This paper considers estimation of moving average (MA) models with non-Gaussian errors. Information in higher-order cumulants allows identification of the parameters without imposing invertibility. By allowing for an unbounded parameter space, the generalized method of moments estimator of the MA(1) model has classical (root-T and asymptotic normal) properties when the moving average root is inside, outside, and on the unit circle. For more general models where the dependence of the cumulants on the model parameters is analytically intractable, we consider simulation-based estimators with two features that distinguish them from the existing work in the literature. First, identification now requires information from the second and higher-order moments of the data. Thus, in addition to an autoregressive model, new auxiliary regressions need to be considered. Second, the errors used to simulate the model are drawn from a flexible functional form to accommodate a large class of distributions with non-Gaussian features. The proposed simulation estimators are also asymptotically normally distributed without imposing the assumption of invertibility. In the application considered, there is overwhelming evidence of non-invertibility in the Fama-French portfolio returns.
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