A negative binomial ingarch model for overdispersed count time series structure, parameter estimation and application to real data.
Date
2025
Authors
Mulamfu, Carols
Journal Title
Journal ISSN
Volume Title
Publisher
The University of Zambia
Abstract
This dissertation develops a negative binomial integer-valued generalized autoregressive conditional heteroscedastic model of order p, q (negative binomial INGARCH(p, q)) for overdispersed count time series data with potential extreme values. The model is formulated such that its conditional distribution follows the negative binomial distribution, allowing the conditional variance to exceed the conditional mean and dynamically adjust for overdispersion based on past observations. Furthermore, the unconditional variance exceeding the unconditional mean demonstrates the model’s capability to capture extreme values in the data. A simulation study evaluates the finite sample performance of the Yule–Walker, conditional least squares, and maximum likelihood estimation methods for the three sparsely parameterized negative binomial INGARCH(p, q) models. Results indicate that maximum likelihood estimation is the most efficient and reliable approach. The conditional log-likelihood function is maximized numerically using MATLAB’s fmincon function, with constraints to ensure stationarity and non-negativity of parameters. Conditional least squares estimates serve as initial values to facilitate convergence and enhance stability. For application, the negative binomial INGARCH(p, q) model is applied to syphilis count data from the R ZIM package, originally sourced from the CDC Morbidity and Mortality Weekly Report CDC MMWR. The dataset consists of weekly syphilis cases in Maryland, United States, from January 2007 to May 2010, with 209 observations. The empirical mean (3.47) and variance (9.28) confirm overdispersion, justifying the use of the negative binomial distribution. Model performance is assessed and compared to the Poisson INGARCH(p, q) and double Poisson INGARCH(p, q) models using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). Additionally, tail probabilities of residuals are analyzed to evaluate the models’ ability to capture extreme values. Results from AIC, BIC, and tail probability analysis indicate that the negative binomial INGARCH(p, q) model outperforms the Poisson INGARCH(p, q) and double Poisson INGARCH(p, q) models.
Description
Thesis of Master of Science in Statistics.