Subspace methods for identification of linear time-invariant dynamical systems typically consist of two main steps. First, a so-called subspace estimate is constructed. This first step usually consists of estimating the range space of the extended observability matrix. Secondly, an estimate of system parameters is obtained, based on the subspace estimate. In this paper, the consistency of a large class of methods for estimating the extended observability matrix is analyzed. Persistence of excitation conditions on the input signal are given which guarantee consistent estimates for systems with only measurement noise. For systems with process noise, it is shown that a persistence of excitation condition on the input is not sufficient. More precisely, an example for which the subspace methods fail to give a consistent estimate of the transfer function is given. This failure occurs even if the input is persistently exciting of any order. It is also shown that this problem can be eliminated if stronger conditions on the input signal are imposed.