Regime-Switching Random Environment INAR(1) Model with Modified Negative Binomial Thinning and Innovation-Free Dynamics

Abstract We introduce a new first-order integer-valued autoregressive model in a random environment. We propose the Regime-Switching Random Environment INAR(1) with Negative Binomial Thinning and Innovation-Free Dynamics, a count time series model capturing regime shifts via a latent control process that enables flexible transitions between autoregressive and memoryless regimes. The control process assigns either INAR(1) properties or i.i.d. characteristics to model elements. At each time, the process follows either a modified INAR(1) structure without innovation or an i.i.d. sequence, as determined by the control process. The thinning operator uses a geometric counting mechanism, producing a negative binomial outcome. We use a known modified negative binomial operator. We derive key properties of the model and estimate parameters using the method of moments, conditional maximum likelihood, and conditional least squares. Estimation quality is assessed via simulations. The regime structure is learned a posteriori via unsupervised clustering of observed trajectories, capturing hidden heterogeneity without fully specifying the environmental process. We illustrate the model’s application on real data. Results show improved fit over conventional INAR-type models in settings with structural shifts and nonstationarity.