Bayesian log-linear beta-negative binomial integer-valued Garch model

When dealing with time series with outlying and atypical data, a commonly used approach is to develop models based on heavy-tailed distributions. The literature coping with continuous-valued time series with extreme observations is well explored. However, current literature on modelling integer-valued time series data with heavy-tailedness is less considered. The state of the art research on this topic is presented by Gorgi (J R Stat Soc Ser B (Stat Methodol) 82:1325–1347, 2020) very recently, which introduced a linear Beta-negative binomial integer-valued generalized autoregressive conditional heteroscedastic (BNB-INGARCH) model. However, such proposed process allows for positive correlation only. This paper develops a log-linear version of the BNB-INGARCH model, which accommodates both negative and positive serial correlations. Moreover, we adopt Bayesian inference for better quantifying the uncertainty of unknown parameters. Due to the high computational demand, we resort to adaptive Markov chain Monte Carlo sampling schemes for parameter estimations and inferences. The performance of the proposed method is evaluated via a simulation study and empirical applications.