Softplus INGARCH Model

Abstract

During the last decades, a large variety of models have been proposed for count time series, where the integer-valued autoregressive moving average (ARMA) and integer-valued generalized autoregressive conditional heteroskedasticity (IN-GARCH) models are the most popular ones. 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, which uses the softplus function as a response function. The softplus function behaves approximately linear, but avoids the drawback of not being differentiable in zero. Stochastic properties of the novel model are derived. The proposed model indeed exhibits an approximately linear structure, which is confirmed by extensive simulations, and 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

Monitoring count time series: Robustness to nonlinearity when linear models are utilized

“…Weiß et al 20 . suggest to modify the linear INGARCH model () to the spINGARCH model by defining $$\begin{equation} \textstyle M_t = s_c{\left(M_t^*\right)} \quad \text{with } M_t^* = \alpha _0+\sum \limits _{i=1}^p\alpha _i\, X_{t-i}+\sum \limits _{j=1}^q\beta _j\, M_{t-j}.$$…”

“…The plots in Figure 1 illustrate that sp 𝑐 (𝑥) and sc 𝑐 (𝑥) become piecewise linear for 𝑐 → 0: sp 𝑐 (𝑥) → max{0, 𝑥} for 𝑐 → 0, and sc 𝑐 (𝑥) → min{1, max{0, 𝑥}} for 𝑐 → 0. 20 The limiting case 𝑐 → 0 corresponds to…”

A Spatially Correlated Model with Generalized Autoregressive Conditionally Heteroskedastic Structure for Counts of Crimes

“…Modeling time series of counts has received important and growing attention since the 1950s [ 1 , 2 , 3 , 4 , 5 ] and over recent decades (see [ 6 , 7 , 8 , 9 , 10 ]). It is known that some well-known discrete distributions, such as Poisson and negative binomial (NB), can only deal with overdispersion; however, generalized Poisson (GP) and double Poisson (DP) distributions can treat both overdispersion and underdispersion.…”

“…These are the DP model [ 13 ] and the GP model [ 14 ]; see also [ 15 ] for a proposal of a Conway–Maxwell (COM) Poisson INGARCH distribution. The reader is referred to the very latest literature in this field [ 8 , 9 , 10 ].…”

Nearly Unstable Integer-Valued ARCH Process and Unit Root Testing

“…Another popular way for dealing with count time series data is the INteger-valued Genenalized AutoRegressive Conditional Heterokedastic (INGARCH) models by Ferland, Latour and Oraichi (2006), Fokianos, Rahbek and Tjøstheim (2009), Fokianos and Fried (2010), Zhu (2011), Fokianos and Tjøstheim (2011), Zhu (2012), Christou and Fokianos (2015), Gonçalves et al (2015), Davis and Liu (2016), Silva and Barreto-Souza (2019), Weiß et al (2020), which constitute in some sense an integer-valued counterpart of the classical GARCH models by Bollerslev (1986). The INGARCH methodology is the focus of this paper.…”