Summary: For univariate time series we suggest a new variant of efficient score tests against fractional alternatives. This test has three important merits. First, by means of simulations, we observe that it is superior in terms of size and power in some situations of practical interest. Second, it is easily understood and implemented as a slight modification of the Dickey-Fuller test, although our score test has a limiting normal distribution. Third, and most important, our test generalizes to multivariate cointegration tests just as the Dickey-Fuller test does. Thus it allows to determine the cointegration rank of fractionally integrated time series.
It does so by solving a generalized eigenvalue problem of the type proposed by S. Johansen [J. Econ. Dyn. Control 12, 231–254 (1988; Zbl 0647.62102)]. However, the limiting distribution of the corresponding trace statistic is \(X^2\), where the degrees of freedom depend only on the cointegration rank under the null hypothesis. The usefulness of the asymptotic theory for finite samples is established in a Monte Carlo experiment.