Cointegration 209
15.49 necessary to establish statistical significance at the 5% level. We do
not reject the null hypothesis. We therefore conclude that there is at least
one cointegrating vector. There is no need to evaluate λtrace (2).
The maximum eigenvalue statistic, denoted λmax, reinforces the con-
clusion from the λ trace test statistic. We can use λmax(0, 1) to test the null
hypothesis that the variables lack (0) cointegration against the alternative
that they are cointegrated with one (1) cointegrating vector. Table 10.6 pres-
ents the value of λmax(0, 1). Again, for pedagogic reasons, we outline the
calculation of λmax(0, 1):
λmax (0,1) = (−T ln(1 − λi*)
= −96 ln (1 − 0.227) = 24.72The computed value of 24.72 exceeds the critical value of 21.13 at the 5%
significance level and has a p-value of 0.01. Once again, this leads to rejec-
tion of the null hypothesis that the stock indices lack cointegration. The
conclusion is that there exists at least one cointegrating vector.
The next step requires a presentation of the cointegrating equation and
an analysis of the error-correction model. Table 10.7 presents both. The
cointegrating equation is a multivariate representation of zt− 1 in the Engle-
Granger cointegration test. This is presented in Panel A of Table 10.7. The
error-correction model takes the following representation:
(^) ∆=ybti 10 +∆by 11 ti−−+∆cxjtjt+−dy 11 ()−−axt 11 +ett
j
n
i
n
= =
∑∑
1 1
(10.11)
The notation of equation (10.11) differs somewhat from the notation of
equations (10.5) and (10.6). The notation used in equation (10.11) reflects
the matrix notation adopted for the Johansen-Juselius cointegration test in
equation (10.9). Nevertheless, for expositional convenience, we did not use
the matrix notation for the error-correction term. The notation Δ means the
first difference of the variable; thus Δy 1 t− 1 means the change in the log of the
French stock index in period t − 1, (y 1 t− 1 − y 1 t− 2 ). Equation (10.11) claims
that changes in the log of the French stock index are due to changes in the
French stock index during the last two periods; changes in the German stock
index during the last two periods; changes in the Netherlands stock index
during the last two periods; and finally deviations of the French stock index
from its stochastic trend with Germany and the Netherlands.^17 An analo-
gous equation could be written for both Germany and the Netherlands.
(^17) Lag length of two periods is determined on the basis of information criteria such as
the Bayesian information criterion and it is provided in statistical software programs.