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The third order cumulant of the output of an ARMA (p,q) model, driven by unobservable non-Gaussian i.i.d. noise, is used to identify the model parameters. The model is assumed to be causal and stable but need not be minimum-phase. The symmetry properties of the third order cumulant are applied to use the cumulant values in the first non-redundant region, where it is proved that the matrices used to solve for the AR parameters are of full rank and have a transpose equivalence that can be used to enhance the efficiency of the estimation process. The estimated AR parameters are then used to estimate the MA order and parameters. The simulation results also show that the AR model order can be estimated from the scattering of the estimated poles in the complex Z-plane.