Model diagnostics and semi-parametric inference for count time series

For modeling the serial dependence in discrete-valued time series, various approaches have been proposed in the literature. In particular, models based on a recursive, autoregressive-type structure such as the integer-valued autoregressive (INAR) models for count time series are very popular in practice. While their estimation typically relies on purely parametric approaches that impose restrictive assumptions on the innovation distribution, we consider semi-parametric estimation techniques that jointly estimate the autoregressive coefficients and the innovation distribution without requiring parametric specification. Building on this, we propose a general semi-parametric bootstrap procedure for INAR models and prove its consistency for general classes of statistics that are functions of the estimated model coefficients and the estimated innovation distribution. This semi-parametric bootstrap approach can be leveraged for various statistical tasks such as goodness-of-fit testing, predictive inference, and dispersion analysis. Additionally, we introduce novel semi-parametric goodness-of-fit tests tailored for the INAR model class. Relying on the INAR-specific shape of the joint probability generating function, our approach allows for model validation of INAR models without specifying the parametric family of the innovation distribution. We derive the limiting null distribution of our proposed test statistics, prove consistency under fixed alternatives and discuss its asymptotic behavior under local alternatives. Moreover, when it comes to predictive inference for discrete-valued time series, this task cannot be implemented through the construction of prediction intervals as they are generally not able to retain a desired coverage level neither in finite samples nor asymptotically. To address this problem, we propose to reverse the construction principle by considering preselected sets of interest and estimating the corresponding predictive probability. The accuracy of this prediction is then evaluated by quantifying the uncertainty associated with the estimation of these predictive probabilities. In this context, we consider
parametric and non-parametric approaches and derive asymptotic as well as bootstrap theory, which also covers the practically important case of model misspecification.