Applied Econometrics and International Development.
AEID.Vol. 6-1 (2006)
EMPIRICAL EVIDENCE FOR A MONEY DEMAND FUNCTION:
A PANEL DATA ANALYSIS OF 27 COUNTRIES 1988-98.
GARCIA-HIERNAUX, Alfredo *
CERNO, Leonel
Abstract
The purpose of this paper is to estimate the money demand function of Cagan (1956)
using a panel data set covering 27 countries with different economic levels over the
period 1988-98. The static and dynamic fixed effects reveal that a money demand
equation exists. However, in contrast to the theory proposed by Cagan, estimates of the
output elasticity of money demand are in the range from 0.18 to 0.20.
Keywords: Money demand, Inflation, Panel data, Dynamic fixed effects
JEL Classification: C23 – C52 – E41
1. Introduction
Inflation has been the focus of attention of a lot of relevant essays in the economic
literature. The analysis of the Money Demand Functions (MDF) is one of the most used
approach to examine it. Recent studies analyze the properties of several MDF in different
countries. Lütkepohl and Wolters (1998) or Beyer (1998) investigated whether the MDF
would remain stable despite the German unification. Dekle and Pradhan (1999) studied
the case of some Asian emergent countries, while Torsen (2002) focused his research on
developing countries as, for example, Mongolia. Therefore, the analysis of the MDF is an
important subject of investigation for both, academic researchers and policy makers. Most
of these previous works and recently others like, e.g., Brand and Cassola (2004) or
Nielsen (2004), use time series (usually ARMA or VAR) or cross section data procedures
in their analyses
The purpose of this paper is to estimate the MDF proposed by Cagan (1956). However,
instead of applying usual analysis with data from one country, we use a panel data model
covering 27 countries with marked economic differences. Starting from these estimates, a
check of the theory is made. In the second part of this paper, we include some lagged
variables in the model by performing the fitting, so we study the MDF in a dynamic
context. Particular panel data techniques are briefly shown for these aims.
The research is structured as follows. Firstly, we present the MDF theory proposed by
Cagan (1956) that originates the main equation of our investigation. Subsequently, we
introduce the econometric framework regarding panel data for both, static and dynamic
*
Alfredo García-Hiernaux is Assistant Professor, Departamento de Fundamentos de Análisis
Económico II, Facultad de Ciencias Económicas y Empresariales, Campus de Somosaguas,
Madrid, Spain. E-mail. agarciah@ccee.ucm.es.
Leonel Cerno is Associate Professor,
Departamento de Fundamentos del Análisis Económico, Universidad Europea de Madrid, Spain.
He gratefully acknowledges financial support from the Ministerio de Educación y Ciencia, project
SEJ2005-07388/ECON.
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AEID.Vol. 6-1 (2006)
Applied Econometrics and International Development.
models. In Section 3, after a short introduction of the data set employed, we examine the
estimation outcomes. Finally, Section 4 discusses previous results and summarizes the
basic conclusions.
2.The Theoretical Model:
From a conceptual point of view, standard theories of money demand postulate that real
money is related to the nominal interest rates, i, and the production output, Y. Defining
the general equation of money demand as L (i ,Y ) , the equilibrium condition for the
monetary market is:
M
= L (i ,Y )
P
(1)
where M is the money stock and P is the price level. Also, we assume that the function
L(i,Y) is decreasing in i and increasing in Y.
On the other hand, the Fisher Identity relates the spread between real (r) and nominal
interest rates (i) to the level of future inflation ( π e ), such as i = r + π e . Whether we are
interested in calculating government returns obtained via money creation, hereafter M,
this would have to be interpreted as the monetary base and L(i,Y) as the monetary-base
demand. Looking at this from the point of view of the equilibrium we have that,
M
= L ( r + g m ,Y
P
)
(2)
where g m is the monetary growth rate which can be defined as:
gm =
M&
M
(3)
Ignoring, for simplicity, the growth rate of the product, the real values will remain
constant in the static situation. This implies that inflation would be the same as the
monetary growth rate.
Thus, the MDF proposed by Cagan (1956) is a good example of the relation between
inflation and what it is known as the seigniorage equilibrium (see, e.g., Özmen, 1998 or
Selcuk, 2001). Specially in an inflation context, it is a useful money demand description
that can be expressed mathematically as,
ln
M
= a − bi + ln y
P
52
(4)
Garcia-Hiernaux, A, Cerno, L.
Empirical evidence for a money demand function
3. Panel Data Econometric Models:
3.1 The Static Model
A widespread representation of the panel data models assumes that discrepancies across
units can be captured in differences in the constant term. These kinds of models are
usually named fixed effects models and can be written as,
yi, t = α i + β T xi, t + ui ,t
(5)
where yi , t is the observable output, xi ,t is a k-regressor vector and ui ,t is a random error
such that: E [ui ] = 0 , E u iT ui = σ u2 IT and E u iTu j = 0 (for all i ≠ j )
Equation (5) may be interpreted as a classical regression model and can be estimated by
ordinary least squares. One may obtain estimates of β in (5) in three possible ways:
Pool estimation
−1
t
βˆt = S txx Sxy
(6)
−1
βˆt = S xxw Sxyw
(7)
−1
b
βˆt = S bxx Sxy
(8)
Within estimation
Between estimation
t
w
b
where S txx , S xy , S xxw , S xy , S bxx and S xy are the matrices of sum-of-squares and crossproducts and are suitably defined in Appendix A.
3.2 The Dynamic Model
To include dynamic effects, we decide to introduce some lagged (endogenous and
exogenous) variables into the equation (5). Thus, the new model can be expressed as,
p
yi, t = ∑ γ yi, t−k + β
k =1
T
( L ) xi,t + λt + αi + ν i ,t
(9)
where λt and α i are specific temporary and individual effects, respectively, and xit is
the regressor vector. Furthermore, β
backshift operator L.
T
( L)
53
is a polinomial vector that includes the
AEID.Vol. 6-1 (2006)
Applied Econometrics and International Development.
Usually, in these cases, the time series size of each individual
( Ti )
is short but the
number of individual ( N ) is large. Here, for simplicity, we present the equation for the i
individual that can be written as follows:
yt = Wtδ + υ tηt + ν t
(10)
where δ is a parameter vector to be estimated (which comprehends α k , β and λ ), Wt
is a matrix containing data (lagged endogenous, exogenous and dummy variables) and υt
is an all-ones Ti ×1 vector.
The estimates are calculated using a class of so-called Generalized Methods of the
Moments (GMM for short):
−1
δˆ = ∑ Wt*T Zt AN ∑ Z tTWt*
t
t
∑W
t
t
*T
Zt AN ∑Wt*T y*t
t
(11)
where
1
AN =
N
∑t Z Ht Z t
−1
T
t
(12)
In equation (11), Wt * and yt* denote some transformation of Wt and yt , respectively,
such as levels, first differences, orthogonal deviations, etc. Moreover, Zt is an
instrumental variable matrix and Ht is an individual weight matrix. When the number of
columns of Zt is the same as Wt * , then AN is irrelevant and δˆ simplifies to:
−1
δˆ = ∑ Z tT Wt* ∑ Z t*T yt*
t
t
(13)
4. Empirical Analysis
In this section we analyze the MDF proposed by Cagan (1956) keeping in mind the
negative influence of the nominal interest rate and the unit value of the coefficient that
multiplies the production output. In a second stage, taking into account the previous
model, we introduce lagged variables in order to study the dynamic of the MDF. We use
the panel data tools mentioned above, with a sample of 27 countries and eleven years
(since 1988 until 1998), for this purpose. The countries included in the sample are
depicted in Table 1. They make up a non-equilibrated panel. Also, variables and data
transformations used in the model estimation are defined in Table 2.
54
Garcia-Hiernaux, A, Cerno, L.
Empirical evidence for a money demand function
Table 1: Countries selected for the analysis
Egypt
South Korea
Spain
Morocco
The Philippines France
South Africa Indonesia
Greece
Argentina
Japan
Holland
Canada
Pakistan
Italy
Colombia
Thailand
Portugal
USA
Germany
UK
Mexico
Australia
Turkey
Venezuela
New Zealand
Belgium
Table 2: Original and Transformed Variables
M1 : Monetary Base, in billions of local currency units.
Y
: GDP measured in prices of 1990, in billions.
IPC : Consumer Price Index (base 1990).
TC : Value of the U.S. Dollar in local currency.
i*
: Nominal interest rates (*).
Data transformations:
m
M
= ln 1
p
IPC
Y
y = ln
TC
Source: National Institute of Statistics except (*), from the International
Monetary Found.
4.1 Econometric Specification and Estimation
The fixed-effect model specified for the MDF for the whole panel is as follows,
m
p = α i + β1ii ,t + β 2 yi,t + ε i,t
i, t
(14)
We estimate equation (14), as starting point, by pool, between and within procedures
shown in Section 3.1. However, we only find consistency with the theory in the last case.
This is not completely unexpected. Indeed, the fixed-effects model become coherent a
priori since there are remarkable differences between many countries used in the analysis.
Thereby this fact lead us to work with the within fixed-effect model in the static
regression. The outcomes obtained by fitting this model are presented in Table 3.
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AEID.Vol. 6-1 (2006)
Applied Econometrics and International Development.
Table 3: Outcomes for the static Fixed-Effects model
Coefficients Estimates t-Statistic Statistics
-0.004
2.9
R2
0.97
β̂
1
β̂2
0.180
6.8
DW* 1.73
This Table show that, the Cagan’s equation of the MDF is partly validated by the sign
of the coefficients. In fact, as in the theory previously described, the real balance demand
is decreasing in the nominal interest rate and increasing in the income. Notice, also, that
parameters related to both variables are statistically significant. Nevertheless, the
coefficient that multiplies the income, β̂2 (known as the income elasticity of the
demand), is not equal to one. Further, the null hypothesis H 0 : β2 = 1 is clearly rejected.
So, the model predicts that the money demand is affected by the income less than
proportionally. This may be caused by the considerable differences between the analyzed
countries.
4.2 The MDF Dynamic Dependence
Starting from the static relation proposed by Cagan (1956), we introduce dynamic
components in the model allowing the existence of lags of endogenous and exogenous
variables in the right side of the equation (14).
After carry out some different estimations using the techniques introduced in Section
3.2, we obtain a dynamic model specification that fits reasonably well to the data set. The
model representation is as follows:
m
m
p = αi + β1ii, t + β 2 yi,t + β3 p + β 4ii ,t −1 + ε i ,t
i, t
i, t−1
(15)
m
and
p i, t−1
Equation (15) contains the previous components and also the real balance,
the nominal interest rate, ii , t −1 , both lagged one period. Any other attempt to introduce
another kind of dynamic in the model did not give good results. The final outcomes of the
dynamic estimation are shown in Table 4.
Table 4 shows that the coefficients β̂1 and β̂2 , that remain from equation (14), do not
vary a lot in this new estimation. Moreover, parameters related to the past of both, the real
balance and the nominal interest rate, are statistically significant. On the other hand,
notice that the negative influence of the one-period lagged interest rate over the money
demand is bigger than the contemporaneous influence. This may be explained by the
possible existence of a time lag in the composition of asset portfolios.
*
The Durbin–Watson statistic value.
56
Garcia-Hiernaux, A, Cerno, L.
Empirical evidence for a money demand function
Table 4: Outcomes for the dynamic Fixed Effects model
β̂
1
Coefficients
Estimates
-0.005
t-Statistic
1.7
F –Wald Statistic
First order autocorrelation
Second order autocorrelation
β̂2
β̂3
β̂4
0.202
2.0
0.573
6.1
91.9
-0.064
-3.610
-0.017
4.2
5. Conclusion
This article presents an empirical analysis of a monetary demand function. Firstly, an
estimation of the specific monetary demand function established by Cagan (1956) with
panel data techniques is made, testing the theory suggested by the author. The results
obtained from this estimation agree in sign with Cagan’s theory, since it is observed that
the estimated parameters for both, the interest rate coefficient and the income coefficient
are negative and positive respectively. However, their magnitudes do not seem to
correspond with the expected values. Concretely, the income elasticity of money demand
is 0.18, lower than the value suggested by the theory.
In a second stage, motivated by previous results, we improve the model-fitting by
introducing dynamic terms. The new model indicates that the money demand depends on
its past and the near past of the interest rate. Nevertheless, we find no relation between
money demand and income lags.
It is important to emphasize that the possibility to introduce dynamics in these kinds of
models not only improves the fit, but also increases the operational capacity of monetary
policy. Therefore, the analysis of these models with panel data techniques seems relevant
not just in the international aspect, but also in a group of countries with a common
monetary policy, such as countries that belong to the European Monetary Union (EMU).
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Appendix A
The matrices of sum-of-squares and cross-products are defined as follows:
S
S
S
S
t
xx
t
xy
w
xx
w
xy
=
=
=
=
n
i=1
t=1
n
T
∑ ∑ (x
i=1
t=1
n
T
∑ ∑ (x
i=1
t=1
n
T
∑ ∑ (x
i=1
S
S
b
xx
=
b
xy
=
T
∑ ∑ (x
− x
)
it
− x
) ( y it
− y
)
it
− xi
) ( xi t
− xi
)
it
− xi
)(
yi t − y i
)
( x it
− x
)( x it
− x
)
T
( x it
− x
) ( y it
− y
)
n
i=1
) ( x it
T
i=1
∑
− x
t=1
n
∑
it
____________________________________
Journal published by the Euro-American
http://www.usc.es/economet/eaa.htm
58
Association
of
T
T
T
T
T
T
Economic
Development.