Score-based bootstrap test for serial dependence in count time series

Testing for serial dependence in count time series is crucial in many applications, especially when adopting widely used Integer Autoregressive (INAR) models. Existing approaches rely on score-based test statistics, typically assuming Poisson innovations. Such tests cannot account for overdispersion, underdispersion, and zero inflation, which are common issues in real data. Existing bootstrap methods have not been thoroughly investigated for their effectiveness under varying dispersion conditions. We propose a novel restricted bootstrap algorithm to enhance the performance of score-based tests for serial dependence in INAR models. Negative Binomial and Generalized Poisson distributions are considered to address overdispersion, underdispersion, and zero inflation. Both parametric and semiparametric bootstraps are implemented, providing a more flexible framework for testing. We demonstrate the effectiveness of the proposed approach through a simulation study and an analysis of six real datasets with varying autocorrelation strengths, dispersion levels, and zero inflation. The bootstrapped Generalized Poisson score statistics provide preferable results, especially for underdispersed and heavily zero-inflated data. In the Poisson case, semiparametric and parametric bootstraps can be compared, providing a detection tool for equidispersion of the arrivals.