Goodness-of-fit testing for INAR models

In recent years, there has been a growing interest in the analysis of time series of counts. Among the various models designed for dependent count data, integer-valued autoregressive (INAR) processes enjoy great popularity. These processes serve as a natural extension of the widely known AR model used in the context of continuous autoregressive time series and have been used extensively in the statistical literature. Typically, statistical inference for INAR models relies on asymptotic theory and tends to rest upon rather stringent (parametric) model assumptions. Notably, the Poisson-INAR(1) model, a prominent example, has received considerable attention in existing literature. We present a novel semiparametric goodness-of-fit test tailored for the INAR model class, without imposing any parametric assumptions on the distribution of innovations. While parametric assumptions streamline the approach and other straightforward testing strategies, they often introduce too restrictive model assumptions. Our proposed procedure relies on the specific structure of the joint probability generating function of INAR models. This approach allows for enhancing the versatility and applicability of INAR models by accommodating a broader array of innovation distributions. We prove the validity of our testing procedure and carefully examine its performance characteristics, including power and size, through diverse simulation scenarios.