11.2 The Econometric Approach to Money Demand


11.2.1 Econometric Estimation of Money Demand Functions

Since the seminal works of Nelson and Plosser (1982), who have shown that relevant macroeconomic variables exhibit stochastic trends and are only stationary after differencing, and Engle and Granger (1987), who introduced the concept of cointegration, the (vector) error correction model, (V)ECM, is the dominant econometric framework for money demand analysis. If a certain set of conditions about the number of cointegration relations and exogeneity properties is met, the following single equation error correction model (SE-ECM) can be used to estimate money demand functions:

$\displaystyle \Delta\ln MR_t$ $\displaystyle =$ $\displaystyle c_t + \alpha\underbrace{(\ln MR_{t-1} - \beta_2\ln Y_{t-1} - \beta_3 R_{t-1} - \beta_4\pi_{t-1})}_{\textrm{\footnotesize error correction term}}$  
    $\displaystyle + \sum_{i=1}^k \gamma_{1i}\Delta\ln MR_{t-i} + \sum_{i=0}^k \gamma_{2i}\Delta\ln Y_{t-i}$ (11.7)
$\displaystyle <tex2html_comment_mark>2192$   $\displaystyle + \sum_{i=0}^k \gamma_{3i}\Delta R_{t-i} + \sum_{i=0}^k \gamma_{4i}\Delta \pi_{t-i}.$  

It can immediately be seen that (11.6) is a special case of the error correction model (11.7). In other words, the PAM corresponds to a SE-ECM with certain parameter restrictions. The SE-ECM can be interpreted as a partial adjustment model with $ \beta_2$ as long-run income elasticity of money demand, $ \beta_3$ as long-run semi-interest rate elasticity of money demand, and less restrictive short-run dynamics. The coefficient $ \beta_4$, however, cannot be interpreted directly.

In practice, the number of cointegration relations and the exogeneity of certain variables cannot be considered as known. Therefore, the VECM is often applied. In this framework, all variables are assumed to be endogenous a priori, and the imposition of a certain cointegration rank can be justified by statistical tests. The standard VECM is obtained from a vector autoregressive (VAR) model:

$\displaystyle x_t = \mu_t + \sum_{i=1}^k A_i x_{t-i} + u_t,$ (11.8)

where $ x_t$ is a $ (n\times 1)$-dimensional vector of endogenous variables, $ \mu_t$ contains deterministic terms like constant and time trend, $ A_i$ are $ (n\times n)$-dimensional coefficient matrices and $ u_t\sim N(0,\Sigma_u)$ is a serially uncorrelated error term. Subtracting $ x_{t-1}$ and rearranging terms yields the VECM:

$\displaystyle \Delta x_{t-1} = \mu_t + \Pi x_{t-1} + \sum_{i=1}^{k-1} \Gamma_i \Delta x_{t-i} + u_t,$ (11.9)

where $ \Pi$ and $ \Gamma_i$ are functions of the $ A_i$. The matrix $ \Pi$ can be decomposed into two $ (n\times r)$-dimensional matrices $ \alpha $ and $ \beta$: $ \Pi=\alpha\beta^\top$ where $ \alpha $ is called an adjustment matrix, $ \beta$ comprises the cointegration vectors, and $ r$ is the number of linearly independent cointegration vectors (cointegration rank). Following Engle and Granger (1987), a variable is integrated of order $ d$, or I($ d$), if it has to be differenced $ d$-times to become stationary. A vector $ x_t$ is integrated of order $ d$ if the maximum order of integration of the variables in $ x_t$ is $ d$. A vector $ x_t$ is cointegrated, or CI($ d,b$), if there exists a linear combination $ \beta^\top x_t$ that is integrated of a lower order $ (d-b)$ than $ x_t$.

The cointegration framework is only appropriate if the relevant variables are actually integrated. This can be tested using unit root tests. When no unit roots are found, traditional econometric methods can by applied.


11.2.2 Modelling Indonesian Money Demand with Econometric Techniques

We use quarterly data from 1990:1 until 2002:3 for our empirical analysis. The data is not seasonally adjusted and taken from Datastream (gross national product at 1993 prices $ Y$ and long-term interest rate $ R$) and from Bank Indonesia (money stock M2 $ M$ and consumer price index $ P$). In the following, logarithms of the respective variables are indicated by small letters, and $ mr=\ln M - \ln P$ denotes logarithmic real balances. The data is depicted in Figure 11.1.

Figure 11.1: Time series plots of logarithms of real balance, GNP, interest rate, and CPI.

\includegraphics[width=1.42\defpicwidth]{plotall.ps}

In the first step, we analyze the stochastic properties of the variables. Table 11.1 presents the results of unit root tests for logarithmic real balances $ mr$, logarithmic real GNP $ y$, logarithmic price level $ p$, and logarithmic long-term interest rate $ r$. Note that the log interest rate is used here while in the previous section the level of the interest rate has been used. Whether interest rates should be included in logarithms or in levels is mainly an empirical question.


Table 11.1: Unit Root Tests
Variable Deterministic terms Lags Test stat. 1/5/10% CV
$ mr$ c, t, s, P89c (98:3) 2 $ -4.55^{**}$ -4.75 / -4.44 / -4.18
$ y$ c, t, s, P89c (98:1) 0 $ -9.40^{***}$ -4.75 / -4.44 / -4.18
$ p$ c, t, s, P89c (98:1) 2 $ -9.46^{***}$ -4.75 / -4.44 / -4.18
$ r$ c, s 2 $ -4.72^{***}$ -3.57 / -2.92 / -2.60
Note: Unit root test results for the variables indicated in the first column. The second column describes deterministic terms included in the test regression: constant c, seasonal dummies s, linear trend t, and shift and impulse dummies P89c according to the model (c) in Perron (1989) allowing for a change in the mean and slope of a linear trend. Break points are given in parentheses. Lags denotes the number of lags included in the test regression. Column CV contains critical values. Three (two) asterisks denote significance at the 1% (5%) level.

Because the time series graphs show that there seem to be structural breaks in real money, GNP and price level, we allow for the possibility of a mean shift and a change in the slope of a linear trend in the augmented Dickey-Fuller test regression. This corresponds to model (c) in Perron (1989), where the critical values for this type of test are tabulated. In the unit root test for the interest rate, only a constant is considered. According to the test results, real money, real GNP and price level are trend-stationary, that is they do not exhibit a unit root, and the interest rate is also stationary. These results are quite stable with respect to the lag length specification. The result of trend-stationarity is also supported by visual inspection of a fitted trend and the corresponding trend deviations, see Figure 11.2. In the case of real money, the change in the slope of the linear trend is not significant.

Figure 11.2: Fitted trends for real money and real GNP.

\includegraphics[width=0.77\defpicwidth]{m2_trend2.ps} \includegraphics[width=0.77\defpicwidth]{gnp_trend2.ps}

Now, let us denote centered seasonal dummies $ s_{it}$, a step dummy switching from zero to one in the respective quarter $ ds$, and an impulse dummy having value one only in the respective quarter $ di$. Indonesian money demand is then estimated by OLS using the reduced form equation (11.4) ($ t$- and $ p$-values are in round and square parantheses, respectively):

$\displaystyle mr_t$ $\displaystyle =$ \begin{displaymath}\begin{array}[t]{c}0.531\\ {(6.79)}\end{array} mr_{t-1}\, + \...
...ray} y_t\, -\begin{array}[t]{c}0.127\\ {(-6.15)}\end{array} r_t\end{displaymath}  
    $\displaystyle -\begin{array}[t]{c}0.438\\ {(-0.84)}\end{array}-\begin{array}[t]...
...7)}\end{array} s_{2t}\, - \begin{array}[t]{c}0.036\\ {(-2.77)}\end{array}s_{3t}$  
    $\displaystyle +\begin{array}[t]{c}0.174\\ {(3.54)}\end{array}di9802_t\, + \begi...
...y} di9801_t\, + \begin{array}[t]{c}0.112\\ {(5.02)}\end{array} ds9803_t\, + u_t$  
       
$\displaystyle T$ $\displaystyle =$ $\displaystyle 50~(1990:2 - 2002:3)$  
$\displaystyle R^2$ $\displaystyle =$ $\displaystyle 0.987$  
$\displaystyle RESET(1)$ $\displaystyle =$ $\displaystyle 0.006~[0.941]$  
$\displaystyle LM(4)$ $\displaystyle =$ $\displaystyle 0.479~[0.751]$  
$\displaystyle JB$ $\displaystyle =$ $\displaystyle 0.196~[0.906]$  
$\displaystyle ARCH(4)$ $\displaystyle =$ $\displaystyle 0.970~[0.434]$  

Here JB refers to the Jarque-Bera test for nonnormality, RESET is the usual test for general nonlinearity and misspecification, LM(4) denotes a Lagrange-Multiplier test for autocorrelation up to order 4, ARCH(4) is a Lagrange-Multiplier test for autoregressive conditional heteroskedasticity up to order 4. Given these diagnostic statistics, the regression seems to be well specified. There is a mean shift in 1998:3 and the impulse dummies capture the fact, that the structural change in GNP occurs two months before the change in real money. The inflation rate is not significant and is therefore not included in the equation.

The implied income elasticity of money demand is 0.47/(1-0.53) = 1 and the interest rate elasticity is -0.13/(1-0.53) = -0.28. These are quite reasonable magnitudes. Equation (11.10) can be transformed into the following error correction representation:

$\displaystyle \Delta mr_t$ $\displaystyle =$ $\displaystyle -0.47\cdot (mr_{t-1}-y_{t-1}+0.28 r_{t-1})$  
    $\displaystyle +\, 0.47\Delta y_t - 0.13\Delta r_t + \textrm{deterministic terms} + u_t.$ (11.10)

Figure 11.3: Stability test for the real money demand equation (11.10).

\includegraphics[width=1.4\defpicwidth]{recur_cusum.ps}

Stability tests for the real money demand equation (11.10) are depicted in Figure 11.3. The CUSUM of squares test indicates some instability at the time of the Asian crises, and the coefficients of lagged real money and GNP seem to change slightly after the crisis. A possibility to allow for a change in these coefficients from 1998 on is to introduce two additional right-hand-side variables: lagged real money multiplied by the step dummy $ ds9803$ and GNP multiplied by $ ds9803$. Initially, we have also included a corresponding term for the interest rate. The coefficient is negative (-0.04) but not significant ($ p$-value: 0.29), such that we excluded the term from the regression equation. The respective coefficients for the period 1998:3-2002:3 can be obtained by summing the coefficients of lagged real money and lagged real money times step dummy and of GNP and GNP times step dummy, respectively. This reveals that the income elasticity stays approximately constant (0.28/(1-0.70)=0.93) until 1998:02 and ((0.28+0.29)/(1-0.70+0.32)=0.92) from 1998:3 to 2002:3 and that the interest rate elasticity declines in the second half of the sample from -0.13/(1-0.70)=-0.43 to -0.13/(1-0.79+0.32)=-0.21:

$\displaystyle mr_t$ $\displaystyle =$ \begin{displaymath}\begin{array}[t]{c}0.697\\ {(7.09)}\end{array} mr_{t-1}\, + \...
...ray} y_t\, -\begin{array}[t]{c}0.133\\ {(-6.81)}\end{array} r_t\end{displaymath}  
    $\displaystyle -\begin{array}[t]{c}0.322\\ {(-2.54)}\end{array} mr_{t-1}\cdot ds9803_t\, + \begin{array}[t]{c}0.288\\ {(2.63)}\end{array} y_t\cdot ds9803_t$  
    $\displaystyle +\begin{array}[t]{c}0.133\\ {(0.25)}\end{array}-\begin{array}[t]{...
...8)}\end{array} s_{2t}\, - \begin{array}[t]{c}0.034\\ {(-2.76)}\end{array}s_{3t}$  
    $\displaystyle +\begin{array}[t]{c}0.110\\ {(2.04)}\end{array}di9802_t\, + \begin{array}[t]{c}0.194\\ {(5.50)}\end{array} di9801_t\, + u_t.$  


$\displaystyle T$ $\displaystyle =$ $\displaystyle 50~(1990:2 - 2002:3)$  
$\displaystyle R^2$ $\displaystyle =$ $\displaystyle 0.989$  
$\displaystyle RESET(1)$ $\displaystyle =$ $\displaystyle 4.108~ [0.050]$  
$\displaystyle LM(4)$ $\displaystyle =$ $\displaystyle 0.619~ [0.652]$  
$\displaystyle JB$ $\displaystyle =$ $\displaystyle 0.428~ [0.807]$  
$\displaystyle ARCH(4)$ $\displaystyle =$ $\displaystyle 0.408~ [0.802]$  

Accordingly, the absolute adjustment coefficient $ \mu$ in the error correction representation increases from 0.30 to 0.62.

It can be concluded that Indonesian money demand has been surprisingly stable throughout and after the Asian crisis given that the Cusum of squares test indicates only minor stability problems. A shift in the constant term and two impulse dummies that correct for the different break points in real money and real output are sufficient to yield a relatively stable money demand function with an income elasticity of one and an interest rate elasticity of -0.28. However, a more flexible specification shows that the adjustment coefficient $ \mu$ increases and that the interest rate elasticity decreases after the Asian crisis. In the next section, we analyze if these results are supported by a fuzzy clustering technique.