Robust Estimation for Poisson Integer-Valued GARCH Models Using a New Hybrid Loss

The Poisson integer-valued GARCH model is a popular tool in modeling time series of counts. The commonly used maximum likelihood estimator is strongly influenced by outliers, so there is a need to develop a robust M-estimator for this model. This paper has three aims. First, the authors propose a new loss function, which is a hybrid of the tri-weight loss for relatively small errors and the exponential squared loss for relatively large ones. Second, Mallows' quasi-likelihood estimator (MQLE) is proposed as an M-estimator and its existence, uniqueness, consistency and asymptotic normality are established. In addition, a data-adaptive algorithm for computing MQLE is given based on a data-driven selection of tuning parameters in the loss function. Third, simulation studies and analysis of a real example are conducted to illustrate the performance of the new estimator, and a comparison with existing estimators is made.