Recently, increasing interest in the issue of fractional cointegration has emerged from theoretical and empirical viewpoints. Here, as opposed to the traditional prescription of unit root observables with weak dependent cointegrating errors, the orders of integration of these series are allowed to take real values, but, as in the traditional framework, equality of the orders of at least two observable series is necessary for cointegration. This assumption, in view of the real-valued nature of these orders, could pose some difficulties, and in the present paper we explore some ideas related to this issue in a simple bivariate framework. First, in a situation of near-cointegration, where the only difference with respect to the usual fractional cointegration is that the orders of the two observable series differ in an asymptotically negligible way, we analyze properties of standard estimates of the cointegrating parameter. Second, we discuss the estimation of the cointegrating parameter in a situation where the orders of integration of the two observables are truly different but their corresponding balanced versions (with same order of integration) are cointegrated in the usual sense. A Monte Carlo study of finite-sample performance and simulated series is included.I thank Adrian Pagan, James Davidson, and seminar participants at the 2004 Econometric Society European Meeting and the 2004 Simposio de An lisis Econ mico for helpful comments. I also thank two referees and a co-editor whose comments led to improvements of the paper. This research was supported by the Spanish Ministerio de Educaci n y Ciencia through a contract Juan de la Cierva and ref. SEJ2005-07657 ECON, and also by the Universidad de Navarra, ref. 16037001.