Nonlinear Equilibrium Correction in U.S. Real Money Balances, 1869-1997
Author(s): Lucio Sarno, Mark P. Taylor, David A. Peel
Source: Journal of Money, Credit and Banking, Vol. 35, No. 5 (Oct., 2003), pp. 787-799
Published by: Blackwell Publishing
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LUCIO SARNO
MARK P. TAYLOR
DAVID A. PEEL
Nonlinear EquilibriumCorrectionin U.S. Real
Money Balances, 1869-1997
Several theoretical models of money demand imply nonlinear functional
forms for the aggregatedemandfor money, characterizedby smoothadjustment toward long-run equilibrium.In this paper,we propose a nonlinear
equilibriumcorrectionmodel of U.S. money demand that is shown to be
stable over the sample period from 1869 to 1997.
OVERTHElast decade or so, a large body of empirical
researchon modeling the demandfor money has developed, mainlyusing cointegration and equilibriumcorrectiontechniques (see, inter alia, Hendry and Ericsson,
1991a, 1991b, Baba, Hendry,and Starr, 1992, Carlson et al., 2000, and references
therein).1One of the issues that has received widespread attentionfrom researchers is concernedwith the dynamicmodeling of money demandusing low-frequency
data for relatively long sample periods. Long samples, however, may potentiallybe
inappropriatebecause of regime shifts, financialinnovation,and structuralchanges,
requiringextremecarein testingandmodeling.In this paper,we proposean empirical
model of U.S. real money balances estimatedusing an updatedversion of the data
set providedby Friedmanand Schwartz(1982), spanningfrom 1869 through1997.2
This paperwas begun when Lucio Samo was on the staff of the University of Oxford and was partly
written while he was a visiting scholar at the Federal Reserve Bank of St. Louis and Washington
University and when Mark P. Taylor was a visiting scholar at the InternationalMonetary Fund. The
authors are grateful to an editor of this journal, Paul Evans, and to three anonymous referees for
constructive criticisms and suggestions and to Bill Barnett, Bob Rasche, Timo Terasvirta,and Dan
Thorntonfor useful conversations or comments on previous drafts. Lucio Saro and David A. Peel
thank the UK Economic and Social Research Council (ESRC) for providing financial support(Grants
L138251044 and L138251004, respectively). The authors alone are responsible for any errors that
may remain.
Lucio SARNO is a professor of finance at the Warwick Business School, University of
Warwick, and research affiliate, Centre for Economic Policy Research, London. E-mail:
MARK P. TAYLORis a professor of macroeconomics at the
[email protected]
Department of Economics, University of Warwick, and research fellow, Centre for Economic
DAVID A. PEEL is a
Policy Research, London. E-mail: [email protected]
professor of economics at Cardiff Business School, University of Cardiff. E-mail: peeld@
cardiff.ac.uk
Journal of Money, Credit, and Banking, Vol. 35, No. 5 (October 2003)
Copyright 2003 by The Ohio State University
788
: MONEY, CREDIT,AND BANKING
Over the sample period examined, the monetary history of the U.S. has seen a
number of fundamentalchanges in exchange rate regimes, institutional structure,
and policy targets, which, in addition to the continuous evolution of the financial
system and various nominal and real shocks, representserious potential pitfalls to
researchersattemptingto find an empiricalmodel of money demand that is stable
over the full sample. This task is addressedby employing a nonlinear equilibrium
correctionmodel for the change in U.S. real money balancesin the form of a smooth
transitionregression of the type popularizedby Grangerand Terasvirta(1993) and
Terasvirta(1994,1998). The model proposedis shownto be stableover the full sample
period examined, and the finding of significantnonlinearityin the empiricalmoney
demandequationis consistentwith severalrelatedempiricalstudies (e.g., inter alia,
Hendryand Ericsson, 199la, Ltitkepohl,Terasvirta,andWolters,1999, Sarno, 1999,
Terasvirtaand Eliasson, 2001).
At a theoreticallevel, nonlinearityin money demand adjustmentis predictedby
the class of target-boundsmodels developed by, inter alios, Miller and Orr (1966)
and Akerlof (1973, 1979), where a representativeagent specifies a target level of
money balances and thresholds above and below the targetthat the balances must
not cross, therebyimplying that short-runnominal adjustmentoccurs within bounds
set by long-runmagnitudes.At the macroeconomiclevel, this class of models is likely
to generate smooth (as opposed to abruptthreshold) adjustmentin the aggregate
money demand function as an effect of time aggregation and nonsynchronous
adjustment by heterogeneous agents (Bertola and Caballero, 1990, Terasvirta,
1994, 1998). Nonlinear adjustmentin money demand equations may similarly be
rationalizedon the basis of buffer-stockmodels (e.g., Gandolfiand Lothian, 1976,
Cuthbertsonand Taylor, 1987), which recognize nonzero costs of adjustmentof
money balances and imply that it may be optimal for agents to allow short-run
deviations of money balances from long-run equilibrium and to adjust only for
relatively large deviations. In general, these types of models imply that the speed
of adjustmentin money demand functions in response to exogenous shocks may
depend nonlinearlyat the aggregate level on the size of the deviation from longrunequilibrium(for a discussion of these issues, see, for example, Milboume, 1987,
1988, Thornton,1990, Mizen, 1994, 1997, Samo, 1999).3
The nonlinearequilibriumcorrectionmodel proposed in this paper allows us to
capturethe predictions of this strandof the theoretical literature,being consistent
with a world where the behavior of fully optimizing agents that allow short-run
deviationsfrom the long-runequilibriumlevel of money balancesgeneratessmooth
adjustmenttoward equilibriumin the aggregatemoney demand function.
The rest of the paper is structuredas follows. In Section 1 we discuss the class
of nonlinearmodels that we employ for modeling the demandfor money and other
aspects of our econometric methods. Section 2 describes our data set and reports
the empiricalresultsof carryingoutunitroot,cointegration,andlinearitytests, as well
as the results of applying linear and nonlinear equilibrium correction modeling
techniques to our data. Section 3 briefly summarizesthis study.
LUCIO SARNO, MARK P. TAYLOR, AND DAVID A. PEEL : 789
1. NONLINEAR EQUILIBRIUMCORRECTIONMODELING
In this paper,we consider a smooth transitionregression (STR) model of money
demand,parameterizedin the form (Grangerand Terasvirta,1993, Terasvirta,1998)
P
A(m
P)t
P
ko + put- 1 +)t+ oAA(m-P)t-j
j=0
j=1
P
j=1
j=0
p
+
- xt-
yARL,
P
ocajA(m- P)t-j ,+
j=o
PjAYt-
p
yjARL,t_ +
j=0
where Axt =
j=0
p
iSARSt_ + kko+ p'tt-l +
+
P
+
+
3Ayt-j
6sARSt-j ([O; zt,-
j=0
] + t,
(1)
V x; m, p, y, RL, and RS denote the logarithm of nominal
money, the logarithmof the implicit price deflator (hence m - p is the logarithm
of real money), the logarithmof real income, and the long- and short-terminterest
denotes a parametrictransition
rates, respectively;?t denotes white noise; and D[1.]
>
for a given level of (Zt - L).
the
of
0
0
transition
with
function,
determining speed
In particular,we consider an STR model where the transition variable zt is the
lagged estimated equilibrium error from a conventional static long-run money
demand model of the form (m - P)t = a + byt + cRLt + ut, i.e., the lagged residuals
=
Ut-d, with the integer d > 0 denoting a delay parameter such that Zt ut_d
The transitionfunctionI4[.] in Equation(1) may be, for example, an exponential
function of the form [1 - exp{ -0( t-d - g)2}] or a logistic function of the form
[1 + exp{-0(i _d -- )}]1-, resulting in an exponential STR (ESTR) or a logistic
STR (LSTR), respectively.The transitionfunction of the LSTR is a monotonically
increasingfunctionof (ut-d - L)and yields asymmetricadjustmenttowardequilibrium, whereas the transitionfunction of the ESTR is symmetricabout g, although
the tendency to move back to equilibriumis stronger,the larger the absolute size
of the deviation from equilibrium it-d -
l1 (for further details see Granger and
Terasvirta,1993, Terasvirta,1994, 1998).
From the discussion in the introduction,it is clear that the nonlinearadjustment
described by target-boundsmodels and buffer-stockmodels for aggregate money
demand functions, where the speed of adjustmenttoward the long-run equilibrium depends on the absolute size of the deviation from equilibrium, might be
well capturedby an exponentialtransitionfunction.5Nevertheless, in selecting the
transitionfunctionin our empiricalanalysis,we employ a purelystatisticalapproach
in that we execute the sequence of nested tests suggested by Grangerand Terasvirta
(1993) and Terasvirta(1998). Hence, as a preliminaryto model specification and
estimation, we executed linearity tests based on the auxiliary ordinary least
squaresregression
^ECM =
+W~,Wttt-d
+
2W
d+
3Wt-d
+ innovations,
(2)
790
: MONEY, CREDIT,AND BANKING
wherevECM
denotes the residualsfrom a generallinearequilibriumcorrectionmodel
- P)t as a function of Wt, which is the vector comprising the exfor
A(m
(ECM)
variables
A(m - P)t-i, A(m P)t-2, Ayt-, ARLty, and ARSt-y for j = 0,
planatory
1, 2, in additionto a constantterm and the lagged estimatedcointegratingresiduals
or equilibriumcorrectionterm it-1;
l1,52, and 53 are vectors of parameters.A
\0,
test
for
linearityagainst STR-typenonlinearityis then the F-test of the null
general
- =
= 0 where 0 is a null vector; HOG can be tested for
hypothesis HOG: = 52
various values of d, say d E {1,2,...,D}. If linearity is rejected for more than one
value of d using such an F-test (say FG),then d is determinedas the value d, which
minimizes the p value of the linearitytest, and we set d = d.
After rejectingHOGusing FG,the choice between LSTR and ESTR formulations
may be based on a sequence of nested tests within HOG.In practice, the following
hypotheses are tested sequentially Ho3: :3 = 0, H02: 2 = 0 I 03= 0, and Hol:
= 53 = O using F-tests termed F3, F2, and F1, respectively. If
1 = 0 I 02
either F3 or F1 yields the strongest rejection of the linearity hypothesis (i.e., the
lowest p value), we select an LSTR model; if F2 yields the strongest rejection
of linearity,however, we select an ESTR model (for furtherdetails see Grangerand
Terasvirta,1993, Terasvirta,1998).
2. DATA AND EMPIRICALRESULTS
Annualtime series for nominalnarrowmoney, real income, implicit price deflator,
short-terminterestrate (call money rate in percentper annum),and long-termbond
rate (percent per annum) were obtained from updating the data set provided by
FriedmanandSchwartz(1982) using the InternationalMonetaryFund'sInternational
Financial Statistics CD. The sample period spans 1869-1997. From these data, we
constructedthe time series of interestfor the empiricalanalysis,namelythe logarithm
of realmoneybalances(m - p), the logarithmof realincome(y), thelong-terminterest
rate (RL), and the short-terminterestrate (RS).
As a preliminaryexercise, we tested for unit root behaviorof each (m - p), y, RL,
and RS by calculatingaugmentedDickey-Fuller (ADF) test statistics (Fuller, 1976,
Dickey and Fuller, 1979). In each case, the number of lags was chosen such that
no residualautocorrelationwas evident in the auxiliaryregressions.In keeping with
a large number of studies in this context, we were, in each case, unable to reject
the unit root null hypothesis at conventionalnominal levels of significance.8On the
otherhand,puttingthe series into first-differenceform did appearto induce stationarity for each of the series. These results confirm the often recordedresult that real
money balances, real income, and interest rates are generatedby processes with a
single unit root (e.g., see, inter alia, Nelson and Plosser, 1982, HendryandEricsson,
1991a,b,Baba, Hendry,and Starr,1992, Carlsonet al., 2000 and referencestherein).
However, the possibility that the time series in question displays structuralbreaks
complicates the interpretation of unit root tests. Thus, we constructed four
constant shift dummies, say D1, D2, D3, and D4, covering World War-I
LUCIO SARNO, MARK P. TAYLOR, AND DAVID A. PEEL : 791
(1914-1918), the interwarperiod (1919-1938), WorldWar-II(1939-1945), and the
postwarperiod (1946-1997), respectively.We then recalculatedthe ADF tests using
an auxiliary regression that also included the four dummies, essentially using a
variantof the procedureemployed by Perron(1989). Nevertheless, the results from
these unit root tests yielded results that were qualitativelyidentical to those of the
conventionalADF tests, in thatwe were unableto rejectthe unit root null hypothesis
for each (m - p), y, RL, and RS.6
Given our unit root tests results, we then formally tested for cointegrationusing
the Johansen(1988, 1995) maximumlikelihoodprocedurein a vector autoregression
involving (m p), y, RL, an unrestrictedinterceptterm, and the four constantshift
dummies D1, D2, D3, and D4 defined above. We assumed a lag length of two,
which was suggested by both the Akaike informationcriterion and the Schwartz
informationcriterion.Using both Johansentest statistics (namely the Xmaxand btrace
statistics)and the appropriatecriticalvalues (both asymptoticand correctedto adjust
for finite sample and for the presence of the four dummies described above), the
cointegrationresultssuggested thatone cointegratingvector exists between (m - p),
y, and RL; afternormalizingthe coefficienton (m - p) to minus unity,the estimated
cointegratingparameterson y and RL were found to be very strongly statistically
significantly different from 0 at conventional nominal levels of significance and
equal to b = 1.441 and c =-0.058, respectively.7These results are fairly satisfactory, in that the cointegratingparametersare correctly signed, althoughit is disappointing that the hypothesis of income homogeneity (b = 1) is rejected at
conventional nominal levels of significance. We then retrieved the cointegrating
residuals, ut, which we use to estimate an ECM and which we also consider as the
potential transitionvariable in our nonlinearequilibriumcorrectionmodeling.
However, before proceeding to estimate an ECM for the change in real money
balances, we tested whether the cointegrating equation describing the long-run
demandfor money has been stable over our sample period. A numberof tests have
been developed for testing whether long-run cointegrating relationships display
structuralbreaks (see, for example, Hansen, 1992, Andrews, 1993, Gregory and
Hansen, 1996). In this paper,we use the tests suggested by Hansen (1992). Given a
staticmodel, Hansenproposesthreetests of the null hypothesisthatthe cointegrating
vector, say 4O, is constant over the sample period (T). All three tests are based on
a sequenceof F-tests, say Fr, for 1 < t < T.The firsttest proposedby Hansen(1992)
tests the null of parameterstability against the alternativehypothesis that there is
a single break point at time s (i.e., H1: Ol,,t
s+i,r,Twhere iij indicates the value
of the cointegrating vector over the interval t = i..., j V i, j, such that i < j).
The problemwith this test is that the breakpoint s is treatedas known (see Hansen
1992). A second test proposed by Hansen (1992), the sup(F) test, treats the break
point as unknown so that the alternativehypothesis is now HI: 1l,t X T+1I,Twhere
X = (t/T) E Z and 5 is a compact subset of (0,1). The test is computed as
sup(F) = supt(FT,). The third test proposed by Hansen (1992) considers the possibil-
ity that the cointegratingvector Otfollows a martingaleprocess so that there is a
constant hazardof parameterinstability,and HI: Ot == t-l + error.This test may
792
:
MONEY, CREDIT, AND BANKING
be computedas mean(F) = (1/T*)EZE FT,t,and T* = 4Si1. The tests are applied
on a region that does not include the end points of the sample, and Hansen (1992)
and Andrews (1993) suggest a trimmingregion S = [0.15,0.85], which in our case
correspondsto the subperiod1888-1978. The results from carryingout the Hansen
(1992) sup(F) and mean(F) tests suggest that the null hypothesis of constancy of
the cointegratingvector over the sample could not be rejected,with p values of 0.32
and0.30 for the sup(F) andmean(F) tests, respectively.9In turn,these resultsindicate
that the long-run money demand function implied by our cointegrationresults is
stable over the sample period examined even when the null of parameterstability
is tested against alternativehypotheses of instability with unknownbreak points.
Priorto testing for linearity,we estimateda linear ECM for A(m - P)t with a lag
length of two and tested down by sequentiallyimposing restrictionson statistically
insignificantparametersin order to obtain the best-fitting parsimoniousmodel reportedin Table 1. The model appearsto be quite adequatein termsof fit and displays
approximatelywhite noise residuals.Also, each of the estimatedparametersis of an
economically plausible sign and magnitude.Nevertheless, the linear ECM does not
pass the diagnostic test for parameter stability, the null of parameter stability
against the alternativeof smoothly changing parameters(see Lin and Terasvirta,
1994, Eithreimand Terasvirta,1996, Terasvirta,1998, Wolters,Terasvirta,Ltitkepohl,
1998) was rejected very strongly.Also, the linear ECM does not pass the RESET
test at conventionalsignificancelevels. Clearly,this may be interpretedas suggestive
of the fact thatnonlinearequilibriumcorrectionmay be a prerequisitefor parameter
stability,consistent with the evidence recently reportedon other long spans of data
by Saro (1999) for Italy and Terasvirtaand Eliasson (2001) for the UK.
Next, applyingthe Akaike and Schwartzinformationcriteriato a linear ECM for
A(m - p)t, the lag length p was set equal to two in order to execute the linearity
tests discussed in Section 2. Panel A of Table 2 reportsp values of the test statistic
FG for d E {1,2,...,5}. Linearityis rejected most strongly for d = 1, suggesting a
ratherfast response of real money balances to deviations from its linear equilibrium
TABLE 1
ESTIMATED LINEAR ECM
A(m - P)t =
ho+ l(m
FOR THE CHANGE IN U.S.
- P)t-
REAL MONEY BALANCES
+ h2Ayt + h3ARSt+ h4ARSt-
+ hsDl + h6D4 + h7it_l,
where
ho = 0.046 (0.007), hi = 0.162 (0.078), h2 = 0.277 (0.060), h3 = -0.004 (0.001), h4 = -0.005 (0.001),
h5 = -0.039 (0.018), h6 = -0.049 (0.010), h7 = -0.075 (0.021).
R2 = 0.340; DW = 2.017; LB = {0.409}; ARCH = {0.648}; RESET = {5 x 10-6}, LM3 = {0.0035};
LM2 = {4 x 10-6}; LM1 = {0.0156}
NOTES:Estimation is by ordinaryleast squares; figures in parentheses are estimated standarderrors. R2 is the adjusted coefficient of
determination;DW is the Durbin-Watsonstatistic; LB and ARCH are the Ljung-Box test for residual autocorrelationup to orderthree
and a test for autoregressiveconditional heteroskedasticityup to order three, respectively. RESET is the Ramsey (1969) test of the null
hypothesisof linearityagainstthe alternativehypothesisof generalmodel misspecification,where the alternativemodel consideredinvolves
a second-orderpolynomial. LM3, LM2, and LM1 are tests of the null hypothesisthat all coefficientsexcept the coefficients of the dummy
variables are constantagainst the alternativehypothesis that they are smoothly changing parameters,constructedas suggested by Lin and
Terasvirta(1994). For LB, ARCH, and RESET as well as for LM3, LM2, and LM1 we only reportthe p value in brackets.
LUCIO SARNO, MARK P. TAYLOR, AND DAVID A. PEEL
:
793
TABLE 2
LINEARITY TESTS RESULTS: p VALUES
Panel A: FG tests
d = 1 1 x 10-4; d = 2: 0.0446; d = 3: 1910; d = 4: 0.0145; d = 5: 0.2449
Panel B: F3, F2, and F1 tests (d =1)
F3: 0.0240; F2: 3 x10-4; FI: 0.2660
NOTES:
The F6, F3, F2, and F1 statisticsare linearitytests constructedas describedin the text; d denotes the delay parameter.The p values
reportedabove were calculated using the appropriateF-distribution.
path. Panel B of Table 2 then reportsthe p values of the test statistics F3, F2, and
F1, assumingd = 1; the resultsclearlysuggest thatan ESTRis the model indicatedby
the proceduredeveloped by Grangerand Terasvirta(1993) and Terasvirta(1998),
consistent with our economic priors and the theoreticalconsiderationsdiscussed in
the introduction.?10'
Assuming p = 2 and d = 1 accordingto the linearity tests results, we then estimated an ESTR model of the form of Equation (1) by nonlinear least squares
(GrangerandTerisvirta1993). We also followed the recommendationof Grangerand
Terasvirta(1993) and Terasvirta(1998) of standardizingthe transitionparameterby
dividing it by the sample variance of the transitionvariable,o, and using a starting
value of 0 = 1 in the estimation algorithm. A parsimoniousnonlinear ECM was
obtainedfor the changein U.S. real money balancesafterimposing variousexclusion
restrictions (see the LR test in Table 3) in applying the conventional general-tospecific procedure(e.g., see Hendry,Pagan, and Sargan1984). The resultingmodel
displays insignificant diagnostics (including the appropriateparameterconstancy
test), yielding a sizable reductionin the residual variancerelative to the best-fitting
TABLE 3
ESTIMATED NONLINEAR ECM
A(m
-
FOR THE CHANGE IN U.S.
REAL MONEY BALANCES
P)t = ko + YoARL,+ [aoA(m -p)1_ + otAyt+ 'Dl
exp{ - (0/{a)t-i}]
p'it-1][l
+ T D4 +
where
ko = 0.048 (4.90 x 10-3), yo = -0.012 (5.23 x 10-3), aot = 0.394 (0.143), Po = 0.537 (0.130),
tz = -0.047 (0.016), T' = -0.103 (0.023), p' = -0.141 (0.032), 0 = 0.889 (0.362), 6a= 0.054
R2 = 0.434; V = 0.857; PC = {0.359}; NRN = {0.641}; NSC = {0.447}; ARCH = {0.825};
LR= {0.466}
NOTES:Estimation is by nonlinearleast squares; figures in parenthesesare estimated standarderrors. R2 is the adjusted coefficient of
determination;V is the ratio of the estimated residual variance from the ESTR model to the estimated residual variance from the bestfittinglinearECM. PC and NRN are LagrangeMultiplier-typetests for parameterconstancyand for no remainingnonlinearity,respectively,
while NSC is a LagrangeMultipliertest for no serial correlationup to order three, constructedas suggested in Eitrheimand Terasvirta
(1996). ARCH is a test for autoregressiveconditionalheteroskedasticityup to orderthree.LR is the likelihood ratio test for the restrictions
imposed in the estimatedESTR model against a general unrestrictedESTR model with p = 2 and d = 1 and is distributedas X2(q),with
q equal to the numberof restrictions.For each of PC, NRN, NSC, ARCH, and LR we only reportthe p value in brackets.
794
: MONEY, CREDIT, AND BANKING
linearECM (about 15%).Nevertheless,using the minimalnesting strategyof Mizon
and Richard (1986) and applying a simplification encompassing test between the
STR model given in Table 3 and the best-fitting linear ECM in Table 1 as, for
example, in Saro (1999) and Terasvirtaand Eliasson (2001), indicated that the
nonlinearECM does not encompass the linear ECM at the 5% significance level
(althoughit does at the 1% level) without being encompassed.12'13
The strongly nonlinearbehavior implied by our empirical model is made clear
by the plot of the estimated transitionfunction against the transitionvariableut-l,
displayed in Panel a of Figure 1. Note that the exponential transitionfunction is
bounded between 0 and unity, (<[.]: -i->[0,1], has the properties ([0] = 0 and
limx+oo,([x] = 1, and is symmetricallyinverse-bell-shapedaround0. These properties of the ESTR model are attractivein the present modeling context because they
allow a smooth transition between regimes and symmetric adjustment of real
money balances for deviations above and below the long-runequilibriumlevel. The
plot shows that the limiting case of the transition function D[.] = 1 is attained,
which indicates that the speed of transitionbetween regimes is, in fact, very fast
for large deviations from long-run equilibrium.The satisfactorygoodness-of-fit of
the nonlinearECM is then highlighted by Panel b of Figure 1, which plots actual
and fitted values of changes in real money balances over the sample and shows that
the fitted values are reasonablyclose to the actual values.'4
A
B solid line = actual; dashed line = fitted
Estimated transition function
-0.6
-0.2
0.2
0.6
FIG. 1. Nonlinear ECM for the change in real money balances
LUCIO SARNO, MARK P. TAYLOR, AND DAVID A. PEEL : 795
3. CONCLUSIONS
A stylized fact in the empiricalliteratureon modeling U.S. money demandis the
difficultyof obtainingstableempiricalequationsover relativelylong sampleperiods.
In this paper,we have appliedrecentlydevelopednonlineareconometrictechniquesto
an updatedversion of the Friedman-Schwartzdata set in orderto model the demand
for money in the U.S. during the period 1869-1997. The results are encouraging
on a number of fronts. We obtained a unique long-run money demand function
relating real money, real income, and the long-term interest rate, which displayed
a plausibleinterestrate semielasticityof -0.058. Also, a dynamicevolutionequation
for the change in real money balances was obtained by estimating a nonlinear
equilibriumcorrectionin the form of an exponential smooth transitionregression,
with the lagged long-runequilibriumerroractingas the transitionvariable,implying
faster adjustmenttoward equilibrium,the greaterthe absolute size of the deviation
from equilibrium.
While our results aid our understandingof nonlineardynamics in the demandfor
money,we view them as a tentativelyadequatecharacterizationof the datawhich may
be improved.Nevertheless,the nonlinearempiricalmoney demandequationproposed
here yields a sizable reductionof about 15%in the residual variancerelative to the
best-fittinglinear equilibriumcorrectionmodel and appearsto be superiorto linear
money demand modeling in several respects, also passing a batteryof diagnostic
tests anddisplayingparameterconstancydespitethe numberof fundamentalchanges
characterizingthe monetaryhistoryof the U.S. over our long sampleperiod.Furthermore, the empirical model proposed has a fairly naturalinterpretationin the light
of the nonlineartype of adjustmentimplied by target-boundsmodels and bufferstock models of money demand. Overall, our results suggest that failure to allow
for nonlineardynamics characterizedby short-rundeviations from long-runequilibrium may contribute to explaining the difficulty of much empirical research in
obtaining stable aggregate U.S. money demandequations.
NOTES
1. Severalauthorshave recentlybegunto use the term"equilibriumcorrection"insteadof the traditional
"errorcorrection"as the latter term now seems to have a differentmeaning in some recent theories of
economic forecasting(e.g., see Clements and Hendry 1998, p. 18). Since the term equilibriumcorrection
conveys the idea of the adjustmentconsideredin the presentcontext quite well, we use this term below.
2. MacDonaldandTaylor(1992) estimatea lineardynamicmodel for the U.S. money demandfunction
using the originalFriedman-Schwartzdatafrom 1869 to 1970 (for a similar approachappliedto postwar
data, see, for example, Rasche, 1987, Hoffmanand Rasche, 1991, see also the earlierwork by Thornton,
1982). Lucas (1988), Stock and Watson (1993), and Andersonand Rasche (1999) used similar data for
somewhat longer sample periods. Nevertheless, the presentpaper represents,to the best of the authors'
knowledge, the first attempt to model the U.S. demand for money over a sample that extends the
Friedman-Schwartzdata set to cover the whole post-BrettonWoods period until the late 1990s.
3. While consistentwith some of the implicationsof the buffer-stockmoney demandliterature,however,
it is perhapsfair to say that the empirical model proposed below does not aim to be a pure test of the
buffer-stockmodel, particularlybecause thereis no attemptto model expectations(e.g., see Cuthbertson
and Taylor, 1987, Sarno, 1999).
4. In this framework,the long-run demand for money is determinedby income and the long-term
interest rate, although the change in the short-terminterest rate is allowed to affect the dynamic
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: MONEY, CREDIT, AND BANKING
adjustmenttowardthe long-runequilibrium.This has proved to be very effective in several cases in the
presentcontext (see, inter alia, Hendryand Ericsson, 1991a, MacDonaldandTaylor,1992, Taylor,1993).
Nevertheless, some authorshave used the short-terminterest rate (either in place of or in addition to
the long-termrate) in the long-runmodel; see, for example, the study by Hoffman,Rasche, and Tieslau
(1995) and the relevant discussion on these issues in Laidler (1993) and Hoffman and Rasche (1996).
Also, note that, as discussed in Section 2, we also include four dummy variables in the long-run
cointegratingmoney demand function from which we retrieve the equilibriumerror.
5. The nonlinearECM (1) is a direct generalizationof the standardlinear ECM employed in a vast
literatureon money demand. The generalizationobviously consists of the nonlinearcomponent of the
model, and the variables included are standardvariablesincluded in money demand models previously
estimatedand reportedin many of the studies cited in the paper.Also, given that the transitionfunction
([.] can, in principle,be any boundednonlineartransitionfunction, our nonlinearmodel is fairly general.
Inevitably,however, we do need to restrictour attentionto some specific parametrictransitionfunctions
for which formal linearitytests exist, and this is one reason why we focus on the logistic and exponential
functions. However, the exponential function seems economically plausible and fairly consistent with
our theoreticalconsiderationsinspired by target-boundsmodels and buffer-stockmodels. The class of
nonlinearmodels is infinite,andwe have chosen to concentrateon the STR formulationprimarilybecause
of these attractive properties, its relative simplicity, and the large amount of previous research on
the estimation of STR models.
6. Moreover, using nonaugmentedDickey-Fuller tests or augmented Dickey-Fuller tests with any
numberof lags in the range between 1 and 5 yielded qualitativelyidenticalresults,regardlessof whether
the auxiliaryregression included a constant or a constant and a deterministictime trend and whetherit
included dummies. To conserve space, the results of the execution of these preliminaryunit root tests
have not been reported.
7. Balke and Fomby (1997) show that the Johansenmethod of estimating the cointegratingvector
may work reasonablywell in the presence of nonlinearthresholdcointegrationand conjecturethatthese
results may also hold for smooth transitionnonlinearitiesin adjustment;see also Corradi,Swanson, and
White (2000). We also employed the procedure suggested by Phillips and Hansen (1990) to test for
cointegrationin our long-runmoney demandmodel and found estimates of the cointegratingparameters
identical to the ones reportedabove up to the second decimal point.
8. Underthe assumptionthatthe rankof the long-runmatrixof the cointegratingvector autoregression
equals unity, we could not reject the hypothesis of weak exogeneity of y and RL in the static longrun model. Establishingweak exogeneity of the regressorsin the long-runmoney demandmodel allows
us to model real money balances within a single-equationequilibrium-correctionframework (Engle,
Hendry,and Richard 1983).
9. The FTt test is asymptoticallydistributedas X2underthe null hypothesis. The asymptoticcritical
values for the sup(F) and mean(F) tests, computed by Monte Carlo simulations, are given in Hansen
(1992). The Hansen(1992) stabilitytests reportedherewere obtainedusing the GAUSS programFM.PRG
available at Hansen's website (http://www.ssc.wisc.edu/-bhansen).
10. The choice of the transitionvariable was initially not restrictedto the lagged equilibriumerror
in that we also experimentedwith a number of other potential transitionvariables, including lagged
changes in real money balances,income, and both short-and long-terminterestrates. However, linearity
tests with the lagged equilibriumerroras the transitionvariable yielded the strongestrejections of the
null of linearity and, hence, to conserve space, we only report these results in the empirical analysis.
Also, note that using the general linear ECM describedabove to obtain the residualstested for linearity
or using the best linear ECM given in Table 1 below yielded qualitativelyidentical results.
11. We addressedthoroughlythe question of the robustness of our linearity tests results. The main
concern involves the possibility of a spuriousrejectionof the linearityhypothesisunderthe test statistic
FG in Equation (2) in finite sample. We addressedthis issue by executing a number of Monte Carlo
experimentsconstructedusing 5000 replicationsin each experimentand with identical randomnumbers
across experiments. Our simulations, based on a sample size comprising T E {50, 75, 100, 125, 150,
175, 200} artificialdata points, suggested that the linearitytest FG does not tend to over-rejectthe null
hypothesisof linearitywhen the truedatageneratingprocessis linearautoregressive,even ifnonstationary,
and rejectionof the null does not occur by chance when the process is even marginallynonlinear,thus
increasingour confidence in the linearitytests results and, hence, on the validity of the nonlinearECM
proposed below. These results are interesting,in that it appearsthat nonstationaritydoes not affect the
size of the linearity tests (full details on the Monte Carlo simulations are available on request).
12. More precisely, the null hypothesis that the STR model in Table 3 encompasses the best-fitting
linear ECM was rejected with a p value of 0.0281, while the null that the best-fitting linear ECM
encompasses the STR model in Table 3 was strongly rejected with a p value of 4.00 X 10-8.
13. Note that the effective importance of the lagged dependent variable (i.e., lagged real-money
balances) shrinks with the size of the deviation from equilibriumin our estimated STR model. This is
interestingbecause a numberof authorshave arguedthatthe sluggish adjustmentimplied by a statistically
significantlagged dependentvariable is hard to rationalizeat a theoreticallevel (see e.g., Goodfriend,
1985, McCallum and Goodfriend, 1988, Laidler, 1990, Taylor, 1994). The implication in the present
LUCIO SARNO, MARK P. TAYLOR, AND DAVID A. PEEL : 797
context is that inertiais reduced as the size of the disequilibriumgrows, consistent with the buffer-stock
and targets-boundsmoney demand models discussed above.
14. One might wish to allow for the possibility that the nonlinearitywas in fact of the threshold
variety,which would also be consistentwith target-boundsmoney demandmodels undercertainaggregation conditions. Therefore,we considereda relatedparameterizationof the STR model with a transition
function that allows for threshold-typenonlinearity(Jansen and Terasvirta,1996, Terasvirta,1998) so
that the correspondingSTR model then becomes a special case of a threshold equilibriumcorrection
model (TECM).In our attemptsto estimatea model of this kind using our data,however,we experienced
severe problems in achieving convergence of the estimation algorithm,which appearsto suggest that
smooth ratherthan discrete adjustmentin regime may be more appropriateon our aggregate data.
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