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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(velocity
shock)
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.)

Austria
Feb. '21 to Aug. '22

19

-3.09
(1.22)

.92
(1.18, 1.40)

.982

.094

.149

.907
(.409)

-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)

1.319

.315

= 0

17.99
co

-2.50

.9455

.205

-4.76

.098

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

x2
(1)

.104

.265
(.556)

SSRU
.139

=t

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)

.

A
Pl
.55(.24ja
.55(.28)
.58(.27)

$2

A
P3
-.48(.31)

-.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)

J=(3)

.00(.29)
.23(.32)

.06(.20)
.07(.20)
.08(.21)

=w)

-.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

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JournalofPoliticalEoncxqy
78 (Novanber/December 1970): 1228-63.
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cagan,Philxp.
in&
Quantity%eoryofmney,p~.
25-117. EklikdbyM.Riedmm.
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&I:

andKincaid,Gmqe.
CQmrsnt." WC

"Jacobs' Estimates of the Hyperinflaticm
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&uiq

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27
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