Integer-Valued First Order Autoregressive (INAR(1)) Model With Negative Binomial (NB) Innovation For The Forecasting Of Time Series Count Data

Abstract

This paper is about the theoretical investigation of integer-valued first order autoregressive (INAR(1)) model with negative binomial (NB) innovation for the forecasting of time series count data. The study makes use of the Conditional Least squares (CLS) estimator to estimate the parameter of INAR(1) model, and Maximum Likelihood Estimator (MLE) to estimate the mean (μ ) and the dispersion parameter (K) of the NB distribution. A simulation experiment based on theoretical generated data were addressed under different parameter values α =0.2, 0.6, 0.8, different sample sizes n=30, 90, 120, 600 for the class of INAR(1) model, and μ  =0.85, 1.5, 2,  K=1,2, 4 for the NB distribution. The Monte Carlo simulations were conducted with codes written in R, all results were based on 1000 runs. The estimation of parameter for the class of INAR(1) model gives a better result when the number of observations is small and the parameter value is high. The NB estimation gives a better result when the number of observations is small and with large K values. The forecasting accuracy of the model at different lead time period l =1, 3, 5, 7, 9, 15 were investigated with codes written in R. The results showed that the minimum mean square error (MMSE) produced when the number of lead times forecasts is between one and five were less than that produced when the numbers of lead times forecast were greater than five. The MMSE increased when the number of lead time periods increases. This result indicates that forecasting with this class of model is better with short time frame of predictions. The study was applied to the number of deaths arising from COVID-19 in Nigeria which consist of count time series data of 48 observations (weekly data), from January 2021 to December 2021.