Multiplicative Error Models for Count Time Series

Multiplicative error models (MEMs) are commonly used for real-valued time series, but they cannot be applied to discrete-valued count time series as the involved multiplication would not preserve the integer nature of the data. We propose the concept of a multiplicative operator for counts as well as several specific instances thereof, which are then used to develop MEMs for count time series (CMEMs). If equipped with a linear conditional mean, the resulting CMEMs are closely related to the class of so-called integer-valued generalized autoregressive conditional heteroskedasticity (INGARCH) models and might be used as a semi-parametric extension thereof. We derive important stochastic properties of different types of INGARCH-CMEM as well as relevant estimation approaches, namely types of quasi-maximum likelihood and weighted least squares estimation. The performance and application are demonstrated with simulations as well as with two real-world data examples.