Y (Ins)
4.4 Cointegration and long-run money demand estimates
We caution in Section 4.3 above that a regression of a non-stationary time series on another non-stationary time series might produce a spurious regression. However, if the residuals from the latter regression are subjected to a test of stationarity (say the unit root test) and found to be stationary, this regression will be meaningful (not spurious). In this case, the variables in question are said to be cointegrated. Economically speaking, two variables (or more) are said to be cointegrated if there is a long-run or equilibrium relationship between them (Charemza and Deadman, 1992; Gujarati, 2003). Therefore, as Granger (1986: 226) puts it, a test for cointegration can be thought of as a pre-test to avoid spurious regressions situations. Table 4.1 indicates that except for the interest rate differential (RM-RO), the variables in our model (m2, y, p, RM, and RO) are individually 1(1). So, in order to avoid the problem of spurious regression, we ought to subject the residuals of our model to unit root test to find out whether our selected variables cointegrate and determine whether there is a long-run relationship between them.
Our long-run model consists in regressing m2 on y and p. The interest rate differential (RM- RO) will not be included in our long-run specification. Following Nell (2003), since the ADF unit root test (in Table 4.1) shows that RM and RO cointegrate with a unit coefficient (because (RM-RO) is 1(0)), in other words, there are two cointegrating vectors in the money demand function when the interest rates enter independently. The net long-run interest rate effect is therefore zero, because a positive shock to RM will be equally offset by an increase in RO, that is, in the long-run, RM=RO. If the interest rate effect is immediately entered as a differential
(RM-RO) the long-run impact is zero. The cointegrating regression results for our model are given below, with ^-statistics in parentheses.
m2 = -1.064 + 0.596y + 1.592p (4) (-1.269) (7.927) (18.916)
R2= 0.958 adjusted/?2 =0.957 d = 0.308 F = 926.7
According to the extremely high F-test, the model is significant. The computed ^-statistics (for y and/?) are highly statistically significant, the adjusted R1 is high, but the Durbin-Watson d value is extremely low indicating that the model is plagued by the problem of autocorrelation.
We know from Gujarati (2003: 792) that, sometimes autocorrelation results because the underlying time series is non-stationary. However, according to Granger and Newbold (1974), an R2> d is a good rule of thumb to suspect that the estimated regression is spurious.
Moreover, since m2, y, and p are individually non-stationary, there is a possibility that our regression of m2 on y and p in equation (4) is spurious. Therefore, to find whether the regression (4) is not spurious, we performed a unit root test on the residuals estimated from (4) to check whether m2, y and p cointegrate, in other words, we performed the Engle-Granger test as follows.
Aw< = -0.151w,_, t = (-2.515)
adjusted R2= 0.060 d = 1.809
The Engle-Granger 5 per cent critical t value is -1.95. Since the computedr value (-2.515) is greater, in absolute value, than the critical Engle-Granger, we reject the null hypothesis (S = 0) that we have a unit root. The residuals from our regression are 1(0) or stationary.
Hence, m2, y and p are cointegrated, and equation (4) represents the long-run money demand in Rwanda. Compared to Nell's results we note that prices adjust less than the change in the money supply.
Nell (2003) contends that the cointegration test in equation (4) has to be taken as an initial estimation because it prevents difficulties associated with other cointegration procedures.
Moreover, Inder (1993) demonstrates that excluding dynamics in finite samples may affect the performance of the estimator and as an alternative suggests the Unrestricted Error Correction Model (UECM), which should reduce bias and also includes dynamics in the estimation of the long-run model. We specified our UECM of order two as follows:
i=l z'=0 (=0 (=0
+ 56D2+ S7t + axmlt_x + a2 v,_, + oc3pt_] + a4 (RM - RO)l_l + et (5)
In equation (5), the variables have the same definition as before except for the dummy variable Z), that takes the value of one in 1994ql-1995q4 and zero otherwise, and D2 which takes the value of one in 1990ql-2000q4 and zero otherwise, whereas Ms a time trend. Nell (2003) insists on the inclusion of dummy variables, and even outlier dummies, since one cannot easily detect outliers on visual displays of the economic data. The omission of relevant dummies may lead to predictive failure, because the Error Correction Model still converges to the old equilibrium (Clement and Hendry, 1997). Our dummy variable Dx captures the dramatic price increase in 1994-1995 following the Rwandan war and genocide, and we expect its coefficient to have a negative sign. D2 captures the structural adjustment reforms that were implemented in Rwanda since 1990. Variables in level form such as (m2r_,) represent the long-run part of
the model, while those in differences such as (Ay,_,) stand for the short-run. For our parsimonious specification, we have decided to eliminate all other insignificant parameters from the model except for the coefficient of the change in the short-run interest rate differential A(RM - RO). The results of the regression for equation (5) are presented below, with ^-statistics in parentheses.
Aw2 = 0.994 + 0.297 Ay,_, + 0.86 Ap + 0.001 A(RM - RO) - 0.063 D2 + 0.005^
(1.49) (3.11) (3.71) (0.16) (-2.31) (2.93)
-0.319/w2,_, +0.121 y,_t +0.289 /?,_, (6)
(-3.79) (2.12) (2.81)
The insignificance of (a4) the coefficient of (RM - RO)t_^ in equation (6) suggests that (RM- RO) has no long-run effect. This is in line with Nell's (2003) assertion that when the unit root test shows that the RM and RO cointegrate with a unit coefficient, the long-run impact of the interest rate differential (RM-RO) is zero. The salient feature of equation (6) is the statistical insignificance of the coefficients for all lagged changes in (RM-RO). Despite having this, we kept A(RM-RO) in our model because the theory suggests that the demand for money is a function of a scale variable and a set of opportunity cost variables to indicate the forgone earnings by not holding assets, which are alternatives to, money. The opportunity cost of holding money involves two ingredients: the own rate of money and the rate of return on assets alternatives to money (Sriram, 1999). In equation (6), the cointegration test is based on the coefficient of mlt_x (the error correction coefficient)9. The computed r value of the latter
9 Tests of cointegration based not directly on the residuals but on the regression coefficients themselves, might have higher power (Banerjee et al., 1993)
is equal to -3.79 which in absolute value is greater than the 5 per cent Dickey-Fuller critical r of-3.45, leading to the rejection of the null hypothesis of no cointegration. The existence of cointegration is also supported by the plot of the residuals from equation (6) in Figure 4.13 below, as well as a unit root test conducted on the same residuals.
1980 . . 1985 . . 1990 . . 1995 . . 2000
YEAR
Figure 4.13 Residuals from the UECM in equation (6)
It is apparent from Figure 4.13 above, which is a plot of the residuals from the regression of equation (6), and their frequent crossing of zero, that our UECM is a cointegrating regression.
Therefore, m2, y, and p have a long-run relation between them. We derived the long-run solution of equation (6) by taking the ratio of the coefficients of the level variables ( yt_x and
pt_x) to the error correction coefficient of 0.319. This gives us the following equation:
ml = 0.3S> + 0.91/7 (7)
CD
•g w
CD OH -a
CD N T3
k_
CO T3
s
wc
= >
0.0-
. 1 •
-.2
There are however differences between the magnitude of the coefficients in equation (4) when compared to equation (7). We now note that in this second estimation, which includes the short-run dynamics, that prices adjust by more than the money supply. As noted in Chapter Two, money is not superneutral.