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The properties of the so-called subspace algorithms, up to now used almost only for stationary processes, are investigated in the context of cointegrated processes of order 1. It is shown for one of these algorithms that it can be adapted to deliver consistent estimates of all system parameters in the case of general I(1) VARMA models and mild conditions on the underlying noise. Estimates of the cointegrating space are derived and several test procedures for the cointegrating rank are proposed. Consistent estimation of the system order is also discussed. A simulation study shows the usefulness of subspace algorithms for estimation of and testing in cointegrated systems.