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In this study, we consider integer-valued autoregressive conditional heteroscedastic (INGARCH) time series models of counts whose conditional density follows an multiple values-inflated one parameter exponential family. We check the stationarity and ergodicity of the models and verify the consistency and asymptotic normality of the conditional maximum likelihood estimator (CMLE) under regularity conditions. Then, as an application, we consider a parameter change test using the cumulative sum (CUSUM) test based on (standardized) residuals and squares of those residuals, then subsequently derive their limiting null distributions. A simulation study is conducted focusing on the 0-1-inflated time series of counts, which affirms the validity of the proposed methods. Moreover, a real data analysis using the number of monthly drug cases in the 32nd police car beat in Pittsburgh is also provided to further strengthen the validity of our methods.

Multiple values-inflated time series of counts: modeling and inference based on INGARCH scheme