Misspecification versus Bubbles in Hyperinflation Data: Monte Carlo and Interwar European Evidence

M VEIRSUSBUBBLES~ HSYPERINFLATIONDATSAP: MOD C-O ~ INTERWAREUROPEANEVIDENCE MarkA.Hooker FederalReserveBoard,MailStop74 20th&C. St.NW Washington,DC20551 tel.(202)452-3424 fax(202)452-2301 mhooker@frb.gov September,1997 Summarv Thispaperanalyzestestsof the Caganhyperinflation-moneydemandmodelwhichhave severaladvantagesrelativetothoseintheliterature. They do not confoundspectilcation errorwithrationalbubbles,areimplementablewithalinearprocedure,andarefrequently abletodetectperiodicallycollapsingbubbleswhichhavechallengedexistingtesfi. Aftera MonteCarlo analysis, the tests are applied to data from hyperinflationsin Austria, Germany,Hungary,andPoland. Strongevidenceof modelmisspecificationis foundfor Austria,whilethemodelwitharational,explosivebubblecomponentwell-characterizetshe Polishdata. InferencesforGermanyandHungaryaremixed. IthankStevenDurlauf,FrankDiebold,twoanonymousreferees,andseminarparticipants at Stanford, the Richmond Federal Reserve Bank, Cambridge, Bristol, LSE, and Southamptonfor helpfti commentsand discussion on earlierdrafts. Chris Edwards providedvaluableresearchassistance. Part ofthisresearchwascompletedattheFederal ReserveBankof Boston,whichI thankfor its hospitiity. Theviewsexpresseddo not necessarilyrepresentthoseoftheFederalReserveBankof BostonortheFederalReserve BoardofGovernors.

I. Introduction TheCaganmodelofmoneydemandunderhyperinfiation(Cagan1956)hasbeenaworkhorse inmonetaryeconomics,comparableinitsuseas a benchmark(andinits geometricallydiscounted expectedvalue structure)to the dividend-stockprice model in fmancid economicsand tie permanentincome-consumptionmodelin macroeconomics. However, its empiricalliterature “differsin an importantway: mostpapersusing the Cagan spectilcationestimatethe model’s parametersandRt pa.rdctiarrestrictions,but do not test the overallvalidityof the model. As Taylor(1991)not=, theCaganmodelliteratureis largelyverificationistratherthmfalsificationist. Perhapsdueto thissituation,thebasicquestionof whetheror notitis a usefulmodelprovokes widedisagreement,evenwhenassessedonacommondataset.* Part of tie confusionis tied to the fact that the Cagan model has mdtiple equilibria: Explosive,rationalbubblesareconsistentwiththemodel’ssolution. In theliteraturethatfocuses onbubbles,thevalidityoftheCaganmodelis generallya maintained,but untested,hypothesis. Thusdirectestimatesofbubblesmaybe biasedifthemodelismisspecified,andtestswhichregard deviationsfromthe“marketfundamentals”solutionas evidenceofbubblesmaybe misclassifying specificationerror. OtherstrandsoftheCaganliteratureruleoutapriori theexistenceofbubbles, d~pite potentiallyseriousimplicationsforestimatesandtests. DurlaufandHooker(1994)developeda methodologyfortestingthe modelwhichdoes not confoundbubblesandspecificationerror. Thatmethodologyemploystwotransformationsofthe data: underthenti ofcorrectspecificationo, netransformationisorthogonalto aninformationset nomatterwhichofthemtitipleequilibriaobtains,whilethe otheris orthogonalonlyif the nobubbles equilibriumis realized. Thus sequential applicationof the tests allows separate falsificationofthemodel’sgeneralsolutionandra[ionalbubblecomponentsofthatsolution. This paperextendsthatworkinthreeways. First,itpresentsMonteCarloevidenceonthesizeandpowerofthetests. Mosthyperinflation work, includingthat of Durlaufand Hooker, uses asymptoticdistributionsand very short, 1MostpapersesdmatethemodelontheGermanhyperitiation samplefromtheearly1920s.

explosivedatasamples.As G. Evans(1991)andWest(1994)havenoted,thereis a shortageof evidenceonthesmall-sampleperformanceofteswwithexplosivedata. Theteswinthispapermay be implementedlinearly-using two-stageleast squares-which greatlyfacilitatessimulation experiments,in contrmt to the many nonlinearand iterativeapproachesin the hyperinflation literature.Particdarattentionis paidtothehard-to-detectbubblesdescribedin G. Evans(1991), “and allegedto be presentin the Germanhyperinflationandthe recentPolishhyperinflationby BlackbumandSola(1992)andFunke,Hall,andSola(1994)respectively. Second,thepaperderivesestimatorsof themodel’sspec~lcationerroras atimeseries. The analogyinthetermstructureliteratureistheestimatedseriesoftermpremiaratherthanjust a scalar measureofwhethertheyarenonzeroor nonconstant. Thisaddeddimensionmayprovideuseful informationonthenatureoftherejections.AsKim(1996)arguesinthecontextof thepermanent incomehypothesis,measuresof specificationerroraremorelikelyto shedlightontheeconomic magnitudesoftherejectionsthanareteststatistics. FinWy, the paper appliesthe measuresand tests to the classicinterwarhyperinflationsin Austria,Germany,Hungary,andPoland,andcomparesthemtoresultsintheliterature. Thepaperis organizedasfollows. SectionII reviewsthe Caganmodeland the tests, and derivesthenoiseestimators.SectionIIIcontainsadiscussionofsomerecentandrelatedliterature. In SectionIV, MonteCarlo results for test size and power against several potentialbubble alternativesarereported.Applicationsoftheteststo theinterwarhyperinflationdataarepresented insectionV,andsectionVIconcludes. II. The Model, Noise Estimators, and Specification Tests TheCaganmodelisastructuralequationformoneydemandwhichdependsupontheexpected inflationrate. Thelinearformofthemodelis mt-pt =~ +aE[(pt+l-pt)lQt] +Et (1) wheremt and pt are logs of the nominalmoney supply and pricelevel at time t, Qt is an informationset comprisedof variablesthat agentsuse to formtimet expectationsof time t+l 2

prices,~isanunrestrictedconstantterm,a isa constantlessthanO,andStis a stochasticmoney demanddisturbance.Theparametera is the(semi-)elasticityofrealmoneydemandwithrespect to inflation,andis oftenthefocusof attentionintheCaganmodel. Otherfactorsconventionally assignedaroleinmoneydemand,likeinterestratesandincome,areresumedtobe of secondary . importancerelativetoinflationandsatisfactorilycapturedintheconstantandstochasticdisturbance terms. It is customaryto substituteforwardrecursivelyin (1)andexpressitintermsofthecurrent pricelevel. Furtherimposingthetransversalitycondition(thatthe discountedexpectationof the limitingfuturepricegoestozero)yieldsthe.fina!amntal pricesolution, (2) it dependsupon the expectedsequenceof currentand futuremoneysupply levels and money demanddisturbances. The general pricesolution.whichwe denotep: . does not imposethe transversalitycondition. The set of generalsolutionsis Mlnite-dimensional;membersmaybe obtainedbyaddingany“bubble”processbtwhichsatisfi= bt=5 E(bt+llQt) (3) tothefundamentalpricesolution.Thebubblesmaybethoughtof as indexingthesolutions,with bt=Oforalltcorrespondingtothefundamentalsolution. Observedprice level data may be partitionedinto three components. They may behave accordingto thefundamentalsolution,mayalsocontaina nonzerobubblecomponent,and may containelementsinconsistentwiththegeneralsolution.Thesethreeelementsdefineanunobserved componentsidentity: whereStdenotesspectilcationerror(whichwe alsoreferto as modelnoise). It is thislastterm, andi~ implicationsfor estimatesandtests, whichhasbeenignoredin mostofthe Caganmodel 3

literature.DurlaufandHooker(1994)usedtransformationsof (4)anditsorthogonalityproperties underrationalexpectationstogeneratetestsofwhetherstandbtarenonzero. Thesetransformationsuse theperfectforesightfundamenti pricefirst employedby Shiner andSiegel(1977)inanothercontext: (5) j=o SinceE@YIQt)=~ , thedifferencebetwmnthefundamentalandperfectforesightfundamental priceis orthogonalto Qt. Denotingthatforecasterrorvt - ~ - p ~ and substitutinginto (4) yields Pt -P; = Vr+ bt +st, (6) thefirsttransformationof(4). Applyingtheforwardquasi-differenceoperator@- -(1-~ L-1) to(6)eliminatesthepredictablepartofthebubbletermandyields ?-~+=l m(pt-p; ) =@(vt ) + 5 ft+l + @(sf), (7) thesecondtransformationof(4);~t+listheinnovationinthebubbleprocess. Thetermrt+lhasthreecomponents,asmaybeseenfromtheright-handsideof (7). Thefirst twooftheseareorthogonaltoS2tbydefinition:@(u) is thedifferenceof aperiodtforecasterror and a (coefficienttimesa) periodt+l forecasterror, whilethe secondcomponentis the same coefficienttimesthenextperiod’sbubbleinnovation. me i-ma.clomponentis atransformationof the model’sspecflcationerror. It cannotbe orthogonalto Qf unless st itselfis, becausethe transformationisexactlytheinverseofthebubblegrowthrate—implyingthatifStis nonzeroand @(st)1 Qt, thenStgrowsattherateof (andthusis observationallyequivalentto) a bubble. In termsofthevariablesin(l), rt+lmaybe written 1 r~+l= — ~ffl Pt+l -Pt + * ‘t - G ‘t” (8) 4

A. Specification Tests The specificationtests are based on the orthogonalityproperti~ of pt - pf and rt+l.z Projectionsofpl - p f ontotime-tinformationsets, which arereferredto as stock tests, give memuresofbt + Sr. Projectionsof rl+lontothe sameinformationsets, calledflow tests, give measuresof(atransformationo~ modelnoise,@(st).Ifthecorrelationsofpt -p: andrt+lwith ‘-” theinformationsetsarebothstatisticallyindistinguishablefromzero,thenwemayconcludethat thepriceseriesobeysthefundamentalsolution. Ifthept -p; projectionisnonzerowhilethert+l projectioniszero,thatisevidencefor thepresenceof a bubble. If therl+lprojectionis nonzero, then thepr -pf projectionshouldbe aswell;inthiscasemisspecflcationis presentandfurther analysisisrequiredtodeterminewhetheranexplosivecomponentexistsin additiontost.3 These twotypesofprojections,therefore,permitdiscriminationbetweendifferenttypesofviolationsof thefundamentalpricesolution,and, as showninDurlaufandHooker (1994),containall of the time-seriesimplicationsofthegeneralandfundamentalntil hypotheses.4 Projectionsofpl -pf andrt+lcannotbedirectlyimplementedbecause&tis unobserved,and thesumin(5)isinfinite.Weworkwiththeirobservableanalogs: (9) and (lo) 2Itshouldbenotedifprivateagentsobservealargerinformationsetthandoeconometricians,thenpt-p; and rt+larenotnecessarilyorthogonaltoagents’informationsets,butareorthogonaltotheeconometrician’s informationset(Sargent1987p.334-335). 3Ifthepric~ differfromthegeneralsblutionbyanexplosiveamoun~wemightcallthatabubblebutnota rafiod bubble. 4Thatpapercontainsproofsof the consistencyof the tests, andconditionsunderwhichlaggedpricesmust k includedintheinformationsettodetectabubble. 5

thatis,weignoreEtandapproximate(5)bytruncatingthesum-s Thetestsusetransformationsof ht+l andpt -p; whichretaintheorthogonalitypropertiesofp~- p J andr~+lunderparametric wsumptionsonSt. Wechoosetwoimportantandcommonidentifyingassumptionsabout&r:first. thatitis equal . tozeroforallt,referredtoastheexactcase,andsecond,whereitfollowsarandomwalk(Et= ~t.l +utwithutL Qt-l). The exactspecificationhasbeenemployedbyGoodfriend(1982);therelated permanentincomeanddividend-stockpficemodelsareexactaswell. Whileitis probablyoverly restrictive,thiscaseprovidesa usefulbenchmark. Therandomwalkis thestandardspectilcation intheliterature,althoughitsuseiscontroversial.P.Evans(1978)givesaneconomicjusttilcation, whileseveralauthorsprovidestatisticalsupport(mostlybasedontheautocorrelationof residuals). Taylor(1991)hascriticizedthisspec~lcation,notingthatitimpliesthatpricesandfundamentals arenotcointegratedevenwhena bubbleis notpresent,whileChristian (1987)forceftilymakes thepointthat“decisionsabouthowtomodeldisturbancetermscanhave a signflcantimpacton parameterestimates.” In the exactcase, whereEt= Ofor all t, ht+l = rt+l, so no transformationis necessary. Similarly,pt -p; equalspl - p 7 upto thetruncationapproximation.Undertherandomwalk assumption,ht+lisnotnecesstily orthogonalto Qt, becauseitinvolvesEtwhichis afunctionof pastdata. However,theproblemmaybesolvedbydifferencing: Aht+lmaythenbe regressedagainstelementsof Qt-l totestfor spec~lcationerrorrelativeto the model’sgeneralsolutionwhenEtfollowsarandomwalk.GNotethatin therandomwalkcase,the 5Wealsoignoretheconstanttermin(10);inthetests,an unrestrictedconstanttermis includedin theinformation set. Whilepi isconstructedusingail availablemoneyandpricedata(Tis thelast observationavailable),in the teststrangesonlyacrossthepre-monetaryreformobservations.Regimeissuesarediscussedbelow. 6Intheparticularcasewherespecificationerrortoofollowsarandomwalk,thetestwillfailtodetectit—aswillany testbasedonthecovariancestructureofthedata-because itisobservationallyequivalenttothedisturbanminthe model(l). Oneimplication,stressedbyHamiltonandWhiteman(1985)andothers,isthatitisimpossibleto distinguishbetweenunobservablefundamentalswhichhavethesamestructureaseitherspec~lcationemoror bubbles.Theapproachtakeninthispaper,andinthemajorityoftheCaganmodelliterature,istomakeparametric errorprocessassumptionsandtotestthemaspartofthemodel. 6

informa~onsetmustbelagged,as contemporaneousvariablesarecorrelatedwithAht+lunderthe null. It shouldbe notedherethatmore general,but stillparametric,casescotid be handledin similarways. Forexample,ifAstfolloweda finiteorderMAprocess,theinformationsetwould needtobelaggedaccordingly,andifA&ftollowedanARprocess,then(11)couldbeappropriately .. quasi-differencedandthoseparametersestimatedalongwitha. Similarly,intherandomwalkcasept -p; isnotnecessarilyorthogonaltoQt,becauseittoo involvesEt.Againtheproblemmaybesolvedbydifferencing: . j Apt - Api =Avt+Abt+Ast -& L Ut+j (12) x( a-l ) j=O is orthogonalto Qt-l underthe fundamentalnuflhypothesis. Each of these projectionterms, pt -pi md ht+landtheirfirstdifferences,maythenbe constructedgivenestimatesofthemodel parameterstxandP. In studyinghyperinflationepisodes,the issue of monetaryregimechangesand coefficient instabilityarises. Onewayof dealingwiththisis tomodeltheregimes,as Blackburnand Sola (1992)dousinga Markovswitchingprocess. Wefollowthemajorityof theliteraturein simply truncatingthedatasamplesbeforeexpectationsof a regimechangebecomesignificant,withthe exceptionthatourpt -pi seriesareconstructedusingtheentiresample,includingpost-mone~ reformdata. Two lines of reasoningsupportthis practice: First, Flood and Hodrick (1986) demonstratethatiftheterminalvalueintheconstructionofpt - pi containsa bubble,thenthe bubblewillbeexactlycanceledoutandstocktestswillneverrevealit. However,inallcasesthe hyperinflationseitherendormoderate,so it is unlikelythata bubbleexistsintheterminalvalue. Second,theestimatesofa implyveryheavydiscountingoffuturevaluesofmoneyandprices:for instance,witha =-3thediscountfactorisonly0.75, whichweigh~ observations48monthsout by 10-6. By contrast,the discountfactorfor monthlydatais nearunity in most financeand macroeconomicsapplications.

Theempiricalrepresentationsofthept -pf andrt+lprojectionsontoQt arethenregressions ofthe objectsabove—pt-p; andht+lintheexactcase,andApt-Ap~ andAht+lintherandom walk case-on constants, lags of (differencesof) money and prices, and possibly other informationassumedto bein Qf. We estimatea and ~ via GMM.using the flowprojections themselvesas theorthogonalityconditions,andby instrumentalvariables. The tests are of the ‘-” Waldformfor thenull hypothesisthat allnonconstantregressioncoefficientsequalzero. It is importanttoincludelaggedprices,becauseifabubbleexistsandthemoneysupplyis exogenous, thenprojectionsonmoneymayindicateorthogonalitywheninfactthoselaggedprices-which are alsopartofQrare correlatedwithpt -pi or Apt- Ap~. InSectionIV,MonteCarloevidence ontheperformanceofthesetestsispresented. B. Model Noise Estimators Thebasicideaforestimatingmodelnoiseis to projectrt+lontoanempiricalmeasureof Q~to get an estimateof @(st),and then to undo the transformation. The inverseof the forwardquasidifferenceoperator0(.) givesthegeometricallydiscountedexpectedfuturesum of the operand, whichcanbesolvedusingexistingtechniquesandsmallmodflcations. Wederiveestimatorsfor boththeexactandtherandomwalkcases.7 1. ExactCase Intheexactcase,notethat r-t+l= ht+l=@(vt ) + ~ gt+l + @(~t)> (13) sothatbydefinition E[rt+j+lIQt]= E[(l -~ L-l)st I~t]. (14) Applyingtheinverseforwardquasi-differenceoperatorO-l(.) to bothsides (andexchangingthe orderofintegrationwiththeexpectationsoperator)yields 7TheideaofmeasuringmodelnoiseasanunobservedcomponentinasignalextractioncontextisduetoDurlanfmd Hall(1989).SeeKim(1996)foranapplicationtothepermanentincomehypothesis. 8

j=() Expression(15) is a vert[onof a Hansen and Sargent(1980)/ Sargent (1987, Ch X1.19) “predictinggeometricdistributedleads”problem. It is a vectorversion,sincetheinformationset .. Qt generallycontainsmorethanonevariable,andthefirsttermis nonstandard,as it beginsone periodforward. HansenandSargentderiveforrntdaefor expressionsme (15) in ~rms of tie elementsofQtforthecasewherethevariablesinQtfollowavectorautoregressiveprocess. Here wealsomakethatVARassumption. ThesecondandthirdtermsontheRHSof(15)arestandard.Lettingxt=[mfpt]’,A(L)xt= (1-AIL -~L2 ----- &Lr)xt = q beitstih orderautoregressiverepresentation,andI denotea conformableidentitymatrix; = 1[ 1 ~1 ~ + ~1 ~~(fi)~-~A~)L~ xr E ~ (fi)Jxt+j Ixf; Xl-l; ... A(5 (16) [mj=() “ j=l ~=j+l where 1(A ) = 1- Al(fi )-A(5 )2- ...-‘~(~ )r- TocomputethenonstandardfirsttermontheRHSof(15),let 1[ 1 Yt= ~ ~ (~~xr+j 1‘t; ‘t-l; .”” andzt=~ ~(fi~xt+j+,lxt;xt-,; ; [mj=() “ j=o thenyt= Xr+ A Zt,implying Zt = (Yt-xt)l(~ ). Thusthefirsttermcanbe obtainedas the secondelementof r-1 r 2(fi)-1 - I + A(fi)-l~ [ ~ (~)k-jlk] Lj) xt. (17) } j=l ~=j+l Summingtogetherthethreeterms,mdtipliedbytheirrespectivecoefficients,yields; t. 2. RandomWalkCase Intherandomwalkcase,andparallelto(14), 9

E(Ahf+lIQt.l) =E[A@(s~I)~f-1]> (18) sinceE[A@(vt)IQt-l]=EIA~t+lIQt-l]=E[ufIQf-l]=O.ThisimpliesthatAstcanbeestimated via w .-. A< =- (~YIEAht+j+l I~t-1] z j=() 00 =-~(~P{E[(~APt+l+‘*j-AA~Pt+tj+)ljQ~-l;l} (19) j=O thesetermsareconstructedwithestimatmof a from the associatedorthogonalityconditionand describedabove.Equation(19)isanotherHansen-Sargentpredictionproblem,solvedinthesame wayas(15);ityieldsanestimateofthefirstdifferenceofst. III. Discussion of Related Literature Thetestsoutlinedaboveweredesignedto combinetheinsightsof Hall (1978), thatmodels withexpectationsyieldorthogonalityconditionswhichcanbeusedas spectilcationtests, and of West(1987)andCasella(1989),thatacomparisonofthegeneralandfundamentalsolutionscodd yieldatestforbubbles.DurlaufandHall(1989)demonstratedhowto extractestimatesofmodel noiseintheexactcase,andshowedthatthevarianceofthenoiseseriesdefinesa lowerboundthat maybeusedin“variancebounds”tests. TheWestiCasellaprocedurecomparesestimatesof theparametera fromthemodel’sgeneral solutiontoanestimatefromthefundamen~ solutionin a Hausmantest. Intheabsenceofmodel noise, it consistentlyidentiles bubbles(althoughit does requirea spectilcationfor the money supplyseries). However,ifobservedpricescontaina nonzerost component,thenbothestimates of a maybe inconsistent,andthusthetestwillalsobe inconsistent. Othertestswhichinterpret deviationsfromthefundamentalsolutionasevidenceofbubblessharethisdrawback. 10

An interesting recent strand of the Cagan model literature which tests for model misspectilcationwasbegunbyTaylor(1991)andPhylaktisandTaylor(1992, 1993).8 Herethe fact that the fundamentalsolution of the model implies cointegrationrelationships(under reasonableassumptionsaboutthedatageneratingprocesses)is exploited. Thisapproachhas the advantagethatit placesrelativelymildrestrictionson expectationalerrors and money demand innovations(that each is stationary),whereas most of the Cagan model literatureincludes rationalityandparametricerrorprocessrestrictionsinthenullhypothesis. Therearepricesto bepaidfortheseadvantages,however. me mainproblemis thatboth stationarycomponentsof observed prices which do not fit the model, and nonstationary componentsunrelatedtomoneythatdofitthemodel(i.e. bubbles),mayexist. Testingthemodel via coirttegrationwill erroneouslyfail to reject in the former case, while in the latter, no transformationof prices(ormoney,if itis endogenousas is usuallythe case)will renderthem stationary,socointegrationtestsareunimplementable.In smallsamples,differencesofexplosive datamayappearstationaryandthusyieldmisleadingresults. G. Evans (1991)has shownthat somerationalbubbleseffectivelymimicstationaryprocesses, which may confoundthese two problems. Thedifficultyis illustratedby the factthatwhilethe cointegrationapproachis essentiallya formalizationofDibaandGrossman’s(1984)method,itis givena differentinterpretation.Diba andGrossmantestedfor anonstationarycomponentinthe responseseries (thepriceof gold in theirapplication)not also in the forcingvariable;Hamiltonand Whiteman(1985)appliedtheir procedureto the Cagan model and the German hyperinflation.9 However, since correct specflcationis amaintainedhypothesisin thesepapers,thesameevidencethatthe cointegration approachwouldconstrueasmlsspecificationis interpretedas a rationalbubble. Again,thetestis 8OtherpaperswhichtestthespecificationofthemodelincludeP.Evans(1978),whichestimatesARIMAmodels formoneyandpricesintheGermanhyperinflationandcomparestheprocessesobtainedtothosetheoretically impliedbyreasonableassumptionsontie datageneratingprocess,andSalemiandSargent(1979)andChristian (1987),whichuselikelihood-basedtests. OnlySalemiandSargentallowsforthepossibilityofbubbles,andthey alluseasymptoticcriticalvaluesintheirtests. 9Bothofthesepaperswerewrittenbeforethedevelopmentof cointegration.CampbellandShiner(1987)andDiba andGrossman(1988)notedhat geometricdiscount-expectationmsodelsofthistypeyieldcointe~ting relations. 11

strictlyunimplementableandthecaveataboveappliesifinfactthereisabubbleandanendogenous moneysupply. Thus, overall, we believe that the rational expectations/parametricapproach and the cointegrationmethodologyhave differentadvantagesand disadvantages,andthus areusefti as alternativeandcomplementaryprocedures. IV. A Monte Carlo Study of the Specification Tesfi ThespecificationtestsdevelopedinDurlaufandHooker(1994)areasymptoticallydistributed m %2randomvariables. Thereareseveralreasonsthatoneshouldbe concernedaboutthe ftite samplepropertiesofthesetesfi, andof hyperinflationmodeltess moregenerally. First,thedata samples tend to be quite short-for example, the German hyperinflation’s44 pre-reform observationsconstitutearelativelylongdataset.Second,theseriesarequiteexplosive;combining theknownlowpowerofunitroottestswithshortdatasamplesmeansthatthereis considerable uncertaintyabouttheappropriatedegreeof differencingor otherdetrendingprocedures,ARIMA modelspecifications,etc. Finally,theseandothertestscommonlyemploytwo-stageprocedures wherethesamplingerroroftheteststatisticitselfiscompoundedwiththatfroma previousstage’s parameterestimation. Inthesimulations,we generatedataaccordingto theexactandtherandomwalkversionsof theCaganmodelwith an exogenousmoneysupply. Themoneysupplyis assumedto be the particularARIMA(1,2,0)process from univariateestimationon the Germandatasetwith the parametersgiveninTable1. Fundamentalpricesarethencomputedfromthemoneysupplyseries via (2), using the Hansen-Sargentformulasfor predictinggeometticdistributedleads. The simulatedmoneysupplyand fundamentalpriceseries arethus both 1(2).10The innovationto moneydemanddisturbances(intherandomwalkcase)istakentobeastandardnormalvariate. Lneachreplication,thesamplelengthis 50 observationswith another50 used to construct (10);thesearesetroughlytocorrespondtotheGermandatasetwherethenumbersare44 and40, 10West(1994)suggeststhatit wotid be desirableto knowhow hyperinflationmodelandbubblestests perform whenthefundamentalequilibriumdatafollowanI(2)process. 12

respectively.Toreflectuncertaintyaboutthecorrectspecification,testsareperformedwherethe informationsetsconsistoffirstdifference andof seconddifference lagsofmoneyand prices. Weperformourtestson datawherepricesobeythe fundamenti solutionand fundamentalplus bubblessolutions.Twodifferentbubbletypesareconsidered,a simplefirstorderautoregressive processwhichgrowsattheappropriaterate,andtheperiodicallycollapsingbubblesstudiedinG. “-- Evans(1991). Todetermmesmall-samplecriticalvalues,wecomputethe90th,95th, and99thpercentilesof the empiricaldistributionsof the flow and the stocktests, for informationsets containingtwo throughfourlagsoffirstandseconddifferencesofmoneyandprices(seeTable1). Tenthousand replicationsare performedfor each Table 1 revealsthat the empiricaldistributionsdeviate signtilcantlyfromtheirasymptotic~zlimits. Intheexactcase,thefirstdifference informationset testsarequitepoorlybehaved,withatendencytoverylargeteststatisticsasthenumberof degrees offreedomrisesorthedistanceoutinthetailincreases. Thistendencyis exacerbatedinthestock test statisticsrelativeto the flow test statistics. With the (appropriate)second difference informationset,thetestsareconsiderablybetterbehaved,althoughin severalcasescriticalvalues aremorethanthreetimestheirasymptoticcounterpm. The same patternof deteriorationin the tails and as the number of degreesof freedom increasesholdsfortherandomwalktestsstatisticswithfirstdifference regressors. Thecritical valuefora IO~otestwithfourelementsintheinformationsetisabouttheasymptoticvalueof 13.3 inboththeflowandstocktests,buttheempiricalvaluesthenrangeup to twoandthreetimesthe asymptoticvalues.Curiously,theteststatisticsforthe randomwalkcasewithseconddifference regressorsareinmanycasessmallerthantheirasymptoticvalues,particularlyfor the 10%tests andlargerinformationsets. In theapplicationsinthenextsection,we focussomewhatmoreon thebetter-behavedexacttestswithseconddifference regressorsandrandomwalktestswithfirst difference regressors. Table2 reportstheresultsof power and TypeI error calctiationswhen a nonfundamental solutionobtains;herewe simtiate dataaddingbubblesto the fundamentalsolutionfor prices. 13

Each of these experimentsis performedfor three different actualvalues of a which are representativeofthoseestimatedintheliteratureandforonethousandreplicationsineachcase. In thefirst sectionof the table,a standardAR(1)processwith coefficient(u-l)/a is addedto the fundamentalpriceseries,andinthesecondsectionperiodicallycollapsingbubblesas studiedby G. Evans(1991)areadded. Thesebubblesare the sameas the AR(1)bubbleswhenthey are belowa threshold;oncetheycrossittheygrow atthefasterrate(a-1)/za withprobabilityn and crashtoameanlevelbelowthethresholdwithprobability(l-z).ll Thetestsuse the95%critical valuesand3lagsofseconddifference regressorsfortheinformationset. me flow andstockteststogetherarequitesuccessfulatdistinguishinga standardbubblein theI(2)data. In 75Y0to99Y0ofthesimulations,theflowtest(correctly)failsto rejectwhilethe stocktest (correctly)rejects. This comparesfavorablyto the simulationresults of Diba and Grossman(1988),whereunitroottestsareappliedto bubbleswhichareassumedto be directly observable.Thesizeis somewhattoosmall,however,withfewerthan5Y0of the flow statistics leadingto(false)rejection. me successofthetestsis bestillustrated,however,intheirabilityto detectthe periodically collapsingbubbles. G.Evans(1991)showedthatwhentheper-periodprobabilitythatthebubble does not crash, n, is less than 0.95, the BhargavaN1 test has virtuallyno abilityto de~ct collapsingbubblesandtheN2testscorrectlyidentifyfewerthan 12Y0of them.12 Furthermore, theseresultsassumethebubblestobedirectlyobserved. Thept - pl /rt+l tests, by contrast, successfullyidentifybubblesabout50Y0ofthetimewhenz is 0.95 (43Y0with a = -1, 47~0with IIInG Evm~(1991),tie bubbleshavenotrendinthesensethatthemeanleveltheyrem to aftera ~ash ‘d *e peaklevelstheyreachareconstantacrosstime(cf.hisFigure2). Whenaddingsucha bubbleto a stronglymded serieslikehyperinflationaryprices,thebubblebecomesnegligiblerelativeto thefundamentalsas timepasses. In thesimulations,weadaptEvans’bubblesbyhavingthemcrashbacktothelevelofthemoneysupply(whichshins thepricelevel’sstochastictrend)ratherthanaconstant.Thebubblesaregeneratedaccordingtotheformulas bt+l = [(a-1)/a]bt + qt+l ifb?<Tt;br+l= {mr+ [(a-1)/na]6t+I(bt - mt/(a-l)} + qr+l ifb~>Tz. whereqtisawhitenoiseshock f3ris an exogenousi.i.d. Bernoulliprocesstakingthevalue1withprobabilityz andOwithprobabilityl-z, andrtisafinite-variancedeviationfromrnf. Mostof theinterestingvariationin these bubblesoccursacrossvaluesofthetwoparametersvariedintie table,a andz. 12TheBhargavatestsusedbyDibaandGrossmanandEvansareunitroot testswhichallowforrejectionof a unit rootinfavorofeitheranexplosiveorastablealternative.Arejectioninfavorofthelatterwouldleadto acceptance ofthefundamentalsolution,sinceitwouldnotaffecttheorderofintegrationofmoneyorprices. 14

a =-3,and54%witha =-5)whiletheyareanunobservedcomponentofI(2)data. Remarkably, thepowerofthetrotsalsodoesnotdeteriorateasnfalls exceptinthea= -1case: witha= -3or- 5,thepercentageofcorrectrejectionsis between45% and55%acrossthefti rangeofvaluesof n. Thesizedoesrisesomewhatasz falls,with 10-20Yoof thebubblesinterpretedas noisewhen thebubblesarecrashingeveryotherperiodormore(n= 0.50or0.25). Thesuccessofthetestsin “-identifyingthese bubbles can be explainedas follows. Rationalbubbles have two salient characteristics:theyareexplosive,andtheyfollowa particulartime-seriespatterngivenby (3). TheBhargavaunitroottestfocuseson bubbles’explosiveness,and so missesthemwhen they crashoftenenoughtoappearstationary.However,thefrequentcrashesdonotsufficientlychange theautocomelationstructure,andsocanbedetectedbyorthogonalitytests. V. Empirical Results for Interwar Hyperinflation Data Havinginvestigatedtheperformanceofthetests,wenowturnto ananalysisofthedatafrom fourinterwarhyperinflationepisodes,inAustria,Germany,Hungary,andPoland. Thesourcefor tie data is Young (1925) (the data source used by Sargent (1986)), with the exceptionof Germany,whereweusethedatainFloodandGarber(1980). TheresultsreportedforGermany arethoseobtainedby DurlaufandHooker(1994)interpretedhereusing criticalvaluesfromthe MonteCarlosimtiations. For Austria,themoneyseriesconsistsof notes in circtiationand deposits,whilethe price indexis aretailpriceindexof 52commodities.Theirintersectionis availablefrom 1921:1-24:6; stabilizationwas achievedthroughinterventionof the Leagueof Nationsand the signingof im ProtocolsinOctober1922,butreactionsbeganinAugustofthatyear. Hencethedataemployedin thetestsrunfrom1921:1-22:7w, hilethedatafrom 1922:8-24:6areemployedintheconstruction of (10). For Germany,theavailabledatarun fromJanuary 1920untilDecember1926;the lmt datapointunaffectedby signKlcantexpectationsof a regimechangeis August 1923 (LaHaye 1985). Datafrom 1923:9throughthe end of 1926are used in the constructionof (10). For Hungary,theintersectionofmoneystock (currencyand deposits)and pricesruns 1921:7-25:3; 15

Sargent(1986)identifies1924:3asthereformdatesothatisusedasthefirstout-of-sampleperiod. Finally,thePolishmoneystockdataconsistof just currency;the intersectionruns 1921:1-24:4 withthereformdatetakenasJanuary1924. A. Specification Tests TheempiricalresulfiforAustriastronglyrejecttheCaganmodelwithrationalexpectationsin boththeexactandrandomwalkerrorspecifications.In theexactcase,reportedinTable3a, two oftheinformationsetsarecorrelatedwiththeht+lat the590levelandanotheris atthe 1070level, andalltheinformationsetsbutonearesignificantlycorrelatedwithpt -p; . The rejectionsofthe modelintherandomwalkcase, in Table3b, are even stronger. The standarderrors arequite large,particularlyfor theexactmodel,so thatin no casecouldthe nullhypothesisof a = Obe rejectedatthe5Y0level. ThereisnoevidencethatabubbleispresentinthedataforAustria. Durlaufand Hooker (1994) showed that using asymptoticcriticalvalues impliesstrong evidenceagainstthemodelforGermany;therestifi inTables4aand4busingtheempiricalcritical values are less clear-cut. In the exact specification,the better-behavedtes~ with second difference regressorssupporttheexistenceof a bubble,whileintherandomwalkcase,thereis someevidenceof misspecificationandno evidenceof a bubble. The estimatesof a are also considerablymoreprecisethanforAustria,withmanyofthet-ratiosabove2inabsolutevalue. TherestitsforHungaryaresimilartothoseforGermany:thereissomeevidencefor a bubble intheexactcase,andtherandomwalkspectilcationis stronglyrejected. However,thestandard errorsfora arelarge,withmostt-ratiosintherangeof-0.5to-1.5. The evidencefor Poland supports the emstenceof a rationalbubble. No significant specificationerrorisdetectedineithertheexactortherandomwalkcase,andwithtwoexceptions theestimateda valuesarein anarrowrange(-0.87to -2.08). Bycontrast,thefundamentalmdl hypothesisisrejectedfornearlyalloftheinformationsets. Theestimatesof a aregenerallymore precisethanforAustriaandHungary,butlessthanforGermany. 16

It is instructiveto comparetheseresultswithsomeof those obtainedby otherauthorswith similar datasets and specifications. Interestingly,the results here are quite different from Christiano’s(1987);he found that imposingthe random walk error assumptionled to large standarderrorsfor the a estimateswith Germandata. However,we do obtainlargestandard errorswiththeotherdatasefi,underbotherrorprocessassumptions. SalemiandSargent(1979)andTaylor(1991)testedtheCaganmodelwithsimilardatasewfor eachofthesefourepisodes.Theformerauthorsemployedthesamerestrictionsthatwe doin the randomwalkversionof themodel,althoughin aVAR/maxirnumlikelihoodframeworkandwith somedifferentassumptionsabouttrends. Taylor’stestis considerablyless restrictive,allowing deviationsfrom rationalexpectations(as long as the forecastingerrors are stationary)and any stationarymoneydemanddisturbanceas well. It shouldbenotedthatacceptanceof the random walkversionofthemodeliswouldcorrespondtoacointegration-approachrejection,althoughthis restdtdoes not obtainfor any of the datasew. Similarly,acceptanceof the modelusing the cointegration-approacchouldbeconsistentwithrejectionof eitherthe exactor the randomwalk version,duetotheexistenceofstationaryspecificationerror.l3 TheresultsforAustriaaresomewhatcontradicto~. In contrastto ourstrongrejectionofthe model,SalemiandSargentdidnotrejecttherandomwalkversion;theydid,however,findthatthe estimatesof a wereveryimpreciseandmostlywrong-signed. Taylor(1991)rejectedthe model forAustriandatausingresidual-basedcointegrationtestsbutaccepteditusingaJohansentest. SalemiandSargentrejectedthemodelforGermanyinmostcases;onlywith theirModel3 representationand second German dataset did they acceptthe restrictionsof the model at conventionalsignificancelevels. Taylorfoundthat residual-basedtests support the modelfor Germany,whiletheJohansentes~acceptsnon-cointegration. 13sale~i ~d Sagent (1979) useasymptoticcfiticd valuesin their analysis. I referto thefi tests of ratio~ expectationsbutnoterogeneityofthemoneysupply.Taylorusesboththe standardcointegrationcriticalvatua in EngleandYOU(1987)andsmall samplevaluesfrom BlangiewiczandCharemza(1990). Thereis considerable disagreemenatboutthethe smallsamplepowerofcointegrationtests,cf. Engleand-ger (1987),Hakkioand Rush(1991),andHooker(1993). 17

ContradictoryresultsarelikewiseobtainedbytheseauthorsforHungary.SalemiandSargent foundthatthemodelwithrationalexpectationsisrejectedinallcasesbutone;Taylorfoundthatthe residual-basedtestsprovideweakevidenceforcointegration,whiletheJohansentestacceptsnoncointegrationatthe5~0and10Yolevelsforhistwodatasets. Theresulmofthwe authorsaremostconsistentwithourresultsinthecaseofPoland. Salemi andSargentfoundthatthemodelfit thesedatawell, and obtainedestimatesof a and standard errorsverycloseto thoseintables6aand6b. Furthermore,theyfoundexplosiveelgenvaluesin eachof theirestimations(consistentwiththeexistenceof abubble). Taylorfound evidencefor cointegrationusingbothresidual-basedandJohansentests. Sincetheselattertests shouldreject fornonstation~, nonfundamentalpricecomponents,itmaybethatthebubbledetectedby Salemi andSargentandthepf -p; testswasoftheperiodicallycrashingvariety. B. Model Noise Estimates fitimates ofthemodelnoisecomponentpl are presentedin Figures2a-d. Sincethemoney andpricedataarequiteexplosive,theestimatednoiseis scaledtothepriceseriesandtheirratiois plotted. This is also the method recommendedby Kim (1996) for gauging the economic magnitudeof specificationerror. fitimates are constructed assuming a 6th order VAR spec~lcationfor the [mtPt] vectorin (16), usingthemeanestimatedvalueof a acrossthe first andseconddifference informationsets,respectively. ForAustria,thenoiseisinitiallyestimatedtocomprise12or20%ofthepricedata,according tothetwoestimates,fallingovertimesothatbyhalfwaythroughthedatasetitisnegligible.Itthen risesbriefly,andisagan negligiblebytheendofthesample.Thetwoestimatesarequiteparallel, withtheseconddifferencesinformationsetindicatinglessnoise. Thisis consistentwiththeflow testresults,whichrejectedmorewithfirstdifference informationsew. Theaveragesize of the noisecomponentislessthan109oofthepriceseries,suggestingthattheCaganmodelwithrational expectationsandarandomwalkerrormaybe a reasonablerepresentationofthedata. Thelarge 18

sizeofthenoisecomponentinthefirstfewmonthscorrespondsto relativelylow andstablerates ofinflation,wheretheCaganmodelmaybeapoorerdescriptor. ForGermany,noiseestimatedusingfustdifferencesisarelativelysteadyandlargefractionof prices,fluctuatingintherangeof 30-50%. Bycontrast,thenoiseseriesfromseconddifference datadropsoffsharplyandisnearzeroforroughlythelastyearofthehyperinflation.Again,these “-resultsareconsistentwiththeflowtestresults,andthenoisecomponenfiareagainlargestin the earlymonthswhenpricesarerelativelylow andstable. ThenoiseestimatesintheHungarydam aremuchsmaller, and exhibita less markeddrift over the sample. The second differences spectilcationgeneratesnoise estimatesthatare nearlyzerothroughoutthe sample,and the first differencesspetilcation yieldsnoiseabout10%of prices. Thissupportsthe modelas a useful datadescription. For Poland,bothspectilcationerror estimatesarein the 5-8% range, and not veryvolatile.Whilethef~stdifferencesestimaterisesabove10%inthelasttwoobservations,the generalsolutionofthemodelseemstoprovideacloseapproximationtothedata,strengtheningthe interpretationofthef~mdamentaslolutionrejectionsasevidenceofabubble. VI. Summary and Conclusions ThispaperhasanalyzedtestsoftheCaganhyperinflation-moneydemandmodelwhichhave severaladvantagesrelativetothoseintheliterature:Theydonotconfoundspecificationerrorwith rationalbubbles,theyareimplementablewitha linearprocedure,andthey are frequentlyableto detecttheperiodicallycollapsingbubbleswhichhavechallengedexistingtestsandinferences. A MonteCarloanalysisshowsthattheftite samplepropertiesof the tests maybe quitedifferent fromtheirasymptoticcounterpm, butthatthetestsmaycontainconsiderablepower. Theempiricalresultsshowstrongevidenceof modelmisspecificationfor Austria,whiletie modelwitharational,explosivebubblecomponentwell-characterizetshePolishdata. Inferences for Germany and Hungary are mixed, varying across spectilcationof the money demand disturbanceandtheinformationsetsusedinthetests. 19

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Goodfriend,M. S. (1982), ‘AnAlternativeMethodof fitirnating the Cagan MoneyDemand FunctioninHyperinflationUnderRationalExpectations’,Journalof MonetaV Economics, 9, 43-57. Hakkio,C. S. andM. Rush (1991), ‘Cointegration:How Shortis the Long Run?’. Journal of International Money andFinance, 10,5’71-81. Hall, R. E. (1978), ‘StochasticImplicationsof the Life Cycle-PermanentIncomeHypothesis: TheoryandEvidence’,Journal ofPolitical Economy, 86.971-987. .. Hamilton,J. D. and C. H. Whiteman(1985), ‘TheObservableImplicationsof Self-fulfiig Expectations’,Journal ofMonetaq Economics, 16,353-973. Hansen,L. P. andT. J. Sargent(1980), ‘Formtiatingand ~timating DynamicLinearRational ExpectationsModels’,Journal ofEconomic Dynamics and Control,2,7-46. Hooker,M. A. (1993), ‘Testingfor Cointegration: PowerversusFrequencyof Observation’, Economics Letters, 41, 359-362. Kim,C. (1996), ‘MeasuringDeviationsfrom the PermanentIncomeHypothesis’,Intematioti Economic Review ,37, 205-225. LaHaye,L.(1985),‘InflationandCurrencyReform’,Journal ofPolitical Economy, 93,537-560. Phylaktis,K. andM. P. Taylor(1992), ‘TheMonetaryDynamicsof SustainedHigh Inflation’, Southern Economic Journal, 58,610-622. Phylaktis,K.andM.P.Taylor(1993),‘MoneyDemand,theCaganModelandtheInflationTax: SomeLatinAmericanEvidence’,Review ofEconomics and Statistics, 75,32-37. Salemi,M. K. andT. J. Sargent(1979), ‘TheDemandfor MoneyDuringHyperinflationUnder RationalExpectations:II’,International Economic Review, 20,741-758. Sargent,T.J.(1986),Rational Expectations andInflation, Harper&Row, NewYork. Sargent,T.J.(1987),Macroeconomic Z’heory,Second Edition, AcademicPress,Orlando. Shiner,R. J. andJ. J. Seigel(1977), ‘TheGibson Paradoxand HistoricalMovementsin Real InterestRates’,Journal ofPolitical Economy, 85,891-907. Taylor, M. P. (1991), ‘TheHyperinflationModel of MoneyDemandRevisited’,Jouml of Money, Credit, and Banking, 23, 327-351. West, K. D. (1987), ‘A SpecKlcationTest for SpeculativeBubbles’, Quarterly JoumaZ of Economics, 102, 553-580. West, K. D. (1994), ‘RationalBubbles During Poland’s Hyperinflation: Implicationsand EmpiricalEvidence:Comment’,European Economic Review, 38, 1282-1285. Young,J. P. (1925),for theCommissionof GoldandSilverInquiry,theUnitedStatesSenate, European Currency and Finance, Volumes 1 and 2, U.S. GovernmentPrinting Office, Washington. 21

Table 1: Critical Values for Wald Flow and Stock Test Statistics A. ExactModelCase.1stDifference Regressors Urf Ws m w w m w M Z-1agin~set 12.97 19.87 95.90 48.59 88.81 1016.44 3-lagin$set 23.98 39.65 311.29 102.60 243.86 9542.24 4-laginfiset 40.11 68.80 797.49 199.39 608.77 44182.82 B. ExactModelCase.2ndDifference Regressors Wf Ws m m m w m m 2-lagin$set 10.58 15.14 36.59 17.79 27.34 55.37 3-laginfiset 18.23 26.32 56.24 30.13 45.41 87.61 4-laginfiset 26.85 37.56 75.80 46.03 66.88 142.57 c< WalkCase.1stD~ 2-laginf sel 14.62 20.23 33.92 17.34 22.90 37.28 3-laginfiset 19.44 26.59 53.70 21.98 29.96 57.80 4-laginfiset 22.78 31.52 69.01 25.66 35.43 82.52 2-lagin. set 11.14 14.80 24.70 13.35 18.23 29.98 3-laginfiset 7.91 11.48 31.19 8.91 13.28 32.45 4-laginJset 6.37 9.46 21.47 7.46 10.98 25.90 Notes: Simulationsbasedon 10000replicationsof length 50 samples (witb an additional50 out-of-sample observationsto constructthe stocktest). The money supply is an exogenousARIMA (1,2,0) processwith autoreg~sive coefficient0.74, driftequalto 0.02, and innovation variance0.12 (estimatedfrom tbe ~ univariatedata). Thefundamentalpriceseriesis constructedaccordingto equation(2) in tie text using Hansen- Sargentpredictionformtiasfortie expectationalterms. “n-Zaginformationset’refm to thecontemporaneousplus n-1lagsof eitherfirstor seconddifferencesofbotbmoneyandpricesin theexactcase,andn lagsin therandom walkcase;e.g.the2 lagsiseconddifferencesinformationsetforthe exactcasecontainsA2mt, A2pt,A2rnr-1,@ Azpt-l. ~1Ses~ated Using2SLSwiththeinformationsetm ins~en~.

Table 2: Power and Twe I Error in Data with Bubbles Power: YOofre~licationswhich Twe IError: %ofreplicationswhich acceptflowandrejectstocktest re’lectflowtest “Stan&rd”Bubbles ~=- I a= -3 ~=- 5 ~=- 1 a= -3 M 75.37 99.79 99.38 5.26 0.21 0.62 PeriodicallyCollapsingBubbles mm ~ = -5 ~ = -1 ~. -3 a =-5 z = .999 94.30 98.02 98.23 0.00 0.62 0.83 n = .99 70.47 81.64 82.56 0.48 2.08 3.41 n = .95 43.43 47.26 54.18 0.66 5.69 7.43 n = .85 33.02 45.40 47.23 1.20 5.23 7.98 n = .75 31.89 47.08 48.59 2.31 5.41 9.95 n =.50 29.56 47.02 51.11 9.12 12.34 12.04 K= .25 29.72 54.39 53.01 18.16 16.58 18.37 Notes: Simulationsbasedon1000replicationsoflength50samples(withanadditional50out-of-sample observationstoconstructthestockte~t).SeeTable1~orconstru~tionofthemoneyandfundamenmlpriceseries. Therandomwalkcase,f~stdifference/threelagsinformationisused;a isesdrnatedusing2SLSwiti that informationsetasinstruments.ThestandardbubblesareAR(1)proasses withcoefHcient(cc-l)/aandshocks NID(0,.12);theperiodicallycollapsingbubblesaremotiled versionsofthoseinEvans(1991)describedinfmmote 10ofthetext.

Table 3a: Estimates and Tests for Exact Model Specification. Austria Znfomtion Set: FirstDi#eremes M– ~s– 2lags -3.37 6.38 4.80 67.31* (14.34) (18.05) 3lags -5.41 6.73 60.71** 842.29** (30.83) (26.21) 4lags -5.37 6.73 74.12** 3019.59** (31.05) (26.32) SecondDifferences ~ L m. ws– 2lags 0.11 5.64 0.16 1.18 (1.53) (9.41) 3lags -2.14 6.07 53.54** 54.98** (9.26) (15.98) 4lags -2.48 6.10 24.61 57.51* (10.55) (16.27) Notes: Standarderrors,consistentforheteroscedasticitym, parentheses.hformationsetsconsistofthespecified numberofcontemporaneoupsluslagsofeitherfirstorseconddifferenc=ofbothmoneyandprices(e.g.the2 lags/seconddifferencesinformationsetcontainsA2rntA, 2pf,A2rnt.1a,ndA2pr.1).~~and Ws areWaldstatisti~ fortheflowandstocktesw;mymptoticallyunderthenulltheyhaveaX2distributionwithdegreesoffreedomequal tothenumberofelementsoftheinformationset(twotimesthenumberoflags);empirid criticalvaluesfrom simulationsonhyperinflationdataaregiveninTable1. *,**,and***denotesignificanceatthe10,5,andIYo levels.

Table 3b: Estimates and Tests for Random Walk Model Specification. Austria Infomtion Set: FirstDifferewes & m– Ws 2lags -0.29 3.73 2.22 (0.32) 3lags -0.82 65.45*** 70.84*** (0.47) 4lags -1.00 27.43* 400.73*** (0.71) SecondDifferences Ws 2lags -0.53 75.67*** 71.31*** (0.52) 3lags -0.59 66.78*** 66.25*** (0.66) 4lags -0.35 283.69*** 139.45*** (0.29) Notes: Standarderrors,consistentforheteroscedasticityi,nparentheses.Informationsetsconsistofthespecified numberoflagsofeitherfwstorseconddifferencesofbothmoneyandprices(e.g.the2lags/seconddifferences informationsetcontains42m1.1,Azpt-l,A2mt-2a,ndA2pf-2).Wf andWs areWaldstatisticsfortie flowand stocktests;asymptoticallyunderthenulltheyhaveaX2distributionwiti degreesoffreedomequaltothenumberof elementsofrheinformationset(twotimesthenumberoflags);empiri~ criticalvaluesfromsimtiationson hyperinflationdataaregiveninTable1. *,**,and***denotesi~n~l~ce atthe10,5,andl~olevels.

Table 4a: Estimates and Tests for Exact Model Specification, Germanv InformationSet: FirstDifferences & L M– Ws 2lags -4.21 1.59 15.51* 58.36* (1.83) (0.35) 3lags -4.20 1.62 27.74* 85.49 (1.76) (0.36) 4lags -4.36 1.68 47.80* 77.64 (1.86) (0.38) SecondDifferences & & M– Ws 2lags -1.07 1.15 0.66 48.75** (5.74) (21.79) 3lags -1.45 1.15 7.19 170.93*** (0.69) (0.28) 4lags -2.26 1.15 32.64* 175.96*** (1.08) (0.31) Notes:Standardemrs, consistentforheteroscedasticitym, parentheses.Informationsetsconsistofthespecified numberofcontemporaneoupsluslagsofeitherfwstorseconddifferencesofbothmoneyandprices(e.g.the2 lags/seconddifferencesinformationsetcontains42mt,A2pr,A2mt.1,andA2P1.1).WfandWs areWaldstatistics fortheflowandstocktests;asymptoticallyunderthenufltiey haveaX2distributionwithdegreesoffreedomequal tothenumberofelemen~oftheinformationset(twotimesthenumberoflags);empiricalcriticalvaluesfrom simulationsonhyperinflationdataaregiveninTable1. *,**,and***denotesignificanceatthe10,5,~d 1~0 levels.

Table 4b: Estimates and Tests for Random Walk Model Specification. Germanv Inforrn4rtioSnet: FirstDifferences & m– Ws 2lags -1.30 8.05 12.48 (0.96) 3lags -0.79 28.46** 24.15* (0.24) 4lags -0.80 30.30* 24.57 (0.18) SecondDifferences Q w- Ws 2lags -0.75 14.12* 6.63 (0.27) 3lags -0.74 14.29** 12.81* (0.54) 4lags -1.03 26.49*** 56.84*** (0.21) Notes: Standarderrors,consistentforheteroscedasticity,inparentheses.Informationsetsconsistofthespecified numberoflagsofeitherFrosotrseconddifferencesofbotbmoneyandprices(e.g.the2lags/seconddifferences informationsetcontainsA2?nr.1A,2pr-1,A2mt-2,andA2pr-2).~Tfand Ws areWaldstatisticsfortheflowand stocktests;asymptoticallyunderthenulltheyhaveaX2distributionwithdegreesoffreedomequaltothenumberof elementsoftheinformationset(twotimesthenumberoflags);empiricalcriticalvaluesfromsimtiationson hyperinflationdataaregiveninTable1. *,**,~d ***denotesignflcaw atthe10,5,ad 170levels.

Table 5a: Estimates and Tests for Exact Model Specification, Hungary InformationSet: m– &.– FirstDifferences 2lags -7.91 1.81 16.28* 110.42** (10.25) (0.69) 3lags -8.08 1.79 17.24 41O.73** (10.26) (0.66) 4lags -6.29 1.45 17.85 451.74* (6.23) (0.52) SecondDifferences k ~- w– %–– 2lags -1.29 0.94 2.88 0.65 (1.43) (0.37) 3lags -1.24 0.88 3.49 1.10 (1.38) (0.35) 4lags -1.30 0.88 6.72 7.18 (1.48) (0.35) Notes: Standarderrors,consistentforheteroscedasticityi,nparentheses.Informationsewconsistofthespecified numberofcontemporaneoupsluslagsofeithert-ustorseconddifferencesofbothmoneyandprices(e.g.the2 lags/seconddifferencesinformationsetcontainsA2mt,A2pt,A2mt.1,andA2pr-1).Wfand Ws areWaldstatistics fortie flowandstocktests;asymptoticallyunderthenulltheyhaveaX2distributionwithdegreesoffreedomequal tothenumberofelementsoftheinformationset(twotimesthenumberoflags);empiricalcriticalvaluesfrom simulationsonhyperirdlationdataaregiveninTable1. *,**,and***denotesignificanceatthe10,5,ad IYo levels.

Table 5b: Estimates and Tests for Random Walk Model Specification. Hungary InformationSet: w– FirszDifferences Ws 2lags -1.68 27.57** 11.14 (1.05) 3lags -1.31 28.62** 11.62 (0.84) 4lags -1.02 50.87** 22.73 (0.49) SecondDifferences & m- Ws 2lags -0.78 21.34** 9.86 (0.68) 3lags -0.78 50.09*** 17.61** (0.49) 4lags -1.05 34.00*** 15.61** (0.59) Notes: Standarderrors,consistentforheteroscedasticityi,nparenthes=. Informationsetsconsistofthespecified numberoflagsofeitherf~storseconddifferencesofbothmoneyandprices(e.g.the2lags/seconddifferences informationsetcontainsA2mt.1,A2pI-1,A2mt-2,~d A2P1-2).Wf andWS MeWdd statisticsfortheflow~d stocktests;asymptoticallyunderthenulltheyhaveaX2distributionwithdegreesoffreedomequaltothenumberof elementsoftie informationset(tsvotimesthenumberoflags);empiricalcriticalvaluesfromsimtiationson hyperinflationdataaregiveninTable1. *,**,~d ***denotesign~lcanceatthe10,5,ad l~olevels.

. Table 6a: Estimates and Tests for Exact Model Specification. Poland InformationSet: FirstDifferemes & L m– Ws 2lags -1.88 1.33 4.67 89.83** (1.16) (0.35) 3lags -1.95 1.37 6.91 131.01* (1.17) (0.35) 4lags -1.98 1.38 5.11 153.16 (1.21) (0.36) m- Ws SecondDifferences & & 2lags -1.97 1.38 1.15 20.40* (1.51) (0.40) 3lags -2.08 1.41 6.25 104.89*** (1.52) (0.39) 4lags -1.86 1.33 7.46 89.22** (1.24) (0.35) Notes: Standarderrors,consistentforheteroscedasticityi,n parentheses. Informationsetsconsistofthe specified numberof contemporaneousplus lagsof eitherfnst or seconddifferencesof both money andprices(e.g. the 2 lags/seconddifferencesinformationsetcontains42mt,A2Pt,A2mt-lTandA2PI-1).Wfmd Ws MeWddstatistics fortheflowandstocktes~;asymptoticallyunderthenulltheyhaveaX2 distributionwith degr= oftiedom equal to thenumberof elementsof theinformationset (twotimes thenumberof lags); empiricalcriticalvaluesfrom sirnuhtionsonhyperinflationdataaregivenin Table1. *, **, ~d ***&notesign~lcaneeatthe 10,5, md IYo levels.

Table 6b: Estimates and Tests for Random Walk A40del Specification, Poland [formation Set: FirfiDifferences & w- Ws 2lags -0.87 5.27 9.82 (1.22) 3lags -1.28 11.10 19.69 (1.18) 4lags -5.94 4.57 44.78** (8.90) SecondDifferences & m- Ws 2lags -0.87 8.00 3.37 (1.09) 3lags -1.03 4.33 7.52 (1.15) 4lags -6.50 3.73 26.30*** (9.96) Notes: Standarderrors,consistentforheteroscedasticity,inparentheses.Informationsetsconsistofthespecified numberoflagsofeitherfwstorseconddifferencesofbothmoneyandprices(e.g.the2lags/seconddifferences informationsetcontainsA2mt.1,A2pr-1A, 2mr-2a,ndA2pt-2).W“fandWs areWaldstatisticsfortheflowand stocktests;asymptoticallyunderthenulltheyhaveaXzdistributionwithdegreesoffreedomequaltothenumberof elementsoftheinformationset(twotimesthenumberoflags);empiricalcriticalvaluesfromsimtitions on hyperinflationdataaregiveninTable1. *,**,and***denotesign~leanceatthe10.5.and1%levels.

Figure la: MonthlyGrowthRatesofPriceLevel andMoneySupply,Austria 14070 120% 100% 8090 60% \ \ \ \ 40% \ \ \ 20YC Figure lb: MonthlyGrowthRatesofPrice LevelandMoneySupply,Germany (1 100% I I I 80~0 I I I ~— Pries I 1—---Money] I 6070 I I I I 40% I i I I \ 20% l/\ \ \ -20%1

FigureIc: MonthlyGrowthRatesofPriceLevelandMoneySupply,Hungary 100% 80q0 60Y0 40% I -20% 1 Figure Id: MonthlyGrowthRatesofPriceLevelandMoneySupply,Poland 300% 250% 2o090 150% \ 100% \ \ \ \ \ \

I I Figure2a: RatioofSpecificationError toPriceSeries,Austria -0.051 Figure2b: RatioofSpeculation Error toPriceSeries,Germany 0.50 0.40 0.30 “\ 0.20 —Estimated witlrfirstdifference information set ~ -- --%tirnated witfrswond differencerl information set 0.10 ‘-- \ \ \ 0.00 -0.10 -0.2C

i Figure2c: RatioofSpecificationError toPriceSeries,Hungary Figure2d: RatioofSpecificationErrortoPrice Series Poland 0.18 0.16 [ I — tima~ withfirstdifference informriuonsti : 0.14 —-—-Estimated witi second difference information set I 0.12 0.10 0.08 0.06 O.w O.M 0.0(