Economists often refer to the money stock as one important
determinant of the price level. Therefore, the evolution of the
amount of money in the economy is also a focus of monetary policy
analysis. A convenient tool for this is to look at the so called
demand for money. The recent economic literature analyzes the
long-run demand for money (denoted ) as a function of
aggregated income (
,) short and long term interest (
) and inflation rates within a cointegration framework. If
there is more than one cointegration relationship and/or the
relationship of interest helps to explain more than just one
variable, these relationships are estimated more efficiently
within a system rather than as a single equation (see
(Ericsson; 1999) on that). The explanatory variables considered in
the money demand equation might cointegrate not only to a money
demand function but also to a stationary spread between long and
short term interest rates and a stationary real interest rate.
Therefore, in a study about European money demand,
Müller and Hahn (2000)
applied a system specification to determine whether
or not there exists a stationary relationship between the money
stock, aggregated income, 3-months interest rates, government
bond yield and a measure of European inflation. All data are
weighted sums of the series of each of the eleven countries
except for the price measure which has been obtained as the ratio
of nominal and real income. In case of the interest rates the
weights are real income shares and in the cases of money and
income the official EURO rates have been used.
Using a system approach suggests to consider a reduced form regression, where no endogenous variables may enter any of the equations on the right hand side. In contrast to that, the change in money stock is often considered to depend e.g. on the current change in inflation (Lütkepohl and Wolters; 1998) and the same is true for relationship between the short and the long term interest rates. That's why the reduced form regression is used to identify the long-run relationships (cointegration relationships) while in a second step the model is re-written to yield a structural form, as described above. Thus, the structural SEQM is:
We have used the following XploRe code to estimate the parameters of equations (4.17) to (4.21):
; reading in the data z=read("eu.raw") ; getting rid of missing values due to lagged variables z=z[4:rows(z),] ; assigning columns of z to variable names dmp = z[,4] dmp1 = z[,5] dy = z[,7] dy1 = z[,8] d2p = z[,12] d2p1 = z[,13] dil = z[,15] dil1 = z[,16] dik = z[,18] dik1 = z[,19] ec11 = z[,21] ec21 = z[,23] ; creating the matrices for seqlist1 lhs=dmp~dy~dil~dik~d2p one= matrix(rows(z),1) z1=one~dmp1~dy1~dil1~dik1~d2p1~d2p~ec11 z2=one~dmp1~dy1~dil1~dik1~d2p1~ec11~ec21 z3=one~dmp1~dy1~dil1~dik1~d2p1 z4=one~dmp1~dy1~dil1~dik1~d2p1~dil z5=one~dmp1~dy1~dil1~dik1~d2p1~ec21 x=one~dmp1~dy1~dil1~dik1~d2p1~ec11~ec21 ; forming seqlist1 as a list of matrices seqlist1=list(lhs,z1,z2,z3,z4,z5,x) ; creating list of string vectors yl="dmp"|"dy"|"dzl"|"dzk"|"d2p" zl1="one"|"dmp1"|"dy1"|"dil1"|"dik1"|"d2p1"|"d2p"|"ec11" zl2="one"|"dmp1"|"dy1"|"dil1"|"dik1"|"d2p1"|"ec11"|"ec21" zl3="one"|"dmp1"|"dy1"|"dil1"|"dik1"|"d2p1" zl4="one"|"dmp1"|"dy1"|"dil1"|"dik1"|"d2p1"|"dil" zl5="one"|"dmp1"|"dy1"|"dil1"|"dik1"|"d2p1"|"ec21" xl="one"|"dmp1"|"dy1"|"dil1"|"dik1"|"d2p1"|"ec11"|"ec21" ; forming seqlist2 as a list of string vectors seqlist2=list(yl,zl1,zl2,zl3,zl4,zl5,xl) ; finally, calling seq to estimate the model {d3sls,cov3,d2sls}=seq(seqlist1,seqlist2)
[ 1,] "=====================================================" [ 2,] " 3stage Least-squares estimates" [ 3,] "=====================================================" [ 4,] " EQ dep. var. R2" [ 5,] "=====================================================" [ 6,] " 1 dmp 0.643 " [ 7,] " 2 dy 0.506 " [ 8,] " 3 dzl 0.279 " [ 9,] " 4 dzk 0.079 " [10,] " 5 d2p 0.299 " [11,] "=====================================================" [12,] "VARIABLE Coef.Est. Std.Err. t" [13,] "--------------------------------------------------" [14,] "one -0.050 0.074 -0.684" [15,] "dmp1 0.680 0.113 6.023" [16,] "dy1 0.124 0.114 1.089" [17,] "dil1 -0.126 0.153 -0.821" [18,] "dik1 0.029 0.149 0.195" [19,] "d2p1 -0.897 0.541 -1.658" [20,] "d2p -1.421 1.078 -1.319" [21,] "ec11 -0.018 0.026 -0.669" [22,] "-----------------------------------------------------" [23,] "one 0.184 0.054 3.403" [24,] "dmp1 0.270 0.087 3.082" [25,] "dy1 0.393 0.106 3.694" [26,] "dil1 0.016 0.129 0.124" [27,] "dik1 0.041 0.098 0.418" [28,] "d2p1 0.922 0.242 3.809" [29,] "ec11 0.061 0.019 3.262" [30,] "ec21 -0.159 0.054 -2.928" [31,] "-----------------------------------------------------" [32,] "one -0.002 0.001 -1.980" [33,] "dmp1 0.148 0.098 1.516" [34,] "dy1 0.192 0.121 1.590" [35,] "dil1 0.425 0.138 3.075" [36,] "dik1 -0.092 0.103 -0.894" [37,] "d2p1 0.136 0.261 0.522" [38,] "-----------------------------------------------------" [39,] "one -0.003 0.002 -1.698" [40,] "dmp1 0.120 0.174 0.686" [41,] "dy1 0.365 0.221 1.656" [42,] "dil1 0.400 0.374 1.069" [43,] "dik1 0.108 0.157 0.687" [44,] "d2p1 -0.022 0.371 -0.060" [45,] "dil -0.270 0.762 -0.354" [46,] "-----------------------------------------------------" [47,] "one -0.004 0.002 -2.472" [48,] "dmp1 0.030 0.047 0.646" [49,] "dy1 0.009 0.057 0.154" [50,] "dil1 -0.077 0.068 -1.144" [51,] "dik1 0.062 0.048 1.296" [52,] "d2p1 -0.323 0.125 -2.574" [53,] "ec21 0.072 0.028 2.596" [54,] "=====================================================" [55,] "INSTRUMENTS Mean Std.Dev. " [56,] "--------------------------------------------------" [57,] "one 1.000 0.000" [58,] "dmp1 0.000 0.005" [59,] "dy1 0.006 0.005" [60,] "dil1 -0.001 0.004" [61,] "dik1 -0.002 0.006" [62,] "d2p1 0.000 0.002" [63,] "ec11 -2.813 0.027" [64,] "ec21 0.052 0.010" [65,] "====================================================="Interpreting the results, two groups of estimators are of particular interest. These are first the structural or contemporaneous explanatory variables' parameters (
Within the first group, we obtain a negative relationship between
real money growth and inflation growth, which is indicated by the
coefficient of . Its sign does not come as a surprise
since an increase in inflation will naturally depreciate the
value of nominal money stock. The coefficient does not seem to be
statistically significant however, which is indicated by the
marginal probability of
. Similarly the changes in the
long-term interest rate do not seem to have a significant impact
on short rate movements of the same period.
The second group of interest provides some insight into the
feasibility and effects of monetary policy as well as to some
extent into some basic economic relationships. To start with, the
error correction term which is given in (4.22) and labeled
as the long-run money demand enters the money and income growth
equations (4.17) and (4.18). This term indicates
what effect money demand has on the respective variables in
excess of the long-run equilibrium. In the first equation we
assumed it to lead to a slow-down in money growth. This feature
should be present if one expects money to be demanded in quite
the same way as many other ordinary commodities. Thus, in such a
case we would observe an inherent tendency to restore equilibrium.
The estimation results suggests however that this adjustment does
not take place. This is because although the corresponding coefficient
yields the correct sign it has too large a standard error
compared to its magnitude. When the true coefficient is zero then
the we would also have to assume that there is no money demand in
Europe altogether and the cointegrating relationship should
better be rewritten in such a way that it is normalised on
a variable in whose equation the error correction term enters
significantly. This could be the income equation for example.
Sticking to the interpretation of a long-run money demand equilibrium
we notice that excess demand of real money will lead to higher
income growth in the next period as indicated by
.
Of course, re-formulating the
term does not affect the
significance of the coefficient but it could change the sign and
will change magnitude and economic interpretation, which will not
be done here since we are investigating the hypothesis of the
existence of a money demand.
The second error correction term has an interpretation as a real
interest rate. When real interest rates are high, the respective
coefficient in the income equation indicates that income growth
will be less in the following period. This, too, is economically
reasonable, because credits are more expensive in that case.
Already in the first step, in the reduced form estimation, we
found no evidence of an endogenous tendency for the long-term
interest rate to adjust to deviations from the long-run real
interest equilibrium level. Therefore the error correction term
for real interest rates have not been included in eq.
(4.21). Instead, as the corresponding coefficient of the
current estimation (
) implies also, these
deviations may help to predict future inflation.
Since we used the error correction terms obtained in the reduced
form regression and applied the zero adjustment coefficient
restrictions identified in this first step, there was not much
more to learn about the effect of excess money demand on prices,
say. Therefore the additional insight from this 3SLS estimation
is mainly the sensitivity of the short-run adjustment estimates
when explicit structural assumptions enter the model. It has to
be pointed out however, that no final conclusions can be drawn
yet because as the t-statistic of the additional structural
explanatory variables indicate, their inclusion might not have
contributed much to explain the underlying data generating
process. That's why it is not quite clear which of variables are
really part of this process and which are not. In order to find
out more about that some model selection procedures could be
applied. A natural extension in that direction would be e.g. to
systematically exclude unnecessary variables due to some criteria
like -values,
-statistics, Akaike or Schwartz criteria to
obtain more efficient estimates of the remaining true model.