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IDEC
The Stability of the Burundian Demand -for- Money Function:Some Further Results
Jean NDENZAKO
Institut de Dveloppement Economique
B.P. 6210. Bujumbura-Burundi
Discussion Paper
ECDI
February 1998.
The Stability of the Burundian Demand-for- Money Function :Some Further Results
Jean NDENZAKO 1
1
* An earlier version of this paper was delivered at a workshop on MoneyDemand and Monetary Policy organized by the Burundi Economic Development
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The Burundi Economic Development Institute
Summary
This paper attempts to develop specifications for the Burundian demand for
money throughout the period 1970-1995 which efficiently track actual movementsin holdings of real money balances around the long -run demand functions usingrecent techniques in the analysis of cointegrating times series relationshipsdeveloped by, inter alia Engle and Granger (1987) , Johansen (1988) andJohansen and Juselius (1990). The results indicate , once again, that the demandfor money is affected not only by changes in domestic factors such as real incomeand expected inflation, but also by fluctuations in exchange rates expectations.The evidence suggests that these variables entering into the demand for moneyequation may not form a cointegrated system as far as narrow money is concernedunless the exchange rate is included. For broad money, inclusion of the exchangerate may strengthen its stability because of the weakness of tests of cointegrationto what could be borderline stationarity when the exchange rate is excluded.Whatever the case for M2, equilibrium in the M1 demand equation requires
inclusion of the exchange rate. Furthermore, formal stability tests failed to indicateany shift in the demand for money equations over the sample period consideredand the estimated results do not allow to discriminate between alternativeformulations of the demand for money functions with broadly (M2) and narrowly(M1) defined money.Concerning the fiscal implications of inflation, the Burundian private sector adjustsat a rate of approximately 19.3 per cent per annum to any disequilibrium. Abovethis rate of inflation, the private sector systematically reduces its real holdings ofbase money more rapidly than the rate of inflation so that the real inflation taxrevenue declines.
Introduction
One of the most important recurring issue in the theory and
application of macroeconomic policy is whether or not the demand
for money is stable . In fact, for money to exert a predictable
influence in the economy so that the central banks control of the
money supply can be a useful instrument of economic policy, there
should be a stable money demand function [1].
The demand for money relation should be highly predictable in a
statistical sense as measured by the goodness-of -fit statistics,
precision of the estimated coefficients and presumably its ability to
forecast out of sample. Without predictability, the central bank
Institute in July 1998. I wish to thank workshop participants for their usefulcomments and suggestions. Of course any remaining errors are entirely mine.
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cannot know of the net expantionary or contractionary effects of a
given change in the money supply.
In an earlier paper [2], we presented some preliminary
estimates of the Burundian demand-for-money function using the
partial adjustment mechanism and concluded that current income,
expected inflation and the parallel market exchange rate are
explanatory variables for both narrow and broad definition of money
although its was more so for the narrow definition of money (M1)
than with the broader definition (M2) as far as expected inflation is
concerned.
The partial adjustment mechanism uses the Koyck-lag structure
whereby the whole adjustment process is represented by the
inclusion of lagged dependent variable. Since this structure has been
criticized on the grounds that it is highly biased and unduly
restrictive [Darrat ,(1985)], this paper attempts to refocus the issue
and presents an error correction specification which provides a more
general lag structure which nests the partial adjustment process and
does not therefore impose too specific a shape on the model [Arize,
(1989)] while allowing at the same time to avoid the spurious
regression problem outlined by Granger and Newbold (1974).
Moreover, we present a battery of specification , diagnostic
tests and additional stability tests as it has been shown that in
applied econometric research, one can estimate a totally
meaningless model and yet obtain correct signs , a high
coefficient of multiple determination and high t-values without there
being any relationships between variables whatsoever [3] . According
to my knowledge, there is no other study on money demand in
Burundi using the error correction methodology.
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The remainder of the paper is organized as follows: in section 2
a summary discussion on the model is presented, the time series
properties of the model variables using the Dickey-Fuller (DF),
Augmented Dickey-Fuller (ADF) , and the Sargan-Bargava Durbin-
Watson (SBDW) tests are presented in section 3. In section 4, we
implement recent techniques in the analysis of cointegrating time
series relationships developed by, inter alia Engle and Granger
(1987) Johansen (1988), Johansen and Juselius (1990) to determine
what variables are necessary to insure the stationarity of money
demand.
An error correction specification (using OLS estimation) which
captures the long run dynamics will be attempted when cointegration
is found to exist. The identification of a stable and well defined
demand-for-money function will allow an analysis of the fiscal
implications of inflation which depend not only on the long run
inflation elasticity of the demand-for-money but also on the dynamics
of the private sector adjustment towards its long run equilibrium.
Section 5 presents stability tests of the estimated money demand
equations and finally some concluding remarks and policy
implications are presented in section 6.
2. The Money Demand Model
The demand-for-money includes generally a scale variable
which takes into account the level of business transactions plus
variables representing the opportunity cost of holding money relative
to other assets.
The aggregate that is most commonly used as a scale variable, is
gross domestic product (GDP) which is based on the national income
accounts. There are some well known problems with such series in
Burundi. Such problems are due to changes in methodology used to
assemble the accounting data, and to potentially large errors in the
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survey data. Nevertheless, as these are the best income data
available at this time, this paper shall use the GDP series as the
measure of business transactions.
An opportunity cost variable in a demand-for-money function is
intended to measure the yield of money against other assets that
might be held. In financially developed economies, this variable is
usually the interest rate. In addition, it is also often argued that
inventories of real assets are an alternative form in which wealth can
be held, and hence the expected inflation rate should enter as a
determinant of money demand.
Given the limited range of financial assets and the pegging of
interest rate which prevailed in Burundi in most of the period under
review, physical assets represent the most common way of wealth
holding. If readily liquidable, they constitute close substitutes for real
cash balances. The return on money and on bonds become in this
case negligible. Many rechearchers haven even suggested that in
developing economies , interest rates should be dropped from the
money demand function because in such countries, they are
inadequate [ Darrat (1985)].
As a result, most empirical studies on money demand in
developing countries have solely used expected inflation to represent
the opportunity cost of holding money. Because observed interest
rates are centrally determined and remain unchanged for long
periods in Burundi, there is insufficient variation in interest rates to
enable its influence on the demand for money to be estimated with
confidence. Thus, the interest rates variable is dropped from the
money demand equation estimated in this paper.
As far as the definition of the money stock is concerned, it is
sometimes held that to be operationally useful, a money stock
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definition should comprise an aggregate that the monetary
authorities can adequately control. In financially developed
economies, this principle is sometimes adduced in support of a
narrow definition of money including currency in circulation and
demand deposits (M1), which tend to be more responsive to open
market and interest rates policies. In many developing countries,
however available policy instruments apply principally to the volume
of credit extended by the banking system. This would tend to make
the total liabilities of the banking system (M2 =M1+time deposits)
easier to control.
The last question is which measure of inflation should be used.
The GDP deflator as a general measure of prices within an economy
seems inappropriate because it captures changes in the price level of
domestic output only while the inflation measure should also include
prices of imported goods in order to be a measure of the opportunity
of holding money compared with buying goods. The consumer price
index (CPI) will therefore be used instead of the GDP deflator as the
measure of the opportunity cost of holding money balances.
The inflation variable will be defined as the rate of change in current
prices lagged one year. The omission of current inflation is meant to
avoid possible spurious correlation since the dependent variable is
deflated by current prices. The model is estimated over the period
1970-1995. Only annual data are available for all variables. A
discussion of the data sources is contained in the appendix, along
with a description of the nature of the data.
Using the following notation
md t* = (Md/P)* the stock of desired real money balances at time t
[4]
y t = a budget constraint such as real income at time t
t e = the expected rate of inflation used to represent the expected
return on physical goods at time t
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exct = the expected rate of depreciation of the parallel market
exchange rate used to represent the return on foreign money at time
t
ut= a (presumably) white noise disturbance term,
a simple econometric model of money demand can be written as
mdt* = f [ (yt) , (et), (exct )] (1)
With log linear specification, equation (1) can be written as
Ln mdt* = 0 +1 ln (yt) +2 et + 3 (exct) + ut (2)
The expected inflation and the exchange rate depreciation variables
must enter linearly since they assume negative values in some
years, in which case the logarithms are undefined.
It is generally accepted that the following conditions hold for the
partial derivatives .
mdt* / yt>0
mdt* / et > 0
mdt* / exct < 0 or > 0
Expression (2) assumes that the long- run demand for real money
balances depend positively on the income level and negatively on
the expected inflation rate. The sign of the exchange rate could be
either positive or negative. According to the currency substitution
literature, the depreciation of domestic currency leads to increase in
money demand implying a negative relationship [Cuddington (1983),
McKinnon (1982), Perera (1993)]. The effects of depreciation on
money demand could be positive through the expectations of future
depreciation thus lessening demand for domestic money (Perera,
1993).
The issue of potential endogeneity of income is first examined
using the Sargent (1976) procedure. Causality regression models for
the effects of income on both definitions of money are respectively:
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ln m1t= 0 + 1 T +i
m
=
1
2 ln m1t-i +j
n
=
1
3 ln y t-j + v t (3)
ln m2t= 0 + 1 T +i
m
=
1
2 ln m2t-i +j
n
=
1
3 ln y t-j + w t (4)
According to Grangers definition of causality, yt causes m1t or m2t if
the past values of yt taken as a group of additional explanatory
variables jointly influence m1t or m2t.
Thus the null hypothesis in (3) and (4) is that income level does not
Granger cause the Burundian money demand.
For Granger test, we apply the ordinary least squares method (OLS)
to estimate the coefficients in (3) and (4). The calculated values are
F=0.322 and F=1.526 for M1 and M2 definitions of money
respectively against a critical value of F(4,22)=2.82. The null
hypothesis that income does not Granger cause the real money
demand cannot therefore be rejected in regression models with two
lags for both narrow money and broad money. Income can thus be
considered as an exogenous variable in both money demand
equations.
To provide a comprehensive analysis of the long-run
equilibrium money demand function and the short run economic
behavior , this paper extends our previous partial adjustment model
deemed to have limited dynamics [Arize, (1992)] and uses the error
correction specification ( ECM ) which happens to be a generalization
to the partial adjustment type models and provides a more general
lag structure
In general logarithmic form, the error correction model can be
represented as follows:
A(L) log mt = B(L) log zt - (log m-log kz) t-1 +t (5)
where A (L) and B(L) are lag polynomials, z is a vector of explanatory
variables, and the second term of the right-hand size is the error
correction term which is the stationary linear ( cointegrating)
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combination of the non stationary levels of the variables log mt and
log zt, where k is a scalar.
Adopting a general to specific procedure [Hendry,1980], we begin
with an over parameterized model with liberal lags on all variables
and estimate a series of dynamic error correction models for the two
monetary aggregates. The dynamic error correction models are
based around the cointegrating vectors reported in table 2 .
These models are of the general form:
A(L) log mdt = 0+B(L) log yt + C(L) t +D(L) exct + ECM t-1 +t
(6)
where A(L) ....... D(L) are polynomials of the form A(L)=i Li in which
L is the lag operator such that Lr Xt = Xt-r, and ECM is the error
correction term.
3. Time Series properties of the model variables
Before proceeding with the estimation, we examine the time
series characteristics of the data in order to ascertain the order of
integration of the variables as to whether they are stationary or non
stationary; and therefore the number of times each variable has to
be differenced to arrive at stationarity. The stationarity of the data is
important since if times series are characterized by non
stationarities, then the classic t-test and F-test are inappropriate
because the limiting distribution of the asymptotic variance of the
parameter is infinite. [5]
Basically a series Xt is said to be integrated of order p if it
becomes stationary after differencing p times. Such a series is
denoted Xt ~ I(p). Using this terminology, a stationary series is an I(0)
series and a nonstationary series with a single unit root is I(1). Many
logarithmic macroeconomic variables are I(1) [Engle and Granger,
(1987)] but some non-stationary series are of order of 2 or higher, in
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which case the first difference or growth rate of the series will be I(1).
[Adam,1991].
To test the null hypothesis that any Xt series is integrated of
order one H0= Xt ~I(1) against the alternative hypothesis that Xt
~I(0), we apply the Dickey Fuller tests and the Sargan- Bhargava
Durbin Watson test as suggested by Sargan and Barghava (1983)
1. The Dickey Fuller Tests
The Dickey-Fuller tests for unit roots consist of estimating first a
model of the form
Xt = 0+ 1 Xt-1+j
q
=
1
jXt-j + t
Where the lag length q is set so as to ensure that any autocorrelation
inXt is absorbed, and the error term is approximately white noise.
Then we calculate a t- ratio as the ratio of the estimated 1 to its
estimated standard error.
To reject the null hypothesis of nonstationarity, i.e. the series is
stationary; the t-statistic must be significantly negative. If q=0, the
test is called the Dickey -Fuller test but if q > 0, it is termed the
Augmented Dickey-Fuller test. If we can not reject the null
hypothesis, i.e. H0= Xt ~I(1), we may conclude that the series
contains at least one unit root. Then, we test whether the first
difference is stationary that is Xt ~I(0) by estimating a model of
the form
2Xt = 0+ 1Xt-1+
j
p
=
1
j2Xt-j + t
Actually, we replaceXt with2Xt as the dependent variable andXt-1
withXt-1 as regressor and so on.
Rejection of the null hypothesis Xt ~I(1), imply Xt ~I(1). If the null
is written as Xt ~I(2), then the alternative is Xt ~I(1) and the
variable is I(1) if the null hypothesis is rejected: that is the coefficient
of the lagged first difference should be significantly less than zero.
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2. The Sargan-Barghava DurbinWatson (SBDW) test
Sargan and Barghava (1983) present a test of the hypothesis
that the errors on a regression equation follow a random walk.
Following this approach, the SBDW test regresses Xt on a constant
and tests the null hypothesis that the residuals follow a random walk.
In other words, for each series Xt , the null hypothesis is that
the first order autocorrelation coefficient is equal to one, that is the
hypothesis that the first order autocorrelation coefficient is equal to
one , that is =1 in the regression
Xt = 0+ Xt-1 + u t ; X0 = 0 u t ~ (0,2)
If the computed SBDW is larger than the critical value given in
Sargan and Barghava (1983, Table, p.157), the null H0 that the series
is a random walk
(that is non stationary) is rejected.
The SBDW test is based on the DW statistic, but it is not applied
to the residuals of the regression as usual but on level of individual
series as follows:
SBDW=t
T
=
2
(Xt-Xt-1)2 /t
T
=
1
(Xt -X)2
Unlike the DF tests, the test is against the series is I(0), in which case
the value of the DW statistic will tend toward a value of 2. If the
statistic is low then there is evidence of an I (1) series. [Adam,
1991].
The results of unit root tests discussed above are reported in table 1
Table 1: Unit root tests on annual data
Levels Differences
Variables DF ADF SBD
W
Variable
s
DF ADF SBD
WLn m1 - - 0.557 Ln m1 - - 3.212
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2.570 1.074 9.635 9.919Ln m2 -
1.799
-
0.663
0.221 Ln m2 -
6.580
-
6.965
2.584
Ln y -
2.652
-
2.650
0.758 Ln y -
7.670
-
6.061
2.868
e - -
2.556
2.511 e -
8.725
-
4.326
3.136
EXC 0.043 -
1.662
0.125 EXC -
6.117
-
6.533
2.447
The results of the DF and ADF tests as summarized in table 1
fail to reject the null hypothesis that the variables are non-stationary
and the SBDW statistic shows that they are of a random walk i.e. I (1). The results therefore indicate that the variables may not be used at
their levels in the regression equation, except in the case of
cointegrating relationships.
We therefore proceed to test for the presence of cointegration
between variables
4. Testing for cointegration
The concept of long run equilibrium economic relationship has
been discussed by Engle and Granger (1987) using the statistical
notion of cointegration.
The vector X of n dimensional times series, each integrated of the
same order, say b, is said to be cointegrated of order ( b-d ) if there
exists a vector such that W=X is I(b-d) d > 0.
If cointegration occurs it must be unique in the bivariate case. Engle
and Granger (1987) have suggested a two step procedure to test the
existence of cointegrating relationship in the bivariate case.
The test proceeds as follows:
(i) Run OLS of Xt on its explanatory variables in levels
(ii) Derive the residuals from the OLS results
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(iii) Conduct unit roots tests of the residuals and find out whether
they are integrated of order zero, or one, based on the following
equation
et= et-1 + i
k
=
0 i et-1+ vt
(iv) If et is integrated of order zero, it means that Xt and its
determinants are cointegrated of order one.
For a multiple case, Johansen (1988) and Johansen and Juselius
(1990) have developed a procedure to examine the question of
cointegration. This study shall adopt the Johansen testing procedure
as it not only allows to test for the number of cointegrated vectors
but also to estimate the cointegration vectors [Perera, 1993].
If there are N endogenous variables, each of which is first order
integrated, (that is, each has a unit root or stochastic trend or
random walk element), there can be from zero to N1 linearly
independent cointegrating vectors. If there are none, the standard
time series such as VAR applies to the first differences of the data.
If there is one cointegrating equation, the VAR will need an error
correction term involving levels of the series, and this term will
appear on the right hand side of each of the VAR equations, which
otherwise will be in first difference.
Table :2 Johansen cointegration Test
Section A: Variables in the cointegrated system: Ln m1 Ln y e
Null Hypothesis
(N of CEs)
LR Test Statistics 5% critical value
r = 0 35.46* 34.91r 1 17.40 19.96r 2 5.16 9.24
Section B: Variables in the cointegrated system: Ln m1 Ln y e EXC
Null Hypothesis
(N of CEs)
LR Test Statistics 5% critical value
r = 0 73.36* 53.12
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r 1 37.02* 34.91r 2 19.76 19.96r 3 5.28 9.24
Section C: Normalized cointegration vector in section B
(-1 .000, 1.413, -0.446, - 0.018)
Section D: Variables in the cointegrated system: Ln m2 Ln y e
Null Hypothesis
(N of CEs)
LR Test Statistics 5% critical value
r = 0 43.98* 34.91r 1 14.55 19.96r 2 3.39 9.24
Section E: Variables in the cointegrated system: Ln m2 Ln y e EXC
Null Hypothesis
(N of CEs)
LR Test Statistics 5% critical value
r = 0 75.86* 53.12r 1 44.56* 34.91r 2 17.34 19.96
r 3 3.40 9.24Section F: Normalized cointegration vector in section E:
( -1.000 , 2.512, -0.136 , -0.014 )
Table 2 reports the results of the cointegration tests .The
findings reported in section A indicate absence of long -run
relationships among narrow money and its determinants, that is the
LR statistics are unable to reject the null hypothesis r = 0 with
respect to narrow money and its determinants. The LR statistics are
close to their critical values. Adding the exchange rate in section B
substantially changes the results for M1. As seen in section B of table
2, m1, y, the inflation rate and the expected depreciation of the
parallel market exchange rate form a cointegrated relation. The LR
test statistics reject the null hypothesis that there are zero
cointegrating vectors against the 5 per cent critical value and
suggest that there are at most two cointegrating vectors for narrow
money.
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The results for broad definition of money are reported in
section D and E and are different than those for narrow money. The
test statistics rejects the null hypothesis r = 0 with respect to M2 and
its determinants whether the exchange rate is included or not.
However, there is only weak evidence of cointegration between m1, y
and the inflation rate in section D. The findings for M2 reported in
section E are markedly strengthened by the addition of the exchange
rate. Moreover, the LR test statistics indicate that the hypothesis that
there are at most two cointegrating vectors is accepted for broad
money.
Section C and F report the estimated normalized cointegration vector
for the narrow money and broad money respectively. The vectors
have been normalized by dividing by the coefficient on money and
appear to be money demand equations. For both M1 and M2 real
money demand is positively related to real GDP and negatively
related to the inflation rate and the exchange rate. The negative sign
on the exchange rate in the normalized M1 and M2 vectors may
indicate currency substitution.
As market participants increase their demand for foreign currencies
relative to the Burundian franc, the Burundian franc depreciates.
The estimated long run income elasticity with respect to M1 is
1.413 and the long run inflation rate and parallel market exchange
rate elasticities are -0.446 and -0.018 respectively. Our estimated
long run elasticity with respect to M2 is 2.512 which is higher than
the income elasticity for narrow money. The long run inflation and
exchange rate elasticities with respect to broad money are -0.136
and -0.014 which are both lower that their counterparts for M1. It is
interesting to mention also that the exchange rate elasticity is
marginally lower than its counterpart for M1.
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Empirical results
In the presence of cointegrating relationships, we start the
model specification search for an appropriate ECM with an over-
parameterized autoregressive distributed model. The model was then
reduced to a more desirable specification using the information
criterion as a guide.
More specifically, our simplification involves restricting to zero
relatively small coefficients and reformulating the lag pattern in
terms of levels and changes. The results of the more preferred
specifications are presented below:
Table 2 : OLS Money Demand equation for Narrow Money (ln m1)t
Sample is 1974 to 1995
Explanatory
Variables
Coefficient Std Error T-Statistics
Constant 0.034084 0.020059 1.699185
(ln yt ) 0.536962 0.123549 4.346147
(e )t-1 -0.21774 0.083628 -2.603636
(EXC)t -1.626979 0.622512 -2.613572
(Ln m1)t-1 -0.211036 0.121590 -1.735636
(ln m1-ln m1*)t-1 -0.476739 0.170377 -2.798150
R2 adj.== 0.795 SEE=0.085 F(5,17) = 17.337 [0.0000] D.W.
=2.189
ARCH (Autoregressive Conditional Heteroscedasticity)2(1) = 0.410
[ 0.521]
Jarque et Bera test for error normality 2 (2) = 0.847 [ 0.654]
Whites heteroscedastic error test 2(10) = 6.871 [ 0.937]
Q(12) = 15.475 [0.216]
Chow Test F (13,9 ) = 1.410 [0.737]
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Ramsey RESET F (2,19) = 0.024 [ 0.975]
Farley Hinich Mc Guire F (10,16) = 2.49
Table 3: OLS Money Demand equation for Broad Money (ln m2)tSample is 1974 to 1995
ExplanatoryVariables
Coefficient Std Error T-Statistics
Constant 0.061820 0.029816 2.073336
(ln yt ) 0.769910 0.160864 4.794590
0 1(e ) -0.663275 0.294512 -4.479269
0 1(EXC)t -1.733421 1.625669 -2.146127
(ln m2)t-1 -0.438131 0.229560 -2.146127
(ln m2-ln m2*)-t-1 -0.270954 0.176005 -1.539467
R2 adj.=0.706 SEE=0.094 F(7,15) = 8.210 [0.000] D.W. = 1.964
ARCH (Autoregressive
Conditional Heteroscedasticity 2(1) = 0.193 [ 0.659]
Jarque-Bera test for error normality 2 (2) = 5.531 [ 0.062]
Whites heteroscedastic error test 2 (14) = 10.938 [0.690]
Q (12) = 5.412 [0.943]
Chow Test F (13,9 ) = 1.919 [0.221]
Ramsey RESET F (2, 17) = 0.513 [0.610]
Farley Hinich Mc Guire F (14, 12) =3.822
0 1 indicates that the shorn -run dynamics of the dependent
variable is captured by a two period lag.
The diagnostic tests reported in this paper in support of the
empirical results are the following:
RESET (Regression Specification Test) is the Ramsey (1969) test for
omitted variables and functional form misspecification.
The test augments the original regression by adding a number of
powers of the fitted values from the original regression. If these extra
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regressors have non zero coefficients, there is evidence of
specification error
Q(12) is the Box-Pierce statistic for residual autocorrelation to twelfth
order. The statistic is used to test the hypothesis that all the
autocorrelations are zero, that is, the series is white noise.
Under the null hypothesis, Q(12) is distributed as Chi-squared
with degrees of freedom equal the number of observations less the
number of estimated ARMA coefficients.
ARCH LM test is Engles (1982) test for Autoregressive conditional
Heteroscedasticity of the residuals. This particular specification of
heteroscedasticity was motivated by the observation that in working
with macroeconomic series the size of residuals appear to be related
to the size of recent residuals. The statistic provides a test of the
hypothesis that the coefficients of the lagged squared residuals are
all zero. The chi-squared statistic corresponds to a Lagrange-
Multiplier (LM) test and has degrees of freedom equal to the number
of lagged, squared residuals.
The Farley et al. test is the Farley-Hinich and McGuire test (1975) for
a gradual shift in the parameters over the full sample period. White
is the Halbert White (1980) test for general forms of
heteroscedasticity. As is well known, a key assumption in the linear
regression model is that the error term should have a constant
variance (that is, an absence of heteroscedasticity). Violation of this
assumption leads to inefficient estimates and invalidate test
statistics. The Chi-squared statistic has degrees of freedom equal to
the number of regressors and squared regressors. J.B. is the test for
normal residuals described in Jarque and Bera (1980). Under the null
hypothesis, the J.B. statistic is distributed as chi-squared with two
degrees of freedom.
As can be seen form tables 2 and 3 the statistical fit to the data
is good as indicated by values of Theils adjusted R2 and the F value
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for testing the null hypothesis that the right hand side variables as a
group except the constant term have a zero coefficient. All of the
three explanatory variables bear the anticipated signs. The short run-
run currency substitution effect is negative and significant for both
narrow and broad definitions of money indicating therein that foreign
money is considered as an attractive alternative to holding domestic
money balances in the Burundian economy, a conclusion which is
similar to the one reached earlier [Ndenzako (1998)] when we used a
more restrictive lag structure.
The estimated coefficients for real income and expected
inflation rate are also satisfactory and consistent with our a priori
theorizing.; t-ratios are significant at the 5 per cent level, indicating
both the importance of the real income and inflation variables in
explaining changes observed in the demand for money.
Values of 0.27 and 0.46 of the ECM coefficients suggests that in
the case of the broader money balances approximately about 27 per
cent of the previous disequilibrium from the long run demand for
money is corrected in one year while for the narrower M1 aggregate
the figure is closer to 46 per cent per year. This results suggest a
relatively intuitive picture in which the narrow aggregate M1 enjoys a
much faster adjustment than the broader aggregate M2 where
possibly higher transaction costs may preclude rapid adjustment.
This speed of adjustment is almost one and a half as slow as the
adjustment speed implied by our previous partial adjustment model.
In terms of the other short- run dynamic effects in the
model a number of features warrant attention. The first is the
relatively consistent short-run inflation effects for both aggregates. In
each case , the growth rate of inflation ( i.e. the acceleration in the
price level) has a significant effect on real money holdings. On the
basis of the above empirical results, and in view of the specification
and diagnostic tests employed, we may conclude that the estimated
equations in table 2 and 3 fit the actual data quite well and
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adequately describe the money demand relationships in the
Burundian economy. The diagnostic tests appear to suggest that the
error correction models fulfill the conditions of serial non-
autocorrelation, no specification error (i.e. zero disturbance mean,
homoscedasticity, normality of residuals and structural stability.
Forecasting performances
To test how the estimated equations can explain movements in
real money demand in period outside the sample, we re-estimated
the error correction models over the period 1970 through 1990 and
used the parameter estimates to forecast the period 1991 through
1995. The forecast results are reported below:
Forecasting performances for MI definition of money
(ln m1)t = 0.046 + 0.525(ln yt) - 0.210(e)t
(-1.699) (4.346) (-2.603)
-1.637 (EXC)t -0.248(ln m1)t-1 - 0.510(ln m1-ln m1*)t-1
(-2.613) (-1.861) (-2.755)
R2 adj.= 0.819 S.E.E=0.085 D.W.=2.082
ARCH 2 (1) = 0.174 [0.675]
White 2 (10) = 8.114 [0.617]
J.B. = 0.858 [0.651]
Q(12) = 7.963 [0.790]
RESET f(1,16) = 0.154 [0.700]
Chow Forecast Test: F( 5,19)=0.825 [0.556]
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Year Forecast
Errors,
percent
RMSE Theil Inequality
Coefficient
1991 0.417 0.027 0.0026
1992 0.661 0.034 0.00331993 0.890 0.048 0.00461994 0.078 0.100 0.00961995 2.778 0.143 0.0037
The Chow forecast test statistic has a value of 0.825 with a marginal
significance level of approximately of 0.556 for the M1 aggregate at
the five percent significance level. The Chow forecast test is a post
sample predictive failure test. It allows to examine whether the next
five observations have been generated by the same model estimated
for 1970-1990. Both estimated equations pass this predictive failure
test.
6. Stability of the money demand equations
As Johnston (1984, p.507) pointed out, the stability of the
parameters over various data sets is a very important indicator of the
quality of a functional specification. The approximate constancy of
the estimated coefficients over time may be tested through several
available statistical tests. This point has been emphasized by
Boughton (1981), who, along with others, argues that each stability
test addresses a somewhat different aspect of stability and thus
suggests that the researcher employs a battery of stability tests.
In this paper, three stability tests are employed, namely, the Chow
break point test, the Chow (1960) test, the Farley- Hinich- McGuire
(1975) test and the Brown-Durbin-Evans (1975) test.
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6.1. The Chow test
Breaking the sample at the midpoint after 1982, and applying
the Chow F test resulted in an F- statistic of 1.410, for M1 and 1.910
for M2 which does not allow one to reject the hypothesis of stability
at the 5 percent level. Thus, we accept the hypothesis of no
structural shift in the model and therefore conclude for the stability
of the demand for money.
In addition, by retaining the last four observations (1992-1995), we
can check for the stability of the parameters, using out of sample
information. Table 4 and 5 the forecast errors from the estimated M1
and M2 equations. Results of the post sample parameter stability test
are excellent and supportive of the Chow test. The root mean
squared errors providing a measure of the forecast accuracy also
corroborate the findings.
6.2. Time trending regressions: The Farley-Hinich-Mc Guire
test
Next we consider a test for a shift in the slopes of linear times
series model, which has been proposed by Falrey, Hinich and
McGuire [1975]. The coefficients bj of the regression equations were
augmented to
bj*=bj + t , t=1,2 .....T., j =1.....n permitting each coefficient to drift
along a linear time trend. Then an appropriate F ratio has been used
to test the null hypothesis that the coefficients on added trend
variables are jointly zero. This procedure tests for a gradual shift ( in
contrast to a single) in the parameters and it is applied to the full
sample.
When the were jointly tested from significance from zero, an
F-value of 2.49 against a critical value of 2.76 and 2.54 against a
critical value of 3.23 was obtained for the specification of M1 and M2
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respectively. None permits the rejection of the hypothesis of
parameter constancy at the 5 percent level
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6.3. Recursive estimates: The Brown-Durbin-Evans test.
The last test of stability is carried out by the Brown-Durbin-
Evans technique that, unlike Chows F-test or dummy variable
method, detects shits, if any, in a regression relationship without ant
prior knowledge about the timings of such shifts.
Basically, the Brown-Durbin-Evans residual check for testing
structural stability of regression relationship consists of calculating Sr
statistics given by Sr=(k
r
+
1
Wr2 ) (k
T
+
1
Wr2 ) r =k+1,......T.
Where K = number of explanatory variables including the constant,
T = total number of observations in the regression, and
Wr = the standardized residual for the r-th observation of the
dependent variable
The statistics Wr is obtained by standardizing the difference between
the actual and the forecasted values of the r-th observation of the
dependent variable. In forecasting the r-th observation of thedependent variable, the regression equation estimated by using r-1
observations is used. The values of Sr lies between 0 ( if r < (k+1=
and 1 if r = T).
The expected value of Sr is (r-k)/ (T-k). Given the null hypothesis that
the regression equation under consideration is stable over the period
of investigation at a specific level of significance, the plot of Sr should
lie between the pair of lines (r-k)/(T-k) c0,For T observations, k explanatory variables (including the intercept )
and given a significance level of , c0 is found by entering the table
at
m=1/2(T-k)-1 when T-k is even and interpolating linearly between
m=1/2(T-k) - 3/2 and 1/2(T-k) - 1/2 when T-k is odd (see Johnston,
1984).
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Since our study covers data from 1970 to 1995 and there are
three explanatory variables in the regression, the value of c0 in the
cusum of squares test is 0.23298 at 5 percent level of significance.
The Brown-Durbin-Evans stability test is illustrated in figure 1 and 2
for both specifications of the demand for money function.
The null hypothesis of stable demand for money function for
each specification will be accepted at the 5 per cent significance level
if the sample plot of the corresponding S r does not cross the 5 per
cent significance level.
The hypothesis of stable money demand is accepted for Burundi
during 1970-1995 for both definitions of money.
Conclusion and policy implications
There are several policy implications arising from results of this
paper. One such policy is concern over the fiscal implications of
inflation, that is the steady-state inflation capacity of the private
sector and the short run dynamics of money holdings in response to
changes in the rate of inflation.
Following Friedman (1971),and Adam(1991), standard analyses of
inflation tax derive the revenue maximizing rate of inflation as
=1/ - (g)
where is (minus) the inflation elasticity of the demand for money,
is the income elasticity of the demand for money and g is the
average rate of growth of income.
From the long-run demand function for narrow money, we
derive an average revenue -maximizing rate of inflation of 19.3 per
cent where
= -0.4466 ; =1.413 and g = 2.2% per annum. It is interesting to
note that the greater the range of substitutes for domestic base
money, and the more rapidly the private sector can adjust their real
holdings, the lower the revenue -maximizing level of inflation will be.
Above a rate of inflation of 19.3% per annum, the private
sector systematically reduces its real holdings of base money more
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rapidly than the rate of inflation, so that the real inflation tax
revenue declines.
The coefficient on the error-correction term is a direct measure of
this speed of adjustment: according to these results, the Burundian
private sector adjusts at a rate of approximately 0.47 per cent per
annum to any disequilibrium (i.e. increase in real base money).
Four other findings deserve special attention:
(i) The results indicate, once again, that the demand for money is not
only affected by domestic variables such as real income and
expected inflation, but also by fluctuations in the exchange rate
expectations.
(ii) Formal stability tests failed to indicate any shift in the demand for
money equations over the sample period considered.
(iii) The specification of the demand for money efficiently tracks
actual movements in holdings of real balances around the long-run
demand functions.
(iv) The variables entering into the demand for narrow money may
not form a cointegrating system unless the exchange rate is
included. Inclusion of the exchange rate may strengthen the stability
of M2.
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Appendix: Data Sources and Description
This study covers the period from 1970 through 1995 on the
basis of annual observations. The empirical definitions of
the variables are as follows:
M1=Currency in circulation plus demand deposits (narrow
definition)M2=M1 plus time and savings deposits (broad definition)
Y = Real gross domestic product
P=Consumer Price Index
Ex=Parallel market exchange rate ( Burundian franc per
dollar)
e=logPt-1 -logPt-2
EXC=logExt-1 -logEXt-2
All data series were obtained form issues of IMF,
International Statistics, except for the parallel market
exchange rates which were derived from Picks Currency
Yearbook as reported by J.D. Nkurunziza (1997).
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Notes
[1] John P.Judd and John Scadding, The search for a stable
money demand function: a survey of the post-1993
literature , Journal of Economic Literature, vol.XX, Sept.
1982, pp.993-1023.
[2] NDENZAKO, J. (1998) The Demand for Money inBurundi: Some Preliminary Results , RIDEC ( Revue de
lInstitut de Dveloppement Economique du Burundi), vol. 2,
n1, p.178-195.
[3] Arize, A, (1989) Exchange rates, Foreign interest rates,
and the demand for money demand in an open economy:
An empirical investigation in Korea , Savings and
Development, 3, , XIII ,p.245.]
[4] The theory imposes the restriction that money demand
is cast in real terms, that is, real demand for money is
homogenous of degree zero in prices because economic
agents are a priori assumed rational so that their demand
for cash balances is a demand for real purchasing power.
[5] Fuller,W.A., Nonstationary Autoregressive Times
Series , in Hananet al. (ed.), Handbook of Statistics,
Elsevier Publishers, Amsterdam,1985.
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