We consider cointegration rank estimation for a p-dimensional Fractional Vector Error Correction Model. We propose a new two-step procedure which allows testing for further long-run equilibrium relations with possibly different persistence levels. The fi?rst step consists in estimating the parameters of the model under the null hypothesis of the cointegration rank r = 1, 2, ..., p-1. This step provides consistent estimates of the cointegration degree, the cointegration vectors, the speed of adjustment to the equilibrium parameters and the common trends. In the second step we carry out a sup-likelihood ratio test of no-cointegration on the estimated p - r common trends that are not cointegrated under the null. The cointegration degree is re-estimated in the second step to allow for new cointegration relationships with different memory. We augment the error correction model in the second step to control for stochastic trend estimation effects from the first step. The critical values of the tests proposed depend only on the number of common trends under the null, p - r, and on the interval of the cointegration degrees b allowed, but not on the true cointegration degree b0. Hence, no additional simulations are required to approximate the critical values and this procedure can be convenient for practical purposes. In a Monte Carlo study we analyze the fi?nite sample properties of different specifi?cations of the correction terms and compare our procedure with alternative methods.
