Solution
David answered on
Dec 26 2021
Introduction
“No proposition in macroeconomics has received more attention than that there exists, at the
level of the aggregate economy, a stable demand for money function- Laider. D (1982)
A stable demand function for money is considered as a pre-requisite for the use of monetary
aggregates in the conduct of monetary policy. Proper management of monetary policy and its
effectiveness to influence the real economy (variables of the economy like inflation, interest rate,
employment rate etc.) requires for a stable relationship between money-demand and is
determinants such as real GDP, prices, real interest rate and exchange rate. At a high level, the
importance of stable money demand function can be understood by the standard IS-LM
framework where interest rate and GDP are endogenous variables. A stable money demand in
this context allows the slope and the position of LM curve to be determined given the knowledge
of quantity of money. Assuming the knowledge of IS curve, changes in money will exert a
predictable influence on GDP. An unstable money demand function will make it impossible to
predict the effect on interest rate and hence, on aggregate GDP (and on aggregate expenditure) of
a given change in money supply. Hence an efficient monetary transmission relies heavily on the
stability of money demand function. Rangarajan (1989) argued that, “if one were to deny the
existence of a reasonable stable money demand function for money, there will be a little scope
for monetary policy to play the role in inflationary management”.
The demand for money in China has received a great deal of importance. Many studies in the
past have employed the co-integration technique to establish the co-integration among the
variables in the money demand function. The main conclusion of most of these studies is that
M2 monetary aggregate is cointegrated with income and interest rate and this integration is
perceived as a sign of stable demand for money. However, Bahmani-Oskooee and Bohl (2000)
have demonstrated that cointegration does not imply constancy of estimated coefficients.
In this paper, we will try to see not only the cointegrating properties of money demand function,
ut also its stability over time. We include the annual data from 1990 to 2015. The Augmented
Dickey Fuller (ADF) unit root test which is prefe
ed over the Dickey Fuller (DF) test due to the
possibility of serial co
elation of the e
ors in the DF model is conducted to examine the
stationary of the time series data. The, the ARDL approach is employed to test cointegrating
elationship in the model. Finally CUSUM and CUSUMSQ tests are employed to investigate the
stability of demand function in China. We will see that although both M1 and M2 measure of
money demand have a long run relationship with other important economic factors of the
economy (the factors may differ), M1 has a stable relationship, while M2 does not.
Please make a note that throughout this paper, I have used log and ln interchangeably.
Methodology
We use the e
or co
ection model (ECM) to check the stability of the money demand function.
For ECM, we need to establish that the variables in the money demand function should be time
integrated of the same order (other than zero which would mean that the variables are not
stationary at levels) and cointegrated. The cointegration here implies that the variables are not
stationary but are I(d), which means when differenced d times they turn out to be stationary and
their linear combination is stationary. Co-integration helps variables in the system to be modeled
in an e
or co
ection way.
In formulating the demand for money, we try to make sure to incorporate variables that could
possibly affect the money demand in the economy.
ln(Mt) = β0 + β1 ln(Yt) + β2 Rt + β1 ln(EXt) + ut (1)
where M is either Real value of M1 or real value of M2, Y is the real GDP, R is the domestic real
interest rate and EX is the nominal exchange rate.
The above is the co-integration equation where if all the variables M, Y, R and EX are I(1) and
are co-integrated , the e
ors would follow standard white noise process. If the variables are not
integrated (or first difference stationary), we cannot conclude that the variables are cointegrated
even if we get the coefficients for each of the variables are statistically significant.
We will check for the stationarity of each of these series before we establish the co-integration
etween these variables. To check for whether the series is a stationary series or not, we use the
Augmented Dickey Fuller (ADF) test.
The augmented Dickey-Fuller (ADF) test is used to determine the presence of unit roots in the
data sets. The null hypothesis is that the variable contains a unit root and the alternative is that
the variable is generated by the stationary process.
Under the null hypothesis, the true process is either a random walk or random walk with drift.
The Dickey fuller test involves fitting the following model
ttt uXtX 1210
The null hypothesis is that δ2 =1. Estimating the above equation by OLS may fail on the account
of serial co
elation of residuals. That is the reason why we move to Augmented Dickey fuller
(ADF) test.
To avoid the problem of serial co
elation of the e
ors (ut), we take the first difference of the
variable as dependent variable and include the lagged value of the first difference along with the
lag-1 values of the variable. Hence the ADF test is based on the estimate of the following
equation:
k
i
iititt uXXtX
1
1210
The above equation helps us to know whether the series Xt is stationary or not via ADF test.
For the possibility of higher order co-integration we run the following regression too which is via
ADF again.
k
i
iititt XXtX
1
2
1210
2
and test whether β2 =0 (No stationarity). This null hypothesis is same as checking for δ2 =1 in the
Dickey Fuller test.
After checking the order of integration, we can check for the possibility of co-integration. The
Engle and Granger (E-G) method tests the stationarity of the residuals of the co-integration
egression. The E-G method of co-integration helps us knowing the relationship between money
demand and its various possible determinants described in equation (1).
After establishing the relationship between the variables, we construct an ECM (E
or Co
ection
Model) suggested by Engle- Granger where we use the e
or co
ection term obtained from the
co-integration (equation 1). The general form of the model is
tt
n
k
ktk
n
k
ktk
n
k
ktk
n
k
ktkt uEXRYMM
1
0
.1
0
.2
0
.1
0
.1 lnlnlnln (2)
Here ut-1 is the lagged stationary e
or term from the co-integration equation (1). The above
model includes n lagged values of each of the variables. The final model may not include all the
lagged terms. It may depend on the statistical significance of the each of the lagged variable.
After running the model, the final model will include only those variables which will be
statistically significant.
The important thing to note here is that all the variables mentioned in the above equation are
stationary at first difference at least (the variable may be stationary at level i.e I(0), but since we
need to take the highest value of the integration , we take the first difference of each. The
variables which are stationary at level would be stationary at first difference too.)
The above model does not apply when any of the variables is non stationary at first difference.
One of the ways around would be to drop those variables and run the model. But if the variables
are of high importance (but are non-stationary), dropping them would cost us as much as running
the model including them. Hence we need a way out so that we can still keep those non-
stationary variables in our model.
There is a method suggested by Pesaran et al. (2001). Under this technique we don’t need a pre-
unit root testing. Under this method, we replace ut-1 in equation (2) with the linear combination
of the lagged level variables as below:
ttttt
n
k
ktk
n
k
ktk
n
k
ktk
n
k
ktkt EXRYMEXRYMM
14131211
0
.1
0
.2
0
.1
0
.1 lnlnlnlnlnlnln
(3)
Before we run the above model, we check where there is a cointegration among the variables or not. We
test the null hypothesis that there is no co-integration among the variables. That is equivalent to checking
1 = 2 = 3 = 4 =0 which is tested against 1 ≠ 2 ≠ 3 ≠ 4 ≠0. The null hypothesis tells us that
there is no possibility of the cointegration. Where we want most parsimonious model, we should
not drop any variables that may be of high importance.
If we are able to reject the null hypothesis, we can say that there is a co-integration among the
variables. Once the co-integration is established, we apply CUSUM and CUSUMSQ tests to the
esiduals of equation (3). The CUSUM and CUSUMSQ tests help us know the parameter
stability. These tests also tell us that stability of the parameters depends on the nature of
structural changing taking place. If the structural
eak is in the intercept of the regression, then
the CUSUM test has higher power. However, if the structural change involves a slope coefficient
or the variance of the e
or term, then the CUSUMSQ (CUSUM squared) test has a high power.
The CUSUM test is based on a plot of the sum of the recursive residuals. If this sum goes outside
a critical bound, we can conclude that there is a structural
eak at the point at which the sum
egan its movement towards the bound. The CUSUMSQ test is similar to the CUSUMSQ test.
Rather than plotting the sum of the recursive residuals, CUSUMSQ plots the sum of the squared
ecursive residuals, expressed as a fraction of these squared residuals summed over all
observations.
Empirical Results
The problem associated with estimating the demand equation is that money demand, real GDP,
price index (CPI), interest rate and exchange rate can all be characterized as non-stationary
variables. Each variable can be explained without any tendency to return to a long run-level.
However there exists a linear combination described above of these non-stationary variables that
is stationary. As mentioned above our empirical results are based on the data from the period
1990 to 2015 ( the yearly data).
As the visual techniques are very appealing, before going for a formal tests on STATA (ADF
tests for the stationarity), we use look at the graphs of each of the variables we would want to
consider in our model. These variables are real M1, real M2, real GDP, real rate of interest and
Effective nominal exchange rate. These graphs will give us a high level view of whether the
series are stationary or not. We include the annual data from 1990 to 2015
These charts have been created on STATA.
Figure1: Series in Levels
4
4
.5
5
5
.5
ln
(
Y
)t
1990 1995 2000 2005 2010 2015
Yea
3
.5
4
4
.5
5
ln
(
R
e
a
l
M
1
)t
1990 1995 2000 2005 2010 2015
Yea
3
.5
4
4
.5
5
5
.5
ln
(
R
e
a
l
M
2
)t
1990 1995 2000 2005 2010 2015
Yea
-1
0
-5
0
5
1
0
R
e
a
l
In
te
e
s
t
R
a
te
(R
)
1990 1995 2000 2005 2010 2015
Yea
The time series graphs for the variables above show that that all the series are not stationary. This
is in expectation of the time series of the macro economic variables
We now take the first difference of each the series and plot their graphs against the time. (Please
note that sign “?” stands for first difference).
Figure2: Series in first difference
-.
0
5
0
.0
5
.1
.1
5
.2
?
ln
(
E
X
)t
1990 1995 2000 2005 2010 2015
Yea
0
.0
5
.1
.1
5
?
ln
(
R
e
a
l
M
1
)t
1990 1995 2000 2005 2010 2015
Yea
.0
4
.0
6
.0
8
.1
?
ln
(
R
e
a
l
M
2
)t
1990 1995 2000 2005 2010 2015
Yea
-1
0
-5
0
5
1
0
?
R
e
a
l
In
te
e
s
t
R
a
te
(
R
)
1990 1995 2000 2005 2010 2015
Yea
.7
.7
5
.8
.8
5
.9
.9
5
ln
(
E
X
)t
1990 1995 2000 2005 2010 2015
Yea
As we can see in the graphs above, the first difference of the real income still does not seem to
e stationary. For all other variables, there might be a possibility of stationarity at the first
difference. Although some formal tests for the stationarity is employed.
Unit root test (ADF test)
We now do the formal ADF test to check the stationarity of the level and first difference of the
series. For each of the variables under study (both dependent and independent). The series with
unit root is called a non-stationary series.
The ADF test on STATA
For ln(Real M1)t
_cons 1.109094 .6064245 1.83 0.081 -.1485531 2.366742
_trend .014312 .0090757 1.58 0.129 -.0045098 .0331339
L1. -.2862645 .1667011 -1.72 0.100 -.6319814 .0594523
lnRealM1t
D.lnRealM1t Coef. Std. E
. t P>|t| [95% Conf. Interval]
MacKinnon approximate p-value for Z(t) = 0.7432
Z(t) -1.717 -4.380 -3.600 -3.240
Statistic Value Value Value
Test 1% Critical 5% Critical 10% Critical
Interpolated Dickey-Fuller
Dickey-Fuller test for unit root Number of obs = 25
.0
2
.0
3
.0
4
.0
5
.0
6
.0
7
?
ln
(
Y
)t
1990 1995 2000 2005 2010 2015
Yea
Since the value of the test statistic is -1.717 is greater than -3.600 (at 5-pecent level of
significance). This means our statistic lies in the non-rejection area. Hence we don’t reject the
null of no stationarity. Hence the given series is not stationary.
For ln(RealM2)t
Since the value of the test statistic is -2.987 is greater than -3.600 (at 5-pecent level of
significance), we don’t reject the null of no stationarity. Hence the given series is not stationary.
_cons 1.719055 .5499915 3.13 0.005 .5784427 2.859668
_trend .0276848 .0095719 2.89 0.008 .007834 .0475356
L1. -.442177 .1480385 -2.99 0.007 -.7491901 -.1351638
lnRealM2t
D.lnRealM2t Coef. Std. E
. t P>|t| [95% Conf. Interval]
MacKinnon...