Quasi-Likelihood Estimation in Volatility Models for Semi-Continuous Time Series

Time series containing non-negligible portion of possibly dependent zeros, whereas the remaining observations are positive, are considered. They are regarded as GARCH processes consisting of non-negative values. Our first aim lies in estimation of the omnibus model parameters taking into account the semi-continuous distribution. The hurdle distribution together with dependent zeros cause that the classical GARCH estimation techniques fail. Two different quasi-likelihood approaches are employed. Both estimators are proved to be strongly consistent and asymptotically normal. The second goal consists in the proposed predictions with bootstrap add-ons. The considered class of models can be reformulated as multiplicative error models. The empirical properties are illustrated in a simulation study, which demonstrates computational efficiency of the employed methods. The developed techniques are presented through an actuarial problem concerning insurance claims.