The problem of closed-loop system identification given noisy input-output measurements is considered. It is assumed that the various disturbances/noise processes affecting the system are zero-mean stationary Gaussian, whereas the closed-loop system operates under an external nonGaussian 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. The various noise sequences affecting the system are not (explicitly) modeled. First the open-loop transfer function is estimated using the integrated polyspectrum and cross-polyspectrum of the time-domain input-output measurements. Then two existing techniques for parametric system identification given consistent estimates of the underlying transfer function, are exploited. The parameter estimators are strongly consistent. A performance analysis is also carried out. A simulation example using an unstable open-loop system is presented to illustrate the proposed approach.