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A quasi-score linearity test for continuous and count network autoregressive models is developed. We establish the asymptotic distribution of the test when the network dimension is fixed or increasing, under the null hypothesis of linearity and Pitman's local alternatives. When the parameters are identifiable, the test statistic approximates a chi-square and noncentral chi-square asymptotic distribution, respectively. These results still hold true when the parameters tested belong to the boundary of their space. When we deal with non-identifiable parameters, a suitable test is proposed and its asymptotic distribution is established when the network dimension is fixed. Since, in general, critical values of such test cannot be tabulated, the empirical computation of the p-values is implemented using a feasible bound. Bootstrap approximations are also provided. Moreover, consistency and asymptotic normality of the quasi maximum likelihood estimator is established for continuous and count nonlinear network autoregressions, under standard smoothness conditions. A simulation study and two data examples complement this work.

We introduce a new R package for analysis and inference of network count time series. Such data arise frequently in statistics and epidemiology and are modelled as multivariate time series by employing either linear or log-linear models. However, nonlinear models have also been successful in several fields but often excluded from the analysis due to their relative fitting complexity. In this paper, we offer users the flexibility to study and specify non-linear network count time series models by providing them with a toolkit that copes with computational issues. In addition, new estimation tools for (log-)linear network autoregressive models of count time series are also developed. We illustrate the methodology to the weekly number of influenza A & B cases in the 140 districts of the two Southern German states Bavaria and Baden-Wuerttemberg, for the years 2001 to 2008. This dataset is publicly available, so that the analysis is easily reproducible.

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.

Statistical inference for discrete-valued time series has not been developed like traditional methods for time series generated by continuous random variables. Some relevant models exist, but the lack of a homogenous framework raises some critical issues. For instance, it is not trivial to explore whether models are nested and it is quite arduous to derive stochastic properties which simultaneously hold across different specifications. In this paper, inference for a general class of first order observation-driven models for discrete-valued processes is developed. Stochastic properties such as stationarity and ergodicity are derived under easy-to-check conditions, which can be directly applied to all the models encompassed in the class and for every distribution which satisfies mild moment conditions. Consistency and asymptotic normality of quasi-maximum likelihood estimators are established, with the focus on the exponential family. Finite sample properties and the use of information criteria for model selection are investigated throughout Monte Carlo studies. An empirical application to count data is discussed, concerning a test-bed time series on the spread of an infection.