Cointegration 207
cointegration vector. The maximum eigenvalue test, so-named because it is
based on the largest eigenvalue, tests the null hypothesis that there are i cointe-
grating vectors against the alternative hypothesis of i + 1 cointegrating vectors.
Johansen and Juselius derive critical values for both test statistics. The criti-
cal values are different if there is a deterministic time trend A 0 included. Enders
provides tables for both critical statistics with and without the trend terms.^14
Software programs often provide critical values and the relevant p-values.
empirical illustration of Johansen-Juselius procedure Many financial advisors
and portfolio managers argue that investors would be able to improve their
risk/return profile by investing internationally rather than restricting their
holding to domestic stocks (i.e., following a policy of international diversi-
fication). If stock market returns in different countries are not highly cor-
related, then investors could obtain risk reduction without significant loss
of return by investing in different countries. But with the advent of global-
ization and the simultaneous integration of capital markets throughout the
world, the risk-diversifying benefits of international investing have been
challenged. Here, we illustrate how cointegration can shed light on this
issue and apply the Johansen-Juselius cointegration test.
The idea of a common currency for the European countries is to reduce
transactions costs and more closely link the economies. We shall use cointe-
gration to examine whether the stock markets of France, Germany, and the
Netherlands are linked following the introduction of the euro in 1999. We
use monthly data for the period 1999–2006.
Although testing for cointegration requires that the researcher test
for stationarity in the series, Johansen states that testing for stationarity is
redundant since stationarity is revealed through a cointegration vector.^15
However, it is important to establish the appropriate lag length for equation
(10.9). This is typically done by estimating a traditional vector autoregres-
sive (VAR) model (see Chapter 9) and applying a multivariate version of
the Akaike information criterion or Bayesian information criterion. For our
model, we use one lag, and thus the model takes the form:
yt = A 0 + A 1 yt− 1 + ut (10.10)
where yt is the n × 3 vector (y 1 t, y 2 t,... , y 3 t)′ of the logs of the stock market
index for France, Germany, and the Netherlands (i.e., element y 1 t is the log
(^14) Enders, “ARIMA and Cointegration Tests of Purchasing Power Parity.”
(^15) Soren Johansen, Likelihood-Based Inference in Cointegrated Vector Autoregres-
sive Models (New York: Oxford University Press, 1995).