Poisson INGARCH models: Linear Bayesian estimation and causality testing

We develop a linear Bayesian estimator (LBE) for the Poisson integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) model that fuses prior information with sample data, yielding a closed-form solution and thereby bypassing posterior integration. Under the mean squared error matrix (MSEM) criterion, we rigorously establish the theoretical superiority of the proposed estimator over conditional maximum likelihood estimation (CMLE). Exploiting the conditional mean structure of the Poisson INGARCH model, we construct a regression framework for testing the causal effects of exogenous covariates. We derive a least-squares test statistic and prove that it asymptotically follows a chi-square distribution under the null hypothesis of no Granger causality. Simulation studies with prior robustness checks demonstrate that the proposed estimator remains predominantly data-driven and significantly outperforms CMLE in small-to-moderate samples. Applying the method to Chicago crime data, we find that it provides crucial numerical stability when the likelihood geometry is unfavorable. Importantly, we underscore that beyond stabilizing computation, routine distributional diagnostics (e.g., negative binomial (NB) checks) are indispensable for addressing potential overdispersion and ensuring reliable causal conclusions.