Abstract
Two negative binomial quasi-maximum likelihood estimates (NB-QMLEs) for a general class of count time series models are proposed. The first one is the profile NB-QMLE calculated while arbitrarily fixing the dispersion parameter of the negative binomial likelihood. The second one, termed two-stage NB-QMLE, consists of four stages estimating both conditional mean and dispersion parameters. It is shown that the two estimates are consistent and asymptotically Gaussian under mild conditions. Moreover, the two-stage NB-QMLE enjoys a certain asymptotic efficiency property provided that a negative binomial link function relating the conditional mean and conditional variance is specified. The proposed NB-QMLEs are compared with the Poisson QMLE asymptotically and in finite samples for various well-known particular classes of count time series models such as the Poisson and negative binomial integer-valued GARCH model and the INAR(1) model. Application to a real dataset is given.