Poisson integer-valued garch model structure, parameter estimation and a real-data example.

In this dissertation, we present an integer-valued generalised autoregressive conditional heteroscedastic (INGARCH)(p, q) model based on the Poisson distribution. This model proves to be effective for modelling overdispersed integer-valued time series with conditional heteroscedasticity. The basic properties of the model are studied, and a condition for the existence of such a process is given. For the case p = 1, q = 1, it is explicitly shown that an INGARCH process is a standard autoregressive moving average (ARMA)(1, 1) process. An approach for parameter estimation of the model is given using the method of conditional maximum likelihood estimation. In cases where analytical results from the likelihood function are not found, we suggest numerical optimisation methods, particularly using the R software package tscount. The model is illustrated on an overdispersed integer-valued time series. Specifically, we analyse the daily number of shares traded on the Lusaka Securities Exchange (LuSE) from October 1, 2021, to May 10, 2022 (excluding weekends and public holidays), comprising a total of 150 observations. This data set is publicly available on the LuSE website at https://luse.co.zm/market-data. The empirical mean and variance of the data are 2.48287 and 5.76448 respectively, indicating overdispersion in the marginal distribution of the data. In addition to fitting the INGARCH model, we assess its predictive performance using one-step-ahead forecasts and compute the corresponding confidence intervals. The model captures both the volatility clustering and the overdispersion present in the data. The
forecasts closely match the actual data, providing reliable predictions for future observations.

Keywords. Integer-valued time series; INGARCH model; heteroscedasticity; overdispersion; one-step-ahead forecasts; predictive accuracy.