The problem of closed-loop system identification given noisy time-domain input-output measurements is considered. It is assumed that the various disturbances affecting the system are zero-mean stationary whereas the closed-loop system operates under an external cyclostationary input which is not measured. Noisy measurements of the (direct) input and output of the plant are assumed to be available. The closed-loop system must be stable, but it is allowed to be unstable in open loop. No knowledge about the linear-feedback mechanism is assumed. Two identification algorithms are investigated using cyclic spectral analysis of noisy input-output data. For both approaches, the open-loop transfer function is first estimated using the cyclic spectrum and cyclic cross-spectrum of the input-output data. These transfer function estimates are then used as "data" for the proposed algorithms. Both classes of parameter estimators are shown to be weakly consistent in any stationary noise (both at input as well as output). Asymptotic performance analysis of the proposed parameter estimators is also provided. Computer simulation examples are presented in support of the proposed approaches.