Abstract :
We consider the cointegration tests of Johansen ~1988, Journal of Economic
Dynamics and Control 12, 231–254; 1991, Econometrica 59, 1551–1580! when a
vector autoregressive ~VAR! process of order k is used to approximate a more
general linear process with a possibly infinite VAR representation+ Traditional methods
to select the lag order, such as Akaike’s information criterion ~AIC! or the
Bayesian information criterion, often lead to too parsimonious a model with the
implication that the cointegration tests suffer from substantial size distortions in
finite samples+ We extend the analysis of Ng and Perron ~2001, Econometrica 69,
1519–1554! to derive a modified Akaike’s information criterion ~MAIC! in this
multivariate setting+ The idea is to use the information specified by the null hypothesis
as it relates to restrictions on the parameters of the model to keep an extra
term in the penalty function of the AIC+ This MAIC takes a very simple form for
which this extra term is simply the likelihood ratio test for testing the null hypothesis
of r against more than r cointegrating vectors+ We provide theoretical analyses
of its validity and of the fact that cointegration tests constructed from a VAR
whose lag order is selected using the MAIC have the same limit distribution as
when the order is finite and known+ We also provide theoretical and simulation
analyses to show how the MAIC leads to VAR approximations that yield tests
with drastically improved size properties with little loss of power+
