Negative binomial quasi-likelihood inference for general integer-valued time series models

Two negative binomial quasi-maximum likelihood estimates (NB-QMLE's) 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-QMLE's 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 GARCH model and the INAR(1) model. Applications to two real datasets are given.