Abstract
Count time series frequently exhibit serial dependence and either overdispersion or underdispersion relative to the Poisson equidispersion assumption. When such counts are observed across multiple locations, spatial dependence often arises, resulting in complex spatio-temporal dependency structures that are not adequately accommodated by standard integer-valued generalized autoregressive conditional heteroskedasticity (INGARCH) models. To address these challenges, this study proposes a spatial Conway–Maxwell–Poisson INGARCH (SP-COM-INGARCH) framework that jointly models temporal dependence, spatial interaction, and varying dispersion. The proposed framework incorporates the Conway-Maxwell-Poisson (COM-Poisson) distribution, which accommodates both overdispersion and underdispersion through an additional dispersion parameter, while a spatially weighted lag structure captures dependence across neighboring locations. We further introduce covariate-adjusted identity-link and softplus-link extensions. In particular, the softplus formulation allows covariates to have negative effects while ensuring positivity of the conditional COM-Poisson centering parameter. For the SP-COM-INGARCH (1,1) process sufficient conditions for stationarity and ergodicity are established, and expressions for the conditional first- and second-order moments are derived. The finite-sample performance of conditional maximum likelihood estimation is evaluated through simulations under several overdispersion and underdispersion scenarios. Parameter recovery is generally satisfactory, with accuracy tending to improve as the length of the time-series increases. We illustrate the framework using weekly influenza-like illness counts from Arkansas, Kansas, Missouri, and Oklahoma with absolute humidity as a time-varying covariate. Across six competing COM-Poisson and generalized Poisson models, the softplus SP-COM-INGARCHX model achieved the lowest joint out-of-sample RMSE, while the baseline SP-COM-INGARCH model achieved the lowest joint out-of-sample MAE, although the differences among several models were small. Overall, the proposed SP-COM-INGARCH framework offers a flexible extension for modeling spatiotemporal count data with covariate effects and flexible dispersion while providing competitive out-of-sample forecasting performance.