Quasi‐likelihood Estimation of Non‐invertible Moving Average Processes

Classical methods based on Gaussian likelihood or least‐squares cannot identify non‐invertible moving average processes, while recent non‐Gaussian results are based on full likelihood consideration. Since the error distribution is rarely known a quasi‐likelihood approach is desirable, but its consistency properties are yet unknown. In this paper we study the quasi‐likelihood associated with the Laplacian model, a convenient non‐Gaussian model that yields a modified L1 procedure. We show that consistency holds for all standard heavy tailed errors, but not for light tailed errors, showing that a quasi‐likelihood procedure cannot be applied blindly to estimate non‐invertible models. This is an interesting contrast to the standard results of the quasi‐likelihood in regression models, where consistency usually holds much more generally. Similar results hold for estimation of non‐causal non‐invertible ARMA processes. Various simulation studies are presented to validate the theory and to show the effect of the error distribution, and an analysis of the US unemployment series is given as an illustration.