On the Comparison of PAR, DARMA, and INAR in Modeling Count Time Series Data

Abstract

This study evaluates the forecasting and fitting performance of three advanced models—Poisson Autoregressive (PAR), Discrete Autoregressive Moving Average (DARMA), and Integer-Valued Autoregressive (INAR) for count time series data exhibiting complex features such as autocorrelation, overdispersion, and zero inflation. Both simulated and empirical datasets were analyzed, and model performance was assessed using Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Root Mean Square Error (RMSE), and Mean Absolute Error (MAE). The results indicate that PAR models significantly outperform DARMA and INAR models, achieving substantially lower AIC (482.53 vs. >5,310,479) and RMSE (3,742 vs. 246,682), highlighting their robustness in handling periodic trends and autocorrelation. In contrast, standard Poisson regression performs poorly under overdispersion, with an AIC approaching 5.3 million, while zero-inflated datasets compromise error metrics such as MAPE due to division by zero. Although DARMA and INAR models perform comparably, they are less effective in capturing extreme fluctuations or sudden spikes. These findings emphasize the limitations of conventional models and point to the need for more flexible approaches, such as hybrid ZIP-INAR models or Bayesian methods, to effectively manage overdispersion and zero inflation. The study concludes with a practical recommendation to prioritize PAR models when modeling autocorrelated count data.