The INAR (integer-valued autoregressive) and INGARCH (integer-valued GARCH)
classes are among the most commonly employed approaches for count time series
modelling, but have been studied in largely distinct strands of literature. In this
paper, a general model class unifying a large number of INAR and INGARCH mod-
els is introduced and its stochastic properties are studied. The starting point is
given by a novel thinning-based representation of compound-Poisson INGARCH pro-
cesses, which is subsequently generalized. Particular attention is given to a gener-
alization of the INAR(p) model which parallels the extension of the INARCH(p) to
the INGARCH(p, q) model. Models from the class have a natural interpretation
as stochastic epidemic processes. Different instances of the class, including both es-
tablished and newly introduced models, are compared in a real-data application on
infectious disease counts from German routine surveillance.

Resampling-based approaches are very helpful to construct prediction intervals when typically questionable distribution assumptions have to be avoided. Recent literature introduces different procedures to construct bootstrap prediction intervals for (continuous) autoregressive models dealing with the conditional nature of predictive inference in a time series setup. We adapt these findings to count data time series. In contrast to continuous time series, the construction of prediction intervals with a pre-determined level is not reasonable. This is because, due to the integer nature of count data, it is generally not possible to have (not even asymptotically) valid prediction intervals with desired coverage. Instead, we propose to consider pre-defined sets and to estimate the conditional probability that a future observation falls in these sets. Then, the accuracy of the predictive inference procedure and other quality criteria can be evaluated by its capability to mimic the distribution of the estimated conditional probabilities. In simulations, we consider different bootstrap-based procedures to account for the various sources of randomness and variability. Additionally, we deal with the aspect of possible model misspecification for the (point) prediction and propose a robustification of our predictive inference procedure by using non- and semiparametric estimation.

Autoregressive (AR) modeling of real-valued time series data has been used for decades, but it is still popular in practice, because they are flexible and enable an explicit estimation of the model parameters. When dealing with integer-valued time series, autoregressive modeling is not straightforward as classical AR models do not respect the integer-valued range. For count data time series, integer-valued AR (INAR) models based on binomial thinning are popular in practice, because they are of autoregressive nature and respect the integer-valued range. INAR models are still easy to estimate, but their parameter range is restricted in comparison to AR models. While the literature that investigates both model classes separately is huge, it lacks a unified joint approach for real-valued and integer-valued time series.

In this paper, we propose a joint vector-autoregressive model for real- and integer-valued time series. By construction, it respects the corresponding time series ranges and allows for an unrestricted range of the AR model parameters, which coincides with that of a classical vector-autoregressive model. We provide explicit estimators of all model parameters, derive their asymptotic properties and discuss suitable bootstrap approaches. The estimation performance is evaluated by simulations and applicability is illustrated on a real data set.

We derive mixing properties for a class of count time series satisfying a certain contraction condition. Using specific coupling techniques, we can deduce absolute regularity at a geometric rate for non-stationary Poisson-GARCH(1,1) processes with a possibly explosive trend and seasonal patterns.

Moreover, we propose a new model for nonstationary integer-valued time series which is particularly suitable for data with a strong trend. In contrast to popular Poisson-INGARCH models, but in line with classical GARCH models, we propose to pick the conditional distributions from nearly scale invariant families where the mean absolute value and the standard deviation are of the same order of magnitude. Again we provide sufficient conditions for absolute regularity of these count process with exponentially decaying coefficients. Finally, we illustrate the statistical use of our results in a statistical application.