4.3 Application: Money Demand

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 $ m$) as a function of aggregated income ($ y$,) short and long term interest ($ i^s,
i^l$) 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:


$\displaystyle \triangle (m-p)_t$ $\displaystyle =$ $\displaystyle \alpha_{1,1} + \alpha_{1,2}\triangle
(m-p)_{t-1} + \alpha_{1,3}\triangle y_{t-1}+
\alpha_{1,4}\triangle i^l_{t-1}+ \alpha_{1,5}\triangle
i^s_{t-1}$  
    $\displaystyle +\alpha_{1,6}\triangle^2 p_{t-1} + \alpha_{1,7}\triangle^2 p_{t}
+ \alpha_{1,8}ec1_{t-1} + \varepsilon_{1,t}$ (4.15)
$\displaystyle \triangle y_t$ $\displaystyle =$ $\displaystyle \alpha_{2,1} + \alpha_{2,2}\triangle
(m-p)_{t-1} + \alpha_{2,3}\triangle y_{t-1}+
\alpha_{2,4}\triangle i^l_{t-1}+ \alpha_{2,5}\triangle
i^s_{t-1}$  
    $\displaystyle +\alpha_{2,6}\triangle^2 p_{t-1} + \alpha_{1,8}ec1_{t-1}
+ \alpha_{2,9}ec2_{t-1} + \varepsilon_{2,t}$ (4.16)
$\displaystyle \triangle i^l_t$ $\displaystyle =$ $\displaystyle \alpha_{3,1} + \alpha_{3,2}\triangle
(m-p)_{t-1} + \alpha_{3,3}\triangle y_{t-1}+
\alpha_{3,4}\triangle i^l_{t-1}+ \alpha_{3,5}\triangle i^s_{t-1}$  
    $\displaystyle +\alpha_{3,6}\triangle^2 p_{t-1} + \varepsilon_{3,t}$ (4.17)
$\displaystyle \triangle i^s_t$ $\displaystyle =$ $\displaystyle \alpha_{4,1} + \alpha_{4,2}\triangle
(m-p)_{t-1} + \alpha_{4,3}\triangle y_{t-1}+
\alpha_{4,4}\triangle i^l_{t-1}+ \alpha_{4,5}\triangle
i^s_{t-1}$  
    $\displaystyle +\alpha_{4,6}\triangle^2 p_{t-1}
+ \alpha_{4,10} \triangle i^l_{t} + \varepsilon_{4,t}$ (4.18)
$\displaystyle \triangle^2 p_t$ $\displaystyle =$ $\displaystyle \alpha_{5,1} + \alpha_{5,2}\triangle
(m-p)_{t-1} + \alpha_{5,3}\triangle y_{t-1}+
\alpha_{5,4}\triangle i^l_{t-1}+ \alpha_{5,5}\triangle
i^s_{t-1}$  
    $\displaystyle +\alpha_{5,6}\triangle^2 p_{t-1} + \alpha_{5,9}ec2_{t-1}
+ \varepsilon_{1,t}$ (4.19)
$\displaystyle ec1_{t}$ $\displaystyle =$ $\displaystyle (m-p)_t - 1.574y_t + 3.405i^l_t -2.061i^s_t$ (4.20)
$\displaystyle ec2_{t}$ $\displaystyle =$ $\displaystyle i^l - 4\triangle p$ (4.21)

Here, $ \triangle = 1 - L$ where $ L$ is the backshift operator (i.e., $ \triangle y_{t-1}= (1 - L)y_{t-1}=y_{t-1}-y_{t-2}$). It can be easily verified that the order condition holds and all parameters are identified. In our case the endogenous variables are $ \triangle m-p,\triangle y_t,\triangle i^l_t,\triangle i^s_t$ and $ \triangle^2 p_t$ (all variables except the interest rates in logarithms). There are no purely exogenous variables but predetermined (lagged endogenous) variables only. The error correction terms are given by the identities (4.22) and (4.23). They are the results of the first step, the reduced form regression. Thus, we re-estimate the model including the short-run adjustment parameters by 3SLS under more general assumptions about the contemporaneous structure than in the reduced rank model.

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)
8445 XAGseq02.xpl

These lines of XploRe code yield the following table in the output window (table of two-stage least squares estimates has been omitted). It shows for each equation the left-hand side variable (top panel), three-stage least squares coefficent estimates, estimated standard errors and $ t$-values. The bottom panel of the table lists the predetermined and exogenous variables that were used as instruments in the estimation procedure, along with their sample means and standard deviations.
  [ 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 ( $ \alpha_{4,10}$ and $ \alpha_{1,7}$) and, second the parameters of the error correction terms ( $ \alpha_{.,8}$ and $ \alpha_{.,9}$), the so called short-run adjustment parameters.

Within the first group, we obtain a negative relationship between real money growth and inflation growth, which is indicated by the coefficient of $ -1.42$. 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 $ 0.187$. 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 $ \alpha_{2,8}$. Of course, re-formulating the $ ec1$ 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 ( $ \alpha_{2,9}$) 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 $ t$-values, $ F$-statistics, Akaike or Schwartz criteria to obtain more efficient estimates of the remaining true model.