A non-stationary NB–INAR(1) model with time-varying beta–binomial thinning: properties and rolling estimation

Abstract

Many count time series exhibit non-stationary behavior such as drifting intensity, regime switching, structural breaks, evolving serial dependence, and changing innovation levels, which can invalidate classical stationary INAR models. We propose a non-stationary Negative Binomial INAR(1) model based on Beta–Binomial thinning, where the thinning variable is randomized through a Beta mixing distribution. This construction yields a transparent decomposition of dynamic persistence and dispersion: the time-varying mean persistence \(\tau_t=\mathbb{E}(\Theta_t)\) governs dependence, while the time-varying precision \(\phi_t=a_t+b_t\) controls survivor-variance inflation induced by mixing. We derive closed-form conditional moment identities and time-varying recursions for the unconditional mean, variance, and autocovariance structure, highlighting how dependence propagates through products of \(\tau_t\). For inference, we develop practical moment/CLS-based procedures under local stationarity, including rolling-window CLS and smooth parametric evolution for \((\tau_t,m_t)\), together with feasibility constraints that preserve the probabilistic interpretation. Monte Carlo experiments under multiple non-stationary regimes demonstrate accurate tracking under gradual evolution, quantify the RMSE–adaptation-delay trade-off induced by window length under breaks and regime switching, and illustrate the volatility impact of precision variation. An application to weekly accident counts shows that a smooth time-varying innovation mean improves one-step-ahead forecasting, residual lag dependence becomes negligible after mean correction, and negative binomial predictive intervals provide improved coverage under mild overdispersion.