Cointegration 205
and dividends deviate from long-run equilibrium. The error- correction
term is included in both equations to guarantee that the variables do not
drift too far apart. Engle and Granger showed that, if the variables are
cointegrated, the coefficient on the error-correction term, (yt− 1 − axt− 1 ),
in at least one of the equations must be nonzero.^12 The t-value of the error-
correction term in equation (10.5) is statistically different from zero. The
coefficient of −0.07 is referred to as the speed of adjustment coefficient. The
estimated value for the coefficient suggests that 7% of the previous month’s
disequilibrium between the stock index and dividends is eliminated in the
current month. In general, the higher the speed of adjustment coefficient,
the faster the long-run equilibrium is restored. Since the speed of adjustment
coefficient for the dividend equation is statistically indistinguishable from
zero, all of the adjustment falls on the stock price.
An interesting observation from Table 10.5 relates to the lag struc-
ture of equation (10.5). The first lag on past stock price changes is statisti-
cally significant. This means that the change in the stock index this month
depends upon the change during the last month. This is inconsistent with the
efficient market hypothesis. On the other hand, the change in dividend lags
is not statistically different from zero. The efficient market theory suggests,
and the estimated equation confirms, that past changes in dividends do not
affect the current changes in stock prices.
Johansen-Juselius Cointegration test
The Engle-Granger cointegration test has some problems. These problems
are magnified in a multivariate (three or more variables) context. In principle,
when the cointegrating equation is estimated (even in a two-variable prob-
lem), any variable may be utilized as the dependent variable. In illustration
of the application of the Engle-Granger cointegration test, this would entail
placing dividends on the left-hand side of equation (10.2) and the S&P 500
index on the right-hand side. As the sample size approaches infinity, Engle
and Granger showed that the cointegration tests produce the same results
irrespective of what variable is used as the dependent variable. The question
is then: How large a sample is large enough?
A second problem is that the errors we use to test for cointegration
are only estimates and the not the true errors. Thus any mistakes made in
estimating the error term, zt , in equation (10.2) are carried forward into the
regression given by equation (10.3). Finally, the Engle-Granger cointegra-
tion test is unable to detect multiple cointegrating relationships.
(^12) Engle and Granger, “Cointegration and Error-Correction.”