Modeling And Inference For Two Kinds Of Integer-Valued Autoregressive Conditional Heteroscedasticity Models

The widespread availability of time series has led to a wide range of modeling and research.For time series with heteroscedasticity,scholars have proposed autoregressive conditional heteroscedasticity(ARCH)and generalized autoregressive conditional heteroscedasticity(GARCH)models.On this basis,statisticians have further investigated integer-valued time series.At the beginning,many kinds of models have been proposed for modeling time series of counts,such as the integer-valued autoregressive(INAR)processes,the integer-valued moving average(INMA)process,the integervalued autoregressive moving average(INARMA)model,the integer-valued GARCH(INGARCH)model,among others.Then they found that in real-life financial time series,there exist integer-valued data which take both positive and negative values,which is called Z-valued time series,and the volatility of such time series has an asymmetric response.A Poisson Z-valued GJR-GARCH(PZG)model has been developed to model this type of data.In addition,overdispersion(variance greater than the mean)is often present in this type of data.The existing integer-valued asymmetric power GARCH(INAPGARCH)model captures the asymmetric response,but is unable to model overdispersed Z-valued time series.In this paper,generalized models are proposed for the PZG and INAPGARCH models.Moreover,a multiplicative thinningbased integer-valued GARCH(MthINGARCH)model is modified,and the saddlepoint maximum likelihood estimation(SPMLE)method is applied in the parameter estimation to solve the problem that the likelihood function of this model can not be given directly.The main content is divided into three parts as follows:1.A geometric Z-valued GJR-GARCH model.GJR-GARCH model is popular in accounting for asymmetric responses in the volatility in the analysis of financial time series,but asymmetric responses in the volatility are also observed in time series of counts or Z-valued time series,such as the daily number of stock transactions or the daily stock returns divided by tick price(1 cent).Two different INGARCH models based on Poisson distribution have been proposed for these two types of discrete data,respectively,like the Poisson INGARCH model and the PZG model.For that the geometric distribution is more flexible than Poisson distribution,whose variance is greater than its mean.In this paper,we propose a GJR-GARCH model based on geometric distribution for Z-valued time series with asymmetric volatility.Basic probabilistic properties of the new model are given,and the maximum likelihood estimation(MLE)method is used to estimate unknown parameters and the asymptotic normality of corresponding estimators is established.A simulation study is presented to illustrate the well performance of estimation method.Finally,we consider a real data of daily stock returns.A detailed analysis of the data fit and forecast results are presented to show the superiority of our proposed model compared with existing models.2.A zero-inflated Poisson Z-valued asymmetric power GARCH model.The asymmetric power ARCH model(APARCH)for the volatility was introduced in 1993 in order to deal with asymmetric responses in the volatility when analyzing financial time series.Furthermore,asymmetric responses in the volatility also exist in time series of counts and Z-valued time series.The INAPGARCH model for analyzing time series of counts has been proposed.As for Z-valued time series,related GJRGARCH models for Z-valued time series have also been studied.Moreover,there exists a zero-inflation phenomenon,then a zero-inflated Poisson INGARCH model has been proposed for time series of counts to model this phenomenon.Combining the data features mentioned above,we propose a zero-inflated Poisson Z-valued asymmetric power GARCH(ZIPZAG)model.Some probabilistic properties of this model are provided,meanwhile,the unknown parameters are estimated in terms of the MLE and the asymptotic normality is given.To demonstrate the estimation method,a simulation study is presented.A real data example based on the daily stock returns divided by tick price is taken into account to illustrate the superiority of the new model compared with existing models.3.A modified multiplicative thinning-based integer-valued ARCH model.Recently,a multiplicative thinning-based INGARCH(MthINGARCH)model with binomial thinning operator was proposed to model persistence and high overdispersion,which combines the INAR and INGARCH modeling techniques together.Therefore,the two-stage weighted least squares estimation(2SWLSE)method was used in the literature to estimate the parameters.In order to solve the problem that the probability mass function cannot be given directly,scholars proposed the saddlepoint approximation method,which has a broad applications with high accuracy.In this article,we propose a modified multiplicative thinning-based integer-valued ARCH(MthINARCH)model and use the SPMLE to estimate parameters.A simulation study is given to show a better performance of the SPMLE.As for the application of the real data,we concern the number of tick changes by minute of the euro to British pound exchange rate.The results of data fit and forecast show the superiority of our modified model and the SPMLE.