The FOMC's Use of Operational Targets: 85 Years and Counting

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) The FOMC’s Use of Operational Targets: 85 Years and Counting Jeffrey Huther, Elizabeth Klee, Kevin Kiernan, and Ethan Rodriguez-Shah 2023-039 Please cite this paper as: Huther, Jeffrey, Elizabeth Klee, Kevin Kiernan, and Ethan Rodriguez-Shah (2023). “The FOMC’s Use of Operational Targets: 85 Years and Counting,” Finance and Economics DiscussionSeries2023-039. Washington: BoardofGovernorsoftheFederalReserveSystem, https://doi.org/10.17016/FEDS.2023.039. 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.

The FOMC’s Use of Operational Targets: 85 Years and Counting JeffreyHuther∗ ElizabethKlee† KevinKiernan‡ EthanRodriguez-Shah§ May14,2023 Abstract This paper uses summaries of the Federal Open Market Committee’s (FOMC’s) meetings toidentifyitsoperationaltargetsandmapthosetooperatingregimes. Wefindthatoperational targetsweremoreoftendiscussedintheearlierpartoftheFOMC’s85-yearhistory,butrecent years have seen a resurgence in discussions. We identify distinct operating regimes and find that regimes with discussions of multiple targets, usually rate and quantity pairs, are more commonthanregimesdominatedbydiscussionsofsingletargets. Wedocumentthatthecurrentperiod(the2007-2009financialcrisistotoday)isanotablebreakinoperationaltargetsfrom earlierperiods. Wealsoshowthatshiftsinoperationaltargetsoccurduringrecoveries,orafter asignificantdownturninthemacroeconomy. Acknowledgments: WewouldliketothankWeiZhengandDaveXiaforthedevelopmentofthedatabaseonwhich ouranalysiswasbased,andtoMarkCarlson,RuthJudson,MichaelKileyandDaveSmall(inmemoriam)forhelpful commentsandsuggestions.Theviewsexpressedherearestrictlythoseoftheauthorsanddonotnecessarilyrepresent thepositionoftheFederalReserveBoard,theFederalReserveSystem,orFannieMae. ∗FederalReserveBoard †FederalReserveBoard ‡FannieMae §GeorgetownUniversity 1

1 Introduction Theconsensusaroundthefederalfundsrateasan—and, formanyyears, the—operationaltarget of the Federal Open Market Committee (FOMC) has been in place long enough that it is largely takenforgranted. Ithasnotalwaysbeenso. TheFOMC’soperationaltarget—thefinancialrateor quantitythattheFOMCtargetstoachieveitsultimateeconomicobjectives—hasevolvedovertime andhasincludedvariablessuchasthelevelofreservesinthebankingsystem,yieldsonTreasury securities,monetaryaggregates,unsecuredinterbanklendingrates,and,currently,amixofmarket rates, administered rates and asset purchase targets.1 Our goal is to provide a readily accessible perspectiveontheFOMC’suseofthesetargetsovertime,toestablishdifferentoperatingregimes, orperiodsinwhichaparticularoperationaltargetortargetswereused,andtoidentifyfactorsthat influenceshiftsinoperatingregimes. We characterize the FOMC’s operational targets and regimes using techniques derived from the literature on natural language processing (NLP). We apply these techniques to the meeting minutesof85yearsofFOMCdeliberations.2 Wehavethreemainfindings. First,weseediscussions of operational targets peak following three events: the separation of monetary from fiscal policy withtheadventoftheTreasury-FedAccordin1951,theaggressiveeffortstoreduceinflationthat began in late 1979, and the global financial crisis. Second, operational regimes identified in the historicalliteraturearereflectedintherelativefrequencyofassociatedoperationaltargetlanguage. Ouruseofselected“bigrams”—two-wordtermsdescribingoperationaltargets—lineupwithwellknownperiodsandshiftsofoperationalregimes,suggestingthatanalysisofwordusageinFOMC documents can be helpful in identifying policy trends. And third, the FOMC often used more than one operational target simultaneously. We find that, quite often, quantity and rate targets werediscussedsimultaneously. MembersoftheFOMChavelongwrestledwiththequestionofhowoperationaltargetsshould be used to achieve the Committee’s economic objectives (for a few select examples of views over time,seeBox1). Priortotheintroductionofverbatimtranscriptsin1976,debatesanduncertainty aboutappropriateoperationaltoolsandtheirusagecanbegleanedfrommeetingminutes,ex-post reports and syntheses of other contemporary records. Garbade (2021), for example, uses a wide 1SeeCapetal(2020)foradefinitionofoperationaltargetsanddescriptionsofhowtargetsfitintomonetarypolicy implementationframeworks. OurapproachdifferssomewhatfromBindseilandFotia(2021)whoincludemonetary aggregatesintheirclassificationofultimate,ratherthanoperational,targets. 2TitleIIoftheBankingActof1935establishedtheFederalOpenMarketCommitteewhichfirstconvenedinMarch 1936.Ourdatasetendsin2021. 2

range of sources to describe the evolution of FOMC policies from 1951 – 1979. In this paper, we turn to the FOMC Minutes and similar summary documents to construct a continuous record of discussions on operational targets since the inauguration of the modern Committee structure in 1936. Athoroughreadingofthesedocumentswouldbechallenginggiventhenumberandlength of the deliberation summaries, so we explore the use of contemporary computational techniques toidentifyunderlyingtrends. ThelargeamountofinformationontheCommittee’sdeliberationsovertimeisbothablessing and a curse for researchers. Analyses of word usage reduce the direness of the curse by providingarelativelyquicklookatthelanguageCommitteememberswereusingastheyshapedpolicy, allowing for easy identification of turning points in policy discussions. We use this approach to complement broad descriptions of the use of operational targets (e.g. Bindseil 2014) in the academicliteratureonmonetarypolicyimplementationandtheinfluenceofacademicworkonpolicy discussions, notablytheevolutionofmonetaryaggregatesfollowingthepublicationofFriedman and Schwartz (1963). In some cases, our approach suggests the need for additional examination intobroad-strokecharacterizationsofpolicyimplementation. 2 Preliminaries 2.1 NaturallanguageprocessingandtheFOMCminutes Weconstructacontinuoussetofdocumentsbyconcatenatingstaff-authoredsummariesofthedeliberationsatFOMCmeetingsthathavebeenapprovedbyCommitteemembersasofficialrecords.3 Despite differences in naming conventions over time, the summaries are similar in content and structure to the modern FOMC Minutes. The resulting database runs from 1936 to 2021. It comprises989textdocuments,withanaverageofnearly9,600wordsperdocument. Our approach applies techniques from the NLP literature.4 We begin by removing common English stop words and proper nouns such as the names of FOMC participants.5 From there, we constructwordandphrasecountsacrossalldocuments,focusingspecificallyontermsthatdescribe 3We use Historical Minutes (March 1936 – May 1967), Memoranda of Discussion (June 1967 – March 16, 1976), Records of Policy Actions (March 29, 1976 – 1992) and Minutes (1993 – 2021). All documents are available at https://www.federalreserve.gov/monetarypolicy/fomc_historical_year.htm. The transition from Memoranda of DiscussionandRecordsofPolicyActionstoMinutesisdiscussedinLindsey(2003).AlsoseeDankerandLuecke(2005). 4Theeconomicliteratureusingnaturallanguageprocessingisexpandingrapidly. Gentzkowetal(2019)provides anoverview.CieslakandVissing-Jorgensen(2021)andothershaveappliedthesetechniquestotheFOMCminutes. 5SeeBird, S., E.Klein, andE.Loper(2009). NaturalLanguageProcessingwithPython: AnalyzingTextwiththe NaturalLanguageToolkit.O’ReillyMedia,Inc. 3

operationaltargets. Our initial exploration of the word and phrase counts leads us to focus on two-word phrases (“bigrams”)thatdescribeoperationaltargets. Individualwordsover-identifyoperationaltargets while trigrams and quadgrams appear overly restrictive. The most common word pairs include terms that we identified as policy implementation phrases while, for example, trigrams have low countsforcommonoperationaltargetssuchasthefundsrateandmoneysupply. After determining that bigrams best capture references to operational targets, we review the most frequent 5,000 bigrams for all variations on operational target word usage. The number of relevantbigramsineachdocumentaveragesalittleoveronehundred. However,thenumbervaries considerably, with a maximum of 448 in June 1973.6 Out of the 5,000 terms in our sample, we identify135termsrelatedtooperationaltargets. Thesebigramsincludecontemporaryterms(e.g. “fundsrate”),termsrelatingtohistoricaltargets(e.g. “freereserves”and“monetaryaggregates”), andtermsthatpopulatethecompletelist(e.g. “makepurchases”). We group the operational target bigrams into categories of familiar operational approaches. We identify nine categories: four are quantity-focused, four are rate-based, and the last has elements of both. The quantity-focused targets are associated with Treasury, Federal Reserve, or bankliabilities. Reservemanagementincludesdiscussionsaboutthestockofreservesanditscomponents. Measurement of reserves evolved with the Fed’s understanding of bank reserves, commercial banks’ capacity to efficiently allocate reserves, and other factors that affected the stock of reserves (autonomous factors and discount window usage ). Asset management encompasses phrases related to both the duration and quantity of Treasury debt held as assets on the balance sheet of the Fed, including the bills preferably period that encompasses most of the 1950s (see Friedman and Schwartz 1963 and Garbade 2021). Money supply includes terms related to measures of the stock of money and to the broad description of supply (“monetary aggregates”). Finally, reserve requirements, while identified as a distinct category, combine conceptual elements ofthereservemanagementandthemoneysupplycategories,andlinkbankreservestothemoney supply. Therate-focusedtargetsareratesonTreasuryorbankliabilities,determinedbythemarketor administeredbytheFederalReserve. Thediscountrateistherateatwhichbanksborrowdirectly from the Federal Reserve. Bill rates are relevant in the late 1950s and the 1960s when, under the 6Intheearlierpartofoursample,onsomeoccasionsseparateMinutesweredraftedfordifferentpartsofthemeeting. Thismayinpartexplaintheinfrequentmentionofourbigramsinsomeoftheearlydocuments. 4

Bretton Woods mechanism, differences in short-dated government rates created significant arbitrage opportunities. The federal funds rate is the rate at which banks and selected institutions borrowandlendreservebalancesinthefederalfundsmarket. YieldcurvecontrolwasanimportantcomponentofFedpolicyduringandimmediatelyafterWorldWarII.7 Finally, external account, as an operational target, was often discussed in the context of currencyswapsbut,alsoincludesdiscussionsofgoldflowsand,insomecases,movementsofspecific exchange rate pairs. These terms are especially common in discussions leading up to the dissolution of the Bretton Woods agreements as well as various foreign financial crises. The bigrams in each category are listed in table 2. For example, “M2 growth” and “monetary aggregates” are both included in the “Money Supply” category. Of note, there appears to be more heterogeneity for terms describing quantity targets than terms for rate target phrases. Part of this observation likelyreflectslongperiodsinwhichpolicyratesvariedlittle,leadingtolittlediscussionofanyoperationaltargets. Similarly,wedonotuncoveratop5,000bigramassociatedwithmonetarypolicy easing,perhapsreflectingthetimingasymmetryinwhichtheFederalReserveeasesmonetarypolicy—oftenquickly,asaresultofashock—versustightensmonetarypolicy—oftenwell-telegraphed andgradually. Another reason for the prominence of quantity terms may have been the relative difficulty of measuring and operating with quantity targets. For example, targeting the aggregate quantity of reserves was complicated by a disaggregated banking system that did not always reallocate reservesefficiently. ImplementationofCommitteedirectivesrequiredoperationsmanagerstoestimate the reserves that would flow from generally reserve-rich smaller banks, located outside of the major markets, to larger, money center banks. In addition, targeting reserves required estimating discount window borrowing, as borrowing also increases reserves. Perhaps reflecting difficulties with quantity targeting, terms describing monetary aggregates also expanded as researcherssoughttocaptureincreasinglycomprehensivemeasures.8 Inidentifyingrelevantterms, wefacetwoproblemsthatapplytoNLPgenerally,vaguenessandambiguity. Totacklevagueness, weexcludedtermsthatdidnotclearlyidentifyanoperationaltoolorcouldbeinterpretedasunrelated to implementation of a specific tool. While the FOMC documents have been written, and 7TheseregimesdonotincludereferencestotheFed’s“evenkeel”policyinthe1950sand1960sinwhichtheFed avoidedchangingitsoperatingstanceduringTreasuryfinancingoperations. Whilethispolicyinfluencedtheuseof operationaltargets,wedidnotviewitasastand-alonetarget. 8Atkinson(1969)writesthat“Theterm‘feelofthemarket’(FOTM)isperhapsthemostcommonlyusedphrasefor describingaconditionofmarkets”suggestinganerawithoutoperationaltargets. Hisviewsmayhaveoverweighted thetimeinwhichhewaswriting–“tone”and“feel”termsarenotwordsthatappearasacomponentofanyofthetop 3,000bigrams. 5

heavilyreviewed,toavoidunintentionalvagueness,theauthorsand,sometimesFOMCmembers themselves, have intentionally avoided precise use of terms that over or under state policy. To tackleambiguity,wedidnotincludetermssuchas“interestrates”becausetheyhavebeenusedin many contexts – interest rates, for example, may be shorthand for financial conditions or market interpretation of the Committee’s future policy actions as well as a reference to the Committee’s operationaltargets. 2.2 Operationaltargetsovertime Figure1displaystherawcountsofouridentifiedoperationaltargetbigramsovertime. Thenumber of bigrams trends upward, from a low of less than 100 counts per meeting in the FOMC’s early years, to a peak of over 400 counts in the mid-1970s. There is pronounced volatility as well, with a notable drop off in the mid-1970s. This drop off likely reflects in part the release of the FOMC meeting transcripts to the public, an event that was associated with a reduction in the length of theMinutesdocumentsaswellasthefrequencyofpublication. Afteraquieterperiodthroughthe mid-2000s,therewasasubsequentriseduringandfollowingthefinancialcrisisof2007-2009. The seasonalityapparentfromthemid-1990stothepresentreflectsorganizationalmattersreviewedat thebeginningoftheyear;thosesummariesareincludedintheminutesfromtheJanuarymeetings. Figure1: Totaloperationaltargetbigramspermeeting While the number of bigrams is instructive, the length of the FOMC documents has changed 6

over time. The word count in the historical records grew slowly through the 1960s, peaked in the 1970s, gradually declined up to the financial crisis, and then rose to present. The pattern reflectsfactorsincludingmeetingfrequency,forwhichthepacehasgenerallyslowed,pressuresfor increased transparency (notably in the late 1960s and early 1970s), and the introduction of transcripts (which may have changed the writers’ perspective on the role of their documentation) as wellastheintensityofdiscussiononoperationaltargets. Theirregularwordcountsandrelevant bigrams led us to use bigrams as a share of total words in a document (excluding stop words), displayed in figure 2. Over time, the share of words in a document that appear in our dictionary ofbigramsrangesfromlessthan2percenttoover6percent. Theuseofrelative,ratherthanabsolute,intensityofoperationaltargetbigramsgivesusabetter picture of trends in operational targets over time. In particular, figure 2 shows a rise in relevant terms following the accord with Treasury in 1951, the increasing discussions of operational tools inthe1970saseffortstoworkwithmonetaryaggregatetargetsproveddifficulttoimplement,the growingcomfortwithpolicytoolsinthe1980sastheFOMCslowlycoalescedaroundthefederal fundsrateasthesoleoperationaltarget,theauto-pilotapproachtorateincreasesinthemid-2000s, andtheincreasingdiscussionofpolicytoolsfollowingthefinancialcrisisactionsbeginningin2008. Figure2: Bigramsasashareoftotalwords In line with our conclusion that relative frequencies are a better indicator of policy delibera- 7

tionsthanabsolutefrequencies,weuseastandardtechniqueintheNLPliteraturetotoweightthe appearance of a bigram in a particular instance of the Minutes relative to overall bigram counts. Moreprecisely,weuseaterm-frequency-inversedocumentfrequency(TF-IDF)measure. Itiscalculatedastheproductoftwoexpressions. ThefirstexpressionisthetermfrequencyTF(b,m),or therelativefrequencyofbigrambinMinutesm: f TF(b,m) = b,m . (1) (cid:80) f b′∈m b′,m Thesecondexpressionistheinversedocumentfrequency,ortheinverseofthelogofthefraction oftotalMinutes(M)thatcontainthebigramb: M IDF(b,m) = log . (2) {m ∈ M : b ∈ m} Note that if the bigram is not in a particular Minutes, TF(b,m) = 0, or if the bigram is in all the Minutes, IDF(b,m) = 0. Inwhatfollows, ourNLPanalyticsrelyonthismeasure. However, our conclusionsarerobusttoothertechniques. 2.3 Rateandquantitypairs WithourTF-IDFmeasuresinhand,wecanprovideevidenceforoneofthekeyfindingsofthispaper,whichisthattheCommitteeoftenusedmorethanoneoperationaltargetsimultaneously,and morespecifically,wefindthattheCommitteeoftendiscussedoperationaltargetsinrate-quantity pairs. Weexaminethedynamicsofthreeofthesepairsandshowthatthepairscanbesubstitutes, complementsorbothovertime. Westartwithreservemanagementandthediscountwindowborrowingrate. Asshowninfigure3,reservemanagementanddiscountratetermswereamongthemostwidelyusedoperational targets from the early 1950s to around 1970. Despite high coincidence of these terms, usage was occasionallynegativelycorrelated, suggestingthattheCommitteeviewedthetoolsassubstitutes ratherthancomplementsattimes. Ofcourse,thesetermsarealsonegativelycorrelatedinconcept, asthepriceofborrowingreservesinthistimeperiodwasthediscountrate,andifthepricewent up,thequantityshouldgodown. Themainpointisthatneitherpolicytoolshouldbeconsidered inisolationgiventhehighfrequencyofpairedusage. 8

Figure3: TF-IDFsforreservemanagementandthediscountrate Forthenextexample,wepairTreasurybillratesandtheexternalaccount. TheBrettonWoods systemcreatedpotentialarbitrageopportunitiesbetweenshortdatedsovereigndebtinstruments. In the immediate aftermath of World War II, financial market participants could not take meaningful advantage of interest rate differentials given the war-orientation of financial For the next example,wepairTreasurybillratesandtheexternalaccount. TheBrettonWoodssystemcreated potential arbitrage opportunities between short dated sovereign debt instruments. In the immediate aftermath of World War II, financial market participants could not take meaningful advantageofinterestratedifferentialsgiventhewar-orientationoffinancialinstitutionsbut,bythelate 1950s, gold outflows needed to facilitate the arbitrage had grown enough to warrant Committee discussion. Specifically,potentialarbitrageprofitsrosesharplyinearly1958asTreasurybillrates declinedfrom2.75percentattheendofDecember1957to1.55percentattheendofJanuary1958. Ofnote,FriedmanandSchwartz(1963)identifyFebruary1958asthebeginningofgoldoutflows withsterilizationthroughopenmarketoperationsbeginninginMarch1958. Figure 4 shows a strong positive correlation and complementarity in the Committee’s use of terms related to Treasury bills and foreign exchange in the 1960s. The correlation broke down following the last sputterings of Bretton Woods in 1973. Committee discussion of Treasury bills dropped off but the use of external account terms grew, peaking with Japan’s asset price bubble 9

in the late 1980s. The discussion of both terms picked up in the years including and following the2007-2009financialcrisis,reflectinginparttheuseofassetpurchasesoverthisperiodandthe globalimpactofthecrisis. Figure4: TF-IDFsforbillratesandtheexternalaccount Ourfinalexamplefocusesonthefederalfundsrateandmonetaryaggregates. Asdisplayedin figure5,whilemoneysupplytermsappearedintheearlyyearsoftheCommittee,theuseofmoney supply and fedfunds together was lowuntil the late 1960swhen mentions of moneysupply and the fed funds rate rose simultaneously. Both regimes rose in the late 1960s as the Bretton Woods constraintsbrokedownandtheFeddevelopedandmonitoredmonetaryaggregates. Ourmetrics showacomplementaryrelationshipbetweenthefederalfundsrateandmonetaryaggregatesthat persistedintothemid-1990s,althoughtheuseoffedfundstermswassomewhatmorevolatilethan moneysupplytermsoverthisperiod. Thecountsofmonetaryaggregatetermspersistedwellafter thefederalfundsratebecametheprimaryindicatoroftheFOMC’scontemporarystanceonmonetarypolicy: onlyinthe2000sdidtheuseofmonetaryaggregatesfallnotablyinprominence. Since 2000,thefedfundsratehasunambiguouslydominatedtheCommittee’sdiscussionsofoperational tools. Figure5: TF-IDFsforthefundsrateandthemoneysupply 10

Why were there so many operational targets in use at any point in time? One conjecture is that there may be potential weaknesses in relying on a single policy instrument. Bindseil (2014) suggests that the use of various quantity targets (he identifies seven between 1920 and 1983) absolved the Fed from responsibility for short-term interest rates. A staff briefing to the FOMC in 1995 reviewed the use of rate targets and pointed to a weakness in fed funds targeting.9 It noted thatdiscrete,transparentchangesinthefedfundstargetcouldbeasourceofinertiaintheFOMC’s policystancebecausethesediscrete,transparentchangeswouldrequireahighburdenofproofof changesineconomicactivity. Incontrast,continuous,opaquetargetssuchasmonetaryaggregates had allowed for small changes in policy stance. An underlying point of the discussion was that theFOMCoperatesinanenvironmentthatisinherentlyuncertainandthatreducingdecisionsto discrete rate targets forces FOMCs to provide unrealistically certain views of the direction of the economy. Asisevidentfromthesethreepairs,untilthefedfundsratecametodominateoperationaltarget discussions in the 1990s, the FOMC generally discussed targets in terms of rates and quantities. The move to a single, rate-based target may have been based on the lack of a strong correlation between monetary aggregates and economic activity, perhaps in part because of the growth of intermediation outside of the banking sector. The rise in discussions of quantity-based targets followingthefinancialcrisiswaslargelyaconsequenceofinterestratesfallingtotheeffectivelower boundratheraneffortbytheFOMCtosimultaneouslymanageratesandquantities. Thepost-crisis managementofquantitieshaslargelybeendiscussedintermsoftheFed’sassetsincontrasttothe earlier focus on liabilities. This change in quantity focus from liabilities to assets may reflect the evolvingnatureoffinancialmarkets–Fedliabilitiesareconcentratedonthebalancesheetsofbanks andasmallgroupofrepomarketcounterpartieswhileassetsheldbytheFederalReservemaybe thesameasthoseonanyintermediarybalancesheet. 3 Regimes In this section, we discuss FOMC operating regimes. We use the term “regime” to highlight the dominanceofaspecificcategoryorcategoriesofoperationaltargetsthatarethefocusoftheFOMC. WefirstidentifyregimesusingNLPtechniques,focusingonrankingTF-IDFs,clusteranalysis,and distancecalculations. Wethencomparetheseregimestohistoricalaccounts. 9AppendixtotheMarch1995FOMCmeetingwhich,intheconcludingparagraph,assertsthat“theappropriatelevel ofinterestratesisimpossibletopinpoint...” 11

3.1 IdentifyingregimesusingNLP Wetakethreeapproachestoidentifyingregimes. Wediscusseachinturn. 3.1.1 Rankingbigrams For our first analysis, we identify regimes by ranking bigrams. We perform this analysis in five steps. First, we divide our corpus by year. Second, we calculate each bigram’s TF-IDF measure using the documents from that year. Third, we rank the TF-IDFs from highest to lowest. Fourth, we map bigrams to their categories. And finally, we define regimes as periods when the same categoryisrankedfirstorsecondforatleasttwoyearsinarow. Ourdefinitionsleadtotheregimeidentificationdisplayedinfigure6.10 Figure6: Operatingregimes—TF-IDFranking Broadlyspeaking,operationaltargetregimeschangedmorefrequentlybefore1960thanafter- 10AcomprehensivelookattheFOMC’soperationaltargethistoryaswellasthatofothercentralbanksisavailablein Bindseil(2014). 12

wards. SomeoftheearlyvariabilitycanbeattributedtothedifficultiesthattheCommitteefacedin dealingwiththeGreatDepressionandWWIIfinancing,butitisalsolikelytoreflectanaturallearningprocess foracommitteein itspre-adolescentyears. Later stabilitymayalsobe attributableto morerelianceontheoreticalapproachesforpolicyimplementation. TheCommittee’searlyadoption of “bills preferably” for the asset side of the Fed balance sheet still left the Committee with questionsofimplementation. WhentheBrettonWoodsagreementbecameabindingconstraintin thelate1950sandearly1960s,necessitymayhaveprovidedoperationalclaritythathadpreviously beenabsent. Thedevelopmentofmonetaristeconomicsinthelate1960sfocusedimplementation discussionseventhoughtherelationshipbetweenmonetaryaggregatesandeconomicactivityultimatelyturnedouttobeunstable. Fromthe1970sand1980stothepresent, thepushforgreater transparencyoftheFOMCmayhavestrengthenedtheneedforconsistent,readilyidentifiableoperationaltargets. 3.1.2 Clusteranalysis For our second analysis, we rely on clustering techniques. We perform the clustering analysis on the same meeting-frequency TF-IDF vectors for each of our nine categories that we used for our ranking procedure. We then use the k-means algorithm to partition meetings into k groups to minimize the variance in each group. We choose the number of groups heuristically based on a marginalimprovementintheminimization. Forourvectors,wefindsixgroupsperformwell. Resultsfromtheclusteringanalysisaredisplayedinfigure7. Ourchartusesthefirsttwoprincipalcomponentsofourninecategoriesofbigramsastheaxes. Acautionarynoteaboutthisdiagram isthattheseprincipalcomponentsexplainlessthanhalfthevariationinthebigramsovertime. As a rough generalization, a positively sloped vector is closed associated with bills preferably, T-bill ratetargetingandfedfundsregimeswhileanegativelyslopedvectorisassociatedwiththemonetary aggregates and asset purchase regimes. In terms of timing within our 85 year sample, the sequenceisred,blue,aqua,purple,green,andfinally,yellow. On these dimensions, one could also surmise that there are primarily three clusters: Pre-M2, M2,andnow. Butbecausethefigurepresentsinformationonlyforthefirsttwoprincipalcomponents,wepositthattheremaybeimportantvariationcapturedbyaddingmoreclusters. 13

Figure7: Clusteringanalysis 3.1.3 Distanceanalysis Our final NLP exercise for identifying regimes relies on distance metrics at a meeting frequency. Boththex-axisandy-axisoffigure8plot989meetingdates;welabelonlytheyearforclarity. The shadingindicatestheEuclideandistancebetweenthevectorofTF-IDFsfordifferentmeetingdates. Lightshadingindicatesshorterdistances;darklongerones. Aswewouldexpect,distancesalong the45degreelinearelightest,asthedistanceofameetingfromitselfiszero. We highlight three takeaways from this exercise. First, the distances between time-proximate meetingsearlierinoursampletendtobegreater. Forexample,before1970,thereisahigherincidenceofdarkershadingclosertothe45degreeline. Thisfirstobservationleadsustooursecond one: thereisaclearperiodofcalmfromabout1980throughthe2007-2009financialcrisis. Interestingly, this period corresponds to the “Great Moderation” for the macroeconomy, consistent with the idea that regime shifts do not occur during calm periods. Looking at this region around the 45 degree line in the second half of the period, the distances between observations are relatively small. Andthird,ourtimeisdifferent. Thedarkshadingthatstartsaround2012indicatesthattoday’s regimeisdifferentfromthosethatcamebefore. ThisperiodstartedwiththeEuropeancrisisand whenquantitativeeasingbecamelargelyfocusedonprovidingmonetarypolicyaccommodation, 14

in contrast to targeting market functioning or crisis management goals. Importantly, we see that theareaaroundthe45degreelinestillhaslightershading. Still,thereisaclearbreakbetweenthe currentperiodandearlierinthe2000s. Figure8: Meetingdistancemetrics 3.2 Identifyingregimesusinghistoricalanalysis Our identifications align well with narrative histories of the FOMC. We use several well-known historical sources for regimes and overlay the broad agreement on operational regimes over our identifiedbigramsinfigure9. We found that Garbade (2021), in particular, provides a thorough assessment of operational toolusagefrom1951-1979;ourregimesmatchhistoalargeextent. Bindseil(2014)usessecondary sources to construct a chronology of policy regimes prior to targeting fed funds: 1931-1952 (excess, butrestrictive, reserves), 1952-1970(freereservestargeting), 1970-1974(reservesonprivate deposits), 1974-1979 (implicit fed funds targeting), and 1979-1982 (non-borrowed reserve targeting). Friedman and Schwartz (1963), writing before clear periods of several of these regimes developed, review FOMC word usage in terms of policy stance rather than implementation (with theexceptionofnotingtheCommittee’suseofforwardguidance,referredtoasan“open-mouth” 15

Figure9: Historicalregimesidentification policy).11 AcommonthemeoftheseauthorsisthatstatedCommitteegoals(e.g. billspreferably, monetarytargets, variousformsofreservemanagement)occasionallyconflictedwithpoliticalor economicneeds. According to both our bigrams and historical investigation, yield curve control captured the FOMC’sattentioninthe1950sand1960s, inthewakeoftheTreasuryaccord. Thismorphedinto the “bills preferably” period in the 1960s. The monetary aggregates regime was evident in the 1970s, and some importance to this regime lasted through the 1990s. Federal funds targeting arguablyrunsfromtheearly1980s(or, possibly, theearly1970sdependingonauthor)tothe2007- 2009financialcrisis. Forexample,Bindseil(2014)describesthesecondhalfofthe1970sasaperiod of implicit fed funds targeting with the fed funds as the sole target beginning in 1994. His delineationisbasedprimarilyonCookandHahn(1989),whodemonstratedthatthereappearedtobe arelativelytransparentviewofwhatratetheDeskwastargetingoverthisperiod.12 Even with this concordance of computer processing and historians, clear-cut regimes may be moreapparentthanreal. Forexample,characterizationsofoperationaltargetsbyCommitteemembers,highlightedbythequotesinthebox“Thoughtsonoperationaltargetsovertime”,suggestthat theadoptionoffedfundstargetingfollowedexperimentationwithalternativetargetsthatextended wellintothe1990s. Inaddition,agenericobservationofourregime-basedapproachisthatregime volatility is higher when regimes are not widely discussed. For the low-count regimes, a single 11Meltzer(2009)offersamodernperspectiveonearlyCommitteedecision-making. 12AlsoseeAnbiletal(2020)foralongertimeseriesofthefederalfundsrate. 16

mentioncanshifttherelativerelationships. Weseethisvolatility,forexample,withfedfundsand money supply terms in the early years of the FOMC and tools that have generally been used to supportothertoolsforpolicyimplementation,suchasreserverequirements. ThoughtsonOperationalTargetsOverTime “In so concentrating operational attention upon day-to-day market rates rather than on the 3-month bill rate, I recognize I am suggesting a departure of sorts.” Robert Holland (staff member for the Board of Governors, later Federal Reserve Board governor), January 12, 1965. Historical Minutes. Governor Partee: “The question is, can we do better with the federal funds rate or with some kind of reserve [aggregate]? I think the answer was that there is not much difference.” March 29, 1976. Transcript. Governor Baughman: “Well, it simply implies, does it not, that total reserves is our overriding target?” January 8-9, 1980. Transcript. President McTeer: “I think it’s dangerous for our future to have the political spectrum from Milton Friedman on the one hand to Paul Samuelson on the other bashing us for tight money based entirely on M2” February 2-3, 1993. Transcript. 4 Quantitative assessments Asourfinalsetofinvestigations,weapplytypicaleconomometrictechniquestoanswertwoquestions. First, what operational targets are most often associated with shifts in the target regime? Andsecond,whatarethemacroeconomicconditionssurroundingtheseregimeshifts. Weanswer eachinturn. 4.1 Preliminaries To motivate our analysis, figure 10 displays how the regimes shift over time. We define a regime shiftasaperiodinwhichtheclusterchanges,wheretheclustersaredefinedasinfigure7. Weconcentrateontheperiodfrom1960topresent,therebyeliminatedthesomewhatnoisierpre-Accord period. “Yes”indicatesachangeinregime,and“no”doesnot. Theinformationinfigure10largelyagreeswiththatin8. Theperiodsofconsensus,illustrated byalackofregimechangeinthechartabove,includethemonetaryaggregatesregimeinthe1970s, 17

thefedfundsregime(punctuatedbyroutinechangesintheearly2000s)andan“autopilot”period inthelate2010s. Figure10: Regimeshiftsovertime What are the operational targets most frequently associated with these shifts? To investigate thisquestionmoreclosely,weevaluateamodeloftheform: Pr(S = 1) = f(X ;ϵ ) (3) t it t where S is an indicator of a shift in regime, and equals 1 with a regime shift, X is the TF-IDF t it foroperationaltargetiattimet,andϵ isanormallydistributederrorterm. Weevaluateaprobit t modelwithrobuststandarderrorstocontrolforpotentialheteroskedasticityorserialcorrelation. Table3displaystheresults. Thefirstcolumnexploresquantitytargetsonly. Weseelittlestatisticaldifferenceinmanyofthecategoriesforsuggestingaregimechange,saveanegativecorrelation formoneysupply. WebelievethatthisisassociatedwiththeperiodoftheGreatModeration,when operationaltargetsdidnotshift. Asdiscussedabove,federalfundsrateandmoneysupplyterms werepairedformuchofthisperiod. The second column reviews results for rates only. Here we see that discount rate targeting is associated with relatively frequent regime shifts, while federal funds rate targeting is associated withrelativelyinfrequentones. DiscountratetargetingwasmoreprevalentbeforetheGreatMod- 18

eration, and as discussed above, was associated with the gold standard and general operational targetshifts. Thenegativecoefficientonthefundsratecoefficientlikelyreflectsthesamefactorsas thatforthemoneysupply. Thethirdandfourthcolumnsbringintheexternalaccount,aswellastestsallfactorssimultaneously. Asdiscussedabove,discussionsoftheexternalaccountusuallyoccurduringtumultuous times;thepositiveandeconomicallymeaningfulcoefficientisconsistentwithhistoricalcontext. In thelastcolumn,ourpreviousresultsforthesubcategorieshold,savethestatisticalsignificanceof thefederalfundsrate. Giventhatmovementsintheimportanceofthefundsratearelikelysimilar tothatofthemoneysupply,thefallinsignificanceofthefundsrateandtheattenuationofthecoefficient on the money supply sharpens our previous conclusions of multiple targets and relative operationalcalmduringtheGreatModeration. Assuggestedbythefigure,operationalWeexamine the relationship between the Committee’s discussions of specific operational targets and the economicconditionsithasconfrontedovertime. withalookattherelationshipbetweeneconomic variablesinchangesinthepreviouslyidentifiedoperatingregimes: Ourfinalexerciseexplorestheassociationbetweenoperationalregimechangesandthemacroeconomy. Were-evaluateaprobitmodelexploringthefactorsthatexplainregimechanges,withthe functionalform: Pr(S = 1) = f(Y ;ϵ ) (4) t it t whereY isthefour-quarterchangeintherelevantmacroeconomicindicator. Weinvestigatethree: it the unemployment rate, the PCE price index, and GDP growth. Again, we incorporate standard errorsthatarerobusttoheteroskedasticityorserialcorrelationofunknownform. Our results are displayed in table 4. Looking at each macroeconomic factor individually in columns(1)through(3),contemporaneousdeclinesintheunemploymentrateandinflation,and increasesinGDPgrowth,areassociatedwithregimechanges. Thecoefficientsareconsistentwith the observation that regime changes seem to occur during improving economic conditions; that is, it is more likely to see a regime change when the unemployment rate is declining, inflation is going down or GDP is improving. Putting these results together in column (4) preserves their robustness,withonlysmallchangesineconomicandstatisticalsignificance. Whataboutrealconditionsayearortwoprior? Wepresentanalysisofthisquestionincolumns (5) through (8). Overall, conditions tend to be relatively strong in the rear view mirror as well. 19

Interestingly, the economic significance of the unemployment rate declines, but that of inflation andoutputpickupsomewhat. Asoutputtendstobemorevolatilethantheunemploymentrate,it maybethat,historically,theCommitteewaiteduntilthelabormarketrecoveredtomakesignificant changestoitsoperationalframework. 5 Concluding remarks WordcountsandNLPareclearlynotasubstituteforathoughtfulexaminationoftheFOMC’shistorical record and are no substitute for the careful work of, for example, Friedman and Schwartz (1963), Meltzer (2004), and Garbade (2021). Nonetheless, NLP can complement existing work by identifying trends that would otherwise require extensive historical research to establish. The historicaltermusagepatternscoincidewiththebroadevolutionoftheFOMC’sfocalpointsofdiscussion and in this regard are good indicators of operational targets. The phrases found in the minutesarealsosupportedbyquotesfromtranscripts,someofwhichhavebeendisplayedabove. Thosequotesareindicativeoftheuncertaintypolicymakershavehadwhenusingoperationaltools toimplementpolicyobjectives. ThetransparencyofFOMCintentmayhavecontributedtoashift in the audience for operational targets. Specifically, the material in the Minutes and from other FOMC documents suggests that operational targets have evolved from guidance to staff for the implementation of monetary policy to guidance to market participants. This shift may have contributed to the use of the fed funds rate as the single operational target and helps to explain why changes in the target rate have, in recent decades, have led to coincident changes in market rates withoutchangesinFedmarketactivity. Transparentoperationaltargets,however,wereoriginally viewed as having drawbacks. Arguments that have been considered for not having publicly announced targets at all include the potential inappropriateness of given targets when conditions change rapidly and, less compelling, that once published, it is very hard to stop publishing (e.g. Kohn1995).13 Wearefortunatetobeabletodrawontheworkofthosedoingextensivehistoricalresearchon FOMCoperationaltools. TheworkofothersgivesussomeconfidencethatourNLPidentification strategyisareasonablerepresentationofFOMCdeliberations. Itsuggeststhatourapproachmay 13Also in the 1995 memo, Kohn points out that there may have been a separation between discussions of specific targetsandpolicyimplementation: “Ithinkwesometimesexaggeratetheroletheaggregatesusedtoplayinpolicy. After1982thesevariablesdidnottriggerautomaticchangesinthestanceoftheSysteminreservemarkets—andthey werefrequentlyallowedtorunoutsidetargetbandsforagoodwhile.Butmovementsintheaggregateswereconsidered inasignificantwayinpolicymaking,andwhentheyandotherindicatorsweretendingtoruninthesamedirection, moneysupplydevelopmentsmayhavepromptedquickerandmoreforcefulaction.” 20

behelpfulforthoseconsideringotherquestionsrelatedtothehistoryoftheFOMCandtheFederal Reserve more generally. Examples could include shifts of focus on specific industrial sectors, the roleofbankcreditintheeconomy,internationalinfluencesondomesticmonetarypolicy,andthe effects of academic research and shifts in intellectual paradigms on changes in Federal Reserve policy. 21

References Anbil, Sriya, Mark Carlson, Christopher Hanes, and David C. Wheelock (2020). “A New Daily FederalFundsRateSeriesandHistoryoftheFederalFundsMarket,1928-1954,”FinanceandEconomicsDiscussionSeries2020-059. BoardofGovernorsoftheFederalReserveSystem(U.S.). Atkinson,Thomas(1969). “ToneandFeeloftheMarketasaGuidefor FederalOpenMarketOperations”inTargetsandIndicatorsofMonetaryPolicy,ed. byKarlBrunner. ChandlerPublishing. Axilrod,S.H., Lindsey,D.E.(1981). FederalReserveSystemImplementationofMonetaryPolicy: AnalyticalFoundationsoftheNewApproach. TheAmericanEconomicReview,71(2),246–252. Bindseil, Ulrich (2014). Monetary Policy Operations and the Financial System. Oxford UniversityPress. Bindseil, Ulrich and Alessio Fotia (2021). Introduction to Central Banking. Springer Publishing. https://link.springer.com/content/pdf/10.1007%2F978-3-030-70884-9.pdf. Cap, Adam, Mathias Drehmann, and Andreas Schrimpf (2020). “Changes in Monetary Policy Operating Procedures over the Last Decade: Insights from a New Database,” BIS Quarterly Review,December,27-39. Cieslak, Anna, and Annette Vissing-Jorgensen (2021). The Economics of the Fed Put, The ReviewofFinancialStudies,34(9),4045–4089. Cook,TimothyandThomasHahn(1989). “TheEffectofChangesintheFederalFundsRateTarget onMarketInterestRatesinthe1970s,”JournalofMonetaryEconomics,24,331-351. Feinman,Joshua(1993). “ReserveRequirements: History,CurrentPractice,andPotentialReform,” FederalReserveBulletin,June. Friedman,MiltonandAnnaJ.Schwartz(1963). “AMonetaryHistoryoftheUnitedStates,1867–1960,” NBERBooks,NationalBureauofEconomicResearch,Inc,December. Garbade,KennethD.(2021). AftertheAccord,AHistoryofFederalReserveOpenMarketOperations,theUSGovernmentSecuritiesMarket,andTreasuryDebtManagementfrom1951to1979. CambridgeUniversityPress. Gentzkow, Matthew, Bryan Kelly, and Matt Taddy (2019). “Text as Data,” Journal of Economic Literature,57(3): 535-74. Kohn,Don(1995). MemototheFOMC,March. https://www.federalreserve.gov/monetarypolicy/fomchistorical1995.htm. Lindsey,David(2003). AModernHistoryofFOMCCommunication: 1975-2002. https://www.federalreserve.gov/monetarypolicy/files/FOMC20030624memo01.pdf 22

Lindsey, David, Athanasios Orphanides and Robert Rasche (2005). The Reform of October 1979: HowItHappenedandWhy. FederalReserveBoard2005-112. Meltzer,Allan(2009). AHistoryoftheFederalReserve,vol. II.TheUniversityofChicagoPress. 23

Table1: Sources Startdate Enddate(inclusive) Includedinanalysis HistoricalMinutes March1936 May1967 MemorandaofDiscussion June1967 March1976 RecordofPolicyActions March1976 January1993 ModernMinutes February1993 Present Excludedfromanalysis RecordofPolicyActions March1936 December1992 MinutesofActions June1967 December1992 Transcripts March1976 December2016 Statement February1994 Present 24

smargibtegratlanoitarepO :2elbaT lanretxE setaRcitsemoD seititnauQ tnuoccA evruCdleiY laredeF slliByrusaerT dnaetaRtnuocsiD evreseR ylppuSyenoM tessA evreseR lortnoC etarsdnuf secnatpeccAsreknaB stnemeriuqeR tnemeganaM tnemeganaM stnemegnarrayc setacfiitrecsllib etarsdnuf noitcuallib etartnuocsid stnemeriuqeresaercni deraeppasetagergga tbedycnega sevresergnibrosba snoitarepoyc stnuomasetacfiitrec dedartsdnuf sgnidlohllib tnuocsidegnahc stnemeriuqernoitcuder yenomsetagergga seussiycnega sevreserlanoitidda stekramegnahcxe etarmret tegratesiar etarllib tnuocsidesaercni sevreserderiuqer 1mroivaheb sbmycnega esahcruperstnemeerga setaregnahcxe sdnufegnar setarllib secnatpeccasreknab setagerggaredaorb egagtromycnega sevreserknab ycnerrucngierof sdnufetar dlehsllib setartnuocsid yenomdenfied esahcruptessa sevreserdeworrob setacfiitrecdlog sdnuftegrat desahcrupsllib tnuocsidnoitcuder yenomdnamed sesahcruptessa esahcruperretne wofltuodlog egnartegrat llibhtnom tnuocsidetar wodniwtnuocsid seitirucesdekcab sevreserssecxe loopdlog sllibsesahcrup tnuocsidknab yenomesae teehsecnalab sevresereerf ecirpdlog sllibseitiruces yenomysae seitirucestnemnrevog tegrateerf selasdlog htnomeerht 2mnoisnapxe ycnegasgnidloh sevreserhtworg kcotsdlog sllibyrusaert yenomnoisnapxe yrusaertsgnidloh sevreseresaercni dlognoillim yenomgnimrfi sesahcrupekam eerflevel ngierofsnoitarepo setagerggahtworg sllibgnirutam elasdehctam dlogecirp 1mhtworg seussignirutam seitirucesgnirutam stnemegnarrapaws 2mhtworg dekcabegagtrom deworrobten sgniwardpaws 3mhtworg sesahcrupthgirtuo eerften enilpaws yratenomhtworg tessaecap sevreserdeworrobnon snoitcasnartpaws yenomhtworg sesahcrupecap stisopedetavirp htworg1m margorpesahcrup sevreseredivorp 2m1m llesesahcrup tnecrep1m nopuocsesahcrup stnemeergaesahcruper htworg2m regnolsesahcrup snoitarepoesahcruper 3m2m selassesahcrup elbaliavasevreser tnecrep2m seitirucessesahcrup gniknabsevreser launna3m yrusaertsesahcrup dedeensevreser gnimoc3m lapicnirpgnitsevnier neposevreser tnecrep3m ycnegaseitiruces dedivorpsevreser doirep3m dlehseitiruces deilppussevreser yenomserusaem sgnidlohseitiruces esahcruperesrever setagerggayratenom gnirutamseitiruces sevreserylppus esabyratenom neposeitiruces sevresergniylppus noisnapxeyratenom ecapseitiruces sevreserlatot htworgyratenom dlosseitiruces tniartseryratenom sgnidlohytiruces htworgyenom seitiruceslles kcotsyenom seitirucesgnilles ylppusyenom seitirucesyrusaert 1megnar 3megnar 1metar setagerggasetar yenomsevreser yenomthgit 2mraey 25

Table3: Regimechangesandoperationaltargets Quantities Rates Externalaccount All Reservemanagement 0.011 0.004 (0.026) (0.034) Assetmanagement -0.020 0.009 (0.017) (0.019) Moneysupply -0.086*** -0.065** (0.021) (0.023) Reserverequirements 0.050 -0.006 (0.038) (0.041) Discountrate 0.087** 0.088* (0.030) (0.034) Billrate -0.017 -0.042 (0.025) (0.029) Federalfundsrate -0.041* -0.037 (0.020) (0.023) Yieldcurvecontrol 0.041 0.037 (0.025) (0.028) Externalaccount 0.177*** 0.161*** (0.017) (0.017) Numberofobservations 629 629 629 629 AIC 783.6 782.4 710.7 695.4 BIC 805.9 804.6 719.6 739.8 Log.Lik. -386.818 -386.180 -353.337 -337.699 Pseudo-R2 0.039 0.045 0.149 0.195 Significantat: +10percentlevel;∗5percentlevel;∗∗1percentlevel. Notes: This table provides from a probit model to explore determinants of changes in the operational target regime. The dependent variable equals 1 if there is a cluster change, with the clusters illustrated in figure 7. An observation is a document included in the analysis and as described in table 1. Parameter estimates are marginal effects of a change in the TF-IDF measure. Robuststandarderrorsareinparentheses. ∗ p < 0.05,∗∗ p < 0.01,∗∗∗ p < 0.001. 26

selbairavlaerdnasegnahcemigeR :4elbaT egnahcemigerfi1slauqE:elbairavtnednepeD )8( )7( )6( )5( )4( )3( )2( )1( srotcafsuoenaropmetnoC ***960.0- ***380.0etartnemyolpmenU )510.0( )310.0( *620.0- *020.0noitaflnI )310.0( )900.0( *120.0 *510.0 htworgPDG )800.0( )600.0( srotcafdeggaL etartnemyolpmenU 600.0- **370.0raeyenodeggaL )630.0( )520.0( +550.0- 110.0 sraeyowtdeggaL )130.0( )420.0( ECP *660.0- 900.0raeyenodeggaL )720.0( )020.0( 130.0 810.0sraeyowtdeggaL )620.0( )910.0( htworgPDG **630.0 *710.0 raeyenodeggaL )210.0( )800.0( 410.0- 210.0sraeyowtdeggaL )110.0( )800.0( 316 316 316 316 926 926 926 926 .sbO.muN 9.157 2.977 3.477 8.657 7.757 7.797 0.997 0.067 CIA 9.287 5.297 6.787 0.077 4.577 6.608 9.708 9.867 CIB 079.863- 326.683- 251.483- 083.573- 038.473- 358.693- 994.793- 410.873- .kiL.goL 160.0 600.0 310.0 040.0 170.0 010.0 800.0 460.0 2R-oduesP .leveltnecrep1∗∗;leveltnecrep5∗;leveltnecrep01+:tatnacfiingiS .emigertegratlanoitarepoehtnisegnahcfostnanimretederolpxeotledomtiborpamorfsedivorpelbatsihT :setoN asinoitavresbonA.7erugfinidetartsullisretsulcehthtiw,egnahcretsulcasierehtfi1slauqeelbairavtnednepedehT niegnahcafostceffelanigramerasetamitseretemaraP .1elbatnidebircsedsadnasisylanaehtnidedulcnitnemucod .sesehtnerapnierasrorredradnatstsuboR.elbairavlortnoceht .100.0<p∗∗∗,10.0<p∗∗,50.0<p∗ 27