Quasi-likelihood estimation of non-invertible moving average processes

Classical methods based on Gaussian likelihood or least-squares cannot iden tify non-invertible moving average processes, while recent non-Gaussian res ults are based on full likelihood consideration, Since the error distributi on Is rarely known a quasi-likelihood approach is desirable, hut its consis tency properties ale yet unknown, In this paper we study the quasi-likeliho od associated with the Laplacian model, a convenient non-Gaussian model tha t yields a modified Li procedure. We show that consistency holds for all st andard heavy tailed errors, but not for light tailed errors, showing that a quasi-likelihood procedure cannot be applied blindly to estimate non-inver tible models, This is an interesting contrast to the standard results of th e quasi-likelihood in regression models, where consistency usually holds mu ch more generally, Similar results hold for estimation of non-causal non-in vertible ARMA processes. Various simulation studies are presented to valida te the theory and to show the effect of the error distribution, and an anal ysis of the US unemployment series is given as an illustration.