We develop a sequence of tests for specifying the cointegrating rank of, possibly fractional, multiple time series. Memory parameters of observables are treated as unknown, as are those of possible cointegrating errors. The individual test statistics have standard null asymptotics, and are related to Hausman specification test statistics: when the memory parameter is common to several series, an estimate of this parameter based on the assumption of no cointegration achieves an efficiency improvement over estimates based on individual series, whereas if the series are cointegrated the former estimate is generally inconsistent. However, a computationally simpler but asymptotically equivalent approach, which avoids explicit computation of the "efficient" estimate, is instead pursued here. Two versions of it are initially proposed, followed by one that robustifies to possible inequality between memory parameters of observables. Throughout, a semiparametric approach is pursued, modelling serial dependence only at frequencies near the origin, with the goal of validity under broad circumstances and computational convenience. The main development is in terms of stationary series, but an extension to nonstationary ones is also described. The algorithm for estimating cointegrating rank entails carrying out such tests based on potentially all subsets of two or more of the series, though outcomes of previous tests may render some or all subsequent ones unnecessary. A Monte Carlo study of finite sample performance is included.
