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Working Paper 79-5 AN ALTERNATE METHOD OF ESTIMATING THE CAGAN MONEY DEMAND FUNCTION IN HYPERINFLATION UNDER RATIONAL EXPECTATIONS Marvin Goodfriend Federal Reserve Bank of Richmond September 1979 The views expressed here are solely those of the author and do not necessarily reflect the views of the Federal Reserve Bank of Richmond. 1. In~uotion ~~~~a~pti~and~l~~~ofanewstrategy fmxstimatingtheca~lmney~-on~ratianalexpectati~. The procedure has three min virtues. First, it is in@-ted inposingrestrictionsontkmney~lyprocess. without Semnd,theprocm%m Third,itadmitsasiqletestofa isextremlysinpleandemnanical. restrictionimpliedbytheGqanmneydemndfunction. Thetechniquepresen&dinthis~utilizesonly (1) theassumd Cagans~of~y~,(2)~assumptionthatanti~~~~ fonnedrationallyinthe sense of M&h [lJ61], (3) the asmption that a~~~~r~eaus~o~ti~~~priceleveland~ystockis availabletoindividuals,and (4) theassunption under these assunptions, the in~portfoliobalancescheduleissmll. pruposedestimatim thatmobcmwblenoise strategydeliversoansistentestimatesofa,the slope ofthelogofthe~forrealbalanceswith~~~toanticipated inflation, in the Cagan xmney dtsnbmdfunctim. Thexnminder ozmtainsa ofthispaperisorganizedin dfzxriptionoftheestimtionstrategy. . four sections. !kction2 Theestimatimis carriedout~the~~hyperinflati~datainsectian3.~~4 relates issues thathavebeenraisedinrecmt@perinflation theestimaticnresultsofthispaper. A s.mmaryfolluws. 1 studies to 2 2. DescriptionoftheEstimatianstrateqy aEeCaganmneydemandfunctionis lnMt-lnQ=X (1) specifiedas follows: - a(hlnP~+l) + vt =thenaturallogofthe~ysupplyinperiodt WherelnMp l-t- =thenaturallogofthepricele~linperiodt Aln$+1 E thesubjectiveanticipationfomzdinperiodt of the period t+l rate of inflation disturbanoe(velocityshock) vt = an cn?csbservable inpexiodt aE the slope of the log of the &mand for xeal balarloeswith respect to anticipated inflation, a>0 Xisaconstant Itisuseful (2) for the follmingdiscussiontowrite mEklz tk subjective -t+1-%+l where AlnP*l E therealizedrateof iIlpe.bdt+l ut+l' Substituting for AhPew inflaticn adcmmardexpostanticipationerror i.IIperiOd~l in (1) with (2) and -gingthe result yields: aregreaterthanoneandnegative,xwpecti~ly. Itisusefult33explain 3 real balanes. TheCaganstructureofmneydemandtogetherwiththe implicit assmption of stock mnetxy equilibrium inplies that the ma&et is satisfiedwithreducedrealbalancesmlyifanticipatedhflationhas 3&33-l. E'ran (2), for ut+l held constant, this mans tis,inturn,lneansthata mre AlnPt+l must be higher. givenriseoflnPtmustbeasscciatedwitha than proportiona* rise of lnPt+l. Hence, the coefficient lnPt isgreaterthanone. Nowconsiderthe rightsidevariables ccefficientonthelaggedmneystmck. kldcmstant, arise Agaiq;other in the conteq0raneous (periodt) ~ystockisassociatedwithariseinoantemporaneousrealbdlanoes. Inthiscaseanti~~~inflati~mustbel~rifthe~etistobe satisfied holding greater real balances. For given ut+l and lnPt, this xIEzmslnPt+lxInlstbel~. Note that& is&cause Hencetknegative sumofcoefficientson ccefficientm lnPtand lnMtis lnMt. cme. This anequiproporticnate rise oftheperiodtpricelevelandrmney stockleavesrealbalan0esunaffected. Therefore,thisdisturbancemust he associatedwithanmchangedanticipatedrate for ut+l held amstant, of inflation. Fran (2), thisir@iesAlnPt+lmstbeunchanged,so thatInPt+lmustrisepraporticnallywithlnPtand3nMt. Canequation(3)kccnsistentlyestimted? !theanswertothis ~sti~dependsantherelativeimportanceofvelocityshocksinthe equaticm. *Y- ~cases,ane~velocityshocksarezeroandanotherwhere nonnegligible, are discussed below. Case #l: No Velocity Shccks (cJ$= 0) Suppose that the velocity shocks (v's) are small, i.e., noise in the porkf0Liobalance schedule is insignificant, then (3) canbe rewritten as: 4 x -bnMt+a a (4) l+Ci lnPt + %+1-l Inpt+l = - Zi Equation (4) canhe anticipatimerrorut+l mnsisl32ntlyestimtedcmlyif is uncorzlated with lnMt and lnPt. for (4) to be cunsistently estimated it is neozmaxy ccinditicmholds. mth's mrket's the expost Specifically, that rational expecbtions asqtim says that the subjectiveanti~.Pati~ofinflati~~dequdlthenrathematical ~ti~conditianal~infonnationavailabletothemarketinthe periodwhentheanticipaticmisfonred. If itisalsoassumdthatthe informatimontheprioelevelandthe markethasaccurateconteqoraneous naneystock,thentheexpostanticipationerrormustbe~~lated withthelaggedpricelevelandxmneysImck. Inotherwords, ut+l lmt, lnPt = 0 as required. !lhelzfore,thedisturbancetermin I F[ ' equation (4)isdistribub3dindependentlyofthe~explanatoryvariables. price level is in the mrbt's since the con- infomation set,the~~~~anti~~~errorisinits~~~setas wzll Theas~~thatanticipatiansarefo~~ti~lytherefore 1 inplies that g u~+~ 1 lnMt, lnPt, ut c disturbances in (4) mustbe The u~+~ expechticn seriallyunazrelatedatall that the coefficient in (4) canbe MEove.r,thepzdicted = 0. absene lags. error must It follms esti.ma~consistentlywithOIS. of serial correlation of the msiduals 5 in (4) is an inportant testable i@icatim of the joint hypotksis uxkrlyingtheequation. ~priceleveland~stockarehighlyoorrelated,especidlly inhyperinflation. 'Ihisrrrulticollinearitycouldleadtolawprecisi~an the ccefficients of InMt and lnPt if estimated separately. Hokzver, Cqan'snuneydemndspecificatimplaces a restriction across the lx3 ccefficients,rrzquiringthemto~tione. coefficients am specific functions of a. Mxetkmthat,both A restricted -ion of equation(4)canbeestimatedsothat~y~parameter,a,needbe remvmzdfrmtk amids ooefficientsof~~~lanatoryvariables. thedifficultiesthatmlticollinearitymes !&is fortheestimation process,sinoetheestimationprocedureisnotrequiredtoextract~ separateeffectsof~~~dprices,but~~roslly~ operatingthroughthe jointeffect singleparamtera. The restriction that Cagan'smneydemndspecificationimposes can serve as the hasis for a test of the joint hypothesis mderlying equation (4). Alikelihcodratiocanbecakulatedtocheckwhetherthe sumof squaredresidualsis ampared significantly larger forthe to the umxkricted fit of equation (4). If it is, this constitutes evidence againstthe restriction andthe hypothesesunderlyingequation restricbd entire set of (4). Whatndcesthisestimation&chniquework? Ifanecunm2trician believesthat~Iilarketfolms~ti~~ti~sratianally,~ino~ to~istentlyestimateamodel~~lvingantici~ti~f~ti~,it seemsthattkemncxe tricianmuldfirsthave therationallyanticipab2dfuturemneysupply todecidehimselfwhat nuvemmts~duringhis 6 sanpleperiod. Evenifthe eazlnomtrician here tilling to maintain stmng as~antheinformatianset(fllchastheass~availabilityof perfectconteqoraneous informationonthepricelevelandmneystock), itseemshewouldstillhavetoplace actualmney restrictions on,andestimte,an supply rule as abasis forrationalpredictims of future mneygrowthandinflationtouseinhisestimatimpmcedure. ~technique~tratesthatbyiMintaining~ass~ti~that 2 =V =O,sothatthrenoiseintheportfoliobalance~~iszero,a greatsi@ificationcanbeachieved.2 willing to believe that changes in real mney changes inanticipa~ratesofinflation.3 balancesaredueentirelyto !&eccnmWriciancanthen obserwltia Yurnthemneydemandfunctionaround"anduse ocmteqoraneous mney stock andpricelevelto thatthemarketanticipa~sineachperiod. believe thatthemarketfoms Ifhefurtkriswillingto anticipations ratianallyasif coefficientvalues in (4) he coulduse unbiasedpredictionsofthe ra* on infer the rate of inflation accurate ccmterqoraneousobsemationsonmneyandprices, knewthe tricianis Inthiscase,theeconom it had then, if he thatequaticmtomake of inflation. mtheotherhand,ifhe doesnotknawtheooefficientvdLuesbutins~adwantstoestimatethem, kcoulduse (4) asa Yeg-ressionequation" toestixm~theparamtersof the~ydepMnd~~itself,withouthavingtodevelopparticular restrictionsonthermneysupplyprocess. 7 right side variables. iszemsothemism Tb illustrate, fllppose51 expostanticipaticmerror. Ekpation(4)couldstillshowanerrorin Thevelocityshockwuldcausesimltaneous periodt+lifvtisnOnzerO. adjusiztzntinthe currentprioelevelas~~asthe~ticipatedprioe If(4)~estirnatedwithO[IS,~oorrelati~be~the lwel. abias unobservabledisturbanceandlnPtcouldi.n~ Velocityshodcswillhgeneralbewrrelated estimated coefficient a. witi the con~raneo usprioe levelandpassiblytheantemparaneous mcneys~as~~,dependinganthe~yssupplyrule. shocksaresignificant,the Whenvelocity prqosedestimaticntechniquewill,ingeneral, notdeli.ver~istentestimtesofthepa.ramzters .3. in-the inequatim (4). nqd35cal~~ Table1 surmarizestheresultsofestimatingequati~(4)~ Cagan's (1956) hyperinflatim data. Thevariable of primary interest is a, theelasticityofthedemandforrealbdlan~swithrespecttoanticipated inflaticn. Anestirrrateofais~fran~ti~(4)byan~ar leastsquaEsproQdure undertherestrictionsthatthea2efficientsof , xemve1y. -t&lnMt@ aare shm~intkoolum& 4n-e Forccaparisson,-an's Sargent~s [1977] estimates of a for the sam estimalzdvalI.YSof [1956] and data are nSporMunTables2 and 3, re~pectively.~ TheestimtesforaxeportedinTablelareall eanaKically re~~andinfact,asagraup,liein~ythes~rangeas rita;m'sand Sargent's e~timates.~ mre Ifan~g,myestimatesaregrowed tightly together than either Cagan's or Sargent's. Fkmpt for ~@ESTIMATIOkJOFEQUATICN (4) FORCAGAN'S DATA restrict&equation unrestrictedequation I 1 ' liu 6 (s.e.) countxv I --? -Rz e SSRr - bye.) bye.) =t Austria Feb. '21 to Aug. '22 19 -3.09 (1.22) .92 (1.18, 1.40) .982 .094 .149 .907 (.409) .265 (.556) -Y Oct. '20 to June '23 33 -5.27 (1.37) 1.75 (1.38, 1.51) .990 .lOl .315 1.18 (.091) -.18+ (.140) 19 -2.34 (.47) 2.19 .991 ,126 .268 .612 (.171) .771 (.245) -4.08 1.38 .986 .078 .103 1.35 (.171) -.40+ (.214) 1.75 .986 .123 .240 ,891 (.328) .245 (.406) .991 .102 .227 1.23 (.lll) -.21+ (.lll) Feb. '43 to Aug. '44 HwmT Aug. '22 to Feb. '24 19 Poland Aug. '22 to Jan. '24 18 -2.78 (1.18) Russia Feb. '22 to Jan. '24 24 -4.75 (2.0) (2.5) .54 (1.27, 1.45) x2(1) .139 1.319 .315 = 0 .104 17.99 co -2.50 -4.76 .098 .9455 .205 2.837 .212 1.641 SSRr In-, whereq-1degEeoffEedansincecmlyone =Ru i 1 belcrweach D-w statistic are appwriate (dl, du) for 5% level of significance; C@MNIS:NOB~n~ofobsemtions;~2(q) restriction is inposed; the nu&ers -5.56 SSRU since thedepndentvariableis inlogs,SEExlOOis is not used because the sa@e size is too small; ZNOB. thepercentestimaticmermr;Hungary (2) frmCagan'sdata !l!?mm2 CAGAN'S ESTBWES OF a Austria Jatl. '21 to ALlg. '22 -8.55 -(4.43, 31.0) WY -5.46 -(5.05, 6.13) -4.09 -(2.83, 32.5+)b ~c=Y July '22 to Feb. '24 -8.70 -(6.36, 42.2+)b EM.and Apr. '22 to Nov. '23 -2.30 -(1.74, 3.94) RUSSid Dec. '21 to Jan. '24 -3.06 -(2.66, 3.76) Sept. '20 to July '23 Jan. '43 to Aug. '44 SCUFCE: Cagan [1956], Table 3, page 43. a(Q b% q E 90 percent conf- band calculated by cagan. exceeds right-hand figure in parenthfxes. 10 SAIGEW'SFSTIMATFSOFaUSINGCAGAN'SDATA StandardEkror 1.570 Austria Feb. '21 to Aug. '22 -5.97 4.615 -4.09 2.970 Hungary Aug. '22 to Feb. '24 -1.84 .3978 Poland May '22 to Nov. '23 -2.53 .8562 Russia Feb. '22 to Jan. '24 -9.75 10.742 WY Wt. '20 to July '23 Feb. '43 to Aug. '44 SOUFCE: Sargent's [1977], Table 2, p. 76. 11 fImg=y, - standarderrorsofmyestirnatesareeitherroughlyequivalent Q1tkotherhand,judgingby toormu&belmthoserepoWbySar~L repmted confidence intervals, Cagai appears to have esthded a mre preciselythanIinsa~casesandlesspreciselyinothers. Inthe importantGexmancase,Cagan's,Saqent's,andnyestimates of a are -5.46, -5.97, and -5.27, res~ectively.~ Tbeestimteofafrm myprocedureismrtainly~lewhenamparedwiththeirs. asprecisionoftheestima~s, juagingbyhis Asfar confidence interval,Cagan's appearstobegreaterthanmine,whileSargent'sappears~~~.8 AsainKYP~appears togivereasombleresultsbycmparison. C&z inportantinplicaticmof the jointhypothesisunderlyingmy estimation strategy is that the residuals fnxnthe restricted fitof equaticn (4) ShoulddisplaynoevideImzofautoazrelation. thepresenceofresidual reporkdunderD+inTable As acheckon autowrrelatim,theDurbin-Watson statistic is I.. The IMrbin-watsan statistichasbeen s'mwntobebiasedtma.rdW,i.e., towaAacceptingthzhypothesisof noserialcorrelati~,whenlaggeddepmdentvariables right side of a regressian.' appearcmtk Thisxttans thattheDurbin+atsonstatistic shouldlaotbetakenasevidenoeagainst~p~senoeofautocorrelated residualsinthiscase. canbe Wvertheless,avaluewidelydifferentfruntwo interpre&dasetidenceofresidual autocorrelation. Ihe Durbin-watsanstatistic showsnoevidenazofresidual autoam&atimintheGe.man,Greek,andPolishcases. ForHungarythe statistic is inu3nclusive. ButintheAustrianandRLlssiancasesthe statisticdoes indicate residual autocorrelation. 12 since the~urbin-mtson statisticisbiasedinthis context,and sinoeitis~lyusefulasa~ckanfirstorderautocorrela~,an additionaltestforresidual estimating first, semnd, autocoxrelationispresenl%?d. andthirdorderautoregressive for the residuals. These estimates, bg&her wsidual aremportedinTable4. only for Austria and Russia. onlyatlagone. coefficients with their standard errors, autocorrelationappears tobe Inbothcases autocoxrelatimis Therefore, takingintoaccountboththe statistic and estimted This involves significant significant Durbin-Watscn autocorrelation ccefficients, the hypothesis thatresidualsarenotseriallyo3mla~cannotbe rejected for four Inparticular,theimportantGennancase oft& sixhyperinflations. isone fortichnoevidenceofresiduala~rrelationisdetected. ~thewhole,autoaxrelationchecksconstitute evidfsme for the joint hypotksis reasombly underlying the estimatim favorable of eqmtion Turnto~ool~inTablelthatreport~sultsofes~~gan 9 G uurestrictedversimofequation (4). Here,=-aud-~areestimat3s theunrestrictedccefficientsof lnPtandlnMt,respectively. (4). of As is apparentfromtheserestrictedrepresentaticHls,thejointhypothesisupan whichtheestimationofequation oflnPtshouldexuzdone,tk (4) isbas&i@ies It is rem&able coefficient coeffici~toflnMtshouldbenegative,and theseunconstrainedcoefficientsshouldsmtoane. in factcloselyborneoutinthe thatthe Thesehypothesesare Gxmm,Hungarian,audFUx3iancases. that these hypotheses areverifiable in three of the six cases inspiteoftk extremlyshortsanples of the prioe level and mney theGexmnandRussiancases, gmwth. andhighmultimllinearity It is even more rmarkablethatin the impliedestim~ofa,&tainedby 13 Austria z=(l) =w) J=(3) . A Pl .55(.24ja .55(.28) .58(.27) $2 A P3 .00(.29) .23(.32) -.48(.31) .06(.20) .07(.20) .08(.21) -.31(.20) -.32(.20) .07(.21) -.21(.24) -.27(.29) -.26(.34) .02(.31) .06(.37) .04(.37) .28(.26) .40(.27) .30(.31) -.44(.26) -.36(.31) -.20(.31) .11(.26) .13(.28) .13(.30) -.07(.28) -.06(.30) -.l5(.30) .74(.16) -94 (22) .84(.24) -.34(.22) -.l2(.32) -.24(.25) Ge.rmnv mland Ftussia 14 invertingtheestirrratedcoefficientofInMt,isveryclosetotheestirnate of a obtained in the restricted estimation of equation (4).l" ~inportanttestofthejointhypothesisunderlyingthe spzcificationofequation (4) involves checkingVhetherrelaXhgthe res~i~~acros~thelnP~andlnMtcoefficientsleadstoasi~ficant inprovemntinthe presentedin hypo&esis "fit" ofthatequation. ZellnerandPalm Alikelihoodratio statistic, [1974],is ei@oyedheretotesttknull thattherestrictedequaticmis correct.~ Thestatisticis distributed as chi-square withonedegreeoffreedanz x2(1) the sumof squared residuals frantherestrictedregression thesmofsquaredresiduals f.mfntheunrestrictedregression thelenqthofthesanIple ofobsek?ations on th& residuals Aish~~oftheteststatisticindicatethatthedata reject the JXStriCtiOIl. In particular, the restriction is rejecbd at a 5% level of significance if x2(1) exceeds 3.84. !Ibechi-square TableL values forthistestarereporb2dunder~2~1) in ExceptforGreece,valUeSaresmall,indicatingthat* restrictioncan'tberejeckdatthe5%leve1.12 case,therestrictedandmrestricted Infact,i.ntheGerman SSRvaheswere identicaloutt0 *nmbe.rofdecimalplacesrepor&dbyTIOLL.TheGreekchi-square ~~isatleastsixtimeslargerthananyoftheothersandindicates aclearrejectionofthe restrictionatverylawsignifican~levels for 15 Greece. Except for Greece then,thechi-squaretestsproKide-ly impressiveevidence supportingthe jointhypothesis underlying the specification of equation (4). The~~fortherestricti~tobeoansistentwithdatafran all~hyperinflatiansexceptGreeoeisinterestinginli~tofa patentidlinadequacyof~Greekdatarelati~toother~~lati~ Cagan's nmey data@nbdoutbyCagan. series for Greece amsists of rmindexof~quantityofbanknotesissuedby~BankofGreece. It ~notincludedata~bankdepositswhichpres~ly~not Thissuggestsa reasombleexplanatimfortheothemisepuzzling ~jectimoftherestrictionintheGreekcase. mlati~lyinadequate coverageoftheGreekrrrmeydatacanparedtothat oollected for other hyperiMlaticms restrictim 4. Itmaybethatthe is responsible for rejection of the forGreekdata. Related Issues in Recent Hyperinflation Studies A convenimt starting point for this discussim is Sargent [1977]. SargentanalyzesCagan'srrPdelofhyperinflati~~circlnnstanoes whichCagan's adaptiveschene in forforminganticipationsofinflationis "rational" in the sense of Muth [1961]. Underthesecmditions,Saryent is able toshowthatCagan's unless there isnonoise estimtorofais intheportfoliobalance generally inconsistent s&edule.14 Admitting 16 miseintheportfoliobalanceschedule,Sargent isabletoderive caglsistentestimatesofa~theassurrpJtianthatdis~~~stothe demndandsupply formneyareuncorrelated. Sargent'sestimatesofaare interesting in thepresentcmtextin First,Sargent's calculations showCagan'sestimates l3m ways. shouldbe dmnwardbiasedifthexewre of a significantnoise intheportfolio balance schedule. Since Sargent's estimates of a are mnsistent, ~~Of~t~setsofestimatesreportedinTables sbwatendency such b3ndenq forCagan'sestimates 15 is apparent. a 2and3should to fall below Sargent's. Butno This suggests that, at least if Sargent's ~~*iscorrect,noiseintheportfoliobalanceschedulernayinfa~ berelativelylow. Thistidencemaybetaken~supporttheas~on ~l~g~es~ti~strategy,thatnoisein~portfolio~~ sckdule is sndll. ale notable chamhxistic of Sargent's estimtes forHmgaryandPoland,theyareacmfparu WXOBwfienoanparedtOthe is that, except 'edbylargeeStimatedstandard standa&errorsofnyestimates. This suggests that Sargent's estimtir of a is less efficient than mine. Givenlittleevidencethatsargent'sprocedureabtainsanyreductionin bias,hisprocedure,asatechniqueforestimatinga,~ynotbe~~ thecostinefficienqwkncuqaredtomine. !%rgmthasappliedhistheoreticalf~ Jambs' rktoevaluating [1975] estimates of the Cagan model in !%.rgent [1976]. For presentpurposes it is sufficienttisaythat SargentshowsJacmbs' estimateslmbeconsistmtonlyifthereisnofeedback --YCl=-h. Since both Sargent and WAlla= frominflation [1973] and Evans 119781 17 is inappropriate. In his reply, Jambs developedunder asqtionis [X876] enphasizes that Sargent's critiqw special restrictions forwhichCagan's is adaptiveexpectations "rational." Jacobsarguasthattheseadhocrestrictions rnaynotbe~~andso~implicatiansofSargient'sanalytical framvmrkcannotbetrusted. FMherthanassmingamneysupplyprocess sufficienttomaketheadhocadaptiveexpectations "ratimal," Jambs arguesthatthgrronq!prnoes~shouldbemsdeleddirectly.'~ Then,if desired,themodeloouldbesolvedandestimatedunderratiandL~~ti~ consistentwiththeestimatedmneysqplyrule. ~letheabavleissues-~~stingandifip?ortant,theyare also difficult. Amajorattributeofxqestimation straw is that it providesanreansofestimatingawi~havingtopayat~tiantothe mneysup@yrule. amdeloftheentire ylestimtimstrategyforaneednotbeembeddedin inflationaryprccess. My t3zchniw therefore . obtamsapotentialseparationoftheproblmofestirnatingainthenaney ~~~frran~farnr>redifficultproblemofItPdelingthedynamic relationshipbetweenprioes andmneyinhyperinflation. b+kdxinKhan [lg75] has recently calculated the Durbin-Watson statistics for Cagan's [lg56] regreSSioplS. !theyare reported in Table 5. ~seDurbin~~statisticsprovideevidenceofresidualoorrelati~ in all of Cagan's regressions except, possibly, Austria.l7 autocorrelatedresidudLsindica~~misspecificatianofei~the ~y~~~~ortheantici~~formaticrm~sminCagan's The 18 TABLE5 DuREml-m STATISTICS FORCAGAN'S REGRESSIONS Austria 1.60 .33 .77 Hungary (1) .37 POlCiIld .68 Russia .76 SOUFCE: Khan's [1975], Table 1, p. 358. mney liesintheanticipatian~chanismratherthaninthe Cagan mney ~estimtimprwzdurecontinuestoemploythe functionas amaintainedhypothesis. demnd Butitreplaces function. demand the adaptive anticipatianshypothesiswiththeas~ti~thatanticipatiansarefo~ rationally in the sense of Muth [1961]. My restricted regressions yield residuals which exhibit virtually no evidence of serial correlation in allexl=epttheAustrianandFIussiancases. FbAhermre,arestriction inpliedbytheCaganmmeydemandspecificationcanbe theGreekElse. This suggests thatatleast andpOlishcases residual autocorrelatkm inCagan's tohismisspecifiedanticipation rejectedonly in fortheGexman,Hungarian, fomWion&pothesis regressiansisdue andnottohismney demandfunctionspecification. Ifonebelieves thatanticipationsare fomk3dratimally,thenthis evidence further implies that adaptive anticipationsGere notin rational in at least three of the Iqperinflations.18 fact Inthesecasesat least,theproperwaytogoabout~~stigatingthehyperinflati~ seems ~torestsi~thelroneysupplyruleapriorisothatadaptive anticipations are %ki.o~I.,~ butrathertiattenpttoidentifythemney slrpplyprooessdirectlyfrunthedata,andthentormdelanticipatians rationally,basedontheestimatedmneysupplyrule. 5- stnmuary Thispaperhasimpl~~a~~ofestimatingtheCagan~y deinandfmctionunderrationalmpectxtions. ~hetechniqueutilizesthe side assmptions that (1) accuratecontenporaneousinformati~onthe 20 pricelevelandmney stock is available to individuals and (2) unobservable noise in the portfolio balance schedule is negligible. assurrptionstheestimtim estimtesofa,the Under these strategy delivers unbiasedandcansistent slopeof thelogof the demand for real balanceswith respecttoanticipatedinflation. Application of this technique yields estimates of a that are very h reasomble bycmparismwiththoseobtainedbyotherwriters. of the six hyperinflatims, the residuals frmtbe regression shownoevidenceof serialcorrelation. four theoretically restricted Atheoretical restri~~impliedby~Cagan~eydemandspecificatiancannotbe rejected for fiveofthe sixhyperinflations. The restriction is clearly rejectedintheGreekcase,butthisispotentially~lainedbythepoor coverageof Cagan'sGreekmney supplydataampaxedwithdata for the otherhyperinflations. Amajor attribute of my estimtion almgwithnorestri&ionsanthermIey theestinntim strategyneednotbe procedure for a is that it gets supplyprorxss. emkddedinamdeloftheentire inflationary process. ~tedmiqueobtainsa ofestimtingainthemneydemand the dynamics of mney Inparticular separationoftkproblem function fruntheproblemofmdeling and prices as a whole. Q3ysubtracting InPt frunbothsidesof Q+1 l"T-=--Inotherwmds,itrelatesperiod x a $I- Mt + pt treal (4),itcanbewritten: ut+l balances toperiodt+linflation. 2Enrorsin(4)areduetoexpostrrnneygmwthpredictionerrors. The forecasterrorcouldbedue forexzmpletonoise inthemney multiplierortoupdatedinfonmticmonfutummneygrowth. Ifthe gwernmnthastofinanceafixed cumentlevelofE!alexpenditlEswith cwrentmneycreatim,thentklatterdisturbmce wouldcausethe currentprioeleveltorise andtherebyraise cumentmneygrwth. otherwxds, zerovelocityshocks~notnileoutthepossibilityof feedbackfmninflationtolmneygrrrwth. In 3Atleast, the econaretricianmstbe pmparedtibelieve that if thexeisnoiseintheportfoliobalance functicn,itmstbeofminor i~~rtanceaxparedtOpredicticmenmrsonperiod t+lmneygrowthand infomtion updates on future lmney g?xlwth. %he mgressionwasruncmtheMITTRfXLsystem. 5Barro's 119701 estimates of ausingdifferentdataare: Austria -4.09 (-3.6, -4.5)a WY -3.79 (-3.3, -4.3) -'3=Y j.53 (-4.6, -6.9) Poland -2.56 (-2.1, -3.3) ass% confidence intervdLs. 6Mysaqleperiodsarealsosimilartotheirs. 7E3arro'sestimateofa forthfzGexmancaseismuchlmerthan 21 these. 22 sahecakdatedSSR Gemancaselookslike: surface foraintherestrictedregressiminthe 20 10 gSee Ekr~allis [1966]. %heonlymuntxyforbhich- usee -a i ,ispositive andsignificantisGreece. zellner and Palm, p. 34. l%MneyJaocbs [l977,p. l24]has saidthat"Cqan's [estimtion] procedureappears~~*forapriceseriesthatisunrelatedtothe~ s~~~becauselnP~cancelsfionbothsidesofthe~ti~forredl m balances." Inotherwds,JacmbsarguesthattheappearanceOf~t onbothsidesoftheequatianCaganestirnatedwouldguaranteeagood Forwfiatit'sworth,theesthatim "fit" even if the n-&&S wcmg. strategyandrestrictimtestenployedhemarenotsubjectt6 Jambs'criticism. =Cagan [1956], p. 106. 14see Sargent (19771, p. 67. %OSUChtendencyiS %i&e apparent in my estimates either. Evans [1978] and Ekiedman [1978]. 23 17The Durbin-Watson statistics frun E%xrro's [1970] estimtes of cagan's~lalsoindicateresidualautocorrelatian. Barr0'sD-W statistics are: Austria .53 CerJMnY .25 HFWY .31 Poland .32 18m' [1978] findinCJSindicate that dE@.ivFI XkiCip&hIlS rational in the GennanhyperhflatiOIl. WWX XlOt APPENDIX Tkrata . TbedatausedhereistakenfromCag~'s studyofhyperinf3ations in Austria, Germany, Greece, Hungary (I), Poland, and Russia. Ihecagandata~istsofmyltNytirreseries~realbalances the rate of inflation. Itisnecessary IIDlbey supply and price level tim OcmsMonofaPrice and forthepresentstudylm0mstmcta series frcmnCagan's series. Level Series If&Qbetkfizstnmtioftheseries. apositive unknownconstant. AssumlogPto=cwherecis Ben logPQ=c Pq)+1 cons- log pk+l " logPQ)+2= =logp to Pt(J+Z + logPQ)+1 logPq)+1 wnstruction of prim Pt()+3 " II ~Pto+3=10gpb+2+~pto+z logPQ)*= series pQw logpto+"-l+logPto+n-l w-c to 109 Pto+" =ccxlsmlogP WhereccPlstructedlogP i =logP to+" 24 t@l+c level 25 cbnstruction of a Maney Supply Series log qo+" = log (;), +n + actual log Q(pl 9 = log ; 0 +cons~logP~+"+c to* = -log (;) + COIlStXU~ %I+" 1Og Pto+n + C SELEEDBIBLIOGRAPHY "Inflation, the Payments Period, and the Demand for Money.' E%arro,RDbertJ. JournalofPoliticalEoncxqy 78 (Novanber/December 1970): 1228-63. "T-heMonetary Dynamics of Hypezwbtion." In studies cagan,Philxp. in& Quantity%eoryofmney,p~. 25-117. EklikdbyM.Riedmm. chicago: u~~versey of chxago Press, 1956. &I: andKincaid,Gmqe. CQmrsnt." WC "Jacobs' Estimates of the Hyperinflaticm Inquiry I.5 (January 1977): 111-18. "Tim-Series Analysis of the &nmn Hyperinflatian." EVans,Paul. International E&manic Review 19 (February 1978): 195-209. %abilityandRikionalityinb&delsof Hype&Elation." Friedmn,BenjaminM. r&ernational EoancanicMew 19 (February 1978): 45-64. Jambs, Rodney L. "A Difficulty with bbnetarist -1s EwnaIlic Inquiry 13 (September 1975): 337-60. of Hyperbflation." "Ecormwtric Eqeneity: A Reply." JournalofMone~ &axics2 (Noverrs3er 1976): 523-27. &uiq "mint and counterpoint: &ply to Cagan and Kincaid." I.5 (January 1977): 119-24. Kmdall,MauriceG.,andStuart,ALan. 'IbeMcedTheoryof 3 vols. Irardan:Charles Griffin & CD., 1961. Fmncxnic Statistics. Khan, Eibhsin. '!i%~netaryDynamicsofHyperbflation:ANote." Journal of Monetary IEonomics 1 (J'uly1975): 355-62. 'RatiodExpectations andthe%fxxy~fPriceMovarrents.' Muth,JohnF. Eaxmetrica 29 (July 1961): 315-35. Nerlove,Marc,andWallis,KennethF. "UseoftheDurbb-WatsonStatistic 34 (January 1966): 235-38. inInappropriati Situations." ~trica Sargent, Thnas J. "95~ DemandforbbneyduringHyperinflationsunder FbtionalExpectations:I.' InternationalEconcmicflwiew18 (February 1977): 59-82. 26 27 "EocnmrretricEScogeneityandAlternativeEstirnatorsofPortfolio Bake SchedulesforHyperhflaticms: ANOte." JournalofMnetary Etxmomics 2 (Nmenhr 1976): 511-21. ,andwallace,Neil, "Rational~tionsandtheDynamicsof . International Ebonanic Review 14 (June 1973): Hypemflaticm." 328-50. 2Mlner,Arnold,andPalm,Franz. Quation~~ticModels." 17-54. "TimSeriesAnalysisandSjmultaneous Jdof -tries 2 (1974):
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