Abstract: Numerous models have been proposed for count time series, including the integer-valued autoregressive moving average (ARMA) and integer-valued generalized autoregressive conditional heteroskedasticity (INGARCH) models. However, while both models lead to an ARMA-like autocorrelation function (ACF), the attainable range of ACF values is much more restricted, and negative ACF values are usually not possible. The existing log-linear INGARCH model allows for negative ACF values, but the linear conditional mean and the ARMA-like autocorrelation structure are lost. To resolve this dilemma, a novel family of INGARCH models is proposed that uses the softplus function as a response function. The softplus function is approximately linear, but avoids the drawback of not being differentiable in zero. The stochastic properties of the novel model are derived. The proposed model exhibits an approximately linear structure, confirmed using extensive simulations, which makes its model parameters easier to interpret than those of a log-linear INGARCH model. The asymptotics of the maximum likelihood estimators for the parameters are established, and their finite-sample performance is analyzed using simulations. The usefulness of the proposed model is demonstrated by applying it to three real-data examples.

Key words and phrases: Count time series, INGARCH models, maximum likelihood estimation, negative autocorrelation, softplus link.