Tests for cointegration rank and choice of the alternative

This paper discusses likelihood-ratio (LR) tests on the cointegrating (CI) rank which consider any possible dimension of the CI rank under the alternative. The trace test and lambda-max test are obtained as special cases. Limit quantiles for all the tests in the class are derived. It is found that any of these tests can be used to construct an estimator of the CI rank, with no differences in asymptotic properties when the alternative is fixed. The properties of the class of tests are investigated by local asymptotic analysis, a simulation study and an empirical illustration. It is found that all the tests in the class have comparable power, which deteriorates substantially as the number of random walks increases. Tests constructed for a specific class of alternatives present minor power gains for alternatives in the class, and require the alternative to be far from the null. No test in this class is found to be asymptotically (in-)admissible. Some of the new tests in the class can also be arranged to give a constrained estimator of the CI rank, that restricts the minimum number of common trends. The power gains that these tests can obtain by constraining the minimum number of common trends appears to be limited and outweighted by the risk of inconsistency induced by the constrains. As a consequence, no value of the CI rank should be left untested, unless it can be excluded beyond any reasonable doubt.