Predictive likelihood for coherent forecasting of count time series

Abstract

A new forecasting method based on the concept of the profile predictive likelihood function is proposed for discrete‐valued processes. In particular, generalized autoregressive moving average (GARMA) models for Poisson distributed data are explored in detail. Highest density regions are used to construct forecasting regions. The proposed forecast estimates and regions are coherent. Large‐sample results are derived for the forecasting distribution. Numerical studies using simulations and two real data sets are used to establish the performance of the proposed forecasting method. Robustness of the proposed method to possible misspecifications in the model is also studied.

A performance analysis of prediction intervals for count time series

“…In view of our later investigations, it is worth pointing out that many of these PIs rely on Gaussian approximations of the actual count distribution, but articles on PIs for count time series are rare. Lambert (1997) and Mukhopadhyay and Sathish (2018) developed predictive-likelihood-based PIs for generalized autoregressive moving-average (ARMA) models, and data applications were reported by Freeland and McCabe (2004) and Bejleri and Nandram (2018). The article by Silva, Pereira, and Silva (2009) proposed a Bayesian PI for the Poisson integer-valued autoregressive process of order 1 (abbreviated "INAR(1)") due to McKenzie (1985), and it also presented some performance analyses.…”