesuccessfulinthefuture.” Our behavioral learning framework builds on Tetlow and von zur Muehlen (2001) and Cogley et al. (2015). We extend their approach to take into account the limited informational content of interest rate observations at the ZLB by explicitly modeling theassociatednon-linearityintheobservationequationofagents’Bayesianstate-space system. Inrelatedwork, Gustetal.(2018)assumethatagentsareonlyuncertainabout the value of the intercept term in the policy rule where the intercept term can take on valuesfromasmall,finite,andpubliclyknownset. Allotherruleparametersareknown andfixedatalltimes. Price-leveltargetingstrategiesarenotconsideredintheirwork. 5

Limitedinformationandlearningalsoplayaroleintheliteratureonimperfectcredibility, which is often modeled as agents not observing parts of the central bank’s policy strategy. For example, Erceg and Levin (2003) and Schorfheide (2005) interpret the shifts in monetary policy in the 1980s and 1990s through the lens of learning and limited credibility. Bodenstein et al. (2012) show that, under imperfect credibility, private sectoragentsmaydoubtthatthecentralbankwillhonoritsannouncementtokeepinterestrateslowforlonger. Asaresult,theinterestratepathexpectedbytheprivatesector lies above the path announced by the central bank. In our paper, the private sector projects future monetary policy to be tighter than intended by the central bank under price-level targeting, leading to outcomes that resemble those under a lack of credibility. Finally,Kryvtsovetal.(2008)modeltheswitchfromaninflationtargetingstrategyto aprice-leveltargetingstrategyunderimperfectcredibility. Theymodelcredibilitytobe independentoftheobservedinterestratepathandtheydonotimposeaZLBconstraint. Theremainderofthispaperisstructuredasfollows. Section2describesourgeneralized model of learning, while Section 3 contains our application to the introduction of price-level targeting. In Section 4 we discuss the best timing for a central bank to switch from inflation targeting to price-level targeting. Section 5 analyzes a temporary price-leveltargetingstrategy. Section6concludes. 2 A Model of Learning the Monetary Policy Strategy Forthepurposeofouranalysis,amonetarypolicystrategyisfullydescribedbyasimple policyrulethatdescribeshowthecentralbankmapsinflation,theoutputgap,andpossiblyothervariablesintoavalueoftheshort-termnominalinterestrate,thepolicyrate. We assume that the private sector agents do not know with certainty the policy rule of the central bank, and we therefore distinguish between the actual policy rule followed by the central bank and the perceived policy rule that agents use to form expectations. Agentsinfertheparametersoftheperceivedpolicyrulesolelyfromobservationsofeconomicdataasdescribedinthefollowing. 6

2.1 The Economy We conduct the analysis in a discrete-time model that is linear except for the ZLB. Excludingthedescriptionofmonetarypolicy,theequilibriumconditionsaresummarized byalinearforward-lookingsystemofequationsoftheform: 0 = F E [x ]+F x +F x +F i +F u (1) 2 t t+1 1 t 0 t−1 i t u t Allvariablesareexpressedindeviationsfromthedeterministicsteadystateofthemodel. The endogenous variables x enter with their current, past, and expected future valt ues into the model. Exogenous disturbances enter through the random vector u . The t shocksareiidandhavemeanzero. Finally,thepolicyinstrumenti isdeterminedbythe t centralbankasdescribedbelow. 2.2 The Central Bank The actual policy rule for the policy rate (in deviations from its deterministic steady state)i isoftheform: t i = max{i,i∗}, i∗ = Ψ(β )x +e (2) t t t t t t where i∗ is the notional interest rate and i is the lower bound on the policy rate. The t notional rate is set according to a linear rule with parameters Ψ(β ) that are a function t of a small set of parameters β . These parameters can vary over time to accommodate t changes in the central bank’s systematic response to economic outcomes. The policy rateisalsoaffectedbyawhitenoiseprocesse . Thisshockrepresentsone-timediscret tionaryadjustmentstothepolicyrate.5 5Itisstraightforwardtoextendourframeworktoincludepersistentmonetarypolicyshockstomodel time-varyingchangesintheintercepttermoftheruleasinGustetal.(2018) 7

2.3 Full information rational expectations equilibrium Inthefullinformationrationalexpectationsequilibrium,privatesectoragentsobserve the sequences of past and current realizations of the endogenous variables x , the ext ogenous disturbances u and the policy rate i . Agents also know the linear economic t t model given in equation (1). We assume that the central bank commits to the values oftheparametersinthepolicyruleβ inequation(2). Privatesectoragentsknowthese t parametersandtheformofthepolicyruleandhencehaveacompleteunderstandingof the monetary policy strategy. Agents also observe the current value ofthe policy shock e . Ateverypointintime,theprivatesectoragentsknowthecorrectpolicyratepaththat t thecentralbankintendstoimplementcontingentonthestateoftheeconomy. Wesolve forthefullinformationrationalexpectationsequilibriumwithanoccasionallybinding ZLBconstraintusingthealgorithmofHolden(2016). 2.4 Learning Equilibrium Underlearning,privatesectoragentsalsoobservethesequencesofpastandcurrentrealizationsoftheendogenousvariablesx ,theexogenousdisturbancesu andthepolicy t t rate i . They also know the linear economic model given in equation (1). While agents t knowthegeneralformoftheactualpolicyruleinequation(2),theydonotobservethe values of the parameters β . Neither do they observe the realizations of the monetary t policy shock e . Instead, agents believe that the transitory shock and changes to the t parametersattimetarenormallydistributedwhitenoisewith:      e σ2 0 t et   ∼ N 0, . (3) β −β 0 Σ t t−1 βt The variances σ2 > 0 and Σ > 0 are subjective and are an exogenous input to the et βt learningprocess. Theassumptionthatpolicyparameterschangeovertimeiscommon in empirical work. Notably, Boivin (2006) assumes that policy rule parameters follow randomwalkstoassesshowtheconductofU.S.monetarypolicyhaschangedovertime. His estimates suggest that the rule parameters evolve gradually and feature wide error 8

bands. Strictly speaking, these beliefs render agents boundedly rational in our model since the true rule parameters are constant except for a one-time discrete jump. We viewthissetupasasimplifiedrepresentationofanenvironmentwithfundamentaluncertaintyabouttheactualpolicyrule. OurformulationofthelearningequilibriumfollowsCogleyetal.(2015)withtheimportant difference that we include the informational constraints arising from the ZLB. Privatesectoragentsenterperiodtwithbeliefsaboutthepolicyruleparametersinherited from t − 1. In formulating decisions plans, agents treat the mean parameter estimatesasifknownwithcertainty. Thenperiodtshocksarerealized. Agentsobservethe realizationsoftheprivate-sectorshocksandthecentralbank’spolicyactionandinfera perceivedpolicyshocke˜. Outcomesaredeterminedinaccordancewiththebeginningt of-periodplans. Finally,agentsupdatetheirestimatesoftheruleparameters. In providing a detailed description of the learning framework, we start by defining ˆ the mean of the posterior distribution of β at the end of period t − 1 with β . Folt−1 t−1 lowingKreps(1998),agentsplanunderanticipatedutilityandviewtheruleparameters asfixedatβ ˆ .6 Theseplansalsodependontheagents’perceivedvaluee˜ oftheactual t−1 t policy shock. Agents solve for state-contingent paths starting in period t denoted by (cid:110) (cid:111) x(t),s ≥ t thatsatisfythesystemofequations(1)andthepolicyrule(2)withβ = β ˆ s s t−1 foralls ≥ t. Mergingthetwoconditions,thesolutionneedstosatisfy (cid:104) (cid:105) (cid:16) (cid:16) (cid:17) (cid:17) 0 = F E x(t) +F x(t) +F x(t) +F max i,Ψ β ˆ x(t) +e(t) +F u (4) 2 s s+1 1 s 0 s−1 i t−1 s s u s for all s ≥ t with the initial condition x(t) = x . In solving this problem, agents take t−1 t−1 theperceivedpolicyshocksequencee(t) asdistributediidN (0,σ2)withe(t) = e˜. Thus, s e t t the solution x(t) represents the agents’ expectations about the future evolution of the s economyattimet. Again,werelyonthecomputationallyefficientalgorithmofHolden 6Anticipatedutilityreferstothewidelyusedassumptioninthelearningliteraturethatagentsderive theirdecisionsandexpectationsaboutfuturedevelopmentsundertheassumptionthattheircurrentperceptionoftheeconomicenvironment,inourcasethepolicyruleparameters,persistsindefinitely. This simplifyingassumptionignoresthat,atthesametime,thepublictreatstheparametersinthepolicyrule asrandomvariablesinthelearningproblem. See(CogleyandSargent,2008)oninterpretinganticipated utilityasanapproximationtoBayesianoptimallearning. 9

(2016)inthisstep. Theperceivedvalueofthepolicyshockforthecurrentperiode˜ reflectstheobserved t valueofthepolicyrate:   i −Ψ(β ˆ )x ifi > i  t t−1 t t (cid:104) (cid:16) (cid:16) (cid:17) (cid:17) (cid:105)  e˜ t = E e t | i t = max i,Ψ β ˆ t−1 x t +e t ,x t = φ (cid:18) i−Ψ(βˆ t−1)xt (cid:19) (5)     −σ et Φ (cid:18) i−Ψ( σ βˆ e t t −1)xt (cid:19) ifi t = i σet where φ and Φ are the standard normal density and cumulative distribution functions, respectively. If the observed policy rate is above the ZLB, e˜ equals the difference bet tween the observed policy rate i (derived from the actual policy rule with parameters t β )andthepolicyrateprojectedbytheprivatesector(derivedfromtheperceivedpolicy t ˆ rule with parameters β ). If the observed policy rate is at the ZLB, e˜ equals the cont−1 t ditional expectation of the policy shock e when the notional rate i∗ is below the lower t t bound ¯i,i.e.,themeanofatruncatednormaldistribution. To obtain for the equilibrium in period t, we solve simultaneously for x = x(t) , i , t t t and e˜ using equations (4), (5), and the actual policy rule in (2). The appendix provides t detailsonthesolutionalgorithm. Havingobservedtheequilibriumoutcomesx ,agentsupdatetheirbeliefsaboutthe t ruleparametersβ bysolvingthefollowingBayesianfilteringproblem: t i = max{i,i∗} (6) t t (cid:0) (cid:1) i∗ = Ψ(β )x +e , e ∼ N 0,σ2 (7) t t t t t et β = β +(cid:15) , ∼ N (0,Σ ). (8) t t−1 βt βt Using the posterior distribution of beliefs about β as the prior distribution in this t−1 filtering problem, agents derive a new posterior distribution of beliefs about β given t the observations of i and x . As in Cogley et al. (2015), agents treat x as exogenous t t t and thus as independent of the shocks e and (cid:15) . By ignoring the correlation between t βt thepolicy shocksand theeconomic outcomes, agents donot makeuse ofall theavailable information. However, form an analytical perspective, this exogeneity assump- 10

tion greatly simplifies the agents’ filtering problem because the problem reduces to a Bayesian regression with truncation in this case. In addition, we make two computationalapproximationstothefilteringproblem: First, wereplacetheposteriordistribution by its Laplace approximation, i.e. we approximate the posterior distribution of β t with a normal distribution. Second, we approximate the potentially non-linear mappingΦ(β )withafirst-orderTaylorexpansionasinextendedKalmanfiltering,butkeep t thenon-linearityarisingfromtheZLB.Again,werefertotheappendixfordetailsonthe solutionalgorithm. Several of the stated behavioral assumptions imply that the private sector agents are boundedly rational in our model. First, agents behave as anticipated utility mod- ˆ elers and treat the current estimates of the rule parameters β as if known with cert−1 taintywhenderivingtheireconomicdecisionsandwhencomputingtheperceivedpolicyshocke˜. Second,agentsupdatetheestimatesoftheruleparametersthroughafiltert ingproblemthat,althoughBayesian,ignorestheendogeneityofthepolicyshockse to t themodeloutcomesx . Third,agentstakeaperceivedlawofmotionforpolicyinnovat tionsinequation(3)asgiven(inparticularthevariancesσ andΣ ), eventhoughthis et βt lawofmotionmaynotcoincidewiththecentralbank’sactualformulationofmonetary policy. 3 Learning a Price-Level Targeting Strategy Weapplythislearningframeworktoasituationinwhichthecentralbankswitchesfrom an inflation targeting strategy to a price-level targeting strategy. A price-level targeting strategy actively seeks to offset passed misses of inflation from the central bank’s inflation goal. As laid out in the introduction, price-level targeting is considered to be bettersuitedthaninflationtargetingtostabilizethemacroeconomy,particularlywhenthe policyrateisattheZLB.Mostofthesubsequentanalysisalsoappliesforothermakeup strategiessuchasaverageinflationtargetingorshadowraterules.7 While our learning framework can be embedded into any linear model of the form 7Forasummaryofmakeupstrategiessee,Bernankeetal.(2019)andHebdenandLo´pez-Salido(2018). 11

inequation(1),weadoptthetextbookNewKeynesianmodeltocharacterizetheunderlying economic environment in our illustration of the impact of learning on the effectivenessofprice-leveltargeting. Thischoiceofmodelseemswellsuitedgiventhatmost theoretical arguments about the benefits of price-level targeting are formulated within thismodel. 3.1 Economic Model ThetextbookNewKeynesianmodelfeaturestwoequations: π = κygap +βE [π ]+v (9) t t t t+1 t 1 ygap = E (cid:2) ygap| (cid:3) − (i −E π −g ). (10) t t t+1 σ t t t+1 t Allvariablesareexpressedindeviationfromtheirnon-stochasticsteadystatevalues. TheNewKeynesianPhillipsCurvein(9)linksinflation(measuredrelativetoitslong-run targetπ∗)π to its expected valueand the output gapygap. Theoutput gap is defined as t t the difference between actual output and the natural level of output, ygap = y − y∗. t t t v is an inefficient cost-push shock. The Aggregate Demand Curve in (10) provides the t connection between the output gap, inflation, the policy rate i and the natural rate of t interest g . E denotes the subjective expectations of the private sector conditional on t t itsinformationsetI . t Wesetthediscountfactorβ equalto0.9956toimplyasteadystaterealinterestrate of 1.75 percent, and set the intertemporal elasticity of substitution σ equal to 1. The slope of the Phillips Curve κ is fixed at 0.1. The demand and supply shocks g and v t t follow first-order autoregressive processes with autocorrelations ρ = ρ = 0.9. The g u standarddeviationsoftheinnovationsareσ = 0.3andσ = 0.03,respectively,inorder g u tomatchthevolatilityofinflationandtheoutputgapinquarterlyU.S.datafrom1984Q1 to2007Q4. Tohighlighttheimportanceoftheprivatesector’sexpectationsaboutmonetarypolicyfortheevolutionoftheeconomy—theexpectationschannelofmonetarypolicy—we iterate equations (9) and (10) forward. The Aggregate Demand Curve can thus be writ- 12

tenas ∞ ∞ 1 (cid:88) 1 (cid:88) ygap = − E [i −π ]− E [g ] (11) t σ t t+s t+s+1 σ t t+j j=s j=s whichrevealsthedependenceoftheoutputgapontheexpectedpathfortherealinterestrate. Similarly,thePhillipsCurveimpliesthatinflationequalsthediscountedsumof currentandfutureexpectedoutputgaps ∞ ∞ (cid:88) (cid:88) π = κ βsE [ygap]+E βs[v ]. (12) t t t+s t t+s s=0 s=0 Asaresult,inflationalsodependsontheexpectedpathoftherealinterestrate. Thus,the ability of the central bank to steer inflation and the output gap into any desired direction depends importantly on the ability of the central bank to steer the private sector’s expectationsaboutcurrentandfutureinterestrates. If the monetary policy strategy—the state-contingent path of the policy rate i —is t known to the private sector, then the central bank can successfully steer the economy in the desired direction. However, if the monetary policy strategy is not known with certaintyor,alternatively,notperceivedascrediblebytheprivatesector,perceivedand actual monetary policies may differ importantly from each other. As a result, the realizedeconomicoutcomesmayendupdifferingsubstantiallyfromthoseintendedbythe centralbank. 3.2 Monetary Policy Strategies Weassumethat,ateverypointintime,thestrategyofthecentralbankisfullydescribed bytheinterestraterule (cid:18) pgap ygap(cid:19) i∗ = ρ i +(1−ρ ) (1+φ )π +φ t +φ t +e (13) t i t−1 i πt t pt 4 y 4 t i = max{i∗,i} (14) t t 13

whichisconsistentwiththesteadystateformgiveninequation(2). Thelowerboundi onthepolicyrateisexpressedrelativetothesteadystateandequalsi = −π∗ wherethe β steadystateinflationrateπ∗ equalsthecentralbank’sinflationtarget. The central bank arrives at a value for the notional interest rate i∗ from the current t valuesofinflation,theoutputgap,and,possibly,thelaggedrealizedvalueofthepolicy rate. Underprice-leveltargeting,thecentralbankalsorespondstoaprice-levelgap. The price-level gap pgap records the cumulative departure of inflation from its target value t from a fixed date in the past to the present period t and evolves according to pgap = t pgap +π . GiventheZLBconstraint,theactualpolicyratei equalsthenotionalratei∗ if t−1 t t t thelatterisabovei,andequaltoiotherwise. The subsequent analysis assumes that before some period τ , the central bank fol- 0 lows an inflation targeting strategy. More precisely, we characterize inflation targeting bytheinertialversionoftheTaylor(1999)rulewhichhasbeenfeaturedinpublicdocumentsoftheFederalReservesuchastheMonetaryPolicyReporttoCongressandBrayton et al. (2014) and accounts for the empirical observation that central banks adjust ratessluggishlyinresponsetoeconomicconditions,seeEnglishetal.(2003).8 The parameters in the inertial Taylor rule assume the values ρ = 0.85, φ = 1, φ = i y π 0.5 and φ = 0. The inflation target π∗ is set to two percent. The central bank always p adheres to the prescriptions from the rule, and in particular the true monetary policy shock is always zero. However, as stated above, the private sector does not necessarily understandthisfeatureofpolicymakingand,attimes,willperceivethemonetarypolicy shocktodifferfromzero. Giventheseassumptions, weconsideraswitchinmonetarypolicyinperiodτ toa 0 price-leveltargetingrulewithφ = 1andφ = 0. Wesetthereferencedateforthepricep π level gap to be period τ − 1 so that upon switching the policy strategy the price-level 0 8Becauseofinterestrateinertia,theinertialTaylorruleallowsforsomehistorydependencewhenthe economyisnotattheZLB.Repeatedsubstitutionsofthelaggedpolicyratetermintheruleimplythatthe currentvalueofthenotionalraterespondstoaveragesofcurrentandpastinflationandcurrentandpast outputgapswithlowerweightsonobservationsfurtherinthepast.However,whiletheeconomyisatthe ZLB,thishistorydependencecomestoapartialhaltastheactualpolicyratenolongerrecordsfullythe deviationsofinflationandtheoutputgapfromtheirlong-runtargetvalues. Nevertheless,forthesame values of inflation and the output gap, the inertial Taylor rule prescribes a shallower rate path than its non-inertialcounterpartwithρ =0. i 14

gap is closed, i.e., pgap = 0. Without loss in generality, we assume the parameters on τ0−1 thelaggedpolicyrateρ andtheoutputgapφ toremainunchangedandthattheprivate i y sectorunderstandsthistobethecase. 3.3 Beliefs Whileprivatesectoragentsdonotunderstandhowtheparametersφ andφ havechanged π p in period τ , agents can infer the new parameter values from the data by solving the 0 Bayesianregressionproblemdescribedabove. Thisproblemdependsonthesubjective beliefs about the rule parameters. Initially, agents know the parameters of the inertial Taylor rule with certainty, as would be the case if that rule had been in place for a sufficiently long period for the beliefs to have converged to the true values. To update theirestimatesoftheruleparametersφ andφ aftertheswitchtoprice-leveltargeting, π p agentsapplythefilteringprobleminequations(6)–(8)with   φ (cid:18) φ φ (cid:19) β t =  πt , Φ(β t )x t = ρ i i t−1 +(1−ρ i ) (1+φ πt )π t + 4 pt pg t ap + 4 y y t gap φ pt given the rule described in equation (13). Since the parameters on the lagged value of the policy rate ρ and the output gap φ remain unchanged throughout the analysis, i y we assume for simplicity that private sector agents do not consider the possibility of changesintheseparameterseither. Inaddition, agentssettheinitialprice-levelgapto zero. Overtime,theparameterbeliefsconvergetothetrueparametervalues. Thespeedof convergencedependsonthesubjectiveprioruncertaintyabouttheruleparametersΣ βt andthesubjectivevarianceofpolicyshocksσ2. TheratioΣ /σ2 canbeunderstoodas et βt et asignal-to-noiseratiointheBayesianregressionproblem. IftheentryinΣ associated βt with a specific parameter is larger, then a given-size forecast error in the policy rate carriesmoreinformationaboutthisspecificparameterandtheresultingupdateinthis parameterwillbelarger. Wethinkitisunclearhowtojudgeempiricallyhowagentswouldadjusttheirexpec- 15

tationsin thewake oftheadoption ofprice-leveltargeting thatis without precedentin recenthistory.9 Inourbenchmarkspecification,wechooseσ2 andΣ suchthatbeliefs et βt converge about half-way to the truth in 20 quarters after the strategy switch. In detail, weset   Σ 0.4 −0.11 σ2 = 0.01, βt =   et σ2 et −0.11 0.05 implying to a perceived standard deviation of the policy shocks of 10 basis points and perceivedstandarddeviationsoftheinnovationsintheparameterφ andφ of0.06and π p 0.02,respectively,andaperceivedcorrelationbetweeninnovationstoφ andφ of-0.25. π p In other words, agents think that the central bank will tend to pursue either inflation targetingorprice-leveltargeting,buttheydonotthinkthattheyaremutuallyexclusive. We also consider a “slower learning” case implying less willingness of the agents to adjusttheirviewsaboutmonetarypolicy. Inthiscase,wereducetheparameterinnovationmatrixΣ byafactoroftenwhichyieldsmuchslowerconvergenceoftheparame- βt terstothetruth. 3.4 Learning in Normal Times We illustrate the learning mechanism using stochastic simulations. Each simulation is initialized by a draw from the ergodic distribution of the variables generated under the inflation targeting rule (with parameters φ = 0.5 and φ = 0. Given these initial π p conditions,policymakersswitchtothenewruleinperiodτ = 0withparametersφ = 0 0 π andφ = 1. p Figure 1 plots how the private sector’s parameter beliefs evolve under full information and under learning, respectively. Under full information (red lines), private sector beliefsimmediatelyadjusttotheirnewtruevalues. Bycontrast,inourbenchmarklearningcase(yellowlines),theparameterbeliefsad- 9OnecouldfollowtheapproachinBoivin(2006)andestimateapolicyrulewithtime-varyingparametersfromhistoricaldataandusetheresultingestimatestodisciplinethebeliefprocessinourlearning model. However,suchananalysisisunlikelytoyieldmuchinformationonthepastevolutionoftheparameterontheprice-levelgapφ sincenocentralbankhaspursuedaprice-leveltarget,andevenlesson pt itsfutureevolution. 16

Figure1: Beliefsofruleparametersinnormaltimes. Note: Solidlinesshowmedianbeliefsφˆ andφˆ across1,000simulations. Shadedareasshow10thand πt pt 90thpercentiles. justslowly, butconvergeovertime. Thepathsoftheparameterbeliefsalongaparticularsimulationdependontherealizationsoftheunderlyingeconomicshocks. Thesolid linesinFigure1showthemedianbeliefsovertheruleparameters. Acrosssimulations, these beliefs can vary considerably as evidenced by the shaded area representing the 10thand90thpercentilesofthedistributionofbeliefs. Initially,thebeliefsmovestrongly into the right direction because, as we discuss further below, there is more variation in observed outcomes to learn from just after the rule has switched. The speed of learningdependsimportantlyonthesubjectiveuncertaintythatagentsplaceonchangesin the rule parameters. In our benchmark case, the median belief has almost converged tothetrueparametersafter15years. Beliefsabouttheparameteroninflationconverge somewhatfasterthanthoseontheprice-levelgapparameter. Underthe“slowerlearning”specification(purplelines),agentsapportionthedifferencesbetweentheobserved policy rate and the rate prescribed under the perceived rule more to the discretionary policyshocke thantothechangesintheruleparametersβ . Hence,theupdatingsteps t t intheparametersaresmaller. Figure2plotsthejointdistributionofbeliefsabouttheparametersφ andφ inour π p benchmarklearningcaseatthreepointsintime. Asagentsupdatetheirbeliefs,alarger ˆ estimateoftheparameterontheprice-levelgaptermφ isassociatedwithalowerestipt 17

Figure2: Jointdistributionofruleparameterbeliefsinnormaltimes. 0.6 t=8 t=20 0.5 t=60 0.4 0.3 0.2 0.1 0 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (cid:16) (cid:17) Note: Foreachtimeperiodtshown,adotcorrespondstoabelief φˆ ,φˆ inoneof1,000simulations πt pt underthebenchmarklearningparameterization. ˆ mateoftheparameterontheinflationtermφ . πt The stabilization benefits of switching to price-level targeting are illustrated in Figure 3. We show the volatilities of inflation and the output gap as measured by the interquartile range of realizations across simulations. By excluding the tails of the distributionthefigurecapturesvolatilitiesin“normaltimes.” Underfullinformation,price-leveltargetingreducesinflationvolatilityimmediately by half relative to inflation targeting, while the interquartile range for the output gap drops by a small amount. This feature highlights the stabilizing properties of pricelevel targeting in forward-looking models: When inflation is low, price-level targeting callsforbringingabouthigherfutureinflationtostabilizethepricelevel. Providedthat thepolicyisunderstoodandcredible, inflationexpectationsrise, throughtheforwardlookingPhillipscurve(9),theinitialshortfallininflationismitigated. By contrast, under learning the volatility of inflation falls slowly, while output gap volatility even rises initially after the switch. The central bank’s ability to stabilize the economythroughtheexpectationschannelofmonetarypolicydeterioratesinitiallyfor two reasons. First, because agents perceive the policy rule to be different from the ac- 18

Figure3: Inflationandoutputgapvolatilityinnormaltimes. (a)Annualizedinflation4π . (b)Outputgapygap. t t 4 4 No switch No switch Full information Full information 3.5 3.5 Learning Learning Slower learning Slower learning 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0 8 16 24 32 40 48 0 8 16 24 32 40 48 Note:Linesshowtheinterquartilerange,i.e.the75thminusthe25thpercentile,ofoutcomesacross1,000 simulationsateachtimeperiodt. tual one, agents expect a lower or higher interest rate path in a given simulation than policymakersintendtopursueundertheactualpolicyrule. Mechanically,suchmisperception has similar effects on the economy as a monetary policy shock. Second, belief updatingisitselfasourceofvolatility. Becauseagents’beliefsaboutthepolicyruleparameters change so do their implied expectations about the future interest rate path. Over time, however, as beliefs have moved close enough to the true parameter values, thevolatilitiesfallbelowtheirpre-switchlevelsandapproachthelevelsunderfullinformation. Thebetteragentsunderstandtheswitchofthecentralbanktoprice-leveltargeting, thesmallerwillbethevariationininflationandtheoutputgapacrosssimulations. Consequently,onceagentshavemadesufficientlearningprogresstowardsthetrueparameter values, the data provides less identifying variation, and parameter learning slows down. Thisnexusbetweentheagents’beliefsandthevariabilityofeconomicvariables explainstheslowdowninthepaceoflearninginthelaterpartofthesimulationsshown inFigure1. 19

3.5 Learning During a Deep Recession Proponents of price-level targeting (or other makeup) strategies have emphasized the stabilizing features of this strategy in particular when the policy rate is at the ZLB, although,asillustratedabove,thebenefitsmayapplymorebroadly.10 If,inadeepdemanddriven recession, the policy rate reaches the ZLB, a price-level targeting central bank provides automatically additional monetary accommodation. In order to make up for theshortfallofinflationfromitslong-runtarget,thecentralbankwillneedtokeepthe interest rate path sufficiently low to induce future catch-up inflation. In many models, the anticipation of higher future inflation and low future nominal interest rates lowers the expected path for real interest rates which in turn can stimulate the economy up frontwithoutanycontemporaneousinterestrateadjustment(becauseoftheZLB).11 For price-level targeting strategies to stabilize the economy through this expectationschannel,privatesectoragentsmustunderstandthepolicystrategyandconsiderit credible. Underlearning, thecentralbankmaynotachievethedesiredoutcomesfrom adopting the price-level targeting strategy because the central bank cannot reveal its commitmenttothenewstrategythroughtheobservedpolicyrateaslongasthepolicy rateisattheZLB.Agentsreceivelittleinformationaboutthenewpolicyrulewhileatthe ZLBandtheyfailtoanticipatethatthecentralbankwillkeepthepolicyratepathlowin the future. As a result, the switch in policy strategy does not provide further monetary stimulus. If, nevertheless, the central bank remains committed to its new strategy, it willsubsequentlyhavetoallowforhigherinflationtoundotheaccumulatedprice-level gap. We illustrate these challenges to price-level targeting by considering a switch from inflation targeting to price-level targeting in a demand-driven recession during which the ZLB binds. Specifically, we choose a combination of demand shocks g and supply t shocksu that,underinflationtargeting,induceinflationtofallbyroughly1.5percentt 10ComparealsotoSvensson(1999)andVestin(2006). 11Under full information, pairing inflation targeting with forward guidance on the path of the policy ratecouldyieldsimilareffectsaspermanentprice-leveltargeting. However, undercrediblepermanent price-level targeting future policy accommodation is automatic at the ZLB; absent an explicit rule for forwardguidancetheadhocnatureofthisapproachmaydelaythecommunicationoffuturepolicyaccommodation.SeealsothediscussioninSection5ontemporaryprice-leveltargeting. 20

age points and the output gap to fall by roughly 5 percentage points, magnitudes of declines that are comparable to those during the Great Recession. The innovations to theshocksstartinperiodt = 0andendinperiodt = 11; afterwardsg andu converge t t backtozeroatspeedsdictatedbytheauto-regressiveparametersρ andρ . g u Figure 4 shows the median outcomes across simulations conditional on this sequence of shocks (i.e., shocks are sampled randomly except for periods t = 0,...,11). Thebluelines(labeled“noswitch”)depicttheoutcomesintherecessionscenariowhen the central bank follows the inflation-targeting rule throughout the entire simulation. The presence of the effective lower bound exacerbates the effects of the recessionary shocksandleadstoalargedropininflationandtheoutputgap. Notably,undertheinflation targeting strategy the central bank does not makeup for any deviations of inflationfromitslong-runtargetduringtheZLBepisode;thereisnoinflationovershooting. When the central bank adopts a price-level targeting rule at the onset of the recession (τ = 0), this new strategy is very effective in mitigating the adverse effects of the 0 recessionunderfullinformation(redlines): Inflationonlydropshalfasmuchasunder inflationtargetingandthefallintheoutputgapisreduced. Laterinthesimulation,inflation overshoots its target, as the price-level targeting rule keeps policy rates low for longer to close the price-level gap. In fact, it is precisely the expectation of this more accommodativepolicystanceandtheaccompanyinginflationovershootthat,through the expectational channels of the New Keynesian Phillips Curve (9) and the Aggregate DemandCurve(10),preventsinflationfromfallingduringtherecession. Bycontrast,underlearning(yellowlinesinFigure4),agentsfailtoanticipatethefull extentoffuturepolicyaccommodationunderprice-leveltargeting. Asaresult,inflation expectations are lower than under full information, the drop in the real interest rate is restrained, and little buffer is provided against the declines in inflation and the output gap. The large and persistent drop in inflation accumulates to a sizable price-level gap overtimeand,consequently,thecentralbankmustkeepthepolicyratelowerforlonger than under full information to close this gap. A sizable overshoot of inflation results fromtheattemptstoclosethegaplaterinthesimulation. Figure 5 shows the evolution of the private sector beliefs that correspond to these 21

Figure4: Outcomesduringadeeprecession. Output gap Inflation 4 0 3 -2 2 No switch -4 Full info 1 Learning, observe actual Learning, observe notional 0 -6 0 8 16 24 32 40 48 0 8 16 24 32 40 48 Real interest rate Price level gap 2 0 0 -5 -2 -10 0 8 16 24 32 40 48 0 8 16 24 32 40 48 Nominal interest rate Notional interest rate 4 4 2 2 0 0 0 8 16 24 32 40 48 0 8 16 24 32 40 48 Note: Solidlinesshowmedianoutcomesandbeliefsacross1,000simulations. Shocksaresampledrandomlyexceptforperiodst = 0,...,11duringwhichtheyarefixedatthesequencedescribedinthetext. The rule followed by the central bank switches from inflation to price-level targeting in period τ = 0. 0 Variable definitions are as follows: “Output gap” is ygap, “Inflation” is 2 + 4π , “Real Interest Rate” is t t (cid:16) (cid:17) 4 1/β+i −E π(t) ,“NominalInterestRate”is2+4(1/β+i ),“NotionalInterestRate”is2+4(1/β+i∗), t t t+1 t t “PriceLevelGap”ispgap. t 22

Figure5: Beliefsduringadeeprecession. 0.7 No switch 1 Full info 0.6 Learning, observe actual Learning, observe notional 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 0 0 8 16 24 32 40 48 0 8 16 24 32 40 48 Note:Solidlinesshowmedianoutcomesandbeliefsacross1,000simulations.Shocksaresampled randomlyexceptforperiodst=0,...,11duringwhichtheyarefixedatthesequencedescribedinthe text.Therulefollowedbythecentralbankswitchesfrominflationtoprice-leveltargetinginperiod τ =0. 0 simulations. Comparedtotheevolutionofbeliefsinnormaltimes(showninFigure1), themedianparameterbeliefshardlymovetowardsthetrueruleparametersinthefirst 14 quarters after the shock as learning is particularly hampered by the presence of the ZLB.Toelaborateonthisfinding,Figures4and5includetheoutcomesandbeliefswhen agents observe the notional rate i∗ instead of the policy rate i (green lines). As the not t tionalinterestrateisnotcensoredandcanassumenegativevalues,agentsreceivemore informationaboutthetrueruleparameterswhiletheactualpolicyrateisattheZLB.As aresult,beliefsadjustearlierthaninthecasethatagentsobservetheactualpolicyrate only. This earlier adjustment of beliefs is sufficient to noticeably stabilize inflation. In otherwords, thelossofstabilizationbenefitsunderlearningisgreatlyamplifiedbythe limitedinformationalcontentoftheactualpolicyrateswhentheZLBisbinding. Overall, thesesimulationshighlightthattheeffectivenessofflexibleprice-leveltargetingdependsimportantlyontheformationofexpectations. Whenagentsdonotunderstand the future effects of current policy changes, the commitment to stabilize the pricelevelrequiresaprolongedperiodofpolicyaccommodationandhighinflationlater on without the benefits of closer-to-target inflation and output during the recession. Thesimulationsprovideanexampleofacommitment-basedpolicythatisdesignedto achieve sizable stabilization benefits by steering expectations, yet may turn out to be 23

undesirableifexpectationsfailtorespondasintended. 4 When to Adopt New Policy Strategies? If a central bank intends to switch from an inflation to a price-level targeting strategy, we advise to do so as early as possible, but not around an episode in which the ZLB becomes or is already binding. To substantiate this recommendation, we vary the timing of the adoption of price-level targeting—before, during, or after the onset of the recession—while keeping fixed the recession scenario introduced above and rank the resultingeconomicoutcomesaccordingtothecentralbank’slossfunction. Neitherthe central bank nor private sector agents are assumed to have any advance information abouttherecessionpriortoitsrealization. Wespecifythelossfunctionofthecentralbanktobe: LT1 = (cid:88) T1 βs−T0 (cid:0) π2 +(ygap)2(cid:1) . (15) T0 s s s=T0 Thislossfunctionplacesequalweightsonsquareddeviationsoftheinflationratefrom its long-run target and of output from its natural level. Period losses are discounted by the factor β. We measure the discounted loss that occurs between periods T and T , 0 1 wherewesetT = −20. Thesequenceofrecessionshocksstartsinperiod0. 0 Figure6plotsthevalueofthelossfunctionunderfullinformationandunderlearning as a function of the timing of adopting price-level targeting t . The loss under in- 0 flation targeting is normalized to 1. The left panel reports losses that accrue into the infinite future (T = ∞) and the right panel considers the losses that accumulate just 1 aroundtherecessionscenario(T = 20). 1 Giventheparameterizationofthepolicyrulesandthelossfunction,price-leveltargeting is always preferred to inflation targeting under full information. The benefits of price-level targeting are larger the further in advance of the recession the central bank adopts its new strategy. These benefits are diminished when price-level targeting is adoptedaftertheeconomyhasalreadyfallenintorecession. 24

Figure6: Expectedlossesandthetimingofintroducingprice-leveltargeting. 1 1 No switch 0.8 Full info 0.8 Learning Slower learning 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -20 -16 -12 -8 -4 0 4 8 12 16 20 -20 -16 -12 -8 -4 0 4 8 12 16 20 Note: Lines show simulated values of L∞ and L20 , conditional on the recession scenario starting in −20 −20 t = 0 and the rule switching at t = τ . 1,000 simulations for each rule switch period τ . Losses are 0 0 normalizedtooneforthe“noswitch”case. Under learning, the same considerations apply, but with the additional challenge thatprivatesectoragentsdonotlowertheirexpectationsaboutfuturenominalandreal interest rates as quickly as under full information. The switch to price-level targeting stillreducestheexpectedlossrelativetoinflationtargetingoverthelongtermregardless of the time of adoption. However, in the near term—spanning periods T = −20 to 0 T = 20shownintherightpanelofFigure6—wefindthatlateadoptionoftheprice-level 1 targeting strategy has no advantage over inflation targeting. Hence, in the case of late adoption the advantage of price-level targeting over the long term simply reflects the optimalityofprice-leveltargetingatandawayfromtheZLBgiventhecentralbank’sloss function. Inlinewithourearlierdiscussion,duringtherecession,policyisperceivedto belessaggressiveinstabilizingpricesthanitactuallyisunderthenewruleandinflation is more volatile. The slower beliefs adjust, the more do the potential benefits of priceleveltargetingevaporate. 25

Figure7: Beliefsandoutcomeswithearlieradoptionofprice-leveltargeting. (a)Outcomes. Output gap Inflation 1 4 0 3 -1 -2 2 -3 1 -4 No switch -5 Full info 0 Learning -6 -16 -8 0 8 16 24 32 40 48 -16 -8 0 8 16 24 32 40 48 Nominal interest rate Price level gap 4 0 3 -2 2 -4 -6 1 -8 0 -10 -1 -16 -8 0 8 16 24 32 40 48 -16 -8 0 8 16 24 32 40 48 (b)Beliefs. 0.7 No switch 1 Full info 0.6 Learning 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 0 -16 -8 0 8 16 24 32 40 48 -16 -8 0 8 16 24 32 40 48 Note: Linesshowmedianoutcomesandbeliefsacross1,000simulations. Shocksaresampledrandomly exceptforperiodst=0,...,11duringwhichtheyarefixedatthesequencedescribedinthetext.Therule followedbythecentralbankswitchesfrominflationtoprice-leveltargetinginperiodτ = −16. Variable 0 definitionsasinFigure4. 26

Table1: Stabilizationgains. (1) (2) (3) (4) (5) (6) ZLB Meanygap Meanπ s.d. ygap s.d. π Loss binding τ = −∞ 3.19% -0.23 2.00 1.23 0.99 0.151 0 τ = −16 4.49% -0.24 1.99 1.41 1.37 0.236 0 L∞ τ = 0 5.76% -0.27 1.96 1.72 1.96 0.449 −20 0 τ = 8 5.86% -0.30 1.90 1.73 1.95 0.452 0 τ = ∞ 10.05% -0.61 1.50 2.59 3.48 1.000 0 τ = −∞ 13.10% -1.23 1.94 1.04 0.83 0.169 0 τ = −16 19.66% -1.35 1.83 1.91 2.53 0.440 0 L20 τ = 0 25.29% -1.75 1.27 2.91 4.03 1.013 −20 0 τ = 8 25.10% -1.91 0.99 2.90 3.91 0.970 0 τ = ∞ 23.16% -2.05 0.79 2.90 3.93 1.000 0 Note: Results based on 1,000 simulations for each rule switch period τ shown. Recession periods are 0 t = 0,...,8. ZLB periods are the fraction of periods across simulations and across time during which theZLBisbinding. τ = −∞referstothefullinformationcaseinwhichprice-leveltargetingisinplace 0 fromthestartofeachsimulation,whileτ = ∞referstothecaseinwhichinflationtargetingisinplace 0 indefinitely. Simulationsarebasedonthebenchmarkparameterizationofsubjectivebeliefuncertainty. Lossfunctionvaluesarenormalizedtooneforthecaseτ =∞. 0 Price-level targeting is more beneficial under learning if adopted well in advance of the recession. When price-level targeting has been in place sufficiently long, private sector agents have had the opportunity to learn the new policy strategy before the recession begins, so that the stabilizing benefits of this strategy come to fruition. Figure 7 shows the evolution of beliefs and outcomes with learning for the case in which the central bank switches in τ = −16. In this case, beliefs have partially adjusted towards 0 the new rule parameters by the onset of the recession. Even with this partial understanding of the policy rule, policymakers already achieve similar outcomes of inflation andoutputasunderfullinformation. Table1reinforcesourmessageaboutthetimingofadoptionbyshowingadditional statisticsforthebenchmarklearningcase. Earlyadoptionofprice-leveltargeting(τ = 0 −16)yieldssimilaroutcomes(meansandstandarddeviations)ofinflationandtheoutput gap and losses as the full information case (τ = −∞), in particular when we con- 0 sider the long horizon. By contrast, adopting price-level targeting at the onset of a recession(τ = 0)resultsingreatervolatilityofinflationandtheoutputgap. 0 27

Overall, our results suggest that a central bank planning to switch to a price-level targetingstrategyshoulddosoasearlyaspossible,unlessitattachesahighprobability to a deep recession in the near future. In that case, it can be beneficial to postpone announcing price-level targeting until after the recession is over to avoid being stuck withacommitmenttomakeupalargeprice-levelgapbutlittleadditionalstabilization oftheeconomyduringtherecession. 5 Temporary Price-Level Targeting So far, we have focused on the adoption of a permanent price-level targeting strategy, under which the central bank seeks to close the price-level gap regardless of the gap sign and the economic conditions. We now turn to the more state-contingent variant of temporary price-level targeting (TPLT). Under TPLT the central bank only seeks to closethenegativeprice-levelgapthatstemsfromaZLBepisode;oncethisnegativegap hasbeeneliminated,thestrategyswitchesbacktoinflationtargeting. Evans(2012)and Bernanke (2017) argue that TPLT can provide the full stabilization benefits of permanentprice-leveltargetingduringsteepdeclinesofaggregatedemandwhile,atthesame time,canhelpavoidingthepotentialdifficultiesassociatedwithcommunicatingtothe publicthattightermonetarypolicyisneededtoreduceapositiveprice-levelgap. Both thesestudiesassumethattheprivatesectorhasfullinformationandthatthestrategyis perfectlycredible. However, the validity of these assumptions seems to be even more questionable in the case of a TPLT than a permanent price-level targeting strategy. Svensson (2019) articulates these concerns by stating that, if price-level targeting strategies “are only appliedoccasionallyandtemporarily,economicagentswillnotbeveryusedtothem,and considerableexplanationandcommunicationmaybenecessary. Butthismaystillnot besufficientforthetemporaryprice-leveltargettobecredible,inwhichcasethefavorableeffectofraisedinflationexpectationsmaynotoccur. Credibilitynormallyneedsto be earned, meaning that economic agents need to see the policy put into practice and its principles obeyed, in order to believe that it will be maintained and be successful 28

inthefuture.” OurlearningframeworkdirectlyspeakstoSvensson’sconcern,asagents understand and believe a price-level targeting strategy only once the strategy can be inferredfromtheobservationsofthepolicyrate. ATPLTstrategydiffersfromapermanentprice-leveltargetingstrategyalongtwodimensions: thedefinitionofthemakeupmeasureandthestate-contingentruleparameters. InourformulationofTPLT,themakeupmeasureaccumulatespastdeviationsof inflationfromitstargetsinceastate-contingentreferenceperiodτ (t): 0 t (cid:88) z = π . (16) t s s=τ0(t) Thereferenceperiodevolvesaccordingto:   t ifi = iand max z ≥ 0 t−1 τ0(t−1)≤s≤t−1 s τ (t) = . (17) 0  τ (t−1) ifi > ior max z < 0 0 t−1 τ0(t−1)≤s≤t−1 s Intuitively,thereferenceperiodisthelasttimethatthepolicyratereachedtheZLB.The referenceperiod,andthereforethemakeupmeasurez ,areresetwhenthepolicyrateis t at the ZLB and the makeup measure has ever turned positive since the previous reference period. By contrast, under permanent price-level targeting the makeup measure isgivenbytheaccumulated(positiveornegative)price-levelgapsinceafixedreference period. We now turn to the weight that the central bank assigns to the makeup measure in itsinterestraterule. Thepolicyrulecontinuestobeoftheforminequation(13): (cid:18) z ygap(cid:19) i∗ = ρ i +(1−ρ ) (1+φ )π +φ t +φ t +e . (18) t i t−1 i πt t zt 4 y 4 t The parameters φ on inflation and φ on the makeup measure are state-contingent πt zt to split the TPLT strategy de facto into an inflation targeting regime and a price-level 29

targetingregime:   (1,0) ifi = ior max z < 0 t−1 τ0(t)≤s≤t−1 s (φ ,φ ) = . (19) πt zt  (0,0.5) otherwise Thecentralbankassignspositiveweighttothemakeupmeasureonlyintheprice-level targeting regime which gets triggered when the policy rate first reaches the ZLB. The regime stays in place until the makeup measure has been made up for. After that, the centralbankswitchesbacktotheinflationtargetingregime. In line with our previous formulation of beliefs under learning, we assume that agents perfectly observe the makeup measure z , but do not observe the parameters t φ and φ . The beliefs about the evolution of these parameters are parameterized in zt πt thesamewayasinSection3.3. Inparticular,agentsthinkofφ andφ astime-varying, zt πt buttheydonothavetheknowledgethattheparametersfollowtwodiscreteregimes. Asforthepermanentprice-leveltargetingstrategy,weconsidertheseveredemanddriven recession scenario for a TPLT strategy to contrast the performance of the economy under learning with its performance under full information (agents observe the true rule parameters and understand their dependence on the economic conditions). TheoutcomesandbeliefsunderlearningandfullinformationareshowninFigure8. Over the course of the recession, the economic outcomes under TPLT are virtually thesameasunderthepermanentprice-leveltargetingstrategy. Initially,theanticipated stabilization benefits of the strategy do not materialize because agents require time to learnthenewstrategy,inparticularwhilethepolicyrateisconstrainedbytheZLB.Consequentlyagentsfailtoanticipatethemoreaccommodativepathofmonetarypolicyin the future. The cumulative shortfall in inflation is larger under learning and, as a resultthereof,thecentralbankstaysintheprice-leveltargetingregimemuchlongerthan underfullinformation. Once the economy has recovered sufficiently, the central bank returns to its inflation targeting regime and the rule parameters switch yet again as shown in the bottom panels of Figure 8. Under learning this change in parameters initiates a new adjust- 30

Figure8: Beliefsandoutcomeswithtemporaryprice-leveltargeting. (a)Outcomes. Output gap Inflation 1 4 0 3 -1 -2 2 -3 1 -4 No switch -5 Full info 0 Learning -6 0 8 16 24 32 40 48 0 8 16 24 32 40 48 Nominal interest rate TPLT shortfall 1 4 0 3 -1 2 1 -2 0 -3 -1 -4 0 8 16 24 32 40 48 0 8 16 24 32 40 48 (b)Beliefs. 0.7 1 0.6 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 0 0 8 16 24 32 40 48 0 8 16 24 32 40 48 Note: Solidlinesshowmedianoutcomesandbeliefsacross1,000simulations. Thedash-dottedlinein the lower panel additionally plots the median actual rule parameters φ and φ under learning. The πt zt simulationsareinitializedattheergodicdistributionofoutcomesobtainedundertheinertialTaylorrule, andinperiodt = 0thecentralbankstartsfollowingtheTPLTstrategy(seetext). Thestartingvaluesfor theruleinputsarez = 0andτ (0) = −1. VariabledefinitionsasinFigure4,exceptfor“TPLTshortfall” t 0 definedasmin{z ,0}. t 31

mentprocessfortheagents. Overthecourseoftheprice-leveltargetingregime,agents’ beliefspartiallyadjustedtothetrueparametersundertheprice-leveltargetingregime. But upon the central bank’s return to the inflation targeting regime of the TPLT strategy,agentswillreversetheirbeliefsandmovetheirparameterestimatestobeyetagain closertotheparametersoftheinflationtargetingregime. Thebeliefsandthedirection inwhichtheparameterestimatesmoveovertimewillagainbereversedonthenextoccasion the central bank is in the price-level targeting regime during a ZLB episode. In our learning formulation, agents will never understand the state-contingent nature of theTPLTstrategy. Asaresult,andincontrasttopermanentprice-leveltargeting,agents will never be in the position to correctly anticipate the central bank’s policy actions. In particularattheZLB,theTPLTstrategywillneverbeaseffectiveinstabilizingtheeconomy as the permanent price-level targeting strategy (which agents will come to fully understandovertime). The ineffectiveness of TPLT is in part the result of our assumptions about the beliefsthatagentscanentertain. Inparticular,agentscannotentertaintheideaofregime switches in the policy rules embedded in the TPLT strategy. Yet, even if we allowed agentstoconsiderthepossibilityofswitchesbetweentworegimes,itwouldstillbedifficultforagentstolearntheTPLTstrategy. Thesimplereasonisthatagentscannotinfer anythingabouttheregimetheeconomyiscurrentlynotin. BeforeaZLBepisodeoccurs, agentshavenoopportunitytolearnaboutthecentralbank’slikelybehaviorduringthat episode;attheZLB,thereisvirtuallynoinformationthatallowstodiscriminatebetween differentrules;andtheperiodoftherecoveryduringwhichthecentralbankstillfollows price-level targeting is short. Depending on the speed of learning, it would presumably take several zero-lower bound episodes before agents would fully understand the contingentbehaviorofthecentralbank. Moreover,agentswouldalsohavetolearnthe conditionsthattriggertheswitchfromoneregimetotheother,furthercomplicatingthe inferenceproblemrelativetoapermanentprice-leveltargetingstrategy. 32

6 Conclusion We have developed a method of learning about the central bank’s policy strategy from observed policy rates that explicitly takes into account the limited informational contentofobservedpolicyratesattheZLB.WehaveappliedthismethodtoasimpleNew- Keynesian model in which the central bank can pursue either an inflation targeting or price-leveltargetingstrategy. Whenthecentralbankswitchestoprice-leveltargetingattheonsetofadeeprecession,thisswitchmitigatesthelossinoutputandtheshortfallininflationunderrational expectations and full information, as is well known. But when agents are learning, the benefits of price-level targeting do not materialize because agents do not understand thenewpolicyregimeimmediately. Thelearningproblemisfurthercomplicatedbythe fact that the policy rate quickly hits the ZLB, at which point agents receive little information about the true parameters of the policy rule. As a result, under learning, the centralbankisleftwithamuchlargernegativeprice-levelgapthanunderfullinformation, and thus has to allow for substantial overshooting of inflation after the recession without having accrued any stabilization benefits in the midst of the recession. In orderforthesebenefitstomaterialize,price-leveltargetingshouldbeintroducedincalm timestogiveagentstheopportunitytolearnthisnewpolicystrategyratherthanbeing deployedasapolicytoolinadeeprecession. Temporaryprice-leveltargetingstrategies arelikelytobemuchlesseffectivethantheirpermanentcounterparts. References Ambler,Steve,“Price-LevelTargetingAndStabilisationPolicy: ASurvey,”JournalofEconomicSurveys,December2009,23(5),974–997. Angeletos,George-MariosandChenLian,“Forwardguidancewithoutcommonknowledge,”AmericanEconomicReview,2018,108(9),2477–2512. 33

Bernanke, Ben S., “Monetary Policy in a New Era,” Technical Report, prepared for the conference on Rethinking Macroeconomic Policy, Peterson Institute, Washington, DC,October12-13,2017,BrookingsInstitution2017. , Michael T. Kiley, and John M. Roberts, “Monetary Policy Strategies for a Low-Rate Environment,” Finance and Economics Discussion Series 2019-009, Board of GovernorsoftheFederalReserveSystem(US)February2019. Bodenstein,Martin,JamesHebden,andRicardoNunes,“Imperfectcredibilityandthe zerolowerbound,”JournalofMonetaryEconomics,2012,59(2),135–149. Boivin,Jean, “Has U.S.Monetary Policy Changed? Evidencefrom DriftingCoefficients andReal-TimeData,”JournalofMoney,CreditandBanking,August2006,38(5),1149– 1173. Brayton, Flint, Thomas Laubach, and David L. Reifschneider, “The FRB/US Model : AToolforMacroeconomicPolicyAnalysis,”FEDSNotes2014-04-03,BoardofGovernorsoftheFederalReserveSystem(U.S.)April2014. Cogley,TimothyandThomasJ.Sargent,“AnticipatedUtilityAndRationalExpectations As Approximations Of Bayesian Decision Making,” International Economic Review, February2008,49(1),185–221. ,ChristianMatthes,andArgiaM.Sbordone,“OptimizedTaylorrulesfordisinflation whenagentsarelearning,”JournalofMonetaryEconomics,2015,72(C),131–147. Eggertsson,GautiBandMichaelWoodford,“Policyoptionsinaliquiditytrap,”AmericanEconomicReview,2004,94(2),76–79. English, William B., William R. Nelson, and Brian P. Sack, “Interpreting the SignificanceoftheLaggedInterestRateinEstimatedMonetaryPolicyRules,”TheB.E.JournalofMacroeconomics,April2003,3(1),1–18. Erceg,ChristopherJ.andAndrewT.Levin, “Imperfect credibility and inflation persistence,”JournalofMonetaryEconomics,May2003,50(4),915–944. 34

Eusepi, Stefano and Bruce Preston, “The Science of Monetary Policy: An Imperfect KnowledgePerspective,”JournalofEconomicLiterature,March2018,56(1),3–59. Evans, Charles L., “Monetary Policy in a Low-Inflation Environment: Developing a State-ContingentPrice-LevelTarget,” JournalofMoney, CreditandBanking, 2012, 44 (s1),147–155. Evans, George W. and Seppo Honkapohja, Learning and Expectations in MacroeconomicsPrincetonUniversityPress,Princeton: PrincetonUniversityPress,2001. Farhi, Emmanuel and Iva´n Werning, “Monetary Policy, Bounded Rationality, and Incomplete Markets,” NBER Working Papers 23281, National Bureau of Economic Research,Inc2017. Gabaix, Xavier, “A Behavioral New Keynesian Model,” CEPR Discussion Papers 11729, C.E.P.R.DiscussionPapersDecember2016. Gust, Christopher J., Edward Herbst, and David Lo´pez-Salido, “Forward Guidance with Bayesian Learning and Estimation,” Finance and Economics Discussion Series 2018-072,BoardofGovernorsoftheFederalReserveSystem(US)October2018. Hatcher,MichaelandPatrickMinford,“StabilisationPolicy,RationalExpectationsAnd Price-Level Versus Inflation Targeting: A Survey,” Journal of Economic Surveys, April 2016,30(2),327–355. Hebden,JamesandDavidLo´pez-Salido, “FromTaylor’sRuletoBernanke’sTemporary Price Level Targeting,” Finance and Economics Discussion Series 2018-051, Board of GovernorsoftheFederalReserveSystem(US)July2018. Holden,Tom,“Computationofsolutionstodynamicmodelswithoccasionallybinding constraints,”Workingpaper2016. Kreps, David, “Anticipated Utility and Dynamic Choice, 1997 Schwartz Lecture,” in Donald P. Jacobs, Ehud Kalai, and Morton Kamien, eds., Frontiers of Research in EconomicTheory,Cambridge,England: CambridgeUniversityPress,1998. 35

Kryvtsov, Oleksiy, Malik Shukayev, and Alexander Ueberfeldt, “Adopting Price-Level TargetingunderImperfectCredibility: AnUpdate,”StaffWorkingPapers08-37, Bank ofCanada2008. Mele, Antonio, Krisztina Molnar, and Sergio Santoro, “On the perils of stabilizing priceswhenagentsarelearning,”DiscussionPaperSeriesinEconomics22/2018,NorwegianSchoolofEconomics,DepartmentofEconomicsOctober2018. Mertens, ThomasM.andJohnC.Williams, “Monetary Policy Frameworks and the Effective Lower Bound on Interest Rates,” AEA Papers and Proceedings, May 2019, 109, 427–432. Negro, Marco Del, Marc Giannoni, and Christina Patterson, “The forward guidance puzzle,”StaffReports574,FederalReserveBankofNewYork2012. Reifschneider, David and John C. Williams, “Three Lessons for Monetary Policy in a Low-InflationEra,”JournalofMoney,CreditandBanking,2000,32(4),936–966. Schorfheide, Frank, “Learning and Monetary Policy Shifts,” Review of Economic Dynamics,April2005,8(2),392–419. Svensson, Lars E O, “Price-Level Targeting versus Inflation Targeting: A Free Lunch?,” JournalofMoney,CreditandBanking,August1999,31(3),277–295. Svensson,LarsE.0.,“MonetaryPolicyStrategiesfortheFederalReserve,”TechnicalReport, Federal Reserve Bank of Chicago June 4-5 2019. prepared for the conference MonetaryPolicyStrategy,Tools,andCommunicationPractices-AFedListensEvent. Tetlow,RobertandPetervonzurMuehlen,“Simplicityversusoptimality: Thechoiceof monetary policy rules when agents must learn,” Journal of Economic Dynamics and Control,2001,25(1-2),245–279. Vestin, David, “Price-level versus inflation targeting,” Journal of Monetary Economics, October2006,53(7),1361–1376. 36

Woodford,Michael,“MonetaryPolicyAnalysisWhenPlanningHorizonsAreFinite,”in “NBER Macroeconomics Annual 2018, volume 33” NBER Chapters, National Bureau ofEconomicResearch,Inc,June2018. 37

A Details on the solution algorithm ThisappendixdescribesthenumericalalgorithmusedtocomputethelearningequilibriuminSection(2). (cid:16) (cid:17) ˆ 1. Startwithapriorforβ thatisnormallydistributedasN β ,P . t−1 t−1 t−1 (cid:16) (cid:17) ˆ ˆ 2. Wecomputex asafunctionofe˜,u ,x andβ : x = f x ,u ,e˜,β . Inpart t t t−1 t−1 t t−1 t t t−1 ticular, we augment equation (1) with anticipated shocks to the policy rule equation. FollowingHolden(2016),weuseamixed-integerlinearprogrammingsolver to determine the sequence of anticipated shocks such that the max operator in equation(2)willholdperiod-by-periodasprojectedunderperfectforesight. 3. Find(x ,i ,e˜)asthesolutiontothesystemofequations: t t t   i −Ψ(β ˆ )x ifi > i  t t−1 t t  e˜ t = φ (cid:18) i−Ψ(βˆ t−1)xt (cid:19) (A.1)     −σ et Φ (cid:18) i−Ψ( σ βˆ e t t −1)xt (cid:19) ifi t = i σet (cid:16) (cid:17) ˆ x = f x ,u ,e˜,β (A.2) t t−1 t t t−1 (cid:110) (cid:111) ˆ i = max i,Ψ(β )x +e . (A.3) t t−1 t t 4. Obtainaposteriorforβ throughthefilteringproblem(6)–(8). Eventhoughwetake t x asexogenous,thenon-linearitystemmingfromtheZLBandfromthepotential t non-linearity of Ψ(·) make this a non-linear filtering problem. To avoid having to use a numerically expensive particle filter, we make some numerical approximations. First,weapproximatethenon-linearityfromΨ(·)bytakingafirst-order ˆ Taylorexpansionofthenotionalratearoundβ : t−1 i = max{i,i∗} (A.4) t t ∂Ψ i∗ ≈ Ψ(β ˆ )x +β(cid:48) (β ˆ )x +e , e ∼ N (cid:0) 0,σ2 (cid:1) (A.5) t t−1 t t∂β t−1 t t t et (cid:16) (cid:17) ˆ β = β +(cid:15) , β ∼ N β ,P , (cid:15) ∼ N (0,Σ ) (A.6) t t−1 βt t−1 t−1 t−1 βt βt 38

Note that, in our application to price-level targeting in the paper, Ψ(·) is already linear,sotheaboveisanequalityratherthananapproximation. Wewillworkwiththesystematicpartofthenotionalrate,whichwedenotebys = t i∗ −e . Thepriorofs givenx isnormallydistributed: t t t t E[s | x ] = m = Ψ(β ˆ )x (A.7) t t t t−1 t V[s | x ] = S = H(cid:48)(P +Σ )H (A.8) t t t t t−1 βt t ∂Ψ ˆ whereH = (β )x . (A.9) t t−1 t ∂β To get to the posterior of β after observing i , we have to distinguish whether the t t ZLBisbindingornot. (a) Ifi > i,thefilteringproblem(A.4)–(A.6)reducestotheextendedKalmanfilter t (cid:16) (cid:17) ˆ (EKF) and the posterior is normally distributed as N β ,P . The filtering t t equationsarestandard: (P +Σ )H t−1 βt t K = (A.10) t S +σ2 t et ˆ ˆ β = β +K (i −m ) (A.11) t t−1 t t t (cid:0) (cid:1) P = P +Σ −K S +σ2 K(cid:48). (A.12) t t−1 βt t t et t IfΨislinear,thentheEKFisjustthestandardKalmanfilterandwehavefound anexactsolutiontotheposterior. (b) Ifi = i,wecomputethemeanandthevarianceoftheposteriorofs givenx t t t andtheobservationthats +e ≤ i. Foranarbitraryintegrablefunctiong,we t t 39

havethattheposteriormeanofg(s )isgivenby: t E[g(s ) | i = i,x ] = E[g(s ) | x ,s ≤ i−e ] t t t t t t t E[g(s )1{s ≤ i−e } | x ] t t t t = P(s ≤ i−e | x ) t t t E[g(s )E[1{s ≤ i−e } | s ,x ] | x ] t t t t t t = P(s ≤ i−e | x ) t t t (cid:20) P(e ≤ i−s | s ,x ) (cid:21) = E g(s ) t t t t | x t P(s +e ≤ i | x ) t t t t (cid:16) (cid:17) (cid:90) ∞ Φ i σ − 2 s 1 (s−mt)2 = g(s) (cid:16) et (cid:17) √ e 2St 2 ds. (A.13) −∞ Φ i−mt 2πS t St+σ e 2 t We compute these expressions using Gaussian quadrature for g(s) = s and g(s) = s2toobtaintheposteriormeanandvarianceofs ,whichwedenoteby t (cid:16) (cid:17) ˜ ˜ m˜ andS . Wenowapproximatetheposteriordistributionofs asN m˜ ,S . t t t t t With this approximation, the posterior for β given x and i = i is normally t t t (cid:16) (cid:17) ˆ distributedasN β ,P ,withtheupdatingformula: t t (P +Σ )H t−1 βt t K = (A.14) t S t ˆ ˆ β = β +K (m˜ −m ) (A.15) t t−1 t t t (cid:16) (cid:17) P = P +Σ −K S −S ˜ K(cid:48). (A.16) t t−1 βt t t t t 40