Cointegration 203
table 10.4 Augmented Dickey-Fuller Tests of Residuals for Cointegration
Panel A. 1962 − 1982 n = 248
Variable Coefficient t-Stat p-Value
zt −0.063 −3.23 0.001
Δzt− 1 0.272 4.32 0.000
Δzt− 2 −0.030 −0.46 0.642
Δzt− 3 0.090 1.40 0.162
t-statistic of p = −3.23; critical values at 5% and 10% are −3.36 and −3.06 respectively.
Panel B. 1962 − 2006 n = 536
Variable Coefficient t-Stat p-Value
zt −0.008 −1.81 0.070
Δzt− 1 0.265 6.13 0.000
Δzt− 2 −0.048 −1.08 0.280
Δzt− 3 0.031 0.71 0.477
t-statistic of p = −1.81; critical values at 5% and 10% are 3.35 and 3.05, respectively.
The critical values of the augumented Dickey-Fuller (ADF) statistic are from Engle
and Yoo (1987). The cointegration equation errors used to perform the ADF test are
based on the following regression:
Δzt = −pzt− 1 + aΔzt− 1 + bΔzt− 2 + cΔzt− 3 + etwhere Δzt is the change in the error term from the cointegration regression and et is
a random error. If p is positive and significantly different from zero, the z residuals
from the equilibrium equation are stationary so we may accept the null hypothesis
of cointegration. In both equations the error terms are white noise, so no further
stationarity tests were performed.
dividends variable because the t-test is not appropriate unless the variables
are cointegrated. This is, of course, the issue that we are investigating.
Once we estimate the regression in Step 2, the next step involves testing
the residuals of the regression, zt, for stationarity from equation (10.3). By
definition, the residuals have a zero mean and lack a time trend. This sim-
plifies the test for stationarity. This is accomplished by estimating equation
(10.4). The null hypothesis is that the variables lack cointegration. If we con-
clude that p in equation (10.4) is negative and statistically significant, then
we reject the null hypothesis and conclude that the evidence is consistent
with the presence of cointegration between the stock index and dividends.
The appropriate lag lengths may be determined by the Akaike information
criterion or theoretical and practical considerations. We decided to use a
lag length of three periods representing one quarter. The results are pre-
sented in Table 10.4. For the 1962–1982 period, the null hypothesis of no