Abstract
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.
Highlights
The analysis of time series that are generated by continuous random variables has a long tradition in statistics and dates back, in the parametric setting, to [42] and [41], who introduced the concept of autoregression, a dynamic model for the conditional mean of a stochastic process
The merit of Box and Jenkins’s work was the specification of a unified class of processes, generalizing ARMA models to account for non-stationarity, seasonality, exogenous regressors, as well as the systematic treatment of all the sub-models belonging to the class, which led to the development of well established inferential procedures
A general modelling framework is introduced which aims (i) to provide a unified specification for a broad class of discrete-valued time series where relevant instances represent special cases, (ii) to provide direct relationships among different models which belong to the framework but are not necessarily nested within each other, (iii) to derive the stochastic properties for first order models which hold simultaneously for the entire class, (iv) to implement Quasi MLE (QMLE) inference that allows us to define model selection criteria across different, and not nested, models, (v) to derive the asymptotic properties of QMLE, and (vi) to make all the models encompassed in the framework fully applicable in practice
Summary
The analysis of time series that are generated by continuous random variables has a long tradition in statistics and dates back, in the parametric setting, to [42] and [41], who introduced the concept of autoregression, a dynamic model for the conditional mean of a stochastic process. A general modelling framework is introduced which aims (i) to provide a unified specification for a broad class of discrete-valued time series where relevant instances represent special cases, (ii) to provide direct relationships among different models which belong to the framework but are not necessarily nested within each other, (iii) to derive the stochastic properties for first order models which hold simultaneously for the entire class (strict stationarity and ergodicity), (iv) to implement QMLE inference that allows us to define model selection criteria across different, and not nested, models, (v) to derive the asymptotic properties of QMLE, and (vi) to make all the models encompassed in the framework fully applicable in practice. With the focus on model comparison, models included in the general framework are applied for the analysis of a test-bed time series in count data analysis, on the spread of an infection, namely Escherichia coli, in the German region of North Rhine-Westphalia
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