The BerG generalized autoregressive moving average model for count time series

•The model considers autocorrelation and it is capable to capture equi/under/over dispersion.•Naturally zero inflated (or deflated), not needing an extra parameter.•Monte Carlo simulation indicates that the estimators are consistent.•An easy method to obtain forecasts and their confidence intervals is presented.•The model outperformed all the competitive models in the real data analyses. In this work, we present a new generalized autoregressive moving average model (GARMA), based on the Bernoulli-geometric (BerG) distribution, for modeling the conditional mean of count time series. The proposed model is able to deal with the equi, under or over-dispersed data. Our main contribution is to suggest a GARMA model with a response variable following a BerG distribution, which also accommodates zero inflated (or deflated) data. The proposed model combines the dispersion flexibility with the inclusion of covariates and lagged terms to model the conditional mean response, inducing an autocorrelation structure (usually relevant in time series). We exhibit the conditional maximum likelihood estimation, the hypothesis testing inference, the diagnostic analysis, and the out-of-sample forecasting procedure. Using the closed-form quantile function of the BerG distribution, the confidence intervals for out-of-sample forecasts are easily obtained. In particular, we provide the closed-form expressions for the conditional score vector and conditional Fisher information matrix. Moreover, we developed a computational study which confirmed that the maximum likelihood estimators are consistent for all dispersion scenarios. And finally, we illustrate the applicability of the postulated model by exploring two real data applications.