A ℤ-valued GJR-GARCH model utilizing discrete normal distribution

Considering the asymmetric volatility in Z-valued financial time series, several types of models have been proposed. Beyond adapting appropriate conditional distributions, another key focus of these models lies in how to construct Z-valued data in model. The current approach to constructing Z-valued data is limited to generating sequences of Z-valued random variables with equal probability of taking positive or negative values, which poses practical constraints in real-world applications. To address this limitation and simplify the model structure, we adopt the discrete normal distribution to represent the conditional distribution, which enables the generation of more generalized Z-valued sequences. We present the fundamental properties of the model, derive parameter estimates through the method of maximum likelihood and subsequently demonstrate their asymptotic normality. The effectiveness of new model is demonstrated through the analysis of real daily stock returns.