Chen, Zhengyang; Valca cel, Vic o J.
A icle — Published Ve sion
A g anula in es iga ion on he s abili y o money demand
Mac oeconomic Dynamics
Sugges ed Ci a ion: Chen, Zhengyang; Valca cel, Vic o J. (2024) : A g anula in es iga ion on he
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Mac oeconomic Dynamics (2024), 1–26
doi:10.1017/S1365100524000427
ARTICLE
A g anula in es iga ion on he s abili y o money
demand
Zhengyang Chen1and Vic o J. Valca cel2
1Da id W. Wilson College o Business, Uni e si y o No he n Iowa, Ceda Falls, IA, USA
2School o Economic, Poli ical and Policy Sciences, Uni e si y o Texas a Dallas, Richa dson, TX, USA
Co esponding au ho : Vic o J. Valca cel; Email: ic o . alca [email p o ec ed]
Abs ac
A la ge li e a u e has shown money demand unc ions cons uc ed om simple-sum agg ega es a e uns a-
ble. We e isi he con o e sy su ounding he ins abili y o money demand by examining coin eg a ing
income-money ela ionships wi h he Di isia mone a y agg ega es o he U.S., and compa e hem wi h
hei simple-sum coun e pa s. We inno a e by conduc ing a mo e g anula analysis o a ious mone a y
asse s and hei associa ed use cos s. We ind cha ac e izing money demand wi h simple-sum measu es
only wo ks well in a pe iod p eceding 1980. Di isia agg ega es, hei componen s, and hei use cos s p o-
ide a mo e eliable in e p e a ion o money demand. Subsample analysis ac oss 1980 and 2008 sugges s
he ins abili y o money demand is a ma e o measu emen a he han a consequence o a s uc u al
change in agen s’ p e e ence o mone a y asse s.
Keywo ds: Money demand; Di isia mone a y agg ega es; coin eg a ion es s; bank deposi s
1. In oduc ion
The s abili y o money demand has ecei ed g ea a en ion o e he yea s because i is consid-
e ed c ucial in o ma ion o mone a y policy de e mina ion. Success ully pinning down money
demand shocks makes he job o a ge ing in e es a es easie , he eby acili a ing he cen al
bank’s mac oeconomic objec i es. Fu he mo e, mone a y agg ega es had also been conside ed
in he pas as po en ial a ge s o mone a y policy. A en ion o money demand in he ield o
mone a y economics began o ade in he ea ly 1990s wi h he publica ion o in luen ial mone-
a y pape s ha ocused exclusi ely on in e es a es. Relying on he indings ha he p e iously
s able ela ionship be ween simple-sum money and o he key mac oeconomic a iables began o
weaken d ama ically a e 1980, F iedman and Ku ne (1992) and Be nanke and Blinde (1992)
show ha changes in he ede al unds a e encapsula e he e ec s o mone a y policy, wi h money
balances mo ing endogenously wi h high, and di icul o p edic , a ia ion.
To a la ge ex en , in o ma ion om mone a y agg ega es os ensibly exi ed mone a y modeling
since he ea ly 1990s. Impo an esea ch has success ully endea o ed o—as Leepe and Roush
(2003) pu i —“b ing money back in mone a y policy” since hen. A g owing numbe o s uc-
u ally sound mone a y models wi h Di isia money ha e eme ged in mo e mode n imes. We
ha e in mind impo an wo k in money demand de e mina ion by Se le is (1991), Se le is and
Gogas (2014), Jadidzadeh and Se le is (2019), and Ba ne e al. (2022). And o mone a y policy
iden i ica ion, con ibu ions oo nume ous o lis by Belongia and I eland [see he ecen wo k
in Belongia and I eland (2016) and Belongia and I eland (2018)] as well as Kea ing e al. (2019).
Howe e , o da e, he majo i y o mone a y esea ch seems o con inue o dismiss he in o ma ion
C
The Au ho (s), 2024. Published by Camb idge Uni e si y P ess. This is an Open Access a icle, dis ibu ed unde he e ms o he
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non-comme cial e-use, dis ibu ion, and ep oduc ion in any medium, p o ided ha no al e a ions a e made and he o iginal a icle is p op-
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a icle.
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
2 Z. Chen and V. J. Valca cel
ha mone a y agg ega es b ing o economic modeling, pe haps clinging o he memo y o he
conclusions su ounding he ins abili y o money demand ha eme ged since 1980 in he U.S.
Two p ongs o esea ch ha e in es iga ed he b eakdown in he s abili y o money demand.
A se o pape s inds he ins abili y o he money demand unc ion is associa ed wi h he sam-
ple pe iod. Fo ins ance, Ball (2001) ex ends he pos -wa sample pe iod o S ock and Wa son
(1993) om 1987 o 1996, and ejec s a s able long- un demand o he con en ional M1 agg e-
ga e, which was claimed by he o me pape . Judson e al. (2014) obse e b eakdowns o he
money demand unc ion in he ea ly 1990s and la e 2000s. Many pape s p o ide es s o s uc-
u al b eaks. G ego y and Hansen (1992) and B eue and Lippe (1996) a e among he i s o
es he s uc u al shi in he coin eg a ion be ween money agg ega es and o he economic a i-
ables. Choi and Jung (2009) de ec wo s uc u al b eaks in he sample om 1959 o 2000, and
o he wise ind a s able in e es elas ici y o money demand in e e y subsample sepa a ed by hose
wo s uc u al b eaks. These s udies examine money demand wi h da a om he Fede al Rese e’s
con en ional ( ypically e e ed o as simple-sum) mone a y agg ega es.
Ano he line o pape s ad ances ha a diminished connec ion be ween money and i s oppo -
uni y cos can be a ibu ed o measu emen e o . Lucas and Nicolini (2015)p oposeanew
agg ega e, which ea s cu ency, demand deposi s, and money ma ke deposi accoun s (MMDA)
as al e na i e means o paymen s. Teles and Zhou (2005) use simple-sum M1 be o e 1980, and
money wi h ze o ma u i y (MZM) a e 1980 as he measu e o ansac ion demand o money.
Bo h new measu es ind a mo e s able long- un ela ionship among money, p ices, a measu e
o economic ac i i y, and a nominal in e es a e. Belongia (1996) and Hend ickson (2014) e-
examine p e ious empi ical models ha ind uns able money demand om simple-sum money
by eplacing he la e wi h he Di isia mone a y agg ega es p oposed by Ba ne (1980). They
ind obus links be ween money g ow h and in la ion, nominal income g ow h, as well as he
ou pu gap.
Closely ela ed o ou wo k, Se le is and Gogas (2014) and Belongia and I eland (2019) adop
aJohansen(1991) amewo k o iden i y signi ican coin eg a ing ela ionships be ween Di isia
mone a y agg ega es and measu es o oppo uni y cos s. We ollow hei app oach and expand
i wi h he Johansen (1995) coin eg a ion analysis wi h es ic ions on de e minis ic ends. We
con ibu e o he analysis o agg ega e money demand es ima ion by inding s able ela ionships
wi h Di isia unde a la ge se o coin eg a ing speci ica ions han p e iously modeled.
Speci ically, we answe wo ques ions by es ing coin eg a ing ela ionships be ween eal money
balances, scaled by nominal ou pu , and he oppo uni y cos s o holding money. Fi s , a e
s uc u al b eaks and he ins abili y o he money demand unc ion endemic in con en ional
simple-sum agg ega es? Second, a e he Di isia mone a y indexes indeed immune o hose b eaks;
and i so, can a mo e g anula in es iga ion on he componen s o he Di isia agg ega e—and hei
associa ed p ice duals—shed ligh on he (in)s abili y o money demand?
To answe hese wo ques ions, we compa e he pe o mance o simple-sum M2 and M3 wi h
Di isia M2 and Di isia M3 in hei abili y o p ope ly cha ac e ize money demand. Fi s , we
compa e he elas ici y o money demand wi h espec o wo measu es o oppo uni y cos s o
holding money: he h ee-mon h T easu y-bill a e (hence o h, T-bill yield) and he eal use cos
o Di isia.1We conside he use cos s associa ed wi h he Di isia M2 and Di isia M3 agg ega es
cons uc ed by Ba ne e al. (2013). We begin om wo p emises: (i) Di isia indices a e in o -
ma i e and (ii) he e is addi ional in o ma ion con en in Di isia use cos s sepa a e om Di isia
agg ega es. We look o s a is ical ela ionships be ween eal use cos s and mone a y asse s.2
Beyond he s a is ical ad an ages we conside , Belongia (2006) and Belongia and I eland (2019)
a gue o a mo e heo e ical in e p e a ion. They ad ance ha when in e es e u ns on mone a y
asse s a e non- i ial, he use cos is a mo e heo e ically cohe en measu e o he oppo uni y
cos han a sho - e m nominal in e es a e. On he o he hand, Se le is and Gogas (2014) es ima e
long- un money demand ela ionships o Di isia quan i y agg ega es wi h he h ee-mon h T-bill
yield.
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
Mac oeconomic Dynamics 3
Admi edly, a heo e ical connec ion be ween he T-bill yield and Di isia agg ega es (o hei
componen s) is mo e enuous han ha o eal use cos s. Thus, Belongia and I eland (2019)make
a pe suasi e case on heo e ical g ounds. Bu we also hink Se le is and Gogas (2014) make a con-
incing case on s a is ical g ounds. F om a s a is ical pe spec i e, a educed- o m analysis o he
ela ionship be ween he holding cos s o holding mone a y asse s and he quan i ies o such asse s
can be open mo e gene ally o a ious in e es a es as is emphasized by Cagan (1956).3The e o e,
we emain agnos ic in ou empi ical sea ch be ween conside ing use cos s as Belongia and I eland
(2019) do, o conside ing he T-bill a e as in Se le is and Gogas (2014), and we indeed e i y s a-
is ically ha eal use cos s bea s onge coin eg a ing ela ionships wi h money demand (scaled
by nominal income) han a mo e ypical sho - e m in e es a e, such as he T-bill yield.
Second, we ocus on a po en ial s uc u al b eak su ounding he second qua e o 1980, when
de egula ion on in e es -bea ing checking accoun s led o many inancial inno a ions in e ail
banking. We hen expand analysis o he b oade agg ega es o Di isia M3 and Di isia M4 o a
second po en ial b eak a ound he G ea Financial C isis (GFC) o 2007 and i s a e ma h. Finally,
we documen he coin eg a ing ela ionships be ween quan i ies demanded o indi idual mone-
a y asse s and hei associa ed use cos s. This analysis is impo an as i acili a es iden i ying
which mone a y asse s con ibu e mos o he (in)s abili y o money demand—and i can be con-
duc ed hanks o he g anula i y o use cos da a o each ype o mone a y asse ha is p o ided
by he Cen e o Financial S abili y (CFS). Ou esul s indica e ha he use cos s o Di isia ca y
in o ma ion con en , no only o he agg ega e indices bu also o hose associa ed wi h hei
componen s.
The es o his pape is o ganized as ollows. Sec ion 2p o ides a b ie backg ound on Di isia
agg ega ion. Sec ion 3ou lines he me hodology we employ o es o long- un coin eg a ing
ela ionships. Sec ion 4in oduces he da a and conduc s uni oo es s o he se ies we conside .
Sec ion 5p esen s he main esul s o he s udy. Sec ion 6ex ends he analysis by en e aining he
possibili y o a s uc u al b eak in 1980. Sec ion 7in es iga es whe he he in o ma ion con en o
b oade (M3 and M4) Di isia agg ega es is diminished by he GFC o 2007. Sec ion 8p o ides a
mo e g anula in es iga ion by ex ending he in es iga ion o an analysis o Di isia componen s.
Sec ion 9concludes.
2. Backg ound on Di isia mone a y agg ega es and he eal use cos s
We conside Di isia mone a y agg ega es and hei p ice duals—o Di isia eal use cos s—as
al e na i e measu es o mone a y quan i ies and oppo uni y cos s o holding mone a y asse s.
All se ies a e epo ed a mon hly equencies. Di isia quan i y and p ice indexes a e undamen-
ally di e en om simple-sum mone a y agg ega es and ma ke sho - e m in e es a es in hei
cons uc ion.
Da a a ailabili y by he CFS dic a es ou sample spans Janua y 1967—Ma ch 2020. The e a e
impo an ins i u ional easons why we s op ou analysis in 2020. The Fede al Rese e imple-
men ed a sudden and unp eceden ed ede ini ion o he money supply a ha ime. His o ically,
M1 had been a na ow and ela i ely liquid mone a y agg ega e. M2 and M3 had adi ionally been
b oade and less liquid agg ega es. Beginning Ap il 2020—bucking o e 100 yea s o adi ion—
he Fede al Rese e began including sa ings deposi s as pa o M1. I also began amalgama ing
o he ime deposi s in ha measu e. Cons ained by his impo an s uc u al change in measu e-
men s anda ds, he CFS began a di e en accoun ing: Sa ings and OCD ( h i s and comme cial)
and MMDAs ( h i s and comme cial) began o be added o sa ings deposi s and we e assigned
a single use cos unde he new designa ion ‘O he Liquid Deposi s.’ This po en ous change
complica es he g anula analysis in his pape .
One way o add ess his ecen b eak in measu emen s anda ds is done by Isakin and Se le is
(2023). They begin wi h hese new de ini ions o mone a y asse s as he componen s o he
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
4 Z. Chen and V. J. Valca cel
agg ega es and hen ex apola e backwa ds by a e aging he use cos s o he a ious disagg ega ed
componen s in o he newly de ined asse s (e.g. O he Liquid Deposi s).
Models a en i e o business cycle equencies—pa icula ly o he pu poses o o ecas ing—
should p obably use he mos ecen and up- o-da e de ini ions as is con incingly done by Isakin
and Se le is (2023). Howe e , ou his o ical in es iga ion o he long- un coin eg a ing ela ion-
ships hinges on he impo an inancial inno a ions o he 1980s ha ga e ise o a la ge po olio
o mone a y asse s such as OCDs and MMDAs, and o he s. All hese asse s had a ying liquidi y
p ope ies wi h use cos s ha di e ed ma e ially in he 1990s and 2000s, became mo e simila
be ween 2008 and 2015, only o ma e ially decomp ess and subs an ially a y a e wa ds. We op
o asc ibe ou sel es o he o iginal de ini ions o he a ious asse s ha emained ele an o hese
las 40 yea s.
In he u u e, economis s may ind ha 2020 could ha e had ou size e ec s in he long- un
coin eg a ing ela ionships be ween mone a y asse s and in e es a es. Howe e , i will no be
ac able o asce ain whe he i was he COVID shock i sel o he change in he measu emen o
mone a y asse s ha would explain his new b eak in he ela ionship. Es ablishing his will equi e
mo e da a han is cu en ly a ailable, as he shock is oo ecen —a he ime o his w i ing— o
collec in e ence on he long- un coin eg a ing ela ionship o a pos -2020 sample.
Seminal wo k in Ba ne (1978)and(1980) demons a ed how economic agg ega ion heo y
could be used o cons uc app op ia e measu es o money balances, whe e liquidi y se ices a e
p o ided h ough an en i e gamu o asse s ha includes a ious ypes o in e es -bea ing asse s
and non-in e es -bea ing deposi s. By compa ison, he simple-sum measu es o money balances
he Fede al Rese e epo s a e p oblema ic. This is because adi ional measu es o M1 and M2
simply add up he nominal alue o all mone a y asse s in ci cula ion while igno ing he ac ha
hei componen s yield di e en lows o liquidi y se ices and, in equilib ium, also di e in he
oppo uni y (o use ) cos s ha households and i ms incu when hey include hem in hei
po olios. Ch ys al and MacDonald (1994) dubbed he essen ial message o Ba ne ’s wo k—
ha simply summing mone a y asse s imposes, un ealis ically, ha hey a e pe ec subs i u es
o each o he e en when hey ende di e en yields—as he “Ba ne C i ique.” Belongia and
I eland (2014) emphasize he Ba ne C i ique is “...as ele an oday as i was 30 yea s ago.”
As Ba ne (1978) emphasizes, he assump ions implici in he da a cons uc ion p ocedu es
should be consis en wi h he assump ions used o gene a e he models in which he da a a e
nes ed. Absen cohe ence be ween he s uc u e o an agg ega o unc ion and he econome -
ic models in which agg ega es a e embedded, measu emen e o likely occu s. Simple-sum
agg ega es lack his cohe ence.
The de i a ion o Di isia mone a y agg ega es is i mly embedded in mic oeconomic he-
o y. Conside a decision p oblem o e mone a y asse s. Le m be he ec o o eal balances o
mone a y asse s in pe iod and deno e he ec o o holding-pe iod yields o hose asse s.
The one-pe iod holding yield o a benchma k asse is deno ed as R . The mone a y se ice his
benchma k asse p o ides is s ic ly due o in es men e u ns and o he wise gene a es negligible
liquidi y se ices. R is ypically e e ed o as he benchma k a e, which co esponds o a no ional
maximum holding yield a ailable o households and i ms du ing pe iod . The decision p oblem
ea u es he maximiza ion o u ili y om mone a y asse s subjec o a es ic ion o o al planned
expendi u e in mone a y se ices y .
Max u(m )
subjec o π
m =y
Ba ne (1978) shows he eal use cos o each weakly sepa able g oup o mone a y asse s is
calcula ed by:
πi =R − i
1+R
(1)
The solu ion o his mic oeconomic p oblem shows ha he exac mone a y agg ega e (M )
should equal he u ili y gene a ed om an op imal alloca ion o mone a y asse s m∗
o a
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
Mac oeconomic Dynamics 5
ep esen a i e agen . The eal use cos in (1) and he quan i y o a gi en mone a y asse join ly
de e mine he expendi u e sha e o he asse ela i e o he o al expendi u e o mone a y se ices.
The g ow h a e o he mone a y agg ega e (and i s p ice dual) is de e mined by he g ow h in i s
unde lying mone a y asse s (and hei own eal use cos s) weigh ed by hei espec i e expendi-
u e sha es, which a e hemsel es unc ions o eal use cos s o he o m desc ibed in (1). The
ollowing desc ibes he agg ega e quan i y index and i s associa ed agg ega e use cos :
dlog M
d =isi
dlog m∗
i
d (2)
dlog
d =isi
dlog πi
d (3)
whe e si is he expendi u e sha e o each mone a y asse in he o al expendi u e.
si =πi m∗
i
y
(4)
The eal use cos dual sa is ies Fishe ’s ac o e e sal in con inuous ime
M =π
m (5)
The Di isia index is a disc e e- ime app oxima ion o equa ion (2) desc ibed as:
log M −log M −1=
n
i=0
¯si (logm∗
i −log m∗
i, −1)(6)
whe e
¯si =1
2(si +si, −1)(7)
desc ibes he expendi u e sha es associa ed wi h each unde lying mone a y asse i. In compa ison,
he simple-sum mone a y index assigns a ime-in a ian equal weigh o each mone a y asse , and
i is cons uc ed as:
log M −log M −1=
n
i=1
1
n(logm∗
i −log m∗
i, −1)(8)
CFS p o ides da a o a ious Di isia quan i y indexes and hei espec i e eal use cos s a
a ious le els o agg ega ion, as well as hei componen s. Di isia M1 includes cu ency (C),
demand deposi s (DD), o he checkable deposi s (OCDs) a comme cial banks, and OCDs a h i
ins i u ions. Gi en i s na owness, we do no conduc ou in es iga ion a he M1 le el o agg e-
ga ion. Di isia M2 (hence o h, DM2) adds he ollowing componen s o DM1: sa ings deposi s
(SDs) a comme cial banks, SDs a h i ins i u ions, e ail money-ma ke unds (Re ailMM),
small ime deposi s (STDs) a comme cial banks, and STDs a h i s. Di isia M3 (hence o h,
DM3) adds la ge ime deposi s (LTDs) and O e nigh and Te m Repu chase Ag eemen s (REPOs)
o DM2. Di isia M4 (hence o h, DM4) is, cu en ly, he b oades mone a y agg ega e—excluding
c edi ca d-augmen ed agg ega es—cu en ly a ailable in he U.S. I adds comme cial pape (CP),
and 3-mon h T-bills o he componen s in DM3.
The simple-sum coun e pa o he M2 mone a y agg ega e, as well as eal pe sonal income,
PCE chain- ype p ice index, and he T-bill yield a e d awn om he Fede al Rese e Bank o S .
Louis’ FRED da abase. Cu en ly, a measu e o M3 is no made a ailable by he Fede al Rese e.
We p oduce ou own measu e o simple-sum M3 om he componen in o ma ion made a ailable
by CFS. The na u al loga i hms o hese mone a y agg ega es scaled by ou pu , as well as hei
associa ed use cos s a e plo ed in Fig. 1.4
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
6 Z. Chen and V. J. Valca cel
Figu e 1. Mone a y agg ega e and in e es a e da a. Bo h he Di isia and simple-sum agg ega es a e log ans o med and
no malized by he p oduc o he PCE p ice index and pe sonal income as he scale a iables. Shaded a eas deno e NBER
ecession da es.
3. Me hodology
Ou es ima es o money demand—and he long- un ela ionship be ween money balances and
in e es a es—a e ob ained by applying he Johansen (1991) maximum likelihood app oach
oge he wi h he Johansen (1995) ex ension o linea es ic ions on he coin eg a ing equa ion
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
Mac oeconomic Dynamics 7
and co esponding ec o e o co ec ion model (VECM). We es he long- un coin eg a ing
ela ionships speci ied in wo ypical unc ional o ms o money demand.
Da ing back o Cagan (1956), he log-le el (semi-log) unc ional o m is commonly used as a
s anda d p ac ice (see, e.g., S ock and Wa son (1993) and Ball (2001)). Ano he unc ional o m
is he double log (log-log) o m p oposed by Mel ze (1963) (see also Ho man and Rasche (1991)
and Belongia and I eland (2019)). In his speci ica ion, he money demand equa ion becomes a
nonlinea unc ion o he oppo uni y cos . Bae e al. (2006) claim he double-log o m can be e
accommoda e he liquidi y ap— ha esul s om ex emely low sho - e m in e es a es— han
he benchma k semi-log o m. The wo unc ional o ms a e gi en by:
Log le el :ln(ma
)=α0+α1( )−α2 (9)
Double log :ln(ma
)=δ0+δ1( )−δ2ln( )(10)
whe e
ma
=Ma
P Y
(11)
deno es he loga i hm o eal money balances scaled by nominal income, and ep esen s he
oppo uni y cos o holding money. The p ice le el (P ) is measu ed by he PCE Chain- ype p ice
index. Y deno es eal pe sonal income, which is heo e ically linked wi h he ou pu and con-
sump ion o households. Ou esul s a e quali a i ely obus o al e na i e measu es o p ice and
ou pu , such as CPI and eal GDP.
We speci y he ollowing VECM:
Y =μ +γβY −1+
p−1
k=1
kY −k+ε (12)
whe e μ consis s o de e minis ic e ms, ka e he sho - un adjus men coe icien s on Y −k,
and he e o co ec ion e m γshows he esponse o Y o a de ia ion om he long- un coin-
eg a ing ela ionship. Inso a as Y and i s lag e ms con ain I(0) p ocesses, he coin eg a ion
ela ions—i hey exis —should be cap u ed in βY −1. The ec o βin equa ion (12) includes he
ele an pa ame e s in he coin eg a ion equa ions. Since he ank o he long- un impac ma ix
(γβ) gi es he numbe o coin eg a ing ela ionships in he sys em, we apply he Johansen (1991)
likelihood a io (LR) es —and he Johansen (1995) ex ension— o he numbe o coin eg a ing
ela ionships om equa ion (12).
We allow de e minis ic ends as unc ions o ime [e.g., α1( )andδ1( ) in equa ions (9–10)]
o en e in o he coin eg a ing equa ion. Johansen (1995) summa izes how de e minis ic ends
in VAR speci ica ions co espond o hose in he coin eg a ing equa ion and he VECM, shown
in Table 1below. Gi en ha he asymp o ic dis ibu ion o he LR es does no ha e he usual χ2
dis ibu ion—and i depends on he assump ions made o de e minis ic ends—in o de o ca y
ou he Johansen coin eg a ing es , we need o make a de e mina ion ega ding i /how ends a e
speci ied in he coin eg a ing equa ion. Following Johansen (1995), he hypo hesis H( ) ha he
numbe o coin eg a ing ela ionships equals o is es ed unde he ollowing assump ions on he
de e minis ic e m (μ ) in equa ion (12):
This able shows i e (i)—( ) assump ions wi h he ollowing conside a ions:
(i).ModelH( ):μ =0 (no cons an ): All he se ies in Y a e I(1) wi hou d i and he
coin eg a ing ela ions βY ha e mean ze o.
(ii).ModelH( ):μ =μ0=γρ0( es ic ed cons an ): The se ies in Y a e I(1) wi hou d i
and he coin eg a ing ela ions βY ha e non-ze o means ρ0.
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
8 Z. Chen and V. J. Valca cel
Table 1. T end assump ions o Johansen (1995) coin eg a ion es
Assump ion T ends in le el da a T end in CE VAR
(i).Nocons an μ =0 No de e minis ic end No in e cep o end No in e cep
...................................................................................................................................................................................................................................................................
(ii).Res .cons an μ =μ0=γρ0No de e minis ic end In e cep only No in e cep
...................................................................................................................................................................................................................................................................
(iii). Un es . cons an μ =μ0Allow o linea end In e cep only In e cep only
...................................................................................................................................................................................................................................................................
(i ).Res .T end μ =μ0+γρ1 Allow o linea end In e cep and end In e cep only
...................................................................................................................................................................................................................................................................
( ).Un es .T end μ =μ0+μ1 Allow o quad a ic end In e cep and end In e cep and end
No e: γis he coe icien o he coin eg a ing componen in he VECM model.
(iii).ModelH( ):μ =μ0(un es ic ed cons an ): The se ies in Y a e I(1) wi h d i ec o μ0
and he coin eg a ing ela ions βY may ha e a non-ze o mean.
(i ).ModelH( ):μ =μ0+γρ1 ( es ic ed end). The se ies in Y a e I(1) wi h d i ec o
μ0and he coin eg a ing ela ions βY ha e a linea end e m ρ1 .
( ).ModelH( ):μ =μ0+μ1 (un es ic ed cons an and end). The se ies in Y a e I(1) wi h
a linea end (quad a ic end in le els) and he coin eg a ing ela ions βY ha e a linea
end.
We impose a cons an in e e y bi a ia e speci ica ion we es ima e in his pape . Thus, we con-
cen a e on he la e ou (ii— ) c i e ia o he Johansen (1995) es , whe e ρjco espond o he
αjcoe icien s in he semi-log o m speci ied in equa ion (9)o heδjcoe icien s in he double-log
o m speci ied in equa ion (10) o j={0, 1, 2}.
As Ball (2001) poin s ou , including ends may yield un eliable es ima es o he pa ame e s
because o he high collinea i y be ween end balances and income. The e o e, Ball (2001) also
es ima es coin eg a ing equa ions wi h he added cons ain ha income elas ici y is uni y. These
es ima es u n ou o be in o ma i e and do no al e he indings ha he demand o money
has become uns able when he sample size is ex ended beyond he la e 1980s and up o he mid-
1990s. We ollow his easoning and we impose he uni a y income elas ici y assump ion in ou
equa ions (9)and(10).
4. The pe sis ence o Di isia agg ega es and hei p ice duals
Ou sample encompasses Janua y 1967—Ma ch 2020 measu ed a mon hly equencies. Fig. 1
shows he se ies we conside in he pape . All mone a y agg ega es a e scaled by he PCE imes eal
pe sonal income and a e all log ans o med. The i s ou cha s in he op ow o Fig. 1deno e
he Di isia agg ega es (DM1–DM4) a he ou le els o agg ega ion he CFS p o ides. The nex
h ee cha s ha ollow in he second ow he simple sum measu es M1–M3, and he las cha on
he second ow is he mon hly a e age o he T-bill yield. The emaining cha s on he bo om ow
a e he p ice duals o each o he ou Di isia quan i y indices; namely, DM1_RUC—DM4_RUC,
eal use cos indices.
In line wi h Johansen and Juselius (1990), Bena i e al. (2016), Ande son, e al. (2017) among
many o he s, he heo e ical money demand amewo k is in e p e ed he e as desc ibing po en-
ial long- un coin eg a ing ela ionships be ween wo a iables: he log money-ou pu a io—as
a measu e o he eal demand o money ela i e o a scale a iable—and he oppo uni y cos
o holding money, exp essed in le els o log le els. Adop ing his app oach equi es ha he wo
a iables be gene a ed by in eg a ed ime se ies p ocesses.
Table 2shows esul s om Dickey and Fulle (1979) uni oo es s o each o he a iables
plo ed in Fig. 1. We use he “local o uni y GLS” de ending p ocedu e in oduced by Ellio e al.
(1996) o gain g ea e powe agains he null o s a iona i y o e wha he s anda d Dickey-Fulle
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
Mac oeconomic Dynamics 15
Table 5. S uc u al b eak es s a unknown b eak poin
log(M/PY)
DM2 DM3
And ews and Plobe ge (1994)
Single b eak 1979:M10 1980:M2
(1979:M8, 1980:M2) (1980:M1, 1980:M3)
..........................................................................................................................................................................................
Bai and Pe on (2003)
B eak 1 1980:M2 1979:M10
(1980:M1, 1980:M3) (1979:M9, 1979:M11)
B eak 2 2012:M12 1988:M6
(2012:M10, 2013:M2) (1987:M11, 1988:M12)
Real Use Cos s
DM2 DM3
And ews and Plobe ge (1994)
Single b eak 2008:M10 2008:M12
(2008:M4, 2009:M6) (2005:M1, 2011:M6)
..........................................................................................................................................................................................
Bai and Pe on (2003)
B eak 1 1978:M6 1977:M12
(1976:M10, 1979:M2) (1977:M4, 1978:M3)
B eak 2 2008:M10 1986:M6
(2008:M6, 2008:M4) (1985:M10, 1987:M5)
Residuals e :log(M /P ∗Y )=α+βRUC +e
DM2 DM3
And ews and Plobe ge (1994)
Single b eak 1982:M6 1981:M8
(1982:M5, 1982:M7) (1981:M7, 1981:M9)
..........................................................................................................................................................................................
Bai and Pe on (2003)
B eak 1 1981:M10 1981:M6
(1981:M9, 1981:M11) (1981:M5, 1981:M7)
B eak 2 2012:M12 2001:M10
(2012:M10, 2013:M1) (2001:M6, 2003:M1)
No e: The 95% con idence in e als a e enclosed in pa en heses. The dis ibu ion unc ion used o com-
pu ing he in e als is gi en in Bai (1997). The p ocedu e o gene a e in e als o mul iple b eakpoin s
is desc ibed in Bai and Pe on (2003).
6.1 Money demand as a unc ion o he T-bill yield
Table 6shows subsample esul s o money demand as a unc ion o he T-bill yield. Essen ially,
he op panel o Table 6es ima es he semi-log speci ica ion shown in he op panel o Table 3while
b eaking he sample in 1980:Q2. The i s column in he op panel shows ha be ween 1967 and
1980, all ou speci ica ions ind a coin eg a ing ela ionship be ween simple-sum M2 and he T-
bill yield. The es ima es o α2in equa ion (9) a e all close in magni ude and s a is ically signi ican
wi h -s a is ics o 6.9 o g ea e .
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
16 Z.ChenandV.J.Valca cel
Table 6. Coin eg a ing money demand ela ionships wi h 3 m T-bill a e: p e- & pos -1980 samples
Log money-ou pu a io: ln(ma
)≡ln(Ma
/(PCE ×Real pe sonal income))—Speci ica ion: ln(ma
)=α0+α1( )−α2
≡3 m T-bill yield 1967:Q1—1980:Q2 1980:Q2—2020:Q1
M2 DM2 M3 DM3 M2 DM2 M3 DM3
Johansen (1995) Speci ica ion:
(ii). Res . cons an 0.021 0.210 . 0.145 . 4.85 . .
[7.9] [6.8] [.] [7.0] [.] [5.0] [.] [.]
...................................................................................................................................................................................................................................................................
(iii). Un es . cons an 0.021 0.212 . 0.147 . 4.26 . −0.009
[7.9] [6.9] [.] [7.0] [.] [5.1] [.] [−1.3]
...................................................................................................................................................................................................................................................................
(i ). Res . T end 0.027 0.063 . 0.113 −0.222 −0.155 −0.172 −0.068
[6.9] [7.7] [.] [6.2] [−5.5] [6.0] [−5.3] [−5.6]
...................................................................................................................................................................................................................................................................
( ). Un es . T end 0.027 0.064 0.412 0.116 −0.220 −0162 −0.174 −0.070
[6.9] [7.7] [3.5] [6.2] [−5.5] [−6.1] [−5.3] [−5.7]
Lags 3 4 3 4 17 17 17 18
Log money-ou pu a io: ln(ma
)≡ln(Ma
/(PCE ×Real pe sonal income))—Speci ica ion: ln(ma
)=δ0+δ1( )−δ2ln( )
≡3 m T-bill yield 1967:Q1—1980:Q2 1980:Q2—2020:Q1
M2 DM2 M3 DM3 M2 DM2 M3 DM3
Johansen (1995) Speci ica ion:
(ii). Res . cons an 0.152 2.41 . 1.54 . 0.261 . 0.017
[8.9] [6.2] [.] [3.4] [.] [.] [.] [1.3]
...................................................................................................................................................................................................................................................................
(iii). Un es . cons an 0.153 2.41 . 1.59 −0.740 0.257 0.095 0.016
[8.8] [6.2] [.] [6.3] [−2.6] [3.4] [4.8] [1.3]
...................................................................................................................................................................................................................................................................
(i ). Res . T end 0.174 7.39 1.30 −2.25 −6.17 0.070 5.14 −0.053
[7.6] [5.5] [4.3] [−5.2] [−2.6] [0.9] [4.5] [−2.0]
...................................................................................................................................................................................................................................................................
( ). Un es . T end 0.173 0.940 1.18 −1.86 −5.00 −1.16 0.469 −.076
[7.5] [5.4] [4.4] [−5.2] [−2.6] [−2.5] [4.49] [−2.4]
Lags 24349933
No e: The op panel and bo om panels con ain he semi-log and he log-log speci ica ions, espec i ely. The numbe o coin eg a ing equa ions
(single in ou bi a ia es) a e es ed a a 5% signi icance le el ollowing Johansen (1995). We i s conduc he maximum eigen alue es o he
ace es o he null o no coin eg a ing ela ionship be ween each pai o a iables shown in he able. The able shows es ima es o α2o δ2
o e e y bi- a ia e speci ica ion ha ails o ejec he null o no coin eg a ion o each o he (ii.— .) speci ica ions o he Johansen (1995) es ,
whe e α2is Johansen (1991) maximum likelihood es ima es o he coe icien s o he i s coin eg a ing ec o , no malized so ha he coe icien
on ln(m ) equals one. Numbe s in b acke s unde nea h each es ima e o α2o δ2deno e he -s a is ic. Any blank (missing alue) in each elemen
deno es no coin eg a ion was ound o ha pa icula pai o a iables unde ha pa icula speci ica ion o he ele an sample. Op imal lag
leng hs a e chosen acco ding o he Akaike (1974) c i e ion.
The second column o Table 6shows ha ou conclusions do no change when conside ing
Di isia M2 in he money demand equa ion (9). The magni udes o he α2coe icien s seem some-
wha la ge o he cons an speci ica ions (ii.—iii.) han he end speci ica ions (i .— .)in he
Johansen (1995) c i e ia, bu all coe icien s a e s a is ically signi ican wi h he co ec sign. The
hi d columns shows simple-sum M3 only coin eg a es wi h he T-bill unde he un es ic ed
end speci ica ion ( .) be o e 1980, whe eas Di isia M3 in he ou h column shows coin eg a ion
and s a is ical signi icance unde all c i e ia.
Looking a columns 5—8 in ha op panel o Table 6, he coin eg a ion ela ionships be ween
he T-bill and mone a y agg ega es la gely b eak down in he pos -1980 sample. Only he end
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
Mac oeconomic Dynamics 17
(i .— .) speci ica ions show coin eg a ion be ween simple-sum M2 and he yield on he h ee-
mon h T-bill. The same holds o simple-sum M3. The α2es ima e o bo h agg ega es is
signi ican ly nega i e con adic ing a money demand in e p e a ion. Di isia M3 su e s om
he same nega i e es ima e issue. Howe e , he cons an (ii.—iii.) speci ica ions show esul s
consis en wi h money demand o Di isia M2 in his sample.
The bo om panel o Table 6shows subsample esul s o money demand as a unc ion o he
log o he T-bill. This is consis en wi h he log-log speci ica ion shown in he op panel o Table 4
o he T-bill yield in samples ha s addle 1980. In he p e-1980 sample, he es ima es o δ2in
equa ion (10) a e, o he mos pa , quali a i ely consis en wi h hose o he semi-log speci i-
ca ion in he op panel. Simple-sum M2 and Di isia M2 signi ican ly coin eg a e wi h he T-bill
yield unde all Johansen (1995) c i e ia wi h he co ec sign. We each he same conclusions o
simple-sum M3 unde end (i .— .) speci ica ions and o Di isia M3 unde cons an (ii.—iii.)
speci ica ions. Columns 5—8 in he bo om panel o Table 6show ha a e 1980, simple-sum M2
shows a nega i e sign o δ2, which is p oblema ic. Di isia M2 only shows a quali a i ely sensible
coin eg a ing ela ionship wi h he T-bill yield unde he un es ic ed cons an (iii.) and Di isia
M3 a ely exhibi s a s a is ically signi ican δ2es ima e and ne e wi h he co ec sign. Puzzlingly,
while he semi-log o m in he op panel o Table 6showed simple-sum M3 ne e has a co ec
sign o δ2a e 1980, he log-log o m shows a co ec sign o M3 and he T-bill yield.
O e all, he pic u e ha eme ges om Table 6is ha , bo h o he semi-log o m ( op panel)
and he log-log o m (bo om panel), he T-bill yield coin eg a es well (i espec i e o he Johansen
(1995) selec ion c i e ia) wi h simple-sum M2 in a way ha is consis en wi h a adi ional money
demand in e p e a ion. The same conclusion holds o Di isia M2. While e idence o coin eg a-
ion is weake o M3, he end speci ica ion o he semi-log and log-log o ms show simple-sum
M3 coin eg a es wi h he T-bill yield wi h a co ec sign o α2o δ2. Di isia M3 coin eg a es wi h
he co ec sign o α2unde all (ii.— .) speci ica ions, and o δ2unde he no- end speci ica-
ions (ii.—iii.). Howe e , he long- un ela ionship be ween he T-bill yield and hese mone a y
agg ega es la gely b eakdown in he pos -1980 sample. This b eakdown could be explained by
he ac ha abou one hi d o he pos -1980 sample encompasses he ZLB pe iod, which com-
p essed he T-bill yield o nea ze o o o e hal a decade and o abno mally low alues o o e
a decade. We es his wi h ano he subsample es ima ion o he p e- and pos -2008 pe iod. Bu
i s , we epea he analysis o Table 6 eplacing he T-bill yield wi h he use cos s o Di isia
M2 and Di isia M3, nei he o which we e d i en o ze o in he a e ma h o he GFC pe iod o
he ea e .
6.2 Money demand as a unc ion o he use cos o Di isia money
Table 7shows subsample esul s o money demand o M2 as a unc ion o he use cos o M2,
as well as money demand o M3 as a unc ion o he use cos o M3. As in he p e ious able,
he op panel shows esul s o he semi-log speci ica ion o equa ion (9) and he bo om panel o
Table 7shows esul s o he log-log o m in equa ion (10). While he magni udes o he es ima es
a e mo e sensi i e among he (ii.— .) speci ica ions o bi a ia e pai s wi h use cos s ( han hey
we e o he T-bill yield), he conclusions a e a mo e obus o ou coin eg a ion analysis wi h
use cos s. In he p e-1980 subsample, bo h simple-sum M2 and Di isia M2 a e coin eg a ed wi h
he use cos o Di isia M2 wi h he co ec sign o α2o δ2. A la gely simila conclusion can be
d awn o a money demand in e p e a ion based on coin eg a ion e idence be ween Di isia M3
and i s associa ed use cos . Howe e , e idence o coin eg a ion be ween simple-sum M3 and he
use cos o M3 is ne e ound.
The use cos esul s om he pos -1980 sample p o ide salien con as wi h he conclusions
we d ew o he T-bill yield. Inspec ing columns 5—8 in he op panel o Table 7 o he α2es ima e
o he semi-log o m—along wi h he bo om panel o he δ2es ima e o log-log o m— e eals
a powe ul message ega ding he in o ma ion con en o Di isia balances and hei p ice duals.
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
18 Z.ChenandV.J.Valca cel
Table 7. Coin eg a ing money demand ela ionships wi h use cos s: p e- & pos -1980 samples
Log money-ou pu a io: ln(ma
)≡ln(Ma
/(PCE ×Real pe sonal income))—Speci ica ion: ln(ma
)=α0+α1( )−α2( )
≡RUCa
1967:Q1—1980:Q2 1980:Q2—2020:Q1
M2 DM2 M3 DM3 M2 DM2 M3 DM3
Johansen (1995) Speci ica ion:
(ii). Res . cons an 0.569 6.33 . 3.85 . 5.47 . 9.57
[7.4] [4.9] [.] [4.6] [.] [6.5] [.] [4.5]
...................................................................................................................................................................................................................................................................
(iii). Un es . cons an 0.572 5.71 . 3.81 . 5.44 . 11.2
[7.4] [5.2] [.] [4.7] [.] [6.5] [.] [4.5]
...................................................................................................................................................................................................................................................................
(i ). Res . T end . 1.42 . . 1.90 4.22 . 11.1
[.] [5.9] [.] [.] [1.8] [5.1] [.] [4.0]
...................................................................................................................................................................................................................................................................
( ). Un es . T end 0.687 1.47 . 2.16 . 6.45 . −4.70
[6.4] [5.8] [.] [4.6] [.] [5.4] [.] [−4.3]
Lags 3 3 3 3 10 8 7 6
Log money-ou pu a io: ln(ma
)≡ln(Ma
/(PCE ×Real pe sonal income))—Speci ica ion: ln(ma
)=δ0+δ1( )−δ2ln( )
≡RUCa
1967:Q1—1980:Q2 1980:Q2—2020:Q1
M2 DM2 M3 DM3 M2 DM2 M3 DM3
Johansen (1995) Speci ica ion:
(ii). Res . cons an 0.144 2.72 . 1.45 . 0.935 . 1.93
[8.7] [4.6] [.] [4.5] [.] [6.9] [.] [4.7]
...................................................................................................................................................................................................................................................................
(iii). Un es . cons an 0.145 2.28 . 1.39 1.95 0.933 . 2.18
[8.6] [4.7] [.] [4.5] [4.1] [7.0] [.] [4.7]
...................................................................................................................................................................................................................................................................
(i ). Res . T end 0.153 . . . . 0.862 . 3.66
[7.5] [.] [.] [.] [.] [5.4] [.] [4.4]
...................................................................................................................................................................................................................................................................
( ). Un es . T end 0.153 0.639 . . . 1.14 −1.84 −2.41
[7.5] [4.9] [.] [.] [.] [5.6] [−4.1] [−4.6]
Lags 4 4 4 4 6 6 6 6
No e: The op panel and bo om panels con ain he semi-log and he log-log speci ica ions, espec i ely. The numbe o coin eg a ing equa ions
(single in ou bi a ia es) a e es ed a a 5% signi icance le el ollowing Johansen (1995). We i s conduc he maximum eigen alue es o he
ace es o he null o no coin eg a ing ela ionship be ween each pai o a iables shown in he able. The able shows es ima es o α2o δ2
o e e y bi- a ia e speci ica ion ha ails o ejec he null o no coin eg a ion o each o he (ii.— .) speci ica ions o he Johansen (1995) es ,
whe e α2is Johansen (1991) maximum likelihood es ima es o he coe icien s o he i s coin eg a ing ec o , no malized so ha he coe icien
on ln(m ) equals one. Numbe s in b acke s unde nea h each es ima e o α2o δ2deno e he -s a is ic. Any blank (missing alue) in each elemen
deno es no coin eg a ion was ound o ha pa icula pai o a iables unde ha pa icula speci ica ion o he ele an sample. Op imal lag
leng hs a e chosen acco ding o he Akaike (1974) c i e ion.
The s ong e idence o coin eg a ion be ween simple-sum M2 and he use cos o M2 we see in
he p e-1980 sample la gely anishes in he pos -1980 pe iod. Th ee o he ou speci ica ions ha
showed a s ong coin eg a ing ela ionship be o e 1980 ail o ind coin eg a ion a e 1980. Only
one speci ica ion o M2 shows coin eg a ion wi h a co ec sign o α2a e 1980, bu he es ima e
has he smalles -s a is ic ac oss all ou explo a ion o M2. Con e sely, Di isia M2 coin eg a es
wi h he use cos o M2 a e 1980 unde all Johansen (1995) c i e ia.
Finally, simple-sum M3 ne e coin eg a es wi h he use cos o Di isia M3 in he pos -1980
sample. On he o he hand, wi h he excep ion o he un es ic ed end ( .)Johansen(1995)
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
Mac oeconomic Dynamics 19
c i e ia, we ind s ong e idence o coin eg a ion be ween Di isia M3 and he use cos o M3—
wi h co ec signs o he δ2es ima es—in he pos -1980 sample.
Taking s ock o ou analysis hus a ende s a ew poignan conclusions. Fi s , he ull sample
analysis o Tables 3and 4p o ide s ong e idence ha he simple-sum M2 and M3 agg ega es
ail o coin eg a e wi h he T-bill yield—bo h when es ima ing semi-log o double-log unc ional
o ms o money demand. On he o he hand, Di isia M2 and Di isia M3 show e idence o s onge
coin eg a ion wi h he T-bill yield unde bo h unc ional o ms in he ull sample. Second, ou
subsample analysis in Table 6sugges s ha he coin eg a ion ailu e o simple-sum M2 in he
ull sample can be explained by a s uc u al change aking place in 1980. We ind ha unde
Regula ion D be o e 1980—when he Boa d o Go e no s o he Fede al Rese e es ic ed in e es
ea nings on checkable accoun s—bo h simple-sum M2 and Di isia M2 show a consis en connec-
ion wi h in e es a es—bo h he T-bill yield as well as he use cos o M2. Thi d, he ela ionship
be ween simple-sum M2 and he T-bill yield la gely b eaks down in he pos -1980 sample, while
simple-sum M3 a ely coin eg a es wi h in e es a es in subsamples s addling 1980, and ne e
coin eg a es in he ull sample. Fou h, simple-sum M2 coin eg a es well wi h he use cos o M2
be o e 1980, bu ha ela ionship again la gely b eaks down a e 1980. Finally, we show s ong
e idence ha Di isia M2 and Di isia M3 coin eg a e wi h hei espec i e use cos s bo h in he
p e- and pos -1980 samples.
7. The in o ma ion con en o b oade Di isia agg ega es in he pos -GFC pe iod
In addi ion o s uc u al b eaks in he in o ma ion con en in mone a y agg ega es, ins abili y in
he unc ional o m o money demand could also a ise om s uc u al change in agen s’ oppo u-
ni y cos o holding money. We now conside al e na i e subsamples allowing o he possibili y
ha he ZLB ha began in Decembe 2008 could ha e in oduced a s uc u al b eak in money
demand ia he in e es a e. The e o e, we u n ou a en ion o wo o he subsamples lanked by
he onse o he GFC and i s a e ma h. The las qua e o 2008 b ough abou wo majo e en s
in he his o y o mone a y policy: he ede al unds a e eached ZLB, and he Fede al Rese e
ini ia ed balance shee policy as a new ool o mone a y policy ma ked by wha would become he
i s ound in a se o Quan i a i e Easing measu es. Mo i a ed by he pe iod o unin o ma i e
ze o nominal in e es a es ha began in la e 2008, Lucas and Nicolini (2015) and Ande son e al.
(2017) u n back o a quan i y- heo e ic app oach o mone a y policy analysis.
Gi en ou esul s in he p e ious sec ion ha simple-sum agg ega es essen ially ail in he pos -
1980 pe iod, we eschew any u he analysis o simple-sum agg ega es o he ola ile pe iod
ollowing he GFC. We eplace Di isia M2 wi h Di isia M4, as he b oades measu e o money
a ailable o he U.S., which augmen s he asse s e lec ed in M3 wi h non-bank-issued liquidi y
asse s, such as comme cial pape and T-bills. Bae e al. (2006) sugges he nonlinea ea u e o
he double-log o m is be e sui ed han he semi-log o m in accommoda ing he liquidi y ap
ha ook place du ing he 2008–2015 pe iod, and he e o e i is a mo e app op ia e app oach o
es ima ing he elas ici y o money demand. We ake his poin and we es ima e he double-log
o m when we look o coin eg a ion be ween Di isia M3 and Di isia M4 and hei espec i e
use cos s. Ma son and Valca cel (2016) show ha , while he use cos s o all Di isia agg ega es
made a ailable by he CFS expe ienced subs an ial comp ession in he a e ma h o he GFC, hey
emained well abo e he ZLB. On he o he hand, he T-bill yield eached and emained a i u-
ally ze o o a subs an ial amoun o ime ollowing GFC. Thus, we es ima e he semi-log o m o
a money demand speci ica ion when conside ing coin eg a ion be ween Di isia M3 and Di isia
M4 and he T-bill yield.9
Table 8shows es ima es consis en wi h an app op ia e cha ac e iza ion o money demand o
Di isia M3 and Di isia M4. The op panel shows coin eg a ion es s o bi a ia e semi-log speci i-
ca ions be ween he T-bill yield and Di isia M3, o Di isia M4, o samples ha s addle he hi d
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
20 Z.ChenandV.J.Valca cel
Table 8. Di isia M3 and M4 money demand ela ionships: p e- & pos -GFC samples
Log money-ou pu a io: ln(ma
)≡ln(Ma
/(PCE ×Real pe sonal income))
Speci ica ion: ln(ma
)=α0+α1( )−α2( )
≡3 m T-bill yield 1967:Q1—2008:Q3 2008:Q4—2020:Q1
DM3 DM4 DM3 DM4
Johansen (1995) Speci ica ion:
(ii). Res . cons an 1.03 0.510 . .
[5.3] [5.3] [.] [.]
.................................................................................................................................................................................................
(iii). Un es . cons an 1.46 0.510 . .
[5.3] [5.3] [.] [.]
.................................................................................................................................................................................................
(i ). Res . T end 0.685 0.416 . .
[5.2] [5.2] [.] [.]
.................................................................................................................................................................................................
( ). Un es . T end 0.833 0.462 . .
[5.3] [5.3] [.] [.]
Lags 7 7 2 2
Log money-ou pu a io: ln(ma
)≡ln(Ma
/(PCE ×Real pe sonal income))
Speci ica ion: ln(ma
)=δ0+δ1( )−δ2ln( )
≡RUCa
1967:Q1—2008:Q3 2008:Q4—2020:Q1
DM3 DM4 DM3 DM4
Johansen (1995) Speci ica ion:
(ii). Res . cons an 1.43 1.43 2.18 1.90
[6.6] [6.2] [4.1] [4.5]
.................................................................................................................................................................................................
(iii). Un es . cons an 1.54 1.55 2.18 1.96
[6.7] [6.2] [4.1] [4.5]
.................................................................................................................................................................................................
(i ). Res . T end 2.56 1.18 3.07 8.84
[6.1] [5.6] [4.2] [4.5]
.................................................................................................................................................................................................
( ). Un es . T end 6.69 1.58 3.54 8.82
[6.2] [5.7] [4.2] [4.5]
Lags 3 3 1 1
No e: The numbe o coin eg a ing equa ions (single in ou bi a ia es) a e es ed a a 5% signi icance le el
ollowing Johansen (1995). We i s conduc he maximum eigen alue es o he ace es o he null o
no coin eg a ing ela ionship be ween each pai o a iables shown in he able. The able shows es ima es
o α2and δ2 o e e y bi- a ia e speci ica ion ha ails o ejec he null o no coin eg a ion o each o
he (ii.— .) speci ica ions o he Johansen (1995) es , whe e α2and δ2a e he Johansen (1991) maximum
likelihood es ima es o he coe icien s o he i s coin eg a ing ec o , no malized so ha he coe icien
on ln(m ) equals one. Numbe s in b acke s unde nea h each es ima e o α2and δ2deno e he -s a is ic.
Any blank (missing alue) in each elemen deno es no coin eg a ion was ound o ha pa icula pai o
a iables unde ha pa icula speci ica ion o he ele an sample. Real use cos s (RUCa
) a e cong uen
wi h a=3, 4 le els o agg ega ion. Op imal lag leng hs a e chosen acco ding o he Akaike (1974) c i e ion.
qua e o 2008. The op le column shows ha o a p e-GFC sample, Di isia M3 coin eg a es
wi h he T-bill yield unde all (ii.— .)Johansen(1995) coin eg a ing c i e ia. The second column
e eals he same conclusion can be d awn o Di isia M4. Con e sely, he nex wo columns in
he op panel o Table 8shows he T-bill yield ails o coin eg a e wi h Di isia M3 o Di isia M4
a e he GFC pe iod. The likely explana ion is he p o ac ed pe iod o a i ual ze o le el o his
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
Mac oeconomic Dynamics 21
a e ollowing 2008. This would sugges he GFC des oys he in o ma ion con en he T-bill yield
ca ies o money demand de e mina ion. The bo om panel o Table 8shows coin eg a ion es s
o bi a ia e log-log speci ica ions be ween Di isia M3 and he use cos o M3, as well as Di isia
M4 and i s associa ed use cos . The le mos column o he bo om panel shows ha , o he p e-
GFC sample, all (ii.— .)Johansen(1995) c i e ia show coin eg a ion be ween Di isia M3 and i s
use cos . All δ2es ima es a e signi ican and ca y he expec ed sign o a co ec cha ac e iza ion
o money demand. The nex column shows simila indings o Di isia M4 in he p e-GFC pe iod.
Impo an ly, he nex wo columns show ha bo h Di isia M3 and Di isia M4 con inue o coin-
eg a e well unde all Johansen (1995) c i e ia a e he GFC. All δ2es ima es emain s a is ically
signi ican . In e es ingly, he magni ude o he δ2coe icien s in he espec i e money demand
equa ions a e somewha highe in he pos -GFC pe iod. This unde sco es ha he use cos s o
Di isia M3 and Di isia M4 do no su e om he in o ma ion loss ha seemed o ensna e he
T-bill yield a e 2008.
8. An in es iga ion o he componen s o he Di isia agg ega e
We now u n o a mo e g anula in es iga ion o he long- un money demand ela ionships
by examining he quan i y o each mone a y asse e lec ed in he Di isia agg ega es and i s
ela ed use cos . This analysis should be pa icula ly in o ma i e o asce ain whe he newe
in e es -bea ing deposi p oduc s s emming om he inancial de egula ion ollowing he passage
o DIDMCA in 1980 ha e made he es ima ion o he money demand unc ion mo e di icul .
Impo an ly, since we look a he quan i y o each disagg ega ed mone a y asse , he esul s will
bind only o in o ma ion on quan i ies and o he wise be silen on he issue o app op ia e liquidi y
weigh ing.
No only does CFS p o ide he use cos o each o he ou mone a y (Di isia M1—Di isia
M4) agg ega es (some o which we employ in ou analysis abo e), bu i also p oduces and
dis ibu es he use cos s o each componen asse including: cu ency, demand deposi s, o he
checkable deposi s, sa ings deposi s, e ail money ma ke s, ins i u ional money ma ke s, STDs, LTDs,
epu chase ag eemen s, comme cial pape , and T-bills. In his sec ion we a e looking mo e g anu-
la ly a balances o each componen , and he e o e, we eplace he use cos s o M2, M3, and M4
Di isia agg ega es wi h he use cos s o hese componen s in ou eg essions.
Table 9shows es ima es o he long- un coin eg a ing ela ionship be ween he quan i y o each
ype o mone a y asse and i s associa ed use cos , o he T-bill yield. We ollow he e he same
easoning in he p e ious sec ion, which led us o es ima e he semi-log o m o money demand
when conside ing he T-bill yield and he double-log o m o money demand when conside ing
use cos s. The op panel o Table 9shows coin eg a ion es ima es and a money demand es ima ion
ha ollows equa ion (9) o he log o each asse (scaled by nominal income) and he T-bill yield.
The bo om panel o Table 9shows coin eg a ion es ima es and a money demand es ima ion ha
ollows equa ion (10) o he log o each asse and he log o i s espec i e use cos . Fo example,
he le mos column in he op panel es ima es a eg ession o he o m o (9) be ween he log o
cu ency balances and he a e o he T-bill—whe eas he le mos column in he bo om panel
es ima es a eg ession o he o m o (10) be ween he log o cu ency balances and he log o he
use cos o cu ency.
The op panel shows ha only cu ency balances coin eg a e well wi h he T-bill yield unde
all o he (ii.— .)Johansen(1995) c i e ia. The nex column in he op panel shows coin eg a ion
be ween demand deposi s and he T-bill yield is only ound o he un es ic ed cons an (iii.)
in Johansen (1995). Resul s om he op panel o Table 9show ha some componen s do no
coin eg a e wi h he T-bill yield (e.g. sa ings) o i hey do, hey show a s a is ically signi ican
es ima e wi h an inco ec sign o α2(e.g. STDs, LTDs, REPOs). O e all, he op panel o Table 9
shows gene ally weak e idence o coin eg a ion be ween mos mone a y asse s and he T-bill yield,
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
22 Z.ChenandV.J.Valca cel
Table 9. Coin eg a ing money demand ela ionships o indi idual mone a y asse s
Log money-ou pu a io: ln(ma
)≡ln(Ma
/(PCE ×Real pe sonal income))—Speci ica ion: ln(ma
)=α0+α1( )−α2( )
≡3 m T-bill yield Cu ency DD OCD Sa ing Re ailMM STD LTD InsMM Repo CP +T-bill
Johansen (1995) Speci ica ion:
(ii). Res . cons an 0.097 . −0.923 . −0.073 . . 0.266 0.044 0.266
[6.9] [.] [−2.7] [.] [−0.8] [.] [.] [2.7] [1.0] [4.7]
.......................................................................................................................................................................................................................................................................
(iii). Un es . cons an 0.097 1.63 −1.04 . −0.047 . . 0.291 0.063 0.258
[7.1] [3.8] [−2.9] [.] [−0.5] [.] [.] [2.9] [1.5] [4.7]
.......................................................................................................................................................................................................................................................................
(i ). Res . T end 0.068 . −0.863 . −0.433 . . −21.7 −1.3 −0.550
[4.9] [.] [−6.9] [.] [-3.8] [.] [.] [−4.9] [−4.8] [−5.7]
.......................................................................................................................................................................................................................................................................
( ). Un es . T end 0.085 . −0.892 . −0.726 −0.361 −0.351 −4.81 −0.827 −0.566
[5.3] [.] [−6.9] [.] [−5.0] [−4.6] [−4.4] [−4.8] [−5.0] [−5.7]
Lags 14 20 7 14 14 14 17 17 17 17
Log money-ou pu a io: ln(ma
)≡ln(Ma
/(PCE ×Real pe sonal income))—Speci ica ion: ln(ma
)=δ0+δ1( )−δ2ln( )
≡RUCa
Cu ency DD OCD Sa ing Re ailMM STD LTD InsMM Repo CP +T-bill
Johansen (1995) Speci ica ion:
(ii). Res . cons an 0.357 148 0.046 1.47 0.218 10.9 . . 3.20 .
[13.5] [6.4] [0.0] [5.4] [0.1] [4.9] [.] [.] [5.5] [.]
.......................................................................................................................................................................................................................................................................
(iii). Un es . cons an 0.356 97.4 2.48 1.48 0.470 10.9 3.84 2.32 3.36 0.871
[13.5] [6.4] [2.4] [5.4] [0.3] [4.9] [3.4] [2.2] [5.6] [4.6]
.......................................................................................................................................................................................................................................................................
(i ). Res . T end 0.333 3.52 . 2.10 4.00 −4.67 . . 20.2 3.10
[7.8] [6.9] [.] [5.2] [1.4] [−5.3] [.] [.] [5.6] [4.7]
.......................................................................................................................................................................................................................................................................
( ). Un es . T end 0.352 3.78 . 2.30 . −5.20 2.50 . 25.5 3.20
[8.0] [7.0] [.] [5.2] [.] [−5.3] [3.2] [.] [5.6] [4.7]
Lags 6 4 2 3 10 5 5 16 2 4
No e: The op panel and bo om panels con ain he semi-log and he log-log speci ica ions, espec i ely. The numbe o coin eg a ing equa ions
(single in ou bi a ia es) a e es ed a a 5% signi icance le el ollowing Johansen (1995). We i s conduc he maximum eigen alue es o he ace
es o he null o no coin eg a ing ela ionship be ween each pai o a iables shown in he able. The able shows es ima es o α2o δ2 o e e y
bi- a ia e speci ica ion ha ails o ejec he null o no coin eg a ion o each o he (ii.— .) speci ica ions o he Johansen (1995) es , whe e α2
o δ2a e he Johansen (1991) maximum likelihood es ima es o he coe icien s o he i s coin eg a ing ec o , no malized so ha he coe icien
on ln(m ) equals one. Numbe s in b acke s unde nea h each es ima e o α2o δ2deno e he -s a is ic. Any blank (missing alue) in each elemen
deno es no coin eg a ion was ound o ha pa icula pai o a iables unde ha pa icula speci ica ion o he ele an sample. Op imal lag
leng hs a e chosen acco ding o he Akaike (1974) c i e ion.
which sugges s hei ela ionship would ail o quali y as a easible cha ac e iza ion o mone a y
demand unc ions.
The bo om panel o Table 9yields s a k con as o he e idence om he op panel. All (ii.—
.) c i e ia o he Johansen (1995) es ind s ong e idence o coin eg a ion be ween: cu ency,
demand deposi s, sa ings, and Repos wi h hei espec i e use cos s—all wi h he co ec sign. Fo
all o he componen s, a leas wo c i e ia o he Johansen (1995) es ind coin eg a ing ela ion-
ships wi h hei espec i e use cos s (excep o ins i u ional money ma ke s, which coin eg a e
wi h hei own use cos unde a single c i e ia). This able p o ides a o al 40 es ima es o δ2 o
all combina ions o 10 pai s imes ou Johansen (1995) c i e ia. No ably, ou o hese 40 es i-
ma es, 29 show he co ec sign consis en wi h a nega i e use cos elas ici y o money demand;
nine ail o ind a coin eg a ing ela ionship; and only wo ind an in e ed sign o δ2(bo h end
speci ica ions o STDs).
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
Mac oeconomic Dynamics 23
The mo e adi ional mone a y asse s ha exis ed well be o e 1980—such as cu ency, demand
deposi s, and a ious ime deposi s—show s ong coin eg a ion wi h hei espec i e use cos s.
This obus e idence o use cos s con as s an appa en lack o a s ong link mos o hese
adi ional asse s ha e wi h he T-bill yield. Less adi ional asse s such as: OCDs, e ail, and ins i-
u ional money ma ke unds gained popula i y as a esul o inancial inno a ions ha ensued
a e 1980. Ou es s show ha OCDs coin eg a e wi h he T-bill yield bu always wi h he inco -
ec sign o α2; and hey coin eg a e wi h hei espec i e use cos s wi h a co ec sign o δ2
unde c i e ia (ii.)and(iii.), bu only one shows a signi ican es ima e o δ2. A simila pic u e
eme ges o e ail money ma ke unds, which coin eg a e wi h he T-bill yield wi h he w ong
sign o α2. While we ind e idence o coin eg a ion be ween e ail money ma ke unds and hei
use cos o mos speci ica ions, none o hem show a s a is ically signi ican es ima e o δ2.
In e es ingly, unde he cons an c i e ia (ii.—iii.)o heJohansen(1995) es , ins i u ional
money ma ke s coin eg a e bo h wi h hei use cos , as well as he T-bill yield. Repu chase ag ee-
men s became popula ehicles o sho - e m deb in he a e ma h o he GFC. We ind s ong
e idence o coin eg a ion wi h hei use cos unde all speci ica ions (see bo om panel o Table 9),
and gene ally weak e idence o a sensible ela ionship wi h he T-bill yield. Finally, we agg e-
ga e ins umen s o p i a e and public deb in he o m o comme cial pape and T-bill balances
and es he coin eg a ion ela ionship wi h he a e age o hei espec i e use cos s and he T-
bill yield. We ind hese balances gene ally coin eg a e wi h bo h in e es a e measu es, wi h he
expec ed sign o i e o he eigh Johansen (1995) c i e ia.
Resul s om ou g anula analysis p o ide e idence ha he use cos s o Di isia ca y in o -
ma ion con en ha is ma e ial and meaning ul o mone a y asse s in a way ha seems o be
so ely lacking in he T-bill yield. This conclusion seems o bu ess ou indings o he supe io
in o ma ion con en o use cos s a he agg ega e le el, ela i e o he T-bill yield.
9. Concluding ema ks
We ind e idence consis en wi h many ea lie wo ks ha he once-s ong connec ion be ween
he oppo uni y cos o holding money and simple-sum money agg ega es expe ienced a sub-
s an ial b eakdown a e 1980 in he U.S. The subs an ial de egula ion in inancial ma ke s ha
esul ed om DIDMCA in 1980 seemed o a es he in o ma ion con en ha was ca ied by
mone a y agg ega es. This in o ma ion ailu e has been used in he ield o essen ially igno e
mone a y agg ega es since hen and, by and la ge, a ionalize aking money ou o mone a y
models in mode n imes—pa icula ly, when dealing wi h mone a y policy de e mina ion. Ou
esul s demons a e ha he ins abili y in money demand documen ed in he li e a u e is inex i-
cably linked o he ailu e o simple-sum mone a y agg ega es o p ope ly accoun o he p ice o
in e es -ea ning bank deposi s.
We ind his lack o in o ma ion om simple-sum agg ega ion ex ends o he GFC pe iod and
beyond. De egula ion on in e es -bea ing checking accoun s and o he inancial inno a ions in
e ail banking since 1980, as well as he pos -2007 GFC pe iod—when a se ies o new Fede al
Rese e balance shee policies was implemen ed—all seem o ha e con ibu ed o e asing he
signal om simple-sum money agg ega es. We ind money demand measu ed wi h simple-sum
agg ega es is ela i ely s able only in he pe iod p eceding 1980. Con e sely, we ind Di isia money
indexes show s able in e es elas ici y o money demand unde wo unc ional o ms o money
demand ac oss mul iple coin eg a ing c i e ia bo h be o e 1980—and mo e impo an —since
DIDMCA.
We also p o ide empi ical e idence ha he use cos o money p o ides be e signal as a dual
p ice o money han he T-bill yield. As a as we know, ou s is he i s g anula in es iga ion o
money demand ela ionships by es ima ing coin eg a ion be ween obse able componen asse s
o he Di isia agg ega es and hei espec i e use cos s in a his o ical con ex . We ind obus
h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
24 Z.ChenandV.J.Valca cel
e idence o he s a is ical ad an ages o di ec ly connec ing mone a y asse s and he use cos s o
hose asse s. We ind ha bo h agg ega e and g anula in o ma ion on he p ice duals o money
highligh ha money demand is much mo e s able han wha is sugges ed by a emp ing o con-
nec imp ope ly weigh ed mone a y agg ega es wi h adi ional nominal in e es a es associa ed
wi h mo e segmen ed ma ke s, such as he T-bill yield. The inescapable conclusion is ha he
ins abili y o money demand is a ma e o measu emen a he han a consequence o a s uc u al
change in agen s’ p e e ence o mone a y asse s.
Acknowledgemen s. We hank Edi o Se le is o he judicious and expedi ious handling o ou manusc ip . We app ecia e
wo anonymous e iewe s o help ul commen s and sugges ions. We hank a endees o he 9 h Con e ence o he Socie y o
Economic Measu emen (SEM 2024) in A lan a o hei eedback, as well as Je an den Noo and he Cen e o Financial
S abili y o da a suppo . And we emain mos g a e ul o William A. Ba ne o being a cons an sou ce o inspi a ion. The
s anda d disclaime applies.
No es
1The eal use cos is essen ially he sp ead be ween he eal a e o e u n on pu e capi al in es men and he asse ’s in e es
a e.
2Valca cel (2018) in es iga es he s a is ical ela ionships be ween use cos s and he ede al unds a e, a he han mone a y
asse s as we s udy in his pape .
3As Cagan (1956)s a es:“The e is a cos o holding cash balances wi h espec o each o he al e na i es o ms o holding
ese es, and in a wide sense, any hing ha can be exchanged o money is an al e na i e o holding ese es in he o m o cash
balances. The cos o holding cash balances [...] is he di e ence be ween he money e u n on a cash balance and he money
e u n on an al e na i e ha is equi alen in alue o he cash balance. The money e u n on bonds includes in e es [...]”
4The Di isia quan i y agg ega es and hei associa ed eal use cos s a e cons uc ed as index numbe s and no malized by
CFS so ha he Di isia agg ega es equal 100 in 1967. In addi ion, we conside nominal low a iables in he denomina o o
he agg ega es as he scale a iables. Bo h o hese poin s sugges ha he alues in he e ical axes o Fig. 1do no p o ide an
exac economic in e p e a ion. Fo example, he bo om le cha in Fig. 1shows he eal use cos o Di isia M1 begins a a
alue o 0.1 in 1967 and i peaks a 0.3 a ound 1980, sugges ing he alue o ha index ipled be ween 1967 and 1980; which
canno , howe e , be ead as an in e es o 10% in 1967 and 30% in 1980.
5We include a cons an in all ou VECM speci ica ions, which emo es op ion (i.) ou lined in Table 1.
6We also conside ed Schwa z (1978) and he Hannan and Quinn (1979) c i e ia o in o m lag selec ion.
7The log o he T-bill yield is well de ined h oughou ou sample. Howe e du ing he ZLB pe iod, i ual—bu no exac ly—
ze o T-bill a es gi e ise o nega i e numbe s when log ans o med. Samples be o e 2008 as well as samples since 2015 show
ou conclusions emain quali a i ely obus ac oss speci ica ions. Thus he poo pe o mance o he T-bill a e in ou analysis
seems endemic h oughou ou his o ical sample and is no necessa ily a consequence o he ZLB pe iod.
8We also employed he Chow (1960) es o a single s uc u al b eak a known change poin and ind ou choice o 1980
p o ides a b eak in bo h se ies.
9Gi en he gene al lack o meaning in he log o a a iable ha emains close o ze o o a meaning ul po ion o a sample.
This is bu essed by a ious esul s in p e ious sec ions o his pape .
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h ps://doi.o g/10.1017/S1365100524000427 Published online by Camb idge Uni e si y P ess
