Essays on discrete valued time series models

Statistical inference for discrete-valued time series has not been developed as systematically as traditional methods for time series generated by continuous random variables. This Ph.D. dissertation deals with time series models for discrete-valued processes. In particular, Chapter 2 is devoted to a comprehensive overview of the literature about observation-driven models for discrete-valued time series. Derivation of stochastic properties for these models is presented. For the inference, general properties of the quasi maximum likelihood estimator (QMLE) are discussed, followed by an illustrative application. In Chapter 3, a general class of observation-driven time series models for discrete-valued processes is introduced. Stationarity and ergodicity are derived under easy-to-check conditions, which can be directly applied to all the models encompassed in the framework. Consistency and asymptotic normality of the QMLE are established, with the focus on the exponential family. Finite sample properties of the estimators are investigated through a Monte Carlo study and illustrative examples are provided. The framework introduced in the paper provides a self-contained background that relates different models developed in the literature as well as novel specifications and makes them fully applicable in practice. Discrete responses are commonly encountered in real applications and are strongly connected to network data. The specification of suitable network autoregressive models for count time series is an important aspect which is not covered by the existing literature. In Chapter 4, we consider network autoregressive models for count data with a known neighborhood structure. The main methodological contribution is the development of conditions that guarantee stability and valid statistical inference. We consider both cases of fixed and increasing network dimension and we show that quasi-likelihood inference provides consistent and asymptotically normally distributed estimators. The work is complemented by simulation results and a data example.