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

Abstract

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.