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This paper considers identification problems for output-error moving average systems with colored noises. The basic idea is, by the auxiliary model identification principle, to replace the unknown noise-free outputs and unmeasurable noise terms in the information vector with the outputs of an auxiliary model and the estimated residuals, and to present an auxiliary model based extended stochastic gradient algorithm. The algorithm proposed has significant computational advantage over existing least squares identification algorithms. The simulation example indicates that the parameter estimation errors become small as the data length increases.