The problem of closed-loop system identification given noisy input-output measurements is considered. It is assumed that the closed-loop system operates under an external non-Gaussian input which is not measured. If the external input has non-vanishing integrated bispectrum (IB) and data IB is used for identification, then the various disturbances/noise processes affecting the system are assumed to be zero-mean stationary with vanishing IB. If the external input has non-vanishing integrated trispectrum (IT) and data IT is used for identification, then the various disturbances/noise processes affecting the system are assumed to be zero-mean stationary Gaussian. 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. Parametric modeling of the various noise sequences affecting the system is not needed. 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. Asymptotic performance analysis is also carried out. A computer simulation example using an unstable open-loop system is presented to illustrate the proposed approach.