Robust Minimum Distance Estimation for Nonlinear Semi-Strong GARCH Models

We develop a class of Minimum Distance Estimators for semi-strong Nonlinear ARMAX-Nonlinear GARCH processes. The estimators are asymptotically normal for possibly very heavy-tailed data due to underlying shocks and/or model parameter values. In particular we only impose trivial moment conditions on the GARCH errors, covering non-stationary GARCH. The MDE class is couched within a Method of Moments framework based on tail-trimming nonlinear functions of the data. The theory applies at least to tail-trimmed versions of Generalized Method of Moments, Quasi-Maximum Likelihood, and Least Absolute Deviations. As opposed to trimming the data or the criterion function directly as is universally done, we trim the estimating equations that govern asymptotics. Finally, we propose a unique method for selecting the trimming proportion based on exploiting the untrimmed criterion.