Inflation at Risk

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Inflation at Risk David Lo´pez-Salido and Francesca Loria 2020-013 Please cite this paper as: L´opez-Salido, David, and Francesca Loria (2020). “Inflation at Risk,” Finance and EconomicsDiscussionSeries2020-013. Washington: BoardofGovernorsoftheFederalReserve System, https://doi.org/10.17016/FEDS.2020.013. 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.

Inflation at Risk DavidLo´pez-Salido FrancescaLoria Federal Reserve Board February6,2020 Abstract Weinvestigatehowmacroeconomicdriversaffectthepredictiveinflationdistributionaswellas theprobabilitythatinflationwillrunaboveorbelowcertainthresholdsoverthenearterm.This iswhatwerefertoasInflation-at-Risk–ameasureofthetailriskstotheinflationoutlook.We findthattherecentmutedresponseoftheconditionalmeanofinflationtoeconomicconditions does notconveyan adequate representation of the overall pattern of inflation dynamics. Analyzing datafrom the 1970s reveals ample variabilityin the conditional predictive distribution ofinflationthatremainsevenwhenfocusingonthepost-2000periodofstableandlowmean inflation.WealsodocumentthatintheUnitedStatesandintheEuroAreatightfinancialconditionscarrysubstantialdownsideinflationrisks,afeatureoverlookedbymuchoftheliterature. Ourpaperoffersanewempiricalperspectivetoexistingmacroeconomicmodels,showingthat changesincreditconditionsarealsokeytounderstandthedynamicsoftheinflationtails. JELClassification:C21,E31. Keywords:QuantileRegression,InflationRisks. SpecialthankstoBenBernanke,SteveCecchetti,DavidCho,JimClouse,DeepaDatta,GiovanniFavara,FelixGalbis- Reig,EdHerbst,PaulLengermann,AntoineLepetit,EdNelson,ClaudiaPacella,JeremyRudd,TatevikSekhposyanand SrecˇkoZimicaswellasseminarparticipantsoftheResearch&StatisticsLunchSeminarattheFederalReserveBoard andoftheDG-EResearchSeminarattheEuropeanCentralBankfortheirusefulcommentsandsuggestionstoearlier versionsofthepaper. WealsothankEugenioCerutti,MichielDePooter,CaitlinDutta,JoaquinGarcı´a-CaboHerrero, LouisaLiles,LukeLillehaugen,MatteoLuciani,PaulTranandRiccardoTrezzifortheirhelpincollectingsomeofthe datausedinthepaper. The usual disclaimer applies: The views expressed in this paper are solely the responsibility of the authors and shouldnotbeinterpretedasreflectingtheviewsoftheBoardofGovernorsoftheFederalReserveSystemorofanyone elseassociatedwiththeFederalReserveSystem.

“Monetary policy responded first in the summer of 2012 by acting to defuse the sovereign debt crisis, whichhadevolvedfromatailriskforinflationintoamaterialthreattopricestability.” MarioDraghi,ECBPresident,Sintra,June2019. 1 1 Introduction Sincetheupheavalsoftheglobalfinancialcrisis,theemergenceofdownsideriskstotheinflation outlookhaveincreasinglybecomeasourceofmacroeconomicconcern. Yet,muchoftheeconomic analysishasbeendevotedtostudyingthefactorsunderlyingthemutedresponseoftheconditional mean of inflation to economic and financial conditions. At the same time, much has been said on theinabilityof pastand currentlabormarketconditions toexplain recentinflation outcomes. The Phillips curve linkages seem to be breaking down. In this paper, we find that some of the macroeconomicfactorscoveredunderthe“Phillipscurveumbrella”arestillatworkinthetailsof theinflationdistribution. Moreover, wealsoshowthatlookingattheentireinflationdistribution uncovers–aftercontrollingforthestateofthelabormarketandinflationexpectations–thattight financialconditionscarrysubstantiallow-inflationrisks,anaspectofinflationbehavioroverlooked bymuchoftheliterature. PresidentDraghi’squoteisanexcellentreminderthat,inthepresenceoftailrisks,theconditionalinflationmeandoesnotnecessarilyadequatelyrepresenttheinflationoutlook.Indeed,ithas beendocumentedthatthedeteriorationincreditmarketconditionsledtoasubstantialdeclinein economicactivityaswellasadeteriorationintheoddsoflowgrowthandofhighunemployment. Tight financial conditions moved the conditional distribution of real GDPgrowth to the left (e.g., Adrian,Boyarchenko,andGiannone,2019)–withitslefttailbeingthemostsensitivetomacroeconomicshocks(seeLoria, Matthes, andZhang,2019a)–andimpliedmedium-termupsiderisksto unemployment(seeKiley,2018).2 Yet,againstthisbackdrop,themodaloutlookforinflationseemed toremainsomewhatinsensitivetothesedevelopments. Weshowthatthecontrastingresponseof thetailsandthemedianoftheinflationdistributionrevealsamorecompletepictureoftheeffects thatrealandfinancialshocksimpingeoninflation. Theobjectiveofthispaperistoseewhatconclusionscanbedrawnfromacloserlookattheentireconditionalinflationdistribution,usingdata bothfortheUnitedStatesandtheeuroarea. Weinvestigatehowmacroeconomicdriversaffectthepredictiveinflationdistributionaswellas the probabilitythatinflation will run above orbelowcertain thresholds overthe nearterm. This is whatwe referto as Inflation-at-Risk – ameasure of the tail risks to the inflation outlook.3 Our econometric strategyfirstconstructs inflation quantiles conditioning on observed economic and financial variables (“risk factors”) using a quantile regression model, where the commonly used approachistofitalinearcurve(e.g.,Koenker,2005).4 1MarioDraghi,“TwentyYearsoftheEuropeanCentralBank’sMonetaryPolicy,”speechdeliveredattheECBForum onCentralBankinginSintraonJune18th,2019(availableathttps://www.bis.org/review/r190618c.htm). 2Inacross-sectionofcountries,Cecchetti(2008)findsthatassetpriceboomsincreasegrowthandinflationrisks. 3Readersmaygleansomelackofnoveltyinthislabel,asweadaptedthisnamefromtheValue-at-Riskliterature. 4Asshownintheliterature,thisisdonemainlybecauselinearmodelsenjoygoodapproximationpropertiesaswell asdesirablecomputationalproperties(e.g.,Chernozhukov,Fernandez-Val,andGalichon,2010). 1

Weframetheeffectsofdifferentriskfactorsoninflationwithinan“augmented”quantilePhillips curve model using data since the 1970s.5 That is, we extend the standard regression analysis – designedtoascertainthedriversoftheconditionalmeanofinflation–todifferentinflationquantiles.6 This setup allows to relate the risks to the inflation outlook to variations in labor market slack,andchangesinthepersistenceofinflationandinflationexpectations,aswellasmovements inrelativeprices(importedgoodsand/oroil).Moreimportantly,weextendtheanalysistoconsider theeffectoffinancialconditionsontheinflationdistributionandontheoddsoflowinflation. The conditional distribution of future inflation is constructed byfitting a flexible distribution on the estimated conditional inflation quantile distribution. Thus, in oureconometric approach, variations in the conditional inflation distribution depend on the evolution of economic and financial factors and howthey(a)symmetricallyaffect the inflation quantiles. These variations are not limited to a change in the mean and in the variance. Indeed, we find that periods featuring a relatively tight and centered inflation distribution evolve into periods in which the tails of the distributionincreasesubstantially,leadingtoachangeinthekurtosisofthedistribution;ortoperiodsinwhichthesymmetryofthedistributionistiltedtowardstheleft(ortheright),leadingtoa changeintheskewness. Finally,ourframeworklinksthepersistenceintheevolutionofeconomic andfinancialconditionstothepersistence,notjustoftheconditionalmean,butalsooftheinflationtails. Thisisillustratedwithhistoricalcontributionsofeconomicandfinancialconditionsto theevolutionofthemedian,thelowerandtheupperquantilesofinflation. As noted above, our “augmented” quantile Phillips-curve model considers changes in credit spreads as an additional factor affecting the entire inflation distribution. In this regard, the recentglobal financial crisis is an ideal case studyforillustrating howboth economic and financial headwindsinfluencedtheinflationoutlookbothintheUnitedStatesandintheeuroarea. IntheUnitedStates, averageinflationexperiencedonlyamodestreductiondespitethefallin outputtriggeredbythefinancialcrisis. Likewise,itfeaturedonlyashyrebounddespitetherecoveryand subsequent growth of the U.S. economy. The inabilityof conventional economic wisdom – derived from the historical Phillips-curve relationship between inflation and economic conditions – to explain this phenomenon is referred to byeconomists as the “missing disinflation and inflation puzzle”. There are two lead explanations supporting this lackof response in average inflation. First,inflationexpectationswerewellanchored(Yellen,2013). Second,asfinancialshocks increased the cost of external finance, liquidity-constrained firms have restrained from cutting pricebelowmarginalcosttosupporttheircash-flowandthushedgeagainsttheserisks(Gilchrist, Schoenle,Sim,andZakrajsˇek,2017). 5Ourapproachhencediffersfrom,andcomplements,studiesthatdefineinflationrisksasthechanceoflostpurchasingpowerresultingfromnegativeinflation-adjustedreturns. Thesestudiesevaluatetheinflationriskpremium associatedwiththecompensationrequiredbyinvestorsforfutureexpectedinflationordeflation–typicallyusinginformationcontainedinfinancialmarketquotes.Animportantdeparturefromourapproachisthat,ingeneral,theylack anexplicitlinkoftheseriskstospecificmacroeconomicoutcomes. 6Korobilis (2017) finds thatthe predictive densities coming from aquantile regression Bayesian model averaging (QR-BMA)modelaresuperiortoandbettercalibratedthanthoseofthetraditionalregressionBMAmodelandthatthis methodologyiscompetitivewithpopularnonlinearspecificationsforU.S.inflation.ManzanandZerom(2013)findthat incorporatingmacroeconomicvariablesintoquantileregressionsimprovestheaccuracyofinflationdensityforecasts. 2

We find that this viewis incomplete, as it ignores howinflation risks moved over this period. Indeed, to anticipate some of our results, we show that although the conditional mean of inflation held up during the crisis, the inflation distribution shifted to the left and was characterized by a substantial left tail. As a result, the odds of very low inflation or even deflation increased. Theseoddsstayedatconsiderablelevelsforsometime: thedownsideriskstotheinflationoutlook were quite persistent, as theyonlyvanished well into the recovery. Mostinterestingly, theirdramaticincreasewasmainlythereflectionofsoaringcreditspreadsduringthefinancialmeltdown. Subsequently,stablelong-terminflationexpectationssustainedtherecoveryofthelefttailofthe inflation distribution, accompanied by easing of credit conditions (as well as, to a lesser extent, improvementsinthelabormarket).7 Thesepatternshavebeenlessbenignintheeuroarea,wherethesovereigndebtcrisistriggered anincreaseintheoddsoflowinflationwhichwasmoreprolongedtothemorelimitedroleofinflationexpectationsincounteractingdownsideriskstoinflationposedbytheeconomicslowdown andfinancialdistress. New research by Christiano, Motto, and Rostagno (2014), Christiano, Eichenbaum, and Trabandt (2015), Del Negro, Giannoni, and Schorfheide (2015), and Gilchrist, Schoenle, Sim, and Zakrajsˇek (2017) argues that, in models with financial frictions, firms’ financial conditions help to explaininflationdynamics. However, thesepapershavealmostexclusivelyfocusedonexplaining the response of the conditional mean of inflation and hardly paid any attention to how financial conditionsaffectthetailsoftheinflationdistribution. Ourpaperoffersanewempiricalperspective on these issues, one that shows that changes in credit conditions are key to understand the tail-riskdynamicsofinflation. Finally,wecheckwhetherthedistributionofUnitedStatesinflationembodiedinfinancialoptions is consistent with some of the conclusions about inflation risks derived from our analysis. Not only do inflation probabilities coming from financial markets and from our quantile Phillips curve model point in the same direction but theyalso share a defining feature, namelythat tight financialconditionscarrysubstantialdownsideinflationrisksandmoststronglysoforthelefttail. Outline The paper is structured as follows. In Section 2 we organize ideas by presenting our theoretical framework and empirical strategy. We then study the role of economic and financial conditionsfortheriskstotheUnitedStatesinflationoutlookinSection3usingourfullsample. As time-variationemergesinthecharacterizationofthedeterminantsoftheinflationdistribution,we illustratesubsampleresultsinSection4andusethemtoshednewlightonthe“missingdisinflation and inflation” debate. In Section 5 we then compare the United States and euro area inflation experiences in the last 20 years and explore the role of financial conditions in affecting the odds oflowinflationduringandaftertheglobalfinancialcrisis. Wethenperformexternalvalidationof ourapproachinSection6. ConcludingremarksandpolicyimplicationsareofferedinSection7. 7AsYellen(2013)noted: “Aftertheonsetofthefinancialcrisis, thesestable[long-runinflation]expectationsalso helpedtheUnitedStatesavoidexcessivedisinflationorevendeflation.” 3

2 Quantile Regressions and Inflation-at-Risk Inmanycircumstancesthestudyofthedeterminantsoftheconditionalmeanofinflationmaybe sufficient to produce a good representation of the modal dynamics of inflation. In other cases, however, studying the response of the tails of the predictive inflation distribution is essential for providing a more complete picture. This is likely to be the case, for instance, in the presence of largerealorfinancialshocks,asitaidsunderstandingtheeffectsthattheseshockshaveoninflation. Because of these considerations, we extend the standard regression analysis – designed to ascertainthedriversoftheconditionalmeanofinflation–totheentireinflationdistribution. Inthissectionwedescribetheeconometricspecificationweusetolinkeconomicandfinancial conditionswithriskstotheinflationoutlook. Wefirstdescribeconditionalinflationquantilesasa functionofobservedeconomicandfinancialvariables(riskfactors).Second,weusethesequantiles toapproximatetheinflationdistribution.Variationsininflationrisksarethenmeasuredaccording tohowmuchthetailsoftheinflationdistributionvarywiththeevolutionofeconomicandfinancial factors. Werefertothese“tailrisks”totheinflationoutlookasInflation-at-Risk(IaR). WeframetheeffectsofdifferentriskfactorsoninflationwithinanaugmentedquantilePhillips curve model. This setup allows us to relate inflation risks to variations in the amount of slack in the labormarket, changes in inflation persistence, variations in inflation expectations, as well as movements in relative prices (usually, imported goods and/oroil). OurPhillips curve model is “augmented” as it also incorporates financial conditions (approximated by credit spreads) as an additionalfactoraffectingnotjustthemean,butmainlythetailsoftheinflationdistribution. 2.1 (Phillips-Curve)QuantileRegressions QuantileregressionmodelsareaflexibletoolforstudyingthedeterminantsofIaR.8 Ourinflation measureofinterestisthe(annualized)averageinflationratebetweenquartertandquartert+4, π¯ .9 Weconsideralinearmodelfortheconditionalinflationquantileswhosepredictedvalue t,t+4 Q(cid:98)τ (π¯ t,t+4 |x t ) = x t βˆ τ , (1) is a consistent linearestimator10 of the quantile function of π¯ conditional on x – where τ ∈ t,t+4 t (0,1),x isa1×k-dimensionalvectorofconditioning(risk)variables,andβˆ isak×1-dimensional t τ vector of estimated quantile-specific parameters. Accordingly, a determinant x may exert nont lineareffectsoninflationdynamicsifitaffectsdifferentlythemedianandthetails. 8Foranintroductiontothequantileregressionmethodology,seeKoenker(2005). 9AsimilarapproachistakeninAdrian,Boyarchenko,andGiannone(2019)fortheaveragegrowthrateofGDP.An alternativeapproachistakenbyGhysels,Iania,andStriaukas(2018)whouseaQuantileAutoregressiveDistributedLag Mixed-FrequencyDataSampling(QADL-MIDAS)regressionmodeltoconstructmeasuresofinflationrisk. 10Formally,thedependencybetweenx andagivenquantileτ ∈(0,1)ofπ¯ ismeasuredbythecoefficientβˆ : t t,t+4 τ T−h βˆ =argmin (cid:88)(cid:0) τ ·1 |π¯ −x β |+(1−τ)·1 |π¯ −x β | (cid:1) , τ (π¯t,t+4≥xtβ) t,t+4 t τ (π¯t,t+4<xtβ) t,t+4 t τ βτ∈Rk t=1 where1 denotestheindicatorfunction,takingthevalueoneiftheconditionissatisfied. (·) 4

OurmodelforconditionalinflationquantilesextendsthePhillips-curvemodelusedintheliterature. In particular, we closelyfollowBlanchard, Cerutti, and Summers (2015) – arecentpaper that nicely summarized a vast empirical literature on inflation dynamics. Formally, the baseline quantileregressionmodelin(1)canbewrittenasanaugmentedPhillipscurvemodel: Q(cid:98)τ (π¯ t,t+4 |x t ) = (1−λˆ τ )π t ∗ −1 +λˆ τ π t LTE +θˆ τ (u t −u∗ t )+γˆ τ (π t R−π t )+δˆ τ F t , (2) whereriskfactorsaffectingthedistributionoffutureinflationcanbedividedindifferentblocks.11 First, the variables π∗ and πLTE respectively represent average inflation over the previous t−1 t four quarters and a measure of long-term inflation expectations. Lagged average inflation captures the role of “intrinsic persistence” or different forms of inertia in the price setting process thatcouldprecipitateupwardordownwarddriftintheaggregateinflationrate.12 Insomemodels, thisvariableproxiesadaptiveornon-rationalexpectationswhereasinothersitisusedtocapture backward-looking or simple rule-of-thumb pricing rules. Long-term inflation expectations approximatetheimportanceofsomefirmssettingpricesinaratherforward-lookingway. Whichof these two elements dominates the persistence observed in the distribution of aggregate inflation dependsonthesizeoftheparameterλ .13 τ Thesecondriskfactorislinkedtovariationsintheamountoflabormarketslack–asmeasured bytheunemploymentgap(u −u∗),whereu isthecivilianunemploymentrateandu∗isthenatural t t t t rateofunemployment. Mostoftherecentliteraturehasconcentratedonthestabilityovertimeof theparametersλandθtoexplaintheevolutionofaverageinflation.Thisliteraturehasfocused,for instance,onunderstandingthefailureofaverageinflationtorespondtounemployment–i.e.,the flattening of the Phillips curve – and on the increasingly dominant role of inflation expectations inexplaininginflationpersistence–i.e.,thewell-anchoringoflong-runinflationexpectations. In thispaperweextendthisanalysisbyexploringtheeffectsofthesevariablesonthetailsofthedistributionofinflation. Theimportanceoftheseeffectsiscapturedbythevariationacrossquantiles oftheparametersλ ,(1−λ )andθ inexpression(2). τ τ τ Thethirdriskfactorin(2)isgivenby(πR−π ),whichreflectsvariationsinrelativeprices. We t t usethequarterlychangeinrelativeimportprices(πI−π ). AsinBlanchard,Cerutti,andSummers t t (2015),thisvariableisusuallyincludedtocapturethepass-throughofbothnominalexchangerates and oil prices into core inflation measures and is perceived as having been a key driver of the run-up of inflation in the late seventies and the eighties. Lately, this variable has been used to approximate a wide range of risk factors, from changes in global commodity prices, taxes and tariffstootherglobalinfluencesondomesticinflation. Itseffectsontheinflationdistributionare capturedbythecross-quantilevariationintheparametersγ inexpression(2). τ 11AfulldescriptionofthedataisprovidedinAppendixA. 12WoltersandTillmann(2015)useaquantileregressionmodelofcoreCPIandcorePCEinflationwhichsolelyconditionsonpastinflationtostudyhowinflationpersistencediffersacrossquantiles. 13Topreservethenotionthatactualinflationpersistentlydeviatesfromlonger-runinflationexpectations,weimposethehomogeneityconstraintinpricesbyconstrainingthetwocoefficientstosumuptoone. Whenλ = 0, the τ quantilemodelbecomesanextensionoftheaccelerationistPhillipscurve,wherechangesininflationareafunctionof theunemploymentgap.Weimpose(1−λ )+λ =1,0≤(1−λ )≤1and0≤λ ≤1followingKoenkerandNg(2005). τ τ τ τ 5

Thelast,butnotleast,riskfactorthatweconsiderisrelatedtofinancialconditions. According to conventional wisdom, economic factors – labor market slack, inflation expectations, and relative prices – have been considered as the major sources of variation in the conditional mean of inflation. However, recent research by Del Negro, Giannoni, and Schorfheide (2015), Christiano, Eichenbaum,andTrabandt(2015),Christiano,Motto,andRostagno(2014)andGilchrist,Schoenle, Sim, and Zakrajsˇek (2017) suggests that changes in firms’ financial conditions (proxied by variations in credit spreads) also helps to explain inflation dynamics. After the financial stress of the fall of 2008, these studies aim at explaining how the sharp contraction in economic activity was accompaniedbyonlyamodestdeclinein(average)inflation. However,theymostlydiscusstherole of financial frictions in amplifying the business cycle and creating adverse feedback loops, while leavingitsimplicationsforinflationnotfullydeveloped. In this sense, these papers do not completely analyze the potential nonlinearities in the responseofinflationtofinancialdistressasausefuldiagnosticofthemodels. Forinstance,Gilchrist andZakrajsˇek(2015)andGilchrist,Schoenle,Sim,andZakrajsˇek(2017)notedthatthereisstillroom forchangesinfirms’financialconditionstoinfluencenotjustaverageinflation,butalsothedistributionofinflation. Inparticular,theseauthorsobservedthatarichheterogeneityinpricesettings will arise from changes in firms’ credit conditions: When external financing is expensive, a liquidityconstrainedfirmmayhavetosacrificecurrentandfuturedemandtogetliquidfundstoday bynotlowering prices in response to tightercreditconditions; butfirms with abundantliquidity maycutpricessubstantially.14 Someofthesedistributionaleffectsmaybereflectednotjustinthe medianbutalsointhetailsoftheaggregateinflationdistributionwhenthemodelissolvedatthird (orhigher) orderso as to capture changes in skewness (and kurtosis). We thus allowforfinancial conditionsF inexpression(2),toaffectdifferentlytheconditionalinflationquantiles.Thisallowsa t testforthepresenceofdifferentialeffectsoffinancialvariablesonthemeanversusthetailsofthe inflationdistribution(e.g.,throughthevariationinδ ). Followingtheseauthors,andasespecially τ recommendedbyGilchristandZakrajsˇek(2012),weapproximateF bythecreditspread,cs . t t 2.2 QuantileFunctionofInflation The estimated conditional quantiles are approximations to the so-called “quantile function”, that is, Q (π¯ |x ) = F−1(π¯ |x ), where F−1(·) is the conditional inverse cumulative distribu- τ t,t+4 t t,t+4 t tion function (CDF) of average future inflation. As noted by Adrian, Boyarchenko, and Giannone (2019),inpracticeitischallengingtomaptheseestimatesintoaprobabilitydistributionfunction (PDF)becauseofapproximationerrorandestimationnoise. Wethereforefollowtheirapproachby smoothingthequantilefunctionusingtheskewedt−distributionproposedbyAzzaliniandCapitanio(2003). Thisflexibledistributionischaracterizedbyfourparametersandgivenby: (cid:32) (cid:115) (cid:33) 2 κ +1 f(π¯ |x ,µ ,σ ,η ,κ ) = ×t(z ;κ )×T η z t ;κ +1 , (3) t,t+4 t t t t t σ t,t+4 t t t,t+4 κ +z2 t t t t,t+4 14SeealsoChevalierandScharfstein(1996)foranearlieranalysisofthismechanism. 6

where z = π¯t,t+4(xt)−µt and t and T respectivelyrepresent the densityand cumulative distrit,t+4 σt butionfunctionofthestudentt-distribution. Theconstantsµ ∈ Randσ ∈ R+ arelocationand t t scaleparameters,whereastheconstantsη ∈ Randκ ∈ Z+ controltheskewnessandthekurtot t sisofthedistribution,respectively. AsinAdrian,Boyarchenko,andGiannone(2019),wecompute theseparametersateachpointintimettominimizethesquareddistancebetweenourestimated quantilefunctionQ(cid:98)τ (π¯ t,t+4 |x t ),obtainedfromthequantilePhillips-curvemodel(2),andthequantilefunctionoftheskewedt−distributionF−1(π¯ |x ,µ ,σ ,η ,κ )tomatchthe5 th,25 th,75 thand t,t+4 t t t t t 95 th quantiles.15 2.3 Inflation-at-Risk Wereferto“Inflation-at-Risk”(IaR)astheprobabilitythatinflationfallsaboveorbelowacertain threshold. These risks are two-sided, with upside risks coming from “excessive inflation” and downsiderisksfromtooloworevennegativeinflation(i.e.,deflation).16 Therearetwokeyelements that characterize our measure of IaR: (i) a pre-specified threshold, i.e., an upper (lower) level of inflationabove(below)whichinflationis“atrisk”and(ii)atimeperiod(say,t+k)overwhichthe risktotheinflationoutlookisassessed. Theseelementsarenecessarytosubstantiatestatements such as: “With (100-τ) percent confidence we shall not experience, on average, inflation below (above)thelevelπ¯∗ overthenextt+k periods.” Theconditionaldownsideinflation-at-risk,PD(π¯ |x ) ≡ Prob(π¯ < π¯∗|x ) 17,istheprobt t,t+4 t t,t+4 t abilitymassbelowπ¯∗ intheconditionaldensityf(π¯ |x ,µ ,σ ,η ,κ ): t,t+k t t t t t (cid:90) π¯∗ PD(π¯ |x ) ≡ f(π¯ |x ,µ ,σ ,η ,κ )dπ¯ , (5) t t,t+4 t t,t+k t t t t t t,t+k −∞ where at (100-τ) percent confidence, inflation will not be, on average, below the level π¯∗ over the next t+k periods.18 In other words, this expression defines the (downside) inflation-at-risk through the integral of the PDFoverthe inflation supportup to a specified threshold or, equivalently,throughtheCDF. 15Theparametersarethusfunctionsoftheconditioningvariablesx . Thisdependenceisnotmadeexplicitfornot tationalconvenience. 16OurapproachdiffersfromtheValue-at-Riskliteratureintwoways.First,inthatliterature,VaR(τ)isnotaprobabilitybutthethresholdsuchthattheprobabilityoffuturereturns(not)exceedingthatthresholdisequaltoτ. Inthat sense,VaR(τ)istheτth quantileoffuturereturns. Formally,accordingtothatdefinition,inflation-at-riskIaR(τ)is thustheτthconditionalinflationquantile,Q (π¯ |x ),implicitlydefinedbytheintegralovertheconditionalinflation τ t,t+4 t densityf(π¯ |x )thatsumsuptoτ: t,t+k t (cid:90) Qτ(π¯t,t+4|xt) f(π¯ |x )dπ¯ =τ. (4) t,t+k t t,t+k −∞ Second,(VaR)analysisandtherecent“Growth-at-Risk”literaturefocusonmeasuringone-sided“downsiderisks”to theeconomicoutlook(e.g.,Adrian,Boyarchenko,andGiannone,2019). 17For simplicity, our notation suppresses the dependence of PD(π¯ |x ) and PU(π¯ |x ) on the parameters t t,t+4 t t t,t+4 t µ , σ , η , κ which,inturn,dependonx . t t t t t 18Similarly, we can define the conditional upside inflation-at-risk PU ≡ Prob(π¯ >π¯∗|x ,µ ,σ ,η ,κ ) as t t,t+4 t t t t t (cid:82)∞f(π¯ |x ,µ ,σ ,η ,κ )dπ¯ = PU(π¯ |x ),whichistheprobabilitythatfutureinflationwillbe,onaverage, π¯∗ t,t+k t t t t t t,t+k t t,t+4 t abovethelevelπ¯∗overthenextt+kperiods. 7

Figure1illustratesthelinkbetweenIaRandthequantilesoftheinflationdistribution. Downsideriskstoinflationcanbecharacterizedbytheprobabilitymasstothelefttailofthedistribution (leftpanel). Theredareaindicatesthatat4percentconfidencelevel,inflationatriskis“zeropercent”. Or, equivalently, that a zero (orbelow) inflation rate corresponds to the 4 th quantile of the inflationdistribution. Similarly,therightpanelillustratesthat,witha15percentprobability,averagefutureinflationcanbeabove3percent–inotherwords,theupside(tail)riskassociatedwith “excessiveinflation”is15percent. Moregenerally,measuringτth-percentIaRisakintoestimating theτ th quantileoftheprobabilitydistributionofinflation(oritsoutlook). Figure1:Inflation-at-Risk. “Deflation”Probability “HighInflation”Probability 0.5 0.5 0.4 0.4 y y tis tis n n e D0.3 e D0.3 y y tilib0.2 tilib0.2 a a b b o o r r P0.1 P0.1 0 0 -4 -2 0 2 4 6 8 -4 -2 0 2 4 6 8 Average One-Year-Ahead Inflation Average One-Year-Ahead Inflation Note:Thefiguredisplayssimulateddistributions.Intheleftpanel,theprobabilityoftheaveragefutureinflationrate fallingbelow0%is4percent.Intherightpaneltheprobabilityofaveragefutureinflationexceeding3%is15percent. 3 Inflation-at-Risk Across Time Inthissection,wefirstpresentfull-sampleestimatesofthePhillips-curvequantilemodeltogauge the influence of the inflation drivers on the tails of the conditional inflation distribution. This naturally leads us to consider the presence (or not) of non-linear inflation dynamics, where the non-linearityis intended as arising from the asymmetryin the importance of inflation determinantsacrossquantiles. Wecomplementtheanalysisbyreportingthehistoricaldecompositionof thelowerandupperinflationtails. Weclosethesectionbyprovidingestimatesoftheconditional distributionofaveragefutureinflation. Ourmeasureofinflationis“coreinflation”. Thismeasureprovidesinformationabouttherate towardwhichheadlineinflationwillconvergeinthemediumtermifpresentpatternscontinue;as volatiletransientshockswillfadeovertime,thecorerateisintendedtobeareliablepredictorof futureheadlineinflation. WefocusoncoreCPIinflation,whereinflationismeasuredasquarterover-quarter annualized growth rates in the underlying price index. In particular, our working measure of inflation is the average inflation rate between t and t + 4 quarters. We also present 8

inAppendixCresultsforcorePCEinflationandforStockandWatson(2019)“CyclicallySensitive Inflation”,whicharequalitativelythesame. Oursamplespanstheperiodfrom1973:Q1to2019:Q1, astheGilchristandZakrajsˇek(2012)creditspreadisonlyavailablestartingintheearly70’s. The fourtop panels of Figure 2 report the estimated slope coefficients θˆ , (1−λˆ ), λˆ and γˆ τ τ τ τ of the quantile regression model (2).19 Theyalso visualize the partial fitted regression lines along with scatterplots of one-year-ahead average inflation against the relevant inflation determinant. In all figures we focus on three partial fitted regression lines, corresponding to the 10th, 50th and 90th quantiles. We also include the partial fitted OLS regression line, which is obtained from the commonly estimated Phillips curve. These slopes are informative about whether economic and financialconditionsaffectthetailsoftheinflationdistributiondifferentlythanthemedian,which isindicativeofthepresenceofnon-linearitiesininflationdynamics.20 The top-left panel of Figure 2 presents the quantile-specific Phillips curve coefficients associated with variations in the unemploymentgap. The results are in line with the recentevidence suggesting a substantial flatness in the Phillips curve, as the conditional median of inflation remainsrelativelymutedinitsresponsetochangesintheunemploymentgap. Thispatterncarries over to the tails, albeit to a lesser extent. Indeed, the lower tail is somewhat more responsive to the unemploymentgap than the median. These results pointto amildlyasymmetric response of inflationtochangesintheunemploymentgap. Asthetop-rightpanelofFigure2reveals,changesinrelativeimportpriceinflationmoststrongly affect the upper tail of inflation. Increases in relative import prices tilt the inflation distribution to the upside, hence substantiallyincreasing the odds of upside inflation risks. However, reductionsinrelativepricesmakethedistributiontighteraroundthemedian,aconsequenceoftheless significantresponseofthelowertail.21 The second row of panels in Figure 2 shows how the inflation quantiles respond to average past inflation and to inflation expectations. Here, we uncover yet another interesting asymmetry: While movements in the median and in the uppertail are mostlydominated byaverage past inflation, the lowertail of the distribution shows the largest response to changes in inflation expectations. Thatis, persistentlyhighpastinflationexperiencestendtotiltthedistributiontothe upside,hencecreatingupsideriskstotheinflationoutlook(andbarelyaffectingthelowertail). In contrast,themodesteffectofpastinflationonthelowertailofthedistributionimpliesthatpersistentlylowinflationexperiencesdonotgeneratesignificantdownsideriskstotheinflationoutlook asthedistributiondoesnotshifttotheleft,butrathergetsmorecompressedaroundthemedian. Conversely,changesinlong-runinflationexpectationstranslateone-for-onetothelefttail,while the effects on the median and the upper tail are smaller. In other words, a sustained decline in longer-runinflationexpectationsposesseriousdownsideinflationrisks,whiletheeffectsofsuch adeclineonupsideriskaremuchmoremuted. 19ThequantileslopesandOLSestimatesaswellastheirconfidenceintervalscanbefoundinFigureB-1. 20Insection5,wecomplementtheinformationinthesefiguresbyshowingtheconfidencebandsoftheestimated slopesconstructedby“blocks-of-blocks”bootstrapping.SeealsoAppendixBfordetails. 21Resultsaresimilarusingrelativeoilpriceinflation(πO−π )(seeAppendixD.1),share-weightedcoreimportprices t t andtherealexchangerate(resultsavailableuponrequest). 9

Figure2: QuantileRegressionSlopes. 14 14 = 0.1 = 0.5 12 = 0.9 12 OLS 10 10 8 8 6 6 4 4 2 = 0.1 2 0 = 0.5 = 0.9 OLS 0 -2 -2 -1 0 1 2 3 4 5 -60 -40 -20 0 20 40 60 θˆ ={θˆ = −0.38,θˆ = −0.15,θˆ = −0.34} γˆ ={γˆ = 0.04,γˆ = 0.04,γˆ = 0.09} τ 0.1 0.5 0.9 τ 0.1 0.5 0.9 14 14 = 0.1 12 = 0.5 12 = 0.9 OLS 10 45o Line 10 8 8 6 6 4 4 = 0.1 2 2 = 0.5 = 0.9 OLS 0 0 0 2 4 6 8 10 12 14 2 3 4 5 6 7 8 λˆ ={λˆ = 0.96,λˆ = 0.47,λˆ = 0.42},whereλ iscoefficientonπLTE and(1−λ )onπ∗ τ 0.1 0.5 0.9 τ t τ t−1 14 = 0.1 = 0.5 12 = 0.9 OLS 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 δˆ ={δˆ = −0.19,δˆ = −0.02,δˆ = −0.19} τ 0.1 0.5 0.9 Note: The figure displays the slope coefficients of the quantile regression of average four-quarter-ahead core CPI inflationdefinedinexpression(2). Thelinesillustratetheslopesassociatedwiththemedian(red),the10th (blue)and the90th(yellow)inflationquantile.TheblacklinesaretheOLSestimates.Circlesindicatescatterplotsofaveragefuture inflationagainstagiveninflationdeterminant. Greycirclesindicatescatterplotsofaveragefutureinflationagainsta givenfinancialvariablepriorto1999:Q4whereasblackcirclesindicatethescatterplotfortheperiodstartingin2000:Q1. 10

For each economic factor, we highlight its relationship with the inflation outlook during the mostrecentperiodwiththeblackcloudofpointswhichfocusesonobservationsfromtheyear2000 onwards. As we will show in Section 4 the roles of the unemployment gap and of relative prices in accounting for variations in average future inflation are considerably dampened. At the same time, we find thatthe abilityof inflation inertiato move the inflation distribution is dramatically reduced,bestowingitspredominantroletolong-runinflationexpectations. ThelowestpanelofFigure2showstheeffectsofchangesincreditspreadsontheinflationquantiles.22 Overall,thenegativesignsuggeststhathighcreditspreads(i.e.,tightfinancialconditions) generatedownsideinflationrisks.Interestingly,creditspreadsaffectbothtailsoftheinflationdistribution. However, as the figure shows, there is substantial subsample instability governing the linkbetweenthetailsandvariationsincreditspreads. Thesub-period1973-1999ischaracterized byrelativelysmallvariationsincreditspreadsinaperiodofhighandvolatileinflationinducedin part by systematic increases in energy prices. This is captured by the light-grey cloud of points. From 2000 onward, lowvariabilityof inflation around 2 percenthas been anotable aspectof the stability of the macroeconomic landscape that has coexisted with substantial variation in credit spreads,aphenomenonamplifiedbytheglobalfinancialcrisis. Thesecombinationscorrespondto theblackcloudofpoints. AswewillshowinSection5,thismorerecentperiodhelpsincorrectly identifying the relationship between the tails of the distribution and the credit spread, which is confoundedbythetimeaggregation. Wefindthatinthepost-2000period,mostofthereduction ininflationfollowinghighcreditspreadsisconcentratedinthelowertailofthedistribution,while theeffectsontheuppertailarepoorlyestimated(i.e.,thepointestimatesareassociatedwithhigh levelsofuncertainty). Thus,theresultspointtoacloserelationshipbetweenatighteningoffinancialconditionsandrisksof“lowinflation”,whileperiodsof“frothiness”andbullyfinancialmarkets havelittleeffectsontheuppertailoftheinflationdistribution;instead,theymakethedistribution ofinflationmoreconcentratedaroundthemedian. 3.1 TheRoleofFinancialConditions Wenowillustratetheinfluenceofcreditspreadsondownsideriskstoinflationandtheirvariations overtime(lateron,wewillfocusonthelastsubsamplestartingin2000bycomparingtheUnited States with the experience in the euro area). To do so we construct the 10th quantile of inflation arisingfromthequantilemodelinitsbaselineversionandinaversioninwhichignorestheroleof financialvariables. Figure 3 displays the evolution overtime of the 10th inflation quantile in the baseline model – whichincludestheeffectsofcreditspreads(straightblueline)–andthe10th quantileconstructed byshuttingdowntheeffectsofthisfinancialvariable(blackdash-dottedline). Thegraphalsoincludesthetimeseriesofthecreditspread(purpledashedline).Itisevidentthatthequantilemodel inwhichtheroleoffinancialvariablesisdisregardedcanbeamisleadingmeasureofdownsideinflationriskiftherearesignificantchangesincreditspreads. Ascreditspreadshavebeengrowing 22InAppendixD.2weshowthattheseresultsarerobusttothechoiceofotherfinancialvariables. 11

over time, so does this model’s miss. Indeed, earlier in the sample the 10th quantile is barely affectedbycreditconditions, whilestartingintheearly2000s–oncethemodelaccountsformore pronounced variations in creditspreads – headwinds coming from financial conditions substantiallyincreasetheoddsoflowinflation. Figure3: TimeEvolutionof10th InflationQuantileAcrossModels. 6 n o i t 4 a l f n I I P C 2 e r o C 0 -2 `74 `78 `82 `86 `90 `94 `98 `02 `06 `10 `14 `18 Note: The figure displays the time evolution of the 10th inflation quantile estimated from the quantile regressions model(2),initsbaselineversion(bluestraight)andinitsversionwheretheeffectofcreditspreadsissettozero(black dash-dotted).Thecreditspread(purpledashed)isalsoreported.ShadedbarsindicateNBER-datedrecessions. Duringthe1990s,thereisaprogressivereductioninthelowertailofthedistributionthatremainedfairlyinsensitivetofinancialdevelopments.Startinginthe2000s,the10thquantileshowed a remarkable resistance to go well below 2 percent. This phenomenon ended at the onset of the global financial crisis and the subsequentzero lowerbound episode. The lowertail of the distributionwassuchthatdownsideinflationrisksmaterialized,withnon-zerodeflationprobabilities. The aftermath of the global financial crisis shows that the lower tail of the distribution exhibits substantial persistence. That is, the tightening in credit conditions tilted the distribution to the downsideforaprolongedperiod. Thereductionindownsideriskswasenabledbyimprovements inthelabormarketandsustainedbyinflationexpectations. Predictive Ability We now formally assess how financial variables influence the accuracy with whichthequantilemodelcharacterizestheactualdistributionofaveragefutureinflation. Inparticular, we test for correct calibration of the conditional predictive distributions implied by the baseline model in one case and by the model which does not condition on financial variables in the other. To do so we use the test of Rossi and Sekhposyan (2019), which evaluates the absolute predictiveabilityofamodelatitsestimatedparametervaluesand,thus,infinitesamples.23 Inthis sense,boththeparametricmodelandtheestimationtechniqueemployedarebeingevaluated. 23SeeRossi(2014)foranexcellentsummaryofdensityforecastevaluations. 12

To run the test, we first define the probability integral transform (PIT), i.e., the conditional quantilez thatcorrespondstotherealizedobservationπ¯∗ : t t,t+4 z ≡ F−1(cid:0) π¯∗ |x (cid:1) = Prob (cid:0) π¯ < π¯∗ |x (cid:1) , (6) t t,t+4 t t,t+4 t,t+4 t where F−1(cid:0) π¯∗ |x (cid:1) refers to the inverse of the conditional CDF or, equivalently, to the condit,t+4 t tional quantile function evaluated atthe realized value π¯∗ . In aperfectlycalibrated model, the t,t+4 predictivedensityshouldfeatureaCDFwhichisuniform,i.e.,equaltothe45◦ line. Thisproperty impliesthattheprobabilitythattherealizedvalueisaboveorbelowthepredictedvalueisthesame (onaverage,acrosstime)irrespectivelyofwhetherhighorlowrealizationsofthepredictedvariable areconsidered. Followingthislogic,iftheempiricalCDFofthePITsliesoutsideofthe5%critical values,thentheRossiandSekhposyan(2019)rejectsthenullhypothesisofcorrectcalibration. Figure4: RossiandSekhposyan(2019)TestforCorrectCalibrationofPredictiveDensity. 1 0.9 0.8 0.7 0.6 F D0.5 C 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Note: ThefigureillustratestheCDFofauniformdistributionalongwiththeempiricalCDFsofout-of-samplePITs obtainedfromthequantileregressionsmodel(2),initsbaselineversion(blue)andintwoversionswhicheitherdonot conditiononfinancialvariablesinestimation(green)orintheconstructionoftheinflationquantiles(black). The5% criticalvaluesforeachmodel(dashed-dotted),arebootstrappedfollowingtheRossiandSekhposyan(2019)procedurefor multi-step-aheadforecasts.AsinAdrian,Boyarchenko,andGiannone(2019),thePITsareconstructedviaanexpanding rollingwindowsestimationinitiallyusing20yearsofdata.Confidencebandsshouldthusbetakenasgeneralguidance sinceRossiandSekhposyan(2019)derivethemforPITscomputedusingafixedrollingwindowscheme. InFigure4weplottheCDFofauniformdistribution(red,dashed)aswellastheempiricalCDFs ofthePITsobtainedfromthebaselinemodel(blueline)andfromtwoversionswhicheitherdonot conditiononthecreditspreadintheestimation(greenline)orintheconstructionoftheinflation 13

quantiles (blackline), along with their5% critical values (dash-dotted lines). These critical values arebootstrappedfollowingtheRossiandSekhposyan(2019)procedureformulti-step-aheadforecasts. AsinAdrian,Boyarchenko,andGiannone(2019),thePITsareconstructedviaanexpanding rollingwindowsestimationinitiallyusing20yearsofdata. Confidencebandsshouldthusbetaken asgeneralguidancesinceRossiandSekhposyan(2019)derivethemforPITscomputedusingafixed rollingwindowscheme.24 Unlikethebaselinemodel,themodelthatdisregardstheroleoffinancialvariablesintheconstruction ofthe quantiles does notpassthe testforcorrectcalibration – asitpoorlyspecifies the predictive inflation distribution by placing too little mass on its lower tail. The model neglecting financial variables in estimation performs worse than the baseline along the entire inflation distributionexceptontheuppertail. 3.2 ThePredictiveDistributionofInflation Figure5displays,forselecteddates,theestimatedconditionalpredictivedensitiesofaverageoneyear-aheadinflationandtheirassociatedfittedinversecumulativedistributionfunctions–shown intheinsetboxes.25 Thetopandthebottompanelsillustratethecontrastbetweentheoddsofhigh inflation,whichcharacterizedtheinflationdistributionduringthefirstpartofthesample,andthe progressiveswitchtowarddownsideriskstotheinflationoutlookwhichbuiltupattheonsetofthe globalfinancialcrisis. As shown in the top panel, during the firstsubsample we selectfourdates. We startourtime travel in early 1975:Q1, right after the recession triggered by the first wave of oil shocks and the easingcycleoftheFederalReserve. Thesecondquarterof1981ischosentocapturetheeffectsof the second OPEC shock. We pick these two dates as representative of the Great Inflation period. Then, we look at the distribution in the mid-eighties, more precisely in 1985:Q2, to capture the effects of the Volcker disinflation – a disinflationary transition period that led the U.S. economy into the so-called Great Moderation. This last period is represented by showing the estimated conditionalinflationdistributioninthelastquarterof1999. Overall,theestimatedquantilemodelsareabletocapturehowtheinflationdistributionmoved fromtheright-withsignificantupsideinflationrisksassociatedwiththepersistenteffectsofthe oilshocksinthemid-70sandearly80s–totheleft,withalmostnegligibleupsiderisksofinflation fallingabove4percentattheeveofthe2000s. Beyondthesegeneralchangesinthedistribution,it isworthnotinghowtheshapeofthedistributionsubstantiallychangedovertime. ThefirstOPEC shock led to an asymmetric inflation distribution, with almost negligible odds of inflation falling below6percentbutalongrighttailcreatingverylargeupsideriskstotheinflationoutlook. These risksmaterializedafterthesecondOPECshock. Thedistributionshiftedfurthertotheright,with 24Nevertheless,ifweusearollingwindowschemewhichuses20yearsofdatawecanconfirmthatthemodelwhich doesnotconsiderthecreditspreadintheconstructionofthequantilesfailstopassthetestbecauseofpoorcalibration ofthelefttail.Also,westillcan’trejectcorrectspecificationofthepredictivedensityofourbaselinemodel(resultsare availableuponrequest). 25AsformallydescribedinSection2.3,weconstructtheskewedt-Studentprobabilitydensityfunctionofinflation usingthequantilesestimatedusingtheregressionmodel(2). 14

upsiderisksbecomingmorebalancedaroundamuchhigheraveragefutureinflationrate. Chairman Volcker’s reaction to the great concern about the rise in long-run inflation expectations led to the aggressive monetary policy reaction designed to curb inflationary pressures and progressivelyhamperinflationexpectations. Theeffectsofthispolicyarereflectedinthenoticeableshiftto-the-left in the estimated inflation distribution, with upside risks substantiallyreduced bythe mid-80s. Duringthoseyears,theinflationdistributionbecamemoresymmetricandsubstantially more concentrated around the median. This disinflationary process continued during the 90s, andbytheendofthemillenniumthedistributionconcentratedaround2.5percent,withthelower tailremainingquiteinsensitivetoeconomicorfinancialdevelopmentsandshowingaremarkable resistancetogobelow2percent,afeaturewhichweanalyzeinSection4. The bottom panel of Figure 5 selects a few dates in the evolution of the inflation distribution duringthelast20years,anditdepictsacompletelydifferentstoryfromthefirstpartofoursample –althoughthereisremarkablesimilaritybetweentheinflationdistributionattheeveoftheGreat Recession,thebluelineinthebottompanelthatcorrespondsto2008:Q4,withtheoneshownfor the lastquarterof 1999 in the top panel. During this more recentperiod the reasons forconcern move from upside inflation risks to low-inflation or even deflation risks – with the ghost of the GreatDepressionfrighteningcentralbanksduringtheaftermathoftheglobalfinancialcrisis. Although we devote Section 5 to develop this issue in depth, the three dates chosen in the bottom panelofFigure5serveasausefulpreambletothatdiscussion. The global financial crisis and the dramatic increase in creditspreads translated into arightskewed(i.e.,fatterleft-tailed)inflationdistribution,withthemedianmovingprogressivelycloser to the lower tail. This phenomenon was exacerbated during the subsequent zero lower bound episode. The lower tail of the distribution was such that downside inflation risks materialized, withnon-zeroprobabilitiesofdeflation(seetheredlinethatdisplaysthedistributionin2009:Q4). Thisemergenceofsubstantialdownsideriskstoinflationhasbeenthemainsourceofincreasing concern among researchers and policymakers. Monetary policy provided accommodation to supportastrongjobmarket,toabatethelingeringheadwindsfromthefinancialcrisis,andtokeep inflation expectations well-anchored. Theseeffects translatedinto asubstantial shiftto theright intheinflationdistribution,curtailingtheoddsofdeflationbytheendof2015(yellowdistribution showninthebottompanelofFigure5). Theseandotherconsiderationsaremoredeeplyexplored inthenextsectionthatdelvesintotheroleplayedbylabormarketdynamics,andespeciallycredit spreadsandinflationexpectationsacrossthetwosubsamplesjustconsidered. 15

Figure5: ConditionalPredictiveInflationDensitiesatSelectedTimeEpisodes. 1973-1999 0.14 15 0.12 0.1 10 0.08 F 5 D P 0.06 0 0.04 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.02 0 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 2000-2019 0.14 10 0.12 0.1 5 0.08 F D 0 P 0.06 0.04 -5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.02 0 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 Note: The figures shows for selected time episodes the estimated skewed-t conditional densities of average fourquarter-aheadcoreCPIinflationassociatedwiththequantileregressionsmodel(2).Theinsetboxreportsthevaluesof averagefutureinflationacrossquantilesatthesameselectedperiods.Moreformally,itdepictstheestimatedskewed-t inverseCDFassociatedwiththeconditionaldensitiesinthemainpanel. 16

4 The Time-Varying Dynamics of Inflation-at-Risk The fullsample resultsanticipatedthattwodistinctsubsamplesemerge whencharacterizing the determinantsoftheinflationdistribution. Thefirstperiod,runningfrom1973to1999,coversthe OPEC shocks, the subsequent Volcker disinflation and the early stages of the Great-Moderation. The presence of large shocks to relative prices and the taming of inflation expectations induced largeswingsintheupperquantile,whilechangesinunemploymentandpastinflationaffectedthe median. Thesewereubiquitousthemesinthedescriptionofinflationdynamics. The first period contrasts with the second subsample, from 2000 to 2019, characterized by large movements in credit spreads, progressively well-anchored inflation expectations but subduedinflationpressures. Thesepatternsarestudiedinmostoftheliteraturediscussingafavorite whipping boy– the flatness of the Phillips curve. Well-anchored long-run inflation expectations Yellen (2013), systematic monetary policy (e.g. Ball and Mazumder, 2019, McLeay and Tenreyro, 2018)andmismeasurementoflabormarketslack(e.g.,StockandWatson,2019)aretheusualsuspectsinexplainingtheobservedmutedresponseinaverageinflation. However,thefinancialcrisis andtheperiodinwhichmonetarypolicyhasbeenconstrainedbythezerolowerbound,havebeen followedbyaperiodofunderperformanceofinflationrelativetoexplicitorimplicitinflationtargets. Thisperiodhasendedwithreductions,ofdifferentsize,oflong-terminflationexpectations. Some authors have pointed out that the risks of persistent below-target inflation are associated with the emergence of this phenomenon and claim that this set the seeds for further downside risks to inflation. Through this section we will show that tight credit conditions arising from financialcrisesalsocontributedtoincreasingoddsoflowinflationorevendeflation,whichpointto agreaterroleforthelabormarketrecoveryandwell-anchoredinflationexpectationsinsupportingaverageinflation. Thispointwillbefurtherinvestigatedinthenextsectionusingcontrasting evidencefromtheUnitedStatesandtheeuroarea. 4.1 SubsampleStabilityandtheMissingDeflation/Inflation To investigate howthe importance of risk factors changed across the two subsamples, we report theirestimatedquantile-specificslopesinFigure6. Threeresultsstandout. Firstinflationinertiahascompletelylostitsgriponinflation,crowning long-run inflation expectations as the decisive inflation determinant among the variables in the modernPhillips curve.26 Second, long-run inflationexpectations exertasymmetric effecton the inflation distribution. In fact, in this context well-anchored long-run inflation expectations lower the response of average inflation to labor market slack, financial conditions and relative pricechanges. However,itwouldbemisleadingtodismisstheroleofthesefactorsfocusingonthe conditionalmeanonly. Instead,financialconditionsand,toalesserextent,labormarketoutcomes are keydrivers of downside inflation risk, which in turn are importantto characterize the entire 26Weobtainlong-terminflationexpectationsfromConsensusEconomics. Resultsaresimilarifweuselong-term inflationexpectationsfromtheSPForMichigansurvey,whicharerespectivelyavailablefrom1987Q1and1981Q1. 17

inflationdistributionanditsdynamics. Finally,relativeimportpricesstillposethreatstotheupperinflationquantile,thoughtoalesserdegreethanpriortotheGreatModeration.27 Figure6:QuantileRegressionSlopesAcrossSubsamples. 0 0.1 -0.2 0.08 0.06 -0.4 0.04 -0.6 0.02 -0.8 0 1973-1999 2000-2019 1973-1999 2000-2019 1 0.6 0.8 0.4 0.6 0.4 0.2 0.2 0 0 1973-1999 2000-2019 1973-1999 2000-2019 0 -0.5 0 -1 -1.5 -0.2 -2 -0.4 1973-1999 2000-2019 Note: The figure displays the estimated slopes of the quantile regression of average four-quarter-ahead core CPI inflationdefinedinexpression(2).Twodifferentsubsamplesareconsidered:(i)1973-1999and(iii)2000-2019.Thebars illustratethecoefficientsassociatedwiththe10thquantile(blue),median(red)and90thquantile(yellow). 27InAppendixCweshowthattheseresultsarerobusttoalternativemeasuresofinflation,eithercorePCEorCSI. UsingcurrentmethodcoreCPIorcorePCEalsodeliverssimilarresults(resultsareavailableuponrequest). 18

InflationProbabilities Wenowaskhowtheinflationdistributionandtheinflation-at-riskprobabilitieswouldhavelookedlikefromtheperspectiveofthesetwodifferentsubsamples. Inparticular,weperformthefollowingexperiment. Wefirstconsiderthesubsamplerunningfrom1973to 1999 and use the estimated relationship between inflation and its economic and financial determinants in that period to compute the “counterfactual” inflation quantiles in the post-2000 era. Thesequantilesarethenusedtoconstructtheinflationdistributionandtheassociatedinflationat-riskprobabilities–asiftheconditionscharacterizinginflationinthefirstsubsamplehadprevailed in the last part of the sample. We then run the opposite experiment. That is, we use the quantileregressionmodelestimatedoverthesamplerangingfrom2000to2019toconstructthe inflationdistributionandinflation-at-riskprobabilitiesprevailingoverthatsamesample. Theassumptionofstabilityintheunderlyingrelationshipscharacterizingtheinflationdistributioninthe firstsampleleadstotheappearanceof“missing”inflationaryanddeflationaryepisodes. InFigure7wepresenttheconditionalpredictivedensities(leftpanel)andinflationprobabilities (rightpanel)associatedwiththesetwo“counterfactual”economies. Twostrikingresultsstandout. First, from the pointof viewof the pre-2000 economy(yellowdash-dotted line), during the zero lower bound episode we should have observed disinflation, and even deflation, with probability one as well as left-skewed distributions. In contrast, the post-2000 economy suggests only tiny deflationanddisinflationprobabilitiesandmoreconcentratedinflationdistributions. Thisresult directly speaks to the debate on the “missing deflation”, i.e., the observed discrepancy between the deflation/disinflation predicted by conventional economic wisdom given the weak inflation fundamentals and the observed resistance of actual inflation falling to negative territory during thezerolowerboundperiod. Second,thepre-2000economy(greenstraightline)supportsarise in inflation with the recoveryand subsequent expansion of the U.S. economy, as reflected bythe increase in the probability of inflation being above 3 percent for the most recent years and by the left-skewed distributions. Conversely, the post-2000 economy wouldn’t have suggested any changeintheinflationodds. Thistensionisrelatedtodebatetheon“missinginflation”, whichis themirrorimageofthe“missingdeflation”conundrum. Theseresultscanbeeasilyrationalizedbyrecallingthedifferentroleofinflationexpectations in the two “counterfactual” economies. While in the pre-2000 economy, the inflation distribution is equally responsive to inflation inertia and inflation expectations as well as very sensitive tochangesintheunemploymentgapandinrelativeimportprices,inthepost-2000economyinflation dynamics are mainly driven by inflation expectations. As the latter have been extremely well-anchoredaround2percentsincetheearly2000s(asopposedtotheperiodpriortothat,see Appendix A), the post-2000 economy would have predicted average future inflation and its tails tostayincheck. Noticethatthetime-varyingsensitivityofinflationtoinflationexpectations(and inflation inertia), as well as its ability to explain the “missing deflation/inflation” puzzle, had alreadybeenexploredandestablishedbyBlanchard,Cerutti,andSummers(2015). Ouranalysisthus extendstheirfindingstotheentiredistributionoftheinflationoutlook. 19

Figure7: “Counterfactual”PredictiveDensities(left)andInflationProbabilities(right). 0.2 0.15 y F D tilib P 0.1 a b o r P 0.05 0 -3 -1 1 3 5 7 0.15 0.1 y F D tilib P a b 0.05 o r P 0 -3 -1 1 3 5 7 0.2 0.15 y F D P 0.1 tilib a b o r P 0.05 0 -3 -1 1 3 5 7 0.25 0.2 y F D 0.15 tilib a P b 0.1 o r P 0.05 0 -3 -1 1 3 5 7 Note: Theleftpanelsshow“counterfactual”skewedt-Studentconditionaldensitiesofaveragefour-quarter-ahead coreCPIinflationcomputedusing“counterfactual”conditionalquantilesover2000-2019whichwereobtainedusing differentsubsampleestimatesofthequantileregressionsmodel(2). Therightpanelsshow“counterfactual”inflation probabilitiesfordifferentcutoffs. Theseprobabilitiesarecomputedfromthe“counterfactual”skewedt-Studentconditionaldensitiesshownintheleftpanels.ShadedbarsindicateNBER-datedrecessions. 20

Whilewell-anchoredinflationexpectationsgoalong-wayinaccountingforthestabilityofthe inflationdistributionthroughouttheGreatRecessionandthesubsequentrecoveryperiod,itwould bemisleadingtothinkthatitwasitssoledriver. Infact,aswepointedoutbefore,downsiderisks were also sensitive to changes in the labormarketand financial conditions. Thus, monetarypolicynotonlyensuredpricestabilityonaveragebykeepinginflationexpectationsincheckbutalso avoideddeflationrisksbysupportingthejobmarketandeasingcreditconditions. Wediscussthe relativeroleoftheseriskfactorsfortheinflationoutlooknext. 5 Euro Area vs. United States During the Great Recession Wenowanalyzetheeffectofinflationdriversontheevolutionoftheinflationdistributionduring the last 20 years of data, comparing the United States experience with that of the euro area. For theeurozone,theanalysisfocusesoneuro-area-widecoreHICPinflation–measuredbyheadline HICPinflation excluding energyand unprocessed food.28 As for the U.S., the quantile regression model(2)usesthesampleperiodavailablefortheeuroarea,thatis,from1999:Q1to2017:Q4. QuantilePhillipsCurve Figure8displaystheestimatedslopesofthequantileregressionmodel (2) forthe euro area (left column) and the United States (right column). The information is organizedasfollows. Boxesineachrowcorrespondtothecovariatesofthequantilemodel. Theblack squares reportthe pointestimates of the 10th, 50th, and 90th quantile-specific slopes. The length oftheverticallinesaroundthepointestimatescorrespondstothe68percentconfidenceintervals constructedby“blocks-of-blocks”bootstrap(seeAppendixB).TheOLSpointestimatesandtheir 95percentconfidenceintervalsaregivenbythehorizontalredlines. Theunemploymentgapgeneratesfairlysimilarresponsesofmedianinflationintheeuroarea andtheU.S.butimportantdifferencesemergewhenlookingatthetailsofinflation.Intheeuroarea theuppertailisthemostsensitivesegmentoftheinflationdistributiontounemployment,whilethe lowertailrespondslittle. Thus,therelativeoddsofhighinflationrisksarisingfromasubstantially tightlabormarketoutweigh the downside risks of lowinflation associated with substantial labor marketslack. ThispatternisreversedintheU.S.,thoughthedegreeofasymmetryandtheroleof unemploymentingeneralismuchmoremuted. Duringthisperiod, changesintherelativepriceofimportedgoodsplayedaminorroleinthe U.S.,andaslightlylargerroleintheeuroarea. Still,therearesomeinterestingdifferencesacross economies. Intheeurozone,themedianandthelowertailofthedistributionseemmoreresponsive than the upper tail of the distribution of inflation. For the U.S., these results are consistent withtheprevioussection,inwhichwepointedtoagreatlyreducedrelevanceofrelativepricesas inflationdeterminantsstartingwiththeGreatModeration.29 28Busetti, Caivano, and Rodano (2015) use dynamic quantile regressions to estimate and forecast the conditional distributionofeuro-areainflation.Tagliabracci(2018)estimatesthisdistributionconditioningonEurocoin. 29Busetti,Caivano,Monache,andPacella(2019)investigatetheroleofdomesticandglobaldeterminantsofeuroarea coreinflationbyestimatingaPhillipscurveusinganexpectileregressionapproach. 21

Figure8: QuantileRegressionSlopesandConfidenceIntervals. EuroAreaCoreHICP UnitedStatesCoreCPI Note: Thefiguredisplaystheslopecoefficientsofthequantileregressionofaveragefour-quarter-aheadeuroarea coreHICP(left)andUnitedStatesCoreCPIinflation(right)definedin(2). Theblacksquarescorrespondtothepoint estimateswhereastheverticallinestothe68%confidenceintervalscomputedvia“blocks-of-blocks”bootstrapusing 10,000replications(seeAppendixB)forthe10thquantile(blue),median(red)and90thquantile(yellow).Theestimation periodis1999:Q1to2017:Q4.TheOLSestimatesandtheir95%confidenceintervalsarerepresentedbytheredlines. 22

Longer-term inflation expectations influence stronglythe overall inflation distribution in the U.S., and to a lesser extent in the eurozone. It is striking that, in the U.S., the estimated slope coefficientisnotstatisticallydifferentfromunityacrossallquantiles. Accordingly,inflationinertia –thebackward-lookinginflationcomponent–playsanegligibleroleincharacterizinginflationin theUnitedStates.Incontrast,intheeuroareatheeffectsoflong-terminflationexpectationsonthe inflationdistributionaremoreasymmetric: Inflationexpectationsplayamajorroleinexplaining the upper tail of inflation, while odds of low inflation are less sensitive to changes in long-run inflation expectations as they are also driven by past inflation. In other words, in the eurozone, unmoored reductions in inflation expectations result in more persistent increases in downside inflation risks as their negative effect is propagated over time through a higher inflation inertia thanintheUnitedStates. The last row in Figure 8 presents the role of credit spreads across inflation quantiles. In the euroareahighercreditspreads(i.e.,tightercreditconditions)shifttheinflationdistributiontothe left as theyhave a fairlysymmetric negative effect across inflation quantiles. This contrasts with the U.S. in which mostof the reduction in inflation following high spreads is concentrated in the lowesttail, while the effects on the uppertail are notverysignificant.30 Mostimportantly, in the U.S.financialconditionsaretheonlysignificantsourceofasymmetryintheinflationdistribution. Inflation Quantiles Figure 9 highlights key aspects of the evolution of the inflation outlook by displaying the time series of the median, the 10th and the 90th inflation quantiles. The top panel showstheevolutionfortheeuroareawhilethebottompanelfocusesontheU.S. In the eurozone, looking at the lower tail, it appears that downside inflation risks have been importantsincetheinceptionoftheeuro. Strikingly, theinflationdistributiontendstotilttothe upside around the three recessionary episodes, while also widening up significantly. By the end of the 2001-2003 recession, downside risks to inflation started to trend down (i.e., the lowertail fell) and after a faint recovery subsequently failed to rebound to the pre-contraction level. This phenomenonisobservedduringtheglobalfinancialcrisisof2008-2009andrepeatedaroundthe 2011-2013recession,whendownsiderisksincreasedfurtherwithoutrecoveringsincethen. The estimated quantiles for the U.S. in the bottom panel of Figure 9 show some salient differences with the eurozone. First, the downward tilts to the distribution associated with the two recessionswereprimarilyaresultofadropinthelefttail,unlikeintheeuroareawherethedownward push was more pronounced for the median and the upper tail. This was particularly acute during the global financial crisis. However, the substantial increase in the odds of low inflation wasfollowedbyasustainedrecoveryuntilthedistributionbecameagaintightlycenteredslightly above2percentwiththe10th quantilemovingbacktocloseto2percent. Thiscontraststheeurozoneexperience,inwhichthelefttailfailedtorecoveraftertheglobalfinancialcrisis. 30Thisresultisalsorobusttolimitingthesampleto1995:Q1-2007:Q4soastolimittheGreatRecessionperiodandto usingtheshort-terminsteadofthelong-termCBONAIRUmeasureforu∗whichfeaturesahigherlevelintheaftermath t oftheGreatRecession. Further,inAppendixCweextendtheseresultstotwoalternativemeasuresofinflation,core PCEandtheStockandWatson(2019)CyclicallySensitiveInflationindex.Theeffectsaremoresymmetricinthecaseof corePCE,whiletheCSImeasureexhibitsasimilarasymmetryascoreCPIalthoughofsomewhatlargermagnitude. 23

Figure9: TimeEvolutionofSelectedConditionalInflationQuantiles. EuroAreaCoreHICP 3 2.5 n o ita 2 lf n 1.5 I P C I 1 H e r 0.5 o C 0 -0.5 `00 `01 `02 `03 `04 `05 `06 `07 `08 `09 `10 `11 `12 `13 `14 `15 `16 `17 UnitedStatesCoreCPI 3 2.5 n o 2 ita lf n 1.5 I I P C 1 e r o 0.5 C = 0.1 0 = 0.5 = 0.9 -0.5 `00 `01 `02 `03 `04 `05 `06 `07 `08 `09 `10 `11 `12 `13 `14 `15 `16 `17 Note:ThefiguredisplaysthetimeevolutionoftheconditionalquantilesofeuroareacoreHICPinflation(top)andof UnitesStatescoreCPIinflation(bottom)estimatedfromthequantileregressiondefinedin(2). Shadedbarsindicate NBER-datedrecessionsfortheUnitedStatesandOECD-basedrecessionindicatorsfortheeuroarea. This opens up the question of which factors contributed to the recovery of the left tail in the caseoftheUnitedStatesandtothefailedrecoveryofthelefttailinthecaseoftheeuroarea. We thusnextinvestigatetheroleoftheinflationdeterminantsfromthePhillips-curvequantilemodel in influencing the inflation tails. In this regard, Figure 10 complements the results in Figure 9 by presenting the contribution of economic and financial factors to changes in the lowerand upper quantilesoftheinflationdistribution. Focusing on the United States (the two charts in the right column of Figure 10) it is striking howlong-terminflationexpectationshaveplayedapredominantroleinsustainingtherecoveryof thelefttail,supportedtosomeextentbyimprovementsinthelabormarketandmoreimportantly by the easing in credit conditions. On average, across time, 66 percent of the variation in the upperquantileofthedistributionisexplainedbychangesinlong-terminflationexpectations,with the residual difference explained by financial conditions (27 percent), the unemployment gap (5 percent)andrelativeimportpriceinflation(2percent). 24

Figure10: HistoricalContributionsofEconomicandFinancialFactors. 10th Quantile 2.5 1.5 2 1.5 1 1 0.5 0.5 0 0 -0.5 `00 `02 `04 `06 `08 `10 `12 `14 `16 `00 `02 `04 `06 `08 `10 `12 `14 `16 90th Quantile 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 `00 `02 `04 `06 `08 `10 `12 `14 `16 `00 `02 `04 `06 `08 `10 `12 `14 `16 EuroAreaCoreHICP UnitedStatesCoreCPI Note: Thefigureshowshistoricalcontributionsofaveragefour-quarter-aheadeuroareacoreHICP(left)andUnited StatescoreCPIinflation(right)associatedwiththequantileregressionsmodel(2).Thecontributionofagiveninflation determinantisobtainedbymultiplyingitstimeserieswithitsestimatedslope. Itsrelativeshareisthenobtainedby weightingthecontributionwithitsrelativemagnitudevis-a`-visthesumofallcontributions(theshareoftheconstant termisdistributedacrosstheinflationdeterminantsbasedontheirrelativesharesoastonotdistortresults). Shaded barsindicateNBER-datedrecessionsfortheUnitedStatesandOECD-basedrecessionindicatorsfortheeuroarea. In the euro area, on the other hand, inflation expectations and labor market conditions had much less grip on downside inflation risks, with an average share of 37 percent and 4 percent respectively. Rather, average past inflation had the predominant role in holding down the lower inflation tail, its average share amounting to 42 percent. As in the U.S., financial factors played an important role also in the eurozone (15 percent share) and relative prices had no meaningful implicationsontheinflationoutlook(2percentshare). AstrikingdifferencetotheU.S.ishowthe lackofrecoveryininflationexpectationshasdrivenmostofthedownwardtrendinthelowertail after2012–atendencythatdiminishedsomewhatduring2017. In the U.S. the upper inflation quantile is mainly dominated by changes in expectations, althoughhighunemploymentandpersistentlytightcreditconditionshavealsocontributedtomake 2percentaneffectiveceiling–yetanotherdentleftbytheglobalfinancialcrisis. Thesamecanbe saidabouttheeurozonewiththeimportantdifferencethatfinancialconditionsexertedastronger downwardpressureontheuppertail,whichthusresultedinalowerimplicitinflationceiling. 25

Wenowturnourattentiontotheincreasingroleplayedbychangesincreditconditionsininfluencing the downside risks of inflation throughout the recovery. In Figure 11, we illustrate the timeevolutionofthe10th conditionalinflationquantileofeuroareacoreHICPinflation(left)and ofUnitesStatescoreCPIinflation(right),bothforthebaselineversionofthemodel(bluestraight) andforaversionwheretheeffectofcreditspreadsissettozero(blackdashed). Thegapbetweenthetwolinescapturesthepartialeffectofcreditspreadsonthe10th quantile. It is evident how tighter financial conditions exert a persistent downward pressure on downside inflationrisksandmorestronglyso,whencreditspreadsarehigh. Itisremarkablehowthelarge spikeincreditspreadsobservedin2008intheU.S.(bottomrightpanelofFigure11)pusheddown thelowerinflationquantile,whichslowlymovedbacktoabout2percentbytheendof2017. Figure11: PartialEffectofCreditSpreadon10th InflationQuantile. 10th Quantile 2 3.5 3 2.5 1.5 2 1.5 1 1 0.5 0 0.5 -0.5 `00 `02 `04 `06 `08 `10 `12 `14 `16 `00 `02 `04 `06 `08 `10 `12 `14 `16 CreditSpreads 3.5 8 3 7 2.5 6 2 5 1.5 4 1 3 0.5 2 0 1 `00 `02 `04 `06 `08 `10 `12 `14 `16 `00 `02 `04 `06 `08 `10 `12 `14 `16 EuroArea UnitedStates Note:Thefiguredisplaysthetimeevolutionofthe10thconditionalinflationquantileofeuroareacoreHICPinflation (left)andofUnitesStatescoreCPIinflation(right)estimatedfromthequantileregressionsmodel(2), initsbaseline version(bluestraight)andinitsversioninwhichtheeffectofcreditspreadsissettozero(blackdash-dotted).Shaded barsindicateNBER-datedrecessionsfortheUnitedStatesandOECD-basedrecessionindicatorsfortheeuroarea. 26

The eurozone is a slightly different story. Financial conditions, which played a more limited role in the lowertail inflation dynamics, became increasinglyimportantafterthe global financial crisis. Toseethis,letusfocusonthebottomleftpanelofFigure11. Thisfigureclearlyshowsthat the tightening in credit conditions occurred in two consecutive waves. The initial tightening in financialconditionshappenedaround2008and2009andmarkedasharpreductioninthelower quantileofthedistributionthat,evenaftersomerecoveryincreditconditionsduring2009,would never rebound. The second wave, linked to the European sovereign debt crisis, exacerbated this change. As of 2012, the deterioration in credit conditions lifted up substantially the odds of low inflation. It is remarkable how, early in 2013, economic conditions pointed to a recovery in the lowerquantile. Accordingtoourmodel,however,thiswouldhaveportrayedamisleadingpicture reflecting the lackof consideration of the substantial downward pressures in place originated by thestillverytightcreditconditionsatthattime. Toseethismoreclearly,wetranslatethevariation inthesequantilesintochangesoftheentiredistributionofinflation,towhichweturnnext.31 TheDistributionofInflation AtaspeechinLondoninJuly2012,MarioDraghi–Presidentofthe EuropeanCentralBankfromNovember2011toOctober2019–announcedtheECB’scommitment of doing “whateverittakes”to preserve the euro. The eurozone was in the throes of crisis, bond yields and credit spreads of weak euro-member governments were soaring, and financial marketsdoubtedthatEuropeaninstitutionscouldavertdisaster. Thisispartofthehistoricalcontext reflected in Figure 12, which plots the estimated euro area core HICP predictive inflation distributions (left column) and their associated quantile functions (right column) across four selected dates. We startatthe dawn of the global financial crisis (2008:Q4), then explore those periods in which downside risks were mostacute (2009:Q4 and 2012:Q4) and finallyzoom in the end of our sample (2017:Q4). The blue straight lines correspond to the baseline quantile model whereas the blackdash-dottedlinesparseouttheeffectofcreditspreads. Theresultsreaffirmthattightfinancialconcernsplayedacrucialroleinshiftingtotheleftthe inflationdistributionduringthelate2009andremarkablysoinlate2012–atthetimeofDraghi’s speech. It is noteworthy how by the end of 2008, inflation-at-risk above 3 percent was virtually non-existent,whiletheleft-tailpointedtosomeminordownsiderisksofinflationrunningbelow 1percent. Overall,creditconditionsbarelyaffectedtheseconclusions. Thetwowavesinwhichthe financialcrisiswasreflectedintightcreditconditionstranslatedintoaremarkablechangeofthe inflationoutlook. Thedistributionshiftedtotheleftandconcentratedaroundamedianinflation littlebelow1percent,withtheoddsoflowinflation–orevendeflation–soaring. Bytheendof2017 the distribution of inflation had fatter tails, with the odds of high inflation above those observed atthe peakof the crisis, butwith substantial downside risks still remaining. The effects of credit conditions on inflation are also illustrated in the rightcolumn of Figure 12, which shows thatthe inflationquantileswhichconditiononcreditspreadsweresignificantlybelowthosethatsolelyrely oneconomicfactors. 31InAppendixCweshowthat,fortheU.S.economytheresultsshowninthetop-rightpanelofFigure11arerobust toalternativemeasuresofinflation,eithercorePCEorCSI. 27

Figure12: SelectedTimeEpisodesofPredictiveDensities(Left)andSkewness(Right). EuroAreaCoreHICPInflation. 0.12 3 0.1 0.08 P 2.5 C F D0.06 IH P 0.04 e ro C 2 0.02 1.5 0 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3 0.12 2.5 0.1 0.08 P C 2 F D IH P0.06 e ro1.5 C 0.04 1 0.02 0 0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.25 2.5 0.2 2 P 0.15 C1.5 F D IH P 0.1 e ro 1 C 0.05 0.5 0 0 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.12 2.5 0.1 2 0.08 P C F D0.06 IH1.5 P e ro 0.04 C 1 0.02 0 0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Note: Theleftpanelsshowtheestimatedskewedt-Studentdensitiesofaveragefour-quarter-aheadeuroareacore HICPinflationforalternativespecificationsofthequantileregressionsmodel(2),initsbaselineversion(bluestraight) andinitsversionwheretheeffectofcreditspreadsissettozero(blackdash-dotted).Therightpanelsshowtheestimated skewedt-inversecumulativeassociatedwiththeconditionaldensitiesintheleftpanels. 28

Figure13: SelectedTimeEpisodesofPredictiveDensities(Left)andSkewness(Right). UnitedStatesCoreCPIInflation. 2.6 0.3 2.4 0.25 2.2 F 0.2 IP C D e 2 P0.15 ro C 1.8 0.1 0.05 1.6 0 1.4 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2.5 0.3 2 1.5 IP F0.2 C D e 1 P ro C 0.5 0.1 0 0 -0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2.5 0.15 2 0.1 IP F C D e P ro C 1.5 0.05 0 1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2.6 0.3 0.25 2.4 0.2 IP F C D P0.15 e ro 2.2 C 0.1 2 0.05 0 1.8 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Note:Theleftpanelsshowtheestimatedskewedt-Studentdensitiesofaveragefour-quarter-aheadUnitedStatescore CPIinflationforalternativespecificationsofthequantileregressionsmodel(2),initsbaselineversion(bluestraight)and initsversionwheretheeffectofcreditspreadsissettozero(blackdash-dotted). Therightpanelsshowtheestimated skewedt-inversecumulativeassociatedwiththeconditionaldensitiesintheleftpanels. 29

In Figure 13, we compare these results with the experience of the United States, forwhich we considersimilardates.Wefocuson2009:Q4asthisiswhendownsiderisksweremostpronounced intheU.S.followingthesharpriseincreditspreadsanddireeconomicconditions. Asinthecase of the eurozone, the effect of financial variables on the inflation distribution is striking during thoseepisodesinwhichfinancialdistresswasmostacute. In2009:Q4,forexample,tightercredit conditionscontributedtopushingtheentireinflationdistributiontotheleft,whilemakingitmore dispersedandpokingdownsubstantiallyitslefttailtothepointofplacingnon-zeroprobabilityof deflationoccurringonaveragewithinthenextyear(aswewillshowbelow).32 Looking attherightcolumns of Figures12 and 13, one importantdifference emerges between the euro areaand the U.S. experience – adifference we had alreadyencountered when analyzing the quantile-specific slopes of credit spreads on average future inflation in Figure 8. While in the eurozone higher credit spreads pushed down the inflation distribution symmetrically across quantiles, in the United States its effects were mostlyreflected in the lefttail. As we showbelow, thetranslationoftheseeffectsintotheprobabilityoflowinflation(i.e.,downsideinflation-at-risk) ismorepronounced,themoretheinflationdistributionisright-skewed(i.e.,thefatteritslefttail). Inflation-at-Risk: TheRoleofSkewness We firstshowthatthe effects of changes in economic and financial conditions on downside inflation-at-risk depend on the skewness of the inflation distribution. To see this, we need to characterize the derivative of PD(π¯ |x ) ≡ Prob(π¯ |x ) < π∗ t t,t+4 t t,t+k t definedin(5),foragivenprobabilitycutoffπ¯∗,withrespecttoaninflationdeterminantx .Formally, t ∂PD(π¯ |x ) ∂ (cid:90) π¯∗ t t,t+4 t = f(π¯ ,µ ,σ ,η ,κ |x )dπ¯ , (7) t,t+k t t t t t t,t+k ∂x ∂x t t −∞ whereweabstractfromthedependenceoftheparametersµ , σ , η , κ onx . t t t t t ApplyingtheLeibnizintegralrule, ∂PD(π¯ |x ) (cid:90) π¯∗ ∂f(π¯ |x ,µ ,σ ,η ,κ )∂π¯ (x ) t t,t+4 t = t,t+k t t t t t t,t+k t dπ¯ , (8) t,t+k ∂x ∂π¯ (x ) ∂x t −∞ t,t+k t t andassumingalinearregressionquantilemodelforthemeanofπ¯ (x )simplifiesto: t,t+k t ∂PD(π¯ |x ) (cid:90) π¯∗ ∂f(π¯ |x ,µ ,σ ,η ,κ ) t t,t+4 t = β t,t+k t t t t t dπ¯ . (9) OLS t,t+k ∂x ∂π¯ (x ) t −∞ t,t+k t From expression (9) it follows that changes in any variable x , besides affecting linearly the t quantileofthedistributionofinflation,introducesa“nonlinear”effectondownsideinflation-atrisk. The first effect captures howa change in x scales (up ordown) the support of the inflation t distribution. Thestrengthofthischannelismeasuredbythecoefficientβ . Thisfirsteffectgets OLS 32InAppendixCweshowthat,fortheU.S.economytheresultsshownFigure13arerobusttoalternativemeasures ofinflation,eithercorePCEorCSI. 30

amplified by the second term that cumulates the derivatives of the conditional density function with respect to its support up to cutoff level π¯∗. It thus follows that the more right-skewed the distributionis(i.e.,themoremassisonitslefttail)atagivenpointintime,thestrongerthedensity changesintheleftpartofthesupportand, inturn, thebiggertheeffectondownsideriskcaused byachangeinx . WeofferanillustrativeexampleofthiseffectinAppendixE. t We now show how credit conditions affect the odds of low and very low inflation. Figure 14 displays,beginningin2000,ourestimatesoftheevolutionoftheprobabilityofobservinginflation rates below 1 or 2 percent over the next four quarters, respectively. The two columns are used to contrastthe eurozone (leftcolumn) with the U.S. (rightcolumn). In each panel, we displaythe probabilities computed using our baseline model (blue straight line) and its version which omits theeffectsattributabletochangesincreditconditions(blackdash-dottedline). Figure14: InflationProbabilitiesforAlternativeCutoffValues. y y tilib tilib a a b b o o r r P P y y tilib tilib a a b b o o r r P P EuroAreaCoreHICP UnitedStatesCoreCPI Note: ThefigureshowsthetimeevolutionofinflationprobabilitiesofeuroareacoreHICPinflation(left)andUnited StatescoreCPIinflation(right)fordifferentcutoffs.Theseprobabilitiesarecomputedfromtheskewedt-Studentconditionaldensitiesoftheaveragefour-quarter-aheadinflationmeasureswhichwerefittedontheestimatedconditional quantilesforalternativespecificationsofthequantileregressionmodel(2).Bothpanelsarereportedforthespecificationwithoutandwiththecreditspread(inbluestraightandblackdash-dottedlines,respectively).Shadedbarsindicate NBER-datedrecessionsfortheUnitedStatesandOECD-basedrecessionindicatorsfortheeuroarea. 31

Several conclusions emerge from these comparisons. Since 2000, the model omitting credit conditions would have assigned zero probability to inflation running below 1 percent in the U.S., whereas accounting for the financial meltdown had profound effects on the inflation outlook – withtheprobabilityofverylowinflation(anddeflation)temporarilyreachingalmost40percentin 2010 (upperrightpanel). Results forthe eurozone are more striking on this account. Changes in the creditspreads in 2009 and especiallyin 2012 induced sharp increases in the odds of verylow inflation and aremarkable divergence between the blue and the dash-dotted lines in the top-left panelofFigure14. Byearly-2014,thisprobabilitywasslightlyabove80percentwhenthemodelincludesfinancialvariables,whileitwasaround30percentinthemodelaccountingfortheeffectsof non-financialvariablesonly. ThebottompanelsofFigure14considersthepartialeffectthatcredit conditionsimpingedontheprobabilitythatinflationcouldrunbelow2percent.Eventhenarrower creditspreadthatprevailedintheU.S.during2015,translatedintoanon-negligibleprobabilityof inflationrunningbelow2percentduring2016and2017,beforesubsidingattheendofthesample. Asfortheeuroarea,thecreditspreadaccountsfortheprobabilityofinflationbeinglowerthan2 percentremaining atsustained levels around the sovereign debtcrisis. Indeed, in the absence of thischannel,duringthatperiodthesameprobabilitywouldhavebeenaround50percentinstead ofabout90percent,onaverage. 6 External Validation: The Case of the United States ThissectionisdevotedtofindanexternalvalidationofthepreviousresultsfortheUnitedStates. Weframetheanalysisonwhetherthedistributionofinflationembodiedinfinancialoptionssupportssomeofourconclusionsaboutinflationrisksderivedfromourquantileregressionanalysis. Specifically,toputtheemphasisonthelowertailoftheinflationdistribution,wefirstcomparethe inflationprobabilitiesimpliedbythe“quantilePhillipscurvemodel”withtheone-year-aheadCPI inflationprobabilitiesderivedfrominflationcapsandfloorscontracts(asdescribedbyKitsuland Wright,2013). Wefocusontheprobabilityoffutureinflationbeingbelow1percent.33 The black-continuous line shown in the top panel of Figure 15 displays the options-implied probabilityofheadlineCPIinflationnextyearbeingbelow1%–frommid-2009untiltheendof2017. A straight reading from this market portraits a quite striking picture. Market participants have systematicallybeenpricingaprobabilityofinflationbelow1%ofaround40percentupuntil2016, andonlyafterthatdatethisprobabilityhasbeenmovingdowntolevelsslightlybelow20percent –tenyearsaftertheglobalfinancialcrisis. However, manyanalyseshaveattributedtheleveland evolution of this probabilityto the high correlation between market-based measures of inflation expectationsandoil-pricerelatedshocks(seeLoria,Matthes,andZhang,2019b)–especiallysince theonsetoftheglobalfinancialcrisis. ThetoppanelofFigure15corroboratesthisclaimbydisplayingtheinflationprobabilitiesalong with3-months-and6-months-aheadoilpricesurprises–computedusingtheoilmarketpriceex- 33Theinflationprobabilitiesarevirtuallyidenticalifweconsiderone-year-aheadinflationinsteadofaverageoneyear-aheadinflation,asinourquantileregressionmodel. 32

pectationsthatBaumeisterandKilian(2016)recoveredfromoilfuturespricesandaftercontrolling forchangesintheriskpremium.34 Asthefigureshows,theoptions-impliedinflationprobabilities exhibitsahighcorrelationwiththeoil-pricesurprises. Indeed,concernsaboutlowinflationassociatedwiththeriseintheprobabilityaroundmid2014tolate2015coincideswithaperiodinwhich financialmarketshavebeensteadilysurprisedtothedownsideintheiroilpriceexpectations. Toimprovecomparabilityoftheoptions-basedheadlineCPIinflationprobabilitywithourmeasure of the tail of the core CPI distribution, we purify the financial markets’ inflation probability from effects of changes in oil, energy and food prices. In particular, we regress it on the two oil pricesurprisesaswellasonenergyandfoodpriceinflation,whichalsocorrelatewiththeoptionsbasedheadlineCPIinflationprobability(seeFigureF-1inAppendixF).35 The dashed red line displayed atthe bottom panel of Figure 15 corresponds to the residual of this regression (where negative values have been set to zero). The bottom panel compares this purified financial-market-based probability with the probability of average future U.S. core CPI inflationbeingbelow1%whicharisefromthequantilePhillipscurvemodel(previouslydisplayed in the top right panel of Figure 14). The figure is very suggestive as it shows how both measures pointin the same direction during mostof the sample period. The probabilityof lowinflation in the U.S. increased immediatelyafterthe global financial crisis and itsubsequentlyfalls to almost zero–remainingclosetozerountil2018,withtheexceptionof2014/2015whenmarketparticipants consistentlyexpected higheroil prices and when energyprices fell considerably. Accordingly, financial markets’ expectations of headline CPI next year being below 1% rose accordingly during thattime(seeFigureF-1inAppendixF). Thesmallremainingdifferencesbetweenthetwomeasurescanbeexplainedbyseveralfactors. First,quantile-regression-basedinflationprobabilitiescomefromastatisticalmodelinwhichrelative prices are estimated to playno role forthe lowerinflation tail whereas market participants seemtopayattentiontothelatter. Moreimportantly,ourregressionpurifiesthefinancialmarkets’ headlineCPIinflationprobabilitiesonlyfromtheiraveragerelationshipwithoil,energyandfood prices–failingtofullycapturetimesinwhichmarketparticipantsstronglyextractedinformation fromthesepricessuchthattheycomovedperfectlywiththeinflationprobabilities(asin2014/2015). Financial Market Inflation Probabilities and the Credit Spread Two defining features of our quantile Phillips curve model are that tight financial conditions carry substantial and persistent downsideriskstoinflationandthattheserisksdiminishasonemovestothelefttotheuppertail of the inflation distribution. To testwhetherthis relationship also holds in financial markets, we run a regression of the options-implied inflation probabilities on the credit spread. We confirm 34The oil price surprises are computed as the difference between the market expectation of oil prices x-months aheadandtherealizedpriceoftheWestTexasIntermediate.Whilethesesurprisesarenoti.i.d.butratherfeaturesome persistence,theystillportraytheactualsurpriseinoilpriceexpectationsoffinancialmarketsparticipants. 35Sincethedependentvariableoftheregressionisaprobabilitywhichfallsbetweenzeroandone, weestimatea generalizedlinearmodelwithalogitlinkandthebinomialfamilytoensurethatthepredictedvaluesarebetweenzero andone.AstandardOLSregressionsdeliversvirtuallyidenticalresults. 33

Figure15: InflationProbabilities,QuantileModelvs.FinancialMarkets. y tilib a b o r P y tilib a b o r P Note:Thetoppanelshowstheoptions-impliedinflationprobabilitiesofUnitedStatesheadlineCPIinflationnextyear beingabove1%alongwiththe3-months-and6-months-aheadoilpricesurprisescomputedusingtheoilmarketprice expectationsthatBaumeisterandKilian(2016)recoveredfromoilfuturesprices(toppanel).Thebottompanelshowsthe probabilityofaverageone-year-aheadcoreCPIinflationbeingbelow1%comingfromthequantileregressionmodelas wellastheprobabilityofone-year-aheadheadlineCPIbeingbelow1%impliedfrominflationcapsandfloorscontracts asinKitsulandWright(2013),purifiedfromoil,energyandfoodpriceeffectsandtransformedtoquarterlyfrequency. thathighercreditspreads are associated with a downward pressure on the inflation distribution andthatasthecutoffsfortheinflationprobabilitiesincreasetherelationshipweakens. Thisresult isestablishedviastandardOLSregressionsandisrobusttotheinclusionoftheregressorsusedto purifytheinflationprobabilitiesfromtheoilpriceeffectsinthepreviousfigure. IntheleftpanelofFigure16wepresentthecoefficientsofthecreditspreadforvariousinflation probabilitiesderivedfromfinancialmarket,alongwiththeir95percentconfidenceintervals. The slopesarerescaledsoastofacilitatethecomparisonwiththosecomingfromourquantilePhillips curvemodel(therightpanelreproducesthebottom-rightboxofFigure8).Inparticular,thecoefficientfortheprobabilityofone-year-aheadinflationbeingbelow1%isrescaledtomatchtheslope estimated on the lowest inflation quantile which arises from the quantile Phillips curve model. Further,thecoefficientfortheprobabilityofinflationbeingbelow1%istransformedfrompositive tonegative–asapositivecorrelationbetweenthecreditspreadandthisprobabilityisequivalent toanegativerelationshipbetweenthecreditspreadandthelowestinflationquantile. Despitethe vast disparities in the construction of the tails of the inflation distribution, the estimated slopes are very similar to each other. Finally, in Appendix F we plot the time series of these financial- 34

market-based probabilities togetherwith the credit spread (see Figure F-2). The graphs confirm the progressive weakening in the relationship between the creditspread and inflation probabilitiesasonemovesfurtherupintheinflationdistributionuntilitcompletelybreaksdownoncethe uppertailisreached. Figure16: CreditSpreadCoefficients,QuantileRegressionvs.FinancialMarkets. FinancialMarkets QuantileRegression Note:Theleftpanelreportstheslopesofseparateregressionsofinflationprobabilitiesonthecreditspread(atmonthly frequency),alongwiththeir95%confidenceinterval. Thecoefficientfortheprobabilityoffutureinflationbeingbelow 1%isrescaledtomatchtheslopeestimatedonthelowestinflationquantilewhicharisesfromthequantilePhillipscurve model(rightpanel,takenfromFigure8).Thecoefficientsaretransformedfrompositivetonegativefortheprobability ofinflationbeingbelow1%–asapositivecorrelationbetweenthecreditspreadandthisprobabilityisequivalenttoa negativerelationshipbetweenthecreditspreadandthelowestinflationquantile. 7 Conclusion Inthispaperweshowthattherecentmutedresponseoftheconditionalmeanofinflationtoeconomic conditions does not necessarily convey an adequate representation of inflation dynamics. Includingdatafromthe1970sshowsamplevariabilityintheentireconditionaldistributionofinflation, a result which we confirm when restricting our analysis to the post-2000 stable and low meaninflationperiod. Inparticular,usingtime-seriesdatafortheUnitedStatesandtheeuroarea, wedocumentthat–aftercontrollingforthestateofthelabormarketandinflationexpectations– tightfinancialconditionsgeneratedtimesofsubstantialandpersistentdownsideriskstoinflation, ananalysisforsakenbymuchoftheliterature. Ourpaperthusoffersanewempiricalperspective toexistingmacroeconomicmodelsandtopolicymakers,showingthatchangesincreditconditions arealsokeytounderstandthetail-riskdynamicsofinflation. In future research, we plan to construct a “risk-adjusted” inflation measure which can be derived from a central bank’s preferences not only about deviations of inflation from its target but also about how tolerable (downside and upside) inflation risks are. Finally, we will further study inflation-at-riskbyexploitingtimevariationinsectoralinflationratesandaskwhetherparticular sectorsmakeinflationparticularlyvulnerable. 35

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A Data Appendix InthissectionweprovidedetailsonthedatafortheUnitedStatesandtheeuroarea. A.1 UnitedStates • CoreConsumerPriceIndex – Source: FRED. – Details: CPILFESL PCA, ConsumerPrice IndexforAll Urban Consumers: All Items Less FoodandEnergy,CompoundedAnnualRateofChange,Quarterly,SeasonallyAdjusted. • StockandWatson(2019)CyclicallySensitiveInflation – Source: Replication Material of Stock and Watson (2019) on Professor Mark Watson’s Website. – Details: QuarterlyCSIinflationrates. • CorePersonalConsumptionExpenditures – Source: FRED. – Details:PCEPILFE PCA,PersonalConsumptionExpendituresExcludingFoodandEnergy (Chain-Type Price Index), Compounded Annual Rate of Change, Quarterly, Seasonally Adjusted. • Long-TermInflationExpectations – Source:Blanchard,Cerutti,andSummers(2015)andConsensusEconomics(providedby thePricesandWagessectionoftheResearch&StatisticsDivisionattheFederalReserve Board). – Details:From1990:Q4onwards,weusesix-to-ten-year-aheadmeanCPIinflationforecastsfromsemiannualsurveysfromConsensusEconomics. Beforethatdateweusethe seriesfromBlanchard,Cerutti,andSummers(2015). • UnemploymentRate – Source: FRED. – Details: UNRATE,CivilianUnemploymentRate,Percent,Quarterly,SeasonallyAdjusted. • NaturalRateofUnemployment – Source: FRED. – Details:NROU,NaturalRateofUnemployment(Long-Term),Percent,Quarterly,NotSeasonallyAdjusted. • ImportPriceIndex – Source: FRED. – Details: B021RG3Q086SBEA CCA,Importsofgoodsandservices(chain-typepriceindex), ContinuouslyCompoundedAnnualRateofChange,Quarterly,SeasonallyAdjusted. 39

• OilPrice – Source: FRED. – Details: WTISPLC CCA, Spot Crude Oil Price: West Texas Intermediate (WTI), ContinuouslyCompoundedAnnualRateofChange,Quarterly,NotSeasonallyAdjusted. • GilchristandZakrajsˇek(2012)CreditSpreadandExcessBondPremium – Source: DataregularlyupdatedinFEDSNotebyFavara,Gilchrist,Lewis,andZakrajsˇek (2016). – Details: Credit spread and excess bond premium as constructed by Gilchrist and Zakrajsˇek(2012). • CorporateBondSpread – Source: FRB/USmodelpackageavailableatthisFederalReserveBoardwebsite. – Details: RBBB minus RG10. RBBB, yield on BBB-rated corporate bonds. RG10, yield on 10-yearTreasurysecurity. • NationalFinancialConditionsIndex – Source: FRED. – Details: NFCI, Chicago Fed National Financial Conditions Index, Index, Quarterly, Not SeasonallyAdjusted. • InflationProbabilitiesfromFinancialMarkets – Source: ProvidedbytheMonetaryandFinancialMarketsAnalysisSectionoftheMonetaryAffairsDivisionoftheFederalReserveBoard. – Details: Probabilities are inferred from inflation caps and floors contracts as in Kitsul andWright(2013). • OilPriceSurprises – Source: DownloadedfromProfessorChristianeBaumeister’sWebsite. – Details:ProbabilitiesareinferredfrominflationcapsandfloorscontractsasinBaumeisterandKilian(2016). 40

FigureA-1:InflationMeasures,UnitedStates. 15 Core CPI CSI Core PCE 10 5 0 `61 `65 `69 `73 `77 `81 `85 `89 `93 `97 `01 `05 `09 `13 `17 15 10 5 0 `62 `66 `70 `74 `78 `82 `86 `90 `94 `98 `02 `06 `10 `14 `18 FigureA-2:Regressors,UnitedStates. Core CPI CSI Core PCE Core CPI CSI Core PCE 41

A.2 EuroArea • HarmonizedIndexofConsumerPrices – Source: Statistical Office of the European Communities and HaverAnalytics (provided bytheAdvancedForeignEconomiessectionoftheInternationalFinanceDivisionatthe FederalReserveBoard). – Details: EA19,Totalexcludingenergy,food,alcoholandtobacco. Quarter-over-quarter annualizedgrowthrates,seasonallyadjusted. • Long-TermInflationExpectations – Source:ConsensusEconomics(providedbytheAdvancedForeignEconomiessectionof theInternationalFinanceDivisionattheFederalReserveBoard). – Details: Six- to-ten-year-ahead mean CPI inflation forecasts from semiannual surveys fromConsensusEconomics. FranceandGermany. • UnemploymentRate – Source: TheArea-wideModel(AWM)database. – Details: Unemployment Rate, Percentage of civilian work force, Total (all ages), Total (maleandfemale),Seasonallyadjusted,butnotworkingdayadjusteddata. • NaturalRateofUnemployment – Source: Authors’Calculations. – Details: HP-filteredtrend(withsmoothingparameterλ = 1600)ofunemploymentrate. • ImportPrices – Source: TheArea-wideModel(AWM)database. – Details: ImportsofGoodsandServicesDeflator,Index,Indexbaseyear1995(1995=1). Defined as the ratio of nominal, and real imports of goods and services. Based on the grossconcept,i.e. bothextra-andintra-areatradeflowsareaccountedfor. • OilPrices – Source: TheArea-wideModel(AWM)database. – Details: OilPrices,UnitedKingdom,Petroleum: UKBrent,USdollarsperbarrel. • CreditSpread – Source: DataregularlyupdatedatthisBanquedeFrancewebsite. – Details: euroareabankcreditspreadsfromGilchristandMojon(2018). • OECDRecessionDates – Source: FRED. – Details: EUROREC, OECD based Recession Indicators for euro area from the Period followingthePeakthroughtheTrough,+1or0,Quarterly,NotSeasonallyAdjusted. 42

FigureA-3:InflationMeasures,EuroArea. FigureA-4:Regressors,EuroArea. 43

B Bootstrap Method To compute confidence bands for the quantile regression model we revert to “blocks-of-blocks” bootstrap. WhilemoredetailsonthismethodologycanbefoundinKilianandLu¨tkepohl(2018)(see Chapter12therein),wehereprovideabriefsummaryofthebootstrapprocedure. “Blocks-of-blocks” bootstrap is used in cases where a researcher is interested in computing confidenceintervalsaroundnonsymmetricstatisticsoftheunderlyingdata(e.g.,autocorrelations orestimatorsofautoregressiveslopecoefficientsinatime-seriescontext). Thisisrelevantinour case since not only the quantile regression slopes are non-linear functions of the data but also, wearedefactorunningah-steppredictiveregressionofinflationonits(past)determinants. The “blocks-of-blocks” bootstrap procedure allows to preserve the (time-series) dependency in the data,whichwouldinmostcasesbedestroyedbyanaivebootstrap. More specifically, the “blocks-of-blocks” bootstrap procedure relies on first dividing the dependentvariableyandtheregressorsX intolconsecutiveblocksofallpossiblem-tuples. Ateach bootstrapreplication,blocksofdataarerandomlydrawntoformanewsampleofthesamesizeas theoriginaldata. Importantly,theblocksareresampledinthesameorderforboththedependent variable y and the regressors X, akeystep which preserves the time-dependencyin the data. In ourparticularapplication,werunthequantileregression(2)andstoretheestimatescorrespondingtoeachbootstrapreplication. Fromthedistributionoftheseestimates,68percentconfidence intervalsareconstructedandcenteredaroundthepointestimateobtainedwiththeoriginalsample. Theprocedureisasymptoticallyvalidforstationaryprocessesiftheblocksizelincreasesata √ suitablerateasT → ∞. FollowingBerkowitz,Biegean,andKilian(1999)wesetl = m = 3 T,where T isthesamplesize. Finally,thisbootstrapprocedurepreservesthequantileregressionfeatureof beingagnosticabouttheunderlyingdistributionoftheerrorterms,asthisisnotaresidual-based. Figure B-1 displays the slope coefficients of the quantile regression of average four-quarteraheadUnitedStatesCoreCPIinflationdefinedin(2).Theblacksquarescorrespondtothepointestimateswhereastheverticallinestothe68%confidenceintervalscomputedvia“blocks-of-blocks” bootstrapusing10,000replicationsforthe10thquantile(blue),median(red)and90thquantile(yellow). The estimation period is 1973:Q1 to 2019:Q1. The OLS estimates and their 95% confidence intervalsarerepresentedbytheredlines. 44

FigureB-1: QuantileRegressionSlopesandConfidenceIntervals. UnitedStatesCoreCPI Note:Thefiguredisplaystheslopecoefficientsofthequantileregressionofaveragefour-quarter-aheadUnitedStates Core CPI inflation defined in (2). The black squares correspond to the pointestimates whereas the vertical lines to the68%confidenceintervalscomputedvia“blocks-of-blocks”bootstrapusing10,000replicationsforthe10thquantile (blue),median(red)and90thquantile(yellow).Theestimationperiodis1973:Q1to2019:Q1.TheOLSestimatesandtheir 95%confidenceintervalsarerepresentedbytheredlines. 45

C Inspecting Other Inflation Measures Inthisappendixwereestimatethequantileregression(2)replacingcoreCPIwithtwoalternative measuresofinflation:corePCEandStockandWatson(2019)“CyclicallySensitiveInflation”,CSI.As inthebaselineanalysis,thedependentvariableistheaverageinflationrateovertheperiodtand t+4quartersahead.TheCSIweights17corePCEcomponentsbytheircyclicalcovariationwithreal activity. Morespecifically,theweightsarecomputedsoastomaximizethecorrelationbetweena compositeindexofcyclicalactivity(developedinthesamepaper)andtheyear-over-yearchange in the CyclicallySensitive Inflation index. The CSI is thus meant to provide a real-time measure of cyclical fluctuations in inflation (see Stock and Watson, 2019 fordetails). The estimated slopes ofthequantileregressionsareplottedinFiguresC-1andFigureC-2,respectively. Theslopesare veryrobustfortheeconomicvariables. TheeffectsofthecreditspreadonCSIareconsistentwith thoseobtainedforcoreCPI,butlesssoforcorePCE.Asnotedinthemaintext,thereissubstantial subsample instabilityin the relationship between creditspreads and inflation. This is confirmed by the subsample results shown in Figure C-3 and Figure C-4. These figures confirm how the importanceofriskfactorschangedacrossthetwosubsamplesandmimictheoneinthemaintext inwhichwereporttheestimatedquantile-specificslopesforcoreCPI(i.e.,Figure6). Importantly, once we control for subsample stability, the results are extremely similar across these different inflationmeasures. FigureC-5mirrorsFigure8ofthemaintextbydisplayingtheestimatedslopesofthequantile regressionmodel(2)fortwomeasuresofinflation: corePCE(leftcolumn)andCSI(rightcolumn), alongwiththeirbootstrappedconfidenceintervalsconstructedasdescribedinAppendixB.First, andnotsurprisingly,CSIisclearlymoreresponsivetochangesinunemployment,whilecorePCE is barely sensitive to labor market slack. The last row presents the role of credit spreads across inflation quantiles and inflation measures. The effects are more symmetric in the case of core PCE,whiletheCSImeasureexhibitsasimilarasymmetryascoreCPIalthoughofsomewhatlarger magnitude. FigureC-6confirmstheimportantinfluenceofcreditspreadsonthe10th quantileofthedistributionbothforcorePCEandforCSIinflation. Thisfiguremimicsthetop-rightpanelofFigure 6inthemaintext. FigureC-7,FigureC-8,andFigureC-9displaysimilarexercisestothosepresentedinthemain textforthesetwoalternativemeasuresofinflation,corePCEandCSI,respectively. 46

FigureC-1: QuantileRegressionSlopes,CorePCE. 12 12 = 0.1 = 0.5 10 = 0.9 10 OLS 8 8 6 6 4 4 2 2 = 0.1 0 = 0.5 = 0.9 OLS 0 -2 -2 -1 0 1 2 3 4 5 -60 -40 -20 0 20 40 60 θˆ ={θˆ = −0.12,θˆ = −0.16,θˆ = −0.33} γˆ ={γˆ = 0.02,γˆ = 0.05,γˆ = 0.07} τ 0.1 0.5 0.9 τ 0.1 0.5 0.9 14 12 = 0.1 12 = = 0 0 . . 5 9 10 OLS 10 45o Line 8 8 6 6 4 4 2 = 0.1 2 = 0.5 = 0.9 OLS 0 0 0 2 4 6 8 10 12 2 3 4 5 6 7 8 λˆ ={λˆ = 0.41,λˆ = 0.75,λˆ = 0.96},whereλ iscoefficientonπLTE and(1−λ )onπ∗ τ 0.1 0.5 0.9 τ t τ t−1 12 = 0.1 = 0.5 10 = 0.9 OLS 8 6 4 2 0 0 1 2 3 4 5 6 7 8 δˆ ={δˆ = 0.09,δˆ = 0.09,δˆ = 0.38} τ 0.1 0.5 0.9 Note: The figuredisplays theslopecoefficients ofthequantileregressionof averagefour-quarter-ahead corePCE inflationdefinedinexpression(2). Thelinesillustratetheslopesassociatedwiththemedian(red),the10th (blue)and the90th(yellow)inflationquantile.TheblacklinesaretheOLSestimates.Circlesindicatescatterplotsofaveragefuture inflationagainstagiveninflationdeterminant. Greycirclesindicatescatterplotsofaveragefutureinflationagainsta givenfinancialvariablepriorto1999:Q4whereasblackcirclesindicatethescatterplotfortheperiodstartingin2000:Q1. 47

FigureC-2: QuantileRegressionSlopes,StockandWatson(2019)CSI. 10 10 = 0.1 = 0.5 8 = 0.9 8 OLS 6 6 4 4 2 2 0 0 = 0.1 = 0.5 = 0.9 OLS -2 -2 -2 -1 0 1 2 3 4 5 -60 -40 -20 0 20 40 60 θˆ ={θˆ = −0.38,θˆ = −0.18,θˆ = 0} γˆ ={γˆ = 0.03,γˆ = 0.04,γˆ = 0.06} τ 0.1 0.5 0.9 τ 0.1 0.5 0.9 10 10 8 8 6 6 4 4 = 0.1 = 0.5 2 2 = 0.9 OLS = 0.1 = 0.5 0 = 0.9 0 OLS 45o Line -2 -2 -2 0 2 4 6 8 10 2 3 4 5 6 7 8 λˆ ={λˆ = 0.63,λˆ = 0.25,λˆ = 0.14},whereλ iscoefficientonπLTE and(1−λ )onπ∗ τ 0.1 0.5 0.9 τ t τ t−1 10 = 0.1 = 0.5 8 = 0.9 OLS 6 4 2 0 -2 0 1 2 3 4 5 6 7 8 δˆ ={δˆ = −0.03,δˆ = −0.13,δˆ = −0.13} τ 0.1 0.5 0.9 Note: Thefiguredisplaystheslopecoefficientsofthequantileregressionofaveragefour-quarter-aheadStockand Watson(2019)CyclicallySensitiveInflationdefinedinexpression(2).Thelinesillustratetheslopesassociatedwiththe median (red), the 10th (blue) and the 90th (yellow) inflation quantile. The black lines are the OLS estimates. Circles indicatescatterplotsofaveragefutureinflationagainstagiveninflationdeterminant.Greycirclesindicatescatterplots ofaveragefutureinflationagainstagivenfinancialvariablepriorto1999:Q4whereasblackcirclesindicatethescatterplot fortheperiodstartingin2000:Q1. 48

FigureC-3:QuantileRegressionSlopesAcrossSubsamples,CorePCE. 0 0.1 -0.2 0.08 0.06 -0.4 0.04 -0.6 0.02 -0.8 0 1973-1999 2000-2019 1973-1999 2000-2019 0.8 1 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 1973-1999 2000-2019 1973-1999 2000-2019 0 -0.5 0 -1 -1.5 -0.2 -2 -0.4 1973-1999 2000-2019 Note: The figure displays the estimated slopes of the quantile regression of average four-quarter-ahead core PCE inflationdefinedinexpression(2).Twodifferentsubsamplesareconsidered:(i)1973-1999and(iii)2000-2019.Thebars illustratethecoefficientsassociatedwiththe10thquantile(blue),median(red)and90thquantile(yellow). 49

FigureC-4:QuantileRegressionSlopesAcrossSubsamples,StockandWatson(2019)CSI. 0 0.1 -0.2 0.08 0.06 -0.4 0.04 -0.6 0.02 -0.8 0 1973-1999 2000-2019 1973-1999 2000-2019 1 1 0.8 0.6 0.5 0.4 0.2 0 0 1973-1999 2000-2019 1973-1999 2000-2019 0 -0.5 0 -1 -1.5 -0.2 -2 -0.4 1973-1999 2000-2019 Note: Thefiguredisplaystheestimatedslopesofthequantileregressionofaveragefour-quarter-aheadStockand Watson(2019)CyclicallySensitiveInflationdefinedinexpression(2).Twodifferentsubsamplesareconsidered:(i)1973- 1999and(iii)2000-2019. Thebarsillustratethecoefficientsassociatedwiththe10th quantile(blue),median(red)and 90thquantile(yellow). 50

FigureC-5: QuantileRegressionSlopesandConfidenceIntervals. CorePCE StockandWatson(2019)CSI Note:Thefiguredisplaystheslopecoefficientsofthequantileregressionofaveragefour-quarter-aheadofcorePCE inflation(left)andStockandWatson(2019)CyclicallySensitiveInflation(right)definedin(2).Theblacksquarescorrespondtothepointestimateswhereastheverticallinestothe68%confidenceintervalscomputedvia“blocks-of-blocks” bootstrap(seeAppendixB)using10,000replicationsforthe10thquantile(blue),median(red)and90thquantile(yellow). Theestimationperiodis1999:Q1to2017:Q4. 51

FigureC-6: PartialEffectofCreditSpreadon10th InflationQuantiles. CorePCEandStockandWatson(2019)CSIInflation. 3 2 2 1.5 1 1 0 0.5 -1 `00 `02 `04 `06 `08 `10 `12 `14 `16 `00 `02 `04 `06 `08 `10 `12 `14 `16 CorePCE StockandWatson(2019)CSI Note: Thefiguredisplaysthetimeevolutionofthe10th conditionalinflationquantileofcorePCEinflation(left)and StockandWatson(2019)CyclicallySensitiveInflation(right)estimatedfromthequantileregressionsmodel(2),inits baselineversion(bluestraight)andinitsversionwheretheeffectofcreditspreadsissettozero(blackdash-dotted). ShadedbarsindicateNBER-datedrecessions. 52

FigureC-7: SelectedTimeEpisodesofPredictiveDensities(Left)andSkewness(Right). CorePCEInflation. 0.2 2.5 0.15 2 E C F D P 0.1 P e ro C1.5 0.05 0 1 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3 0.2 2.5 2 0.15 E C F D P 0.1 P e ro 1.5 C 1 0.05 0.5 0 0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2.5 0.15 2 E 0.1 C F D P P e ro C1.5 0.05 0 1 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2.4 0.25 2.2 0.2 0.15 E C 2 F D P 0.1 P e ro C 1.8 0.05 1.6 0 1.4 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Note:Theleftpanelsshowtheestimatedskewedt-Studentdensitiesofaveragefour-quarter-aheadcorePCEinflation foralternativespecificationsofthequantileregressionsmodel(2),initsbaselineversion(bluestraight)andinitsversion wheretheeffectofcreditspreadsissettozero(blackdash-dotted).Therightpanelsshowtheestimatedskewedt-inverse cumulativeassociatedwiththeconditionaldensitiesintheleftpanels. 53

FigureC-8: SelectedTimeEpisodesofPredictiveDensities(Left)andSkewness(Right). StockandWatson(2019)CSIInflation. 3.5 0.5 3 0.4 F D 0.3 IS2.5 P C 0.2 2 0.1 0 1.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 1.5 0.8 1 0.6 F D IS 0.5 P C 0.4 0 0.2 -0.5 0 -1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2.5 0.4 2 0.3 F D P 0.2 IS C 1.5 1 0.1 0 0.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2.8 0.8 2.6 0.6 2.4 F D IS2.2 P0.4 C 2 0.2 1.8 0 1.6 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Note:Theleftpanelsshowtheestimatedskewedt-Studentdensitiesofaveragefour-quarter-aheadStockandWatson (2019)CyclicallySensitiveInflationforalternativespecificationsofthequantileregressionsmodel(2), initsbaseline version(bluestraight)andinitsversionwheretheeffectofcreditspreadsissettozero(blackdash-dotted). Theright panelsshowtheestimatedskewedt-inversecumulativeassociatedwiththeconditionaldensitiesintheleftpanels. 54

FigureC-9: InflationProbabilitiesforAlternativeCutoffValues. CorePCEandStockandWatson(2019)CSIInflation. y y tilib tilib a a b b o o r r P P y y tilib tilib a a b b o o r r P P y y tilib tilib a a b b o o r r P P CorePCE StockandWatson(2019)CSI Note: ThefigureshowsthetimeevolutionofinflationprobabilitiesofcorePCEinflation(left)andStockandWatson (2019)CyclicallySensitiveInflation(right)fordifferentcutoffs. Theseprobabilitiesarecomputedfromtheskewedt- Studentconditionaldensitiesoftheaveragefour-quarter-aheadinflationmeasureswhichwerefittedontheestimated conditionalquantilesforalternativespecificationsofthequantileregressionmodel(2).Bothpanelsarereportedforthe specificationwithoutandwiththecreditspread(inbluestraightandblackdash-dottedlines,respectively).Shadedbars indicateNBER-datedrecessions. 55

D Robustness InthisAppendixwepresentadditionalrobustnessresultsfortheU.S.economy. ThedataaredescribedinAppendixA.First,weshowthatconditioningonenergypricesinsteadofimportedgoods yieldsverysimilarresults(seeFigureD-1).FigureD-2displayshowchangesinthefinancialvariable affecttheestimatedslopesinthebaselinequantileregressionmodel–forcoreCPIinflation–over the full sample period. We considerthree alternative financial variables: corporate bond spreads (top-leftpanel), excessbondpremiumasconstructedbyGilchristandZakrajsˇek(2012)(top-right panel),andthenationalfinancialconditionsindex(bottom-centerpanel). Theresultsarestriking. Thelowertailofthedistributionofinflationishighlynegativelyresponsivetochangesinfinancial conditions,buttheuppertailofdistributionisnot. Asnoted,thesubstantialsubsampleinstability isresponsibleforfuzzingtheeffectsontheuppertail. Thisisconfirmedbythesubsamplestability resultsshowninFigureD-3,FigureD-4,andFigureD-5. D.1 OilvsImport FigureD-1: QuantileRegressionsSlopesAcrossRelativePriceMeasures. 14 14 = 0.1 12 = 0.5 12 = 0.9 OLS 10 10 8 8 6 6 4 4 2 = 0.1 0 = 0.5 2 = 0.9 OLS -2 0 -60 -40 -20 0 20 40 60 -300 -200 -100 0 100 200 300 400 SlopeγI on(πI −π ) SlopeγO on(πO −π ) τ t t τ t t Note: Thefiguredisplaystheslopecoefficientsonrelativepricesofthequantileregressionofaveragefour-quarteraheadcoreCPIinflationdefinedinexpression(2). Thelinesillustratetheslopesassociatedwiththemedian(red),the 10th (blue)andthe90th (yellow)inflationquantile. TheblacklinesaretheOLSestimates. Circlesindicatescatterplots ofaveragefutureinflationagainstagiveninflationdeterminant. Theleftpanelcorrespondstothemodelwherethe relativepricemeasureisrelativeimportpriceinflation,whereastherightpanelconsidersthemodelwheretherelative pricemeasureisrelativeoilpriceinflation.Greycirclesindicatescatterplotsofaveragefutureinflationagainstagiven financialvariablepriorto1999:Q4whereasblackcirclesindicatethescatterplotfortheperiodstartingin2000:Q1. 56

D.2 DifferentFinancialVariables FigureD-2: QuantileRegressionsSlopesAcrossFinancialVariables. 14 14 = 0.1 = 0.1 12 = = 0 0 . . 5 9 12 = = 0 0 . . 5 9 OLS OLS 10 10 8 8 6 6 4 4 2 2 0 0 0 1 2 3 4 5 6 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Slopesδ oncbs Slopesδ onebp τ t τ t 14 = 0.1 = 0.5 12 = 0.9 OLS 10 8 6 4 2 0 -1 0 1 2 3 4 5 Slopesδ onnfci τ t Note:Thefiguredisplaystheestimatedcoefficientsofthequantileregressionofaveragefour-quarter-aheadcoreCPI inflationdefinedin(2),usingcorporatebondspreads(top,left),theGilchristandZakrajsˇek(2012)excessbondpremium (top,right)andtheNationalFinancialConditionsIndex(center,bottom)andthesamesampleperiodasthebaseline.The linesillustratetheslopesassociatedwiththemedian(red),the10th (blue)andthe90th (yellow)inflationquantile. The blacklinesaretheOLSestimates.Greycirclesindicatescatterplotsofaveragefutureinflationagainstagivenfinancial variablepriorto1999:Q4whereasblackcirclesindicatethescatterplotfortheperiodstartingin2000:Q1. 57

FigureD-3: QuantileRegressionSlopesAcrossSubsamples,CorporateBondSpreads 0 0.1 -0.2 0.08 0.06 -0.4 0.04 -0.6 0.02 -0.8 0 1973-1999 2000-2019 1973-1999 2000-2019 0.8 1 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 1973-1999 2000-2019 1973-1999 2000-2019 0 -0.5 0 -1 -1.5 -0.2 -2 -0.4 1973-1999 2000-2019 Note: The figure displays the estimated slopes of the quantile regression of average four-quarter-ahead core CPI inflation defined in expression (2), using corporate bond spreads and the same sample period as the baseline. Two differentsubsamplesareconsidered: (i)1973-1999and(iii)2000-2019. Thebarsillustratethecoefficientsassociated withthe10thquantile(blue),median(red)and90thquantile(yellow). 58

FigureD-4: QuantileRegressionSlopesAcrossSubsamples,ExcessBondPremium. 0 0.1 -0.2 0.08 0.06 -0.4 0.04 -0.6 0.02 -0.8 0 1973-1999 2000-2019 1973-1999 2000-2019 0.8 1 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 1973-1999 2000-2019 1973-1999 2000-2019 0 -0.5 0 -1 -1.5 -0.2 -2 -0.4 1973-1999 2000-2019 Note: The figure displays the estimated slopes of the quantile regression of average four-quarter-ahead core CPI inflationdefinedinexpression(2),usingtheGilchristandZakrajsˇek(2012)excessbondpremiumandthesamesample periodasthebaseline. Twodifferentsubsamplesareconsidered: (i)1973-1999and(iii)2000-2019. Thebarsillustrate thecoefficientsassociatedwiththe10thquantile(blue),median(red)and90thquantile(yellow). 59

FigureD-5: QuantileRegressionSlopesAcrossSubsamples,NFCI. 0 0.1 -0.2 0.08 0.06 -0.4 0.04 -0.6 0.02 -0.8 0 1973-1999 2000-2019 1973-1999 2000-2019 0.8 1 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 1973-1999 2000-2019 1973-1999 2000-2019 0 -0.5 0 -1 -1.5 -0.2 -2 -0.4 1973-1999 2000-2019 Note: The figure displays the estimated slopes of the quantile regression of average four-quarter-ahead core CPI inflationdefinedinexpression(2),usingtheNFCIandthesamesampleperiodasthebaseline.Twodifferentsubsamples areconsidered: (i)1973-1999and(iii)2000-2019. Thebarsillustratethecoefficientsassociatedwiththe10th quantile (blue),median(red)and90thquantile(yellow). 60

E The Role of Skewness FigureE-1belowillustratestheeffectofachangeinaninflationdeterminantontheprobabilityof averageone-year-aheadinflationfallingbelow1%(downsideinflation-at-risk).Theinitial(normal) densityisillustratedinthetoppanel.Inthisthoughtexperiment,thechangeineconomic/financial conditions induces achange in the mean (centerpanel) and then also in the skewness of the distribution (bottom panel). Itis evidenthowthe effecton downside inflation-at-riskis amplified if thechangeintheinflationdeterminantincreasestheright-skewnessofthedistribution. FigureE-1: InflationProbabilitiesandTheRoleofSkewness. InitialDistribution y tisn e D y tilib a b o rP PerturbedDistribution:ChangeinMean y tisn e D y tilib a b o rP PerturbedDistribution:ChangeinMeanandSkewness y tisn e D y tilib a b o rP Note:Thefiguredisplaysthethreestatesassociatedwithachangeinaninflationdeterminantthatcausestheinitial normaldensity(toppanel)tofeaturealowermean(centerpanel)andthenalsoaright-skew(bottompanel). 61

F Financial Market Probabilities FigureF-1: FinancialMarkets’InflationProbabilitiesvs.Oil,EnergyandFoodPriceMeasures. y tilib a b o r P y tilib a b o r P y tilib a b o r P Note: Thefigureshowsthemonthlyoptions-impliedinflationprobabilitiesofheadlineCPIinflationnextyearbeing above1%alongwiththe3-months-and6-months-aheadoilpricesurprisescomputedusingtheoilmarketpriceexpectationsthatBaumeisterandKilian(2016)recoveredfromoilfuturesprices(toppanel),thenegativeofenergyprice inflation(midpanel)andthenegativeoffoodpriceinflation(bottompanel). 62

FigureF-2: InflationProbabilitiesfromFinancialMarketsvs.CreditSpread. y tilib a b o r P y tilib a b o r P y tilib a b o r P Note: Thefiguresshowsthecreditspreadagainstquarterlyoptions-impliedinflationprobabilitiesofheadlineCPI inflationnextyearbeingabove1%(toppanel),between2%and3%(midpanel)andabove4%(toppanel). 63